{"id":16571,"date":"2024-01-03T15:16:15","date_gmt":"2024-01-03T09:46:15","guid":{"rendered":"https:\/\/envisionmathanswerkey.com\/?p=16571"},"modified":"2024-01-04T10:37:34","modified_gmt":"2024-01-04T05:07:34","slug":"envision-math-common-core-grade-6-answer-key-topic-2","status":"publish","type":"post","link":"https:\/\/envisionmathanswerkey.com\/envision-math-common-core-grade-6-answer-key-topic-2\/","title":{"rendered":"Envision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers"},"content":{"rendered":"

Go through the\u00a0enVision Math Common Core Grade 6 Answer Key<\/a> Topic 2 Integers and Rational Numbers<\/strong>\u00a0regularly and improve your accuracy in solving questions.<\/p>\n

Envision Math Common Core 6th Grade Answers Key Topic 2 Integers and Rational Numbers<\/h2>\n

?Topic Essential Question\u00a0<\/strong>What are integers and rational numbers? How are points graphed on a coordinate plane?
\nAnswer:
\nAn “Integer” can be written as a fraction by giving it a denominator of one. So, any integer is a rational number
\n“Rational numbers” are those numbers that are integers and can be expressed in the form of \\(\\frac{x}{y}\\) where both numerator and denominator are integers
\nTo graph or plot points, we use two perpendicular lines called the x-axis and the y-axis. The horizontal number line is the x-axis and the vertical line is the y-axis.\u00a0 Every point in the coordinate plane is represented by an ordered pair of x and y coordinates.<\/p>\n

3-ACT MATH<\/strong><\/p>\n

The ULTIMATE THROW<\/strong>
\n\"Envision
\nThe Ultimate Throw<\/strong>
\nHave you ever played ultimate? It’s a team sport played with a flying disc. The goal is to score the most points by passing the disc to your opponent’s end zone. Ultimate is played by millions of people across the globe, from casual games to professional leagues.
\n\"Envision
\nThere are many ways to throw a flying disc. It takes a lot of practice to learn each type of throw. If you want the disc to travel a specific path and distance, you need to try different throws with different amounts of spin and power. Think about this during the 3-Act Mathematical Modeling lesson.<\/p>\n

enVision STEM Project<\/strong><\/p>\n

Did You Know?<\/strong>
\n\"Envision<\/p>\n

Your Task: Improve Your School<\/strong><\/p>\n

Now that you have defined the problem, or improvement needed, you and your classmates will apply the engineering design process to propose solutions.
\n\"Envision<\/p>\n

Topic 2 GET READY!<\/h3>\n

Review What You Know!<\/strong><\/p>\n

Vocabulary<\/strong><\/p>\n

Choose the best term from the box to complete each definition.<\/strong><\/p>\n

decimal
\ndenominator
\nfraction
\nnumerator<\/p>\n

Question 1.
\nA ____ names part of a whole, part of a set, or a location on a number line.
\nAnswer:
\nWe know that,
\nA “Fraction” names part of a whole, part of a set, or a location on a number line.
\nHence, from the above,
\nWe can conclude that
\nThe best term that is suitable for the given definition is: Fraction<\/p>\n

Question 2.
\nThe number above the fraction bar that represents the part of the whole is the ____
\nAnswer:
\nWe know that,
\nThe number above the fraction bar that represents the part of the whole is the “Numerator”
\nHence, from the above,
\nWe can conclude that
\nThe best term that is suitable for the given definition is: Numerator<\/p>\n

Question 3.
\nThe number below the fraction bar that represents the total number of equal parts in one whole is the ____
\nAnswer:
\nWe know that,
\nThe number below the fraction bar that represents the total number of equal parts in one whole is the “Denominator”
\nHence, from the above,
\nWe can conclude that
\nThe best term that is suitable for the given definition is: Denominator<\/p>\n

\"<\/p>\n

Fractions and Decimals<\/strong>
\nWrite each fraction as a decimal.<\/p>\n

Question 4.
\n\\(\\frac{2}{5}\\)
\nAnswer:
\nThe given fraction is: \\(\\frac{2}{5}\\)
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given fraction in the form of a decimal is:
\n\"\"<\/p>\n

Question 5.
\n\\(\\frac{3}{4}\\)
\nAnswer:
\nThe given fraction is: \\(\\frac{3}{4}\\)
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given fraction in the form of a decimal is:
\n\"\"<\/p>\n

Question 6.
\n\\(\\frac{10}{4}\\)
\nAnswer:
\nThe given fraction is: \\(\\frac{10}{4}\\)
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given fraction in the form of a decimal is:
\n\"\"<\/p>\n

Question 7.
\n\\(\\frac{12}{5}\\)
\nAnswer:
\nThe given fraction is: \\(\\frac{12}{5}\\)
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given fraction in the form of a decimal is:
\n\"\"<\/p>\n

Question 8.
\n\\(\\frac{3}{5}\\)
\nAnswer:
\nThe given fraction is: \\(\\frac{3}{5}\\)
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given fraction in the form of a decimal is:
\n\"\"<\/p>\n

Question 9.
\n\\(\\frac{15}{3}\\)
\nAnswer:
\nThe given fraction is: \\(\\frac{15}{3}\\)
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given fraction in the form of a decimal is:
\n\"\"<\/p>\n

Division with Decimals<\/strong><\/p>\n

Divide.<\/strong><\/p>\n

Question 10.
\n1.25 \u00f7 0.5
\nAnswer:
\nThe given Division Expression is: 1.25 \u00f7 0.5
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n1.25 \u00f7 0.5 = 2.5<\/p>\n

Question 11.
\n13 \u00f7 0.65
\nAnswer:
\nThe given Division Expression is: 13 \u00f7 0.65
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n13 \u00f7 0.65 = 20<\/p>\n

Question 12.
\n12.2 \u00f7 0.4
\nAnswer:
\nThe given Division Expression is: 12.2 \u00f7 0.4
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n12.2 \u00f7 0.4 = 30<\/p>\n

Ordered Pairs<\/strong>
\nWrite the ordered pair for each point shown on the graph.<\/strong><\/p>\n

\"Envision<\/p>\n

Question 13.
\nJ
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nWe know that,
\n“Ordered pairs” are often used to represent two variables.\u00a0The number which corresponds to the value of x is called the x-coordinate and the number which corresponds to the value of y is called the y-coordinate.
\nHence, from the above,
\nWe can conclude that
\nThe ordered pair J is: (4, 3)<\/p>\n

Question 14.
\nK
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nWe know that,
\n“Ordered pairs” are often used to represent two variables. The number which corresponds to the value of x is called the x-coordinate and the number which corresponds to the value of y is called the y-coordinate.
\nHence, from the above,
\nWe can conclude that
\nThe ordered pair K is: (0, 6)<\/p>\n

Question 15.
\nL
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nWe know that,
\n“Ordered pairs” are often used to represent two variables. The number which corresponds to the value of x is called the x-coordinate and the number which corresponds to the value of y is called the y-coordinate.
\nHence, from the above,
\nWe can conclude that
\nThe ordered pair L is: (6, 8)<\/p>\n

Question 16.
\nM
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nWe know that,
\n“Ordered pairs” are often used to represent two variables. The number which corresponds to the value of x is called the x-coordinate and the number which corresponds to the value of y is called the y-coordinate.
\nHence, from the above,
\nWe can conclude that
\nThe ordered pair M is: (7, 1)<\/p>\n

\"<\/p>\n

Plot each point on the coordinate plane.<\/strong><\/p>\n

Question 17.
\nA(6, 2)
\nAnswer:
\nThe given point is: A (6, 2)
\nHence,
\nThe representation of the given point in a coordinate plane is:
\n\"\"<\/p>\n

Question 18.
\nB(1, 3)
\nAnswer:
\nThe given point is: B (1, 3)
\nHence,
\nThe representation of the given point in a coordinate plane is:
\n\"\"<\/p>\n

Question 19.
\nC(5, 7)
\nAnswer:
\nThe given point is: C (5, 7)
\nHence,
\nThe representation of the given point in a coordinate plane is:
\n\"\"<\/p>\n

Question 20.
\nD(3, 4)
\nAnswer:
\nThe given point is: D (3, 4)
\nHence,
\nThe representation of the given point in a coordinate plane is:
\n\"\"<\/p>\n

Explain<\/strong><\/p>\n

Question 21.
\nLes said that the quotient of 3.9 \u00f7 0.75 is 0.52. Explain how you know Les is incorrect without completing the division.
\nAnswer:
\nIt is given that
\nLes said that the quotient of 3.9 \u00f7 0.75 is 0.52
\nNow,
\nThe given Division Expression is: 3.9 \u00f7 0.75
\nNow,
\nBy using the Long Division,
\n\"\"
\nNow,
\nWhen we observe the numbers after the decimal point in the given division expression,
\nWe can say that there are 2 numbers after the decimal point n the numerator and 1 number after the decimal point in the denominator
\nSo,
\nThe numbers after the decimal point in the quotient will be only 1 number
\nBut,
\nThere are 2 numbers after the decimal point
\nHence, from the above,
\nWe can conclude that Les is incorrect<\/p>\n

Language Development<\/strong><\/p>\n

Use the graphic organizer to help you understand new vocabulary terms.<\/p>\n

\"Envision<\/p>\n

\"Envision<\/p>\n

\"Envision
\nAnswer:
\n\"\"
\n\"\"
\n\"\"<\/p>\n

Topic 2 PICK A PROJECT<\/strong><\/p>\n

PROJECT 2A<\/strong>
\nWhat places would you like to visit in the United States?
\nPROJECT: MAKE A TRAVEL BROCHURE
\n\"Envision<\/p>\n

PROJECT 2B<\/strong>
\nIf you were to solve a puzzle, what type would you choose?
\nPROJECT: DESIGN A CONNECT-THE-DOTS PUZZLE
\n\"Envision<\/p>\n

PROJECT 2C<\/strong>
\nWhat are some exercises for staying fit and having fun?
\nPROJECT: RECORD AN EXERCISE VIDEO
\n\"Envision<\/p>\n

PROJECT 2D<\/strong>
\nIf you were going to make a commercial, what type of product would you feature?
\nPROJECT: WRITE YOUR OWN COMMERCIAL
\n\"Envision<\/p>\n

Lesson 2.1 Understand Integers<\/h3>\n

ACTIVITY<\/strong><\/p>\n

Explain It!<\/strong>
\nSal recorded the outdoor temperature as -4\u00b0F at 7:30 A.M. At noon, it was 22\u00b0F. Sal said the temperature changed by 18\u00b0F because 22 – 4 = 18.<\/p>\n

\"Envision
\nA. Critique Reasoning<\/strong> is Sal right or wrong? Explain.
\nAnswer:
\nIt is given that
\nSal recorded the outdoor temperature as -4\u00b0F at 7:30 A.M. At noon, it was 22\u00b0F. Sal said the temperature changed by 18\u00b0F because 22 – 4 = 18.
\nSo,
\nThe total temperature from 7:30 A.M to noon = (The temperature at 7:30 A.M) + (The temperature at noon
\n= -4 + 22
\n= 18\u00b0 F
\nHence, from the above,
\nWe can conclude that Sal is right<\/p>\n

B. Construct Arguments<\/strong> What was the total temperature change from 7:30 A.M. until noon? Use the thermometer to help justify your solution.
\nAnswer:
\nIt is given that
\nIt is given that
\nSal recorded the outdoor temperature as -4\u00b0F at 7:30 A.M. At noon, it was 22\u00b0F. Sal said the temperature changed by 18\u00b0F because 22 – 4 = 18.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nUsing the thermometer,
\nThe temperature at 7:30 A.M is: -4\u00b0 F
\nNow,
\nThe temperature at noon is: 18\u00b0 F
\nSo,
\nThe change in temperature from 7: 30 A.M to until noon by using the thermometer is: 18\u00b0 F
\nHence, from the above,
\nWe can conclude that
\nThe total temperature change from 7:30 A.M. until noon is: 18\u00b0 F<\/p>\n

\"<\/p>\n

Focus on math practices<\/strong>
\nReasoning<\/strong> 0\u00b0C is the temperature at which water freezes. Which is colder, 10\u00b0C or -10\u00b0C? Explain.
\nAnswer:
\nIt is given that
\n0\u00b0C is the temperature at which water freezes
\nNow,
\nWe know that,
\nA temperature below 0\u00b0C is a negative temperature
\nHence, from the above,
\nWe can conclude that
\n-10\u00b0C is colder than 0\u00b0C<\/p>\n

Visual Learning<\/strong><\/p>\n

? Essential Question<\/strong> What are integers and how are they used to represent real-world quantities?
\nAnswer:
\n“Integers” are a set of numbers that include the positive whole numbers (1, 2, 3, 4, 5, …), their opposites (-1, -2, -3, -4, -5, …) and zero.
\nYou can use integers to help represent many real-world situations, such as Increases and decreases in temperature<\/p>\n

Try It!<\/strong>
\nLabel the integers on the number line.
\nThe opposite of 4 is\"Envision. The opposite of -4 is\"Envision.<\/p>\n

\"Envision
\nAnswer:
\nWe know that,
\nThe opposite of any integer ‘a’ is\u00a0‘<\/b>\u2212a‘<\/b>\u00a0Similarly, the opposite of any integer ‘\u2212a’ is ‘\u2212(\u2212a)’ = ‘a’
\nNow,
\nThe completed number line is:
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nBy using the given definition,
\nThe opposite of 4 is: -4
\nThe opposite of -4 is: 4<\/p>\n

Convince Me!<\/strong> How do you know that two numbers are opposites?
\nAnswer:
\nTwo numbers are opposites\u00a0if they have the same absolute value but different signs. Opposites are the same distance from 0 on a number line, and they are on opposite sides of 0<\/p>\n

Try It!<\/strong>
\nWhich number is greater, -4 or -2? Explain.
\nAnswer:
\nThe given numbers are: -4, and -2
\nNow,
\nWe know that,
\nIn the positive numbers i.e., the numbers that are to the right side of 0,
\n4 > 2
\nSo,
\nIn the negative numbers, i.e., the numbers that are to the left side of 0,
\n-4 < -2
\nHence, from the above,
\nWe can conclude that,
\nThe greater number is: -2<\/p>\n

Try It!<\/strong>
\nWhich integer represents each situation?
\na. A $10 debt
\nAnswer:
\nWe know that,
\nA “Debt” is a negative value
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given situation as an integer is: -$10<\/p>\n

b. Six degrees below zero
\nAnswer:
\nWe know that,
\nIn a vertical number line,
\nThe numbers above zero are positive
\nThe numbers below zero are negative
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given situation as an integer is: -6\u00b0 C<\/p>\n

c. Deposit of $25
\nAnswer:
\nWe know that,
\nA “Deposit” is a positive value
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given situation as an integer is: $25<\/p>\n

KEY CONCEPT<\/strong>
\nIntegers are all of the counting numbers, their opposites, and 0. Opposites are integers that are the same distance from 0 and on opposite sides of 0 on a number line.
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n? Essential Question<\/strong> What are integers and how are they used to represent real-world quantities?
\nAnswer:
\n“Integers” are a set of numbers that include the positive whole numbers (1, 2, 3, 4, 5, …), their opposites (-1, -2, -3, -4, -5, …) and zero.
\nYou can use integers to help represent many real-world situations, such as Increases and decreases in temperature<\/p>\n

Question 2.
\nReasoning<\/strong> What do you know about two different integers that are opposites?
\nAnswer:
\nLet us consider two integers a and b such that a > b (or) a < b
\nNow,
\nThe opposites of the two integers are: -a and -b
\nNow,
\nIn the number line,
\nThe numbers after 0 increase from left to right
\nThe numbers before 0 decrease from right to left
\nSo,
\nWe can conclude that
\n-a < -b (or) -a > -b
\nHence, from the above,
\nWe can conclude that
\nThe two different integers that are opposites can be represented as:
\n-a < -b or -a > -b<\/p>\n

Question 3.
\nHow do you read -17?
\nAnswer:
\nThe given number is: -17
\nHence,
\nWe can conclude that
\n-17 can read as:
\na. The opposite of 17
\nb. The number that is on the left side of the number line<\/p>\n

Question 4.
\nConstruct Arguments<\/strong> which amount represents a debt of two hundred fifty dollars, $250 or -$250? Explain.
\nAnswer:
\nWe know that,
\nA “Debt” is a negative number
\nHence, from the above,
\nWe can conclude that
\nThe representation of a debt of two hundred fifty dollars in the form of an integer is: -$250<\/p>\n

Question 5.
\nGeneralize<\/strong> when comparing two negative integers, how can you determine which integer is the greater number?
\nAnswer:
\nWe know that,
\nIn the number line,
\nThe numbers that are on the left side of 0 are called “Negative Numbers”
\nThe “Negative Numbers” decrease when moves from right to left
\nHence, from the above,
\nWe can conclude that
\nWhen comparing two negative integers,
\nThe\u00a0 negative number that is nearest to zero is the greater number<\/p>\n

Do You Know How?<\/strong><\/p>\n

In 6-17, write the opposite of each integer.<\/strong><\/p>\n

Question 6.
\n1
\nAnswer:
\nThe given integer is: 1
\nNow,
\nWe know that,
\nThe opposite of any integer ‘a’ is\u00a0‘<\/b>\u2212a‘<\/b>\u00a0Similarly, the opposite of any integer ‘\u2212a’ is ‘\u2212(\u2212a)’ = ‘a’
\nHence, from the above,
\nWe can conclude that
\nThe opposite of 1 is: -1<\/p>\n

Question 7.
\n-1
\nAnswer:
\nThe given integer is: -1
\nNow,
\nWe know that,
\nThe opposite of any integer ‘a’ is\u00a0‘<\/b>\u2212a‘<\/b>\u00a0Similarly, the opposite of any integer ‘\u2212a’ is ‘\u2212(\u2212a)’ = ‘a’
\nHence, from the above,
\nWe can conclude that
\nThe opposite of -1 is: 1<\/p>\n

Question 8.
\n-11
\nAnswer:
\nThe given integer is: -11
\nNow,
\nWe know that,
\nThe opposite of any integer ‘a’ is\u00a0‘<\/b>\u2212a‘<\/b>\u00a0Similarly, the opposite of any integer ‘\u2212a’ is ‘\u2212(\u2212a)’ = ‘a’
\nHence, from the above,
\nWe can conclude that
\nThe opposite of -11 is: 11<\/p>\n

Question 9.
\n30
\nAnswer:
\nThe given integer is: 30
\nNow,
\nWe know that,
\nThe opposite of any integer ‘a’ is\u00a0‘<\/b>\u2212a‘<\/b>\u00a0Similarly, the opposite of any integer ‘\u2212a’ is ‘\u2212(\u2212a)’ = ‘a’
\nHence, from the above,
\nWe can conclude that
\nThe opposite of 30 is: -30<\/p>\n

Question 10.
\n0
\nAnswer:
\nThe given integer is: 0
\nNow,
\nWe know that,
\nThe opposite of any integer ‘a’ is\u00a0‘<\/b>\u2212a‘<\/b>\u00a0Similarly, the opposite of any integer ‘\u2212a’ is ‘\u2212(\u2212a)’ = ‘a’
\nThere is not any opposite value for 0
\nHence, from the above,
\nWe can conclude that
\nThe opposite of 0 is: 0<\/p>\n

Question 11.
\n-16
\nAnswer:
\nThe given integer is: -16
\nNow,
\nWe know that,
\nThe opposite of any integer ‘a’ is\u00a0‘<\/b>\u2212a‘<\/b>\u00a0Similarly, the opposite of any integer ‘\u2212a’ is ‘\u2212(\u2212a)’ = ‘a’
\nHence, from the above,
\nWe can conclude that
\nThe opposite of -16 is: 16<\/p>\n

Question 12.
\n-(-8)
\nAnswer:
\nThe given integer is: -(-8)
\nNow,
\nWe know that,
\nThe opposite of any integer ‘a’ is\u00a0‘<\/b>\u2212a‘<\/b>\u00a0Similarly, the opposite of any integer ‘\u2212a’ is ‘\u2212(\u2212a)’ = ‘a’
\nHence, from the above,
\nWe can conclude that
\nThe opposite of -(-8) is: -8<\/p>\n

Question 13.
\n28
\nAnswer:
\nThe given integer is: 28
\nNow,
\nWe know that,
\nThe opposite of any integer ‘a’ is\u00a0‘<\/b>\u2212a‘<\/b>\u00a0Similarly, the opposite of any integer ‘\u2212a’ is ‘\u2212(\u2212a)’ = ‘a’
\nHence, from the above,
\nWe can conclude that
\nThe opposite of 28 is: -28<\/p>\n

Question 14.
\n-(-65)
\nAnswer:
\nThe given integer is: -(-65)
\nNow,
\nWe know that,
\nThe opposite of any integer ‘a’ is\u00a0‘<\/b>\u2212a‘<\/b>\u00a0Similarly, the opposite of any integer ‘\u2212a’ is ‘\u2212(\u2212a)’ = ‘a’
\nHence, from the above,
\nWe can conclude that
\nThe opposite of -(-65) is: -65<\/p>\n

Question 15.
\n98
\nAnswer:
\nThe given integer is: 98
\nNow,
\nWe know that,
\nThe opposite of any integer ‘a’ is\u00a0‘<\/b>\u2212a‘<\/b>\u00a0Similarly, the opposite of any integer ‘\u2212a’ is ‘\u2212(\u2212a)’ = ‘a’
\nHence, from the above,
\nWe can conclude that
\nThe opposite of 98 is: -98<\/p>\n

Question 16.
\n100
\nAnswer:
\nThe given integer is: 100
\nNow,
\nWe know that,
\nThe opposite of any integer ‘a’ is\u00a0‘<\/b>\u2212a‘<\/b>\u00a0Similarly, the opposite of any integer ‘\u2212a’ is ‘\u2212(\u2212a)’ = ‘a’
\nHence, from the above,
\nWe can conclude that
\nThe opposite of 100 is: -100<\/p>\n

Question 17.
\n-33
\nAnswer:
\nThe given integer is: -33
\nNow,
\nWe know that,
\nThe opposite of any integer ‘a’ is\u00a0‘<\/b>\u2212a‘<\/b>\u00a0Similarly, the opposite of any integer ‘\u2212a’ is ‘\u2212(\u2212a)’ = ‘a’
\nHence, from the above,
\nWe can conclude that
\nThe opposite of -33 is: 33<\/p>\n

In 18-20, write the integers in order from least to greatest.<\/strong><\/p>\n

Question 18.
\n2, -3, 0, -4
\nAnswer:
\nThe given integers are: 2, -3, 0, -4
\nNow,
\nWe know that,
\nIn a number line,
\nThe numbers after zero increase from left to right i.e., the farthest number after zero will be a greater number
\nThe numbers before zero decrease from right to left i.e., the nearest number to zero will be a greater number
\nHence, from the above,
\nWe can conclude that
\nThe given integers from the least to the greatest is: -4, -3, 0, 2<\/p>\n

Question 19.
\n4, 12, -12, -11
\nAnswer:
\nThe given integers are: 4, 12, -12, -11
\nNow,
\nWe know that,
\nIn a number line,
\nThe numbers after zero increase from left to right i.e., the farthest number after zero will be a greater number
\nThe numbers before zero decrease from right to left i.e., the nearest number to zero will be a greater number
\nHence, from the above,
\nWe can conclude that
\nThe given integers from the least to the greatest is: -12, -11, 4, 12<\/p>\n

Question 20.
\n-5, 6, -7, -8
\nAnswer:
\nThe given integers are: -5, 6, -7, -8
\nNow,
\nWe know that,
\nIn a number line,
\nThe numbers after zero increase from left to right i.e., the farthest number after zero will be a greater number
\nThe numbers before zero decrease from right to left i.e., the nearest number to zero will be a greater number
\nHence, from the above,
\nWe can conclude that
\nThe given integers from the least to the greatest is: -8, -7, -5, 6<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Scan for Multimedia<\/strong><\/p>\n

In 21-24, use the pictures at the right.
\n\"Envision
\n<\/strong><\/p>\n

Question 21.
\nGeneralize<\/strong> Which integer represents sea level? Explain.
\nAnswer:
\nWe know that,
\nWe will represent the real-world situations always like above the sea level and below the sea level
\nNow,
\nWhen we observe a number line,
\nWe will say the numbers above zero and the numbers below zero
\nNow,
\nWhen we compare the 2 definitions,
\nWe can say that
\nThe integer that represents the sea level is: zero
\nHence, from the above,
\nWe can conclude that
\nThe integer that represents the sea level is: Zero<\/p>\n

Question 22.
\nUse a negative integer to represent the depth to which a dolphin may swim.
\nAnswer:
\nFrom the given figure,
\nWe can observe that
\nA dolphin can swim to 150 feet below the sea level
\nNow,
\nWe know that,
\nThe values below the sea level are all “Negative numbers”
\nHence, from the above,
\nWe can conclude that
\nThe representation of the depth to which a dolphin may swim in the form of an integer is: -150 feet<\/p>\n

Question 23.
\nWhich of these animals can travel at the greatest distance from sea level?
\nAnswer:
\nWe know that,
\nThe values that are above the sea level are “Positive values” and the value that is farthest from the sea level is the greatest value
\nThe values that are below the sea level are “Negative values” and the value that is nearest to the sea level is the greatest value
\nHence, from the above,
\nWe can conclude that
\nThe animal that can travel at the greatest distance above sea level is: Griffon
\nThe animal that can travel at the greatest distance below sea level is: Sperm whale<\/p>\n

Question 24.
\nOrder the elevations of the animals as integers from least to greatest.
\nAnswer:
\nThe given figure is:
\n\"Envision
\n<\/strong>Now,
\nWe know that,
\nWe know that,
\nThe values that are above the sea level are “Positive values” and the value that is farthest from the sea level is the greatest value
\nThe values that are below the sea level are “Negative values” and the value that is nearest to the sea level is the greatest value
\nHence, from the above,
\nWe can conclude that
\nThe elevations of the animals as integers from the least to the greatest is:
\nSperm whale, Dolphin, Migrating bird, Griffon<\/p>\n

In 25-30, plot each point on the number line below.<\/strong><\/p>\n

\"Envision<\/p>\n

Question 25.
\nG(-10)
\nAnswer:
\nThe given point is: G (-10)
\nHence,
\nThe representation of the given point on the number line is:
\n\"\"<\/p>\n

Question 26.
\nH(8)
\nAnswer:
\nThe given point is: H (8)
\nHence,
\nThe representation of the given point on the number line is:
\n\"\"<\/p>\n

Question 27.
\nI(-1)
\nAnswer:
\nThe given point is: I (-1)
\nHence,
\nThe representation of the given point on the number line is:
\n\"\"<\/p>\n

Question 28.
\nJ(9)
\nAnswer:
\nThe given point is: J (9)
\nHence,
\nThe representation of the given point on the number line is:
\n\"\"<\/p>\n

Question 29.
\nK(6)
\nAnswer:
\nThe given point is: K (6)
\nHence,
\nThe representation of the given point on the number line is:
\n\"\"<\/p>\n

Question 30.
\nL(-3)
\nAnswer:
\nThe given point is: L (-3)
\nHence,
\nThe representation of the given point on the number line is:
\n\"\"<\/p>\n

In 31-36, write the integer value that each point represents. Then use the number line to help write its opposite<\/strong>.
\n\"Envision<\/p>\n

Question 31.
\nA
\nAnswer:
\nThe given number line is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe value of A is: -7
\nNow,
\nWe know that,
\nThe opposite of any integer ‘a’ is\u00a0‘<\/b>\u2212a‘<\/b>\u00a0Similarly, the opposite of any integer ‘\u2212a’ is ‘\u2212(\u2212a)’ = ‘a’
\nSo,
\nThe opposite value of A is: 7
\nHence, from the above,
\nWe can conclude that
\nThe integer value that A represents is: -7
\nThe opposite value of -7 is: 7<\/p>\n

Question 32.
\nB
\nAnswer:
\nThe given number line is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe value of B is: 4
\nNow,
\nWe know that,
\nThe opposite of any integer ‘a’ is\u00a0‘<\/b>\u2212a‘<\/b>\u00a0Similarly, the opposite of any integer ‘\u2212a’ is ‘\u2212(\u2212a)’ = ‘a’
\nSo,
\nThe opposite value of B is: -4
\nHence, from the above,
\nWe can conclude that
\nThe integer value that B represents is: 4
\nThe opposite value of 4 is: -4<\/p>\n

Question 33.
\nC
\nAnswer:
\nThe given number line is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe value of C is: 0
\nNow,
\nWe know that,
\nThe opposite of any integer ‘a’ is\u00a0‘<\/b>\u2212a‘<\/b>\u00a0Similarly, the opposite of any integer ‘\u2212a’ is ‘\u2212(\u2212a)’ = ‘a’
\nThe opposite of 0 is 0 i.e., there is not any opposite value for 0
\nSo,
\nThe opposite value of C is: 0
\nHence, from the above,
\nWe can conclude that
\nThe integer value that C represents is: 0
\nThe opposite value of 0 is: 0<\/p>\n

Question 34.
\nD
\nAnswer:
\nThe given number line is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe value of D is: -2
\nNow,
\nWe know that,
\nThe opposite of any integer ‘a’ is\u00a0‘<\/b>\u2212a‘<\/b>\u00a0Similarly, the opposite of any integer ‘\u2212a’ is ‘\u2212(\u2212a)’ = ‘a’
\nSo,
\nThe opposite value of D is: 2
\nHence, from the above,
\nWe can conclude that
\nThe integer value that D represents is: -2
\nThe opposite value of -2 is: 2<\/p>\n

Question 35.
\nE
\nAnswer:
\nThe given number line is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe value of E is: 2
\nNow,
\nWe know that,
\nThe opposite of any integer ‘a’ is\u00a0‘<\/b>\u2212a‘<\/b>\u00a0Similarly, the opposite of any integer ‘\u2212a’ is ‘\u2212(\u2212a)’ = ‘a’
\nSo,
\nThe opposite value of E is: -2
\nHence, from the above,
\nWe can conclude that
\nThe integer value that E represents is: 2
\nThe opposite value of 2 is: -2<\/p>\n

Question 36.
\nF
\nAnswer:
\nThe given number line is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe value of F is: -5
\nNow,
\nWe know that,
\nThe opposite of any integer ‘a’ is\u00a0‘<\/b>\u2212a‘<\/b>\u00a0Similarly, the opposite of any integer ‘\u2212a’ is ‘\u2212(\u2212a)’ = ‘a’
\nSo,
\nThe opposite value of F is: 5
\nHence, from the above,
\nWe can conclude that
\nThe integer value that F represents is: -5
\nThe opposite value of -5 is: 5<\/p>\n

Question 37.
\nWrite the opposite of each integer.
\nA. 5
\nAnswer:
\nThe given integer is: 5
\nNow,
\nWe know that,
\nThe opposite of any integer ‘a’ is\u00a0‘<\/b>\u2212a‘<\/b>\u00a0Similarly, the opposite of any integer ‘\u2212a’ is ‘\u2212(\u2212a)’ = ‘a’
\nHence, from the above,
\nWe can conclude that
\nThe opposite of 5 is: -5<\/p>\n

B. -13
\nAnswer:
\nThe given integer is: -13
\nNow,
\nWe know that,
\nThe opposite of any integer ‘a’ is\u00a0‘<\/b>\u2212a‘<\/b>\u00a0Similarly, the opposite of any integer ‘\u2212a’ is ‘\u2212(\u2212a)’ = ‘a’
\nHence, from the above,
\nWe can conclude that
\nThe opposite of -13 is: 13<\/p>\n

C. -(-22)
\nAnswer:
\nThe given integer is: -(-22)
\nNow,
\nWe know that,
\nThe opposite of any integer ‘a’ is\u00a0‘<\/b>\u2212a‘<\/b>\u00a0Similarly, the opposite of any integer ‘\u2212a’ is ‘\u2212(\u2212a)’ = ‘a’
\nHence, from the above,
\nWe can conclude that
\nThe opposite of -(-22) is: -22<\/p>\n

D. -31
\nAnswer:
\nThe given integer is: -31
\nNow,
\nWe know that,
\nThe opposite of any integer ‘a’ is\u00a0‘<\/b>\u2212a‘<\/b>\u00a0Similarly, the opposite of any integer ‘\u2212a’ is ‘\u2212(\u2212a)’ = ‘a’
\nHence, from the above,
\nWe can conclude that
\nThe opposite of -31 is: 31<\/p>\n

E. -50
\nAnswer:
\nThe given integer is: -50
\nNow,
\nWe know that,
\nThe opposite of any integer ‘a’ is\u00a0‘<\/b>\u2212a‘<\/b>\u00a0Similarly, the opposite of any integer ‘\u2212a’ is ‘\u2212(\u2212a)’ = ‘a’
\nHence, from the above,
\nWe can conclude that
\nThe opposite of -50 is: 50<\/p>\n

F. -(-66)
\nAnswer:
\nThe given integer is: -(-66)
\nNow,
\nWe know that,
\nThe opposite of any integer ‘a’ is\u00a0‘<\/b>\u2212a‘<\/b>\u00a0Similarly, the opposite of any integer ‘\u2212a’ is ‘\u2212(\u2212a)’ = ‘a’
\nHence, from the above,
\nWe can conclude that
\nThe opposite of -(-66) is: -66<\/p>\n

Question 38.
\nCompare the integers and write the integer with the greater value.<\/strong>
\nA. -5, 1
\nAnswer:
\nThe given integers are: -5, 1
\nNow,
\nWe know that,
\nIn a number line,
\nThe numbers that are after zero are “Positive numbers” and the numbers that are farthest to zero are the greater numbers
\nThe numbers that are before zero are “Negative numbers” and the numbers that are nearest to zero are the least numbers
\nThe “Positive numbers” are always greater than “Negative numbers”
\nHence, from the above,
\nWe can conclude that
\n1 > -5
\nThe greater value is: 1<\/p>\n

B. -6, -7
\nAnswer:
\nThe given integers are: -6, -7
\nNow,
\nWe know that,
\nIn a number line,
\nThe numbers that are after zero are “Positive numbers” and the numbers that are farthest to zero are the greater numbers
\nThe numbers that are before zero are “Negative numbers” and the numbers that are nearest to zero are the least numbers
\nThe “Positive numbers” are always greater than “Negative numbers”
\nHence, from the above,
\nWe can conclude that
\n-6 > -7
\nThe greater value is: -6<\/p>\n

C. -9, 8
\nAnswer:
\nThe given integers are: -9, 8
\nNow,
\nWe know that,
\nIn a number line,
\nThe numbers that are after zero are “Positive numbers” and the numbers that are farthest to zero are the greater numbers
\nThe numbers that are before zero are “Negative numbers” and the numbers that are nearest to zero are the least numbers
\nThe “Positive numbers” are always greater than “Negative numbers”
\nHence, from the above,
\nWe can conclude that
\n8 > -9
\nThe greater value is: 8<\/p>\n

D. -12, -(-10)
\nAnswer:
\nThe given integers are: -12, -(-10)
\nSo,
\nThe given integers are: -12, 10
\nNow,
\nWe know that,
\nIn a number line,
\nThe numbers that are after zero are “Positive numbers” and the numbers that are farthest to zero are the greater numbers
\nThe numbers that are before zero are “Negative numbers” and the numbers that are nearest to zero are the least numbers
\nThe “Positive numbers” are always greater than “Negative numbers”
\nHence, from the above,
\nWe can conclude that
\n10 > -12
\nThe greater value is: 10<\/p>\n

E. -(-9), 11
\nAnswer:
\nThe given integers are: -(-9), 11
\nSo,
\nThe given integers are: 9, 11
\nNow,
\nWe know that,
\nIn a number line,
\nThe numbers that are after zero are “Positive numbers” and the numbers that are farthest to zero are the greater numbers
\nThe numbers that are before zero are “Negative numbers” and the numbers that are nearest to zero are the least numbers
\nThe “Positive numbers” are always greater than “Negative numbers”
\nHence, from the above,
\nWe can conclude that
\n11 > 9
\nThe greater value is: 11<\/p>\n

F. -(-4), 3
\nAnswer:
\nThe given integers are: -(-4), 3
\nSo,
\nThe given integers are: 4, 3
\nNow,
\nWe know that,
\nIn a number line,
\nThe numbers that are after zero are “Positive numbers” and the numbers that are farthest to zero are the greater numbers
\nThe numbers that are before zero are “Negative numbers” and the numbers that are nearest to zero are the least numbers
\nThe “Positive numbers” are always greater than “Negative numbers”
\nHence, from the above,
\nWe can conclude that
\n4 > 3
\nThe greater value is: 4<\/p>\n

Question 39.
\nThe display at the right shows the daily low temperatures for several consecutive days in a New England city. Write the temperatures in order from least to greatest. On which day was it the coldest?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe display at the right shows the daily low temperatures for several consecutive days in a New England city.
\nNow,
\nThe given temperatures are:
\n\"Envision
\nNow,
\nWe know that,
\nIn a number line,
\nThe numbers that are after zero are “Positive numbers” and the numbers that are farthest to zero are the greater numbers
\nThe numbers that are before zero are “Negative numbers” and the numbers that are nearest to zero are the least numbers
\nThe “Positive numbers” are always greater than “Negative numbers”
\nSo,
\nThe temperatures in order from the last to the greatest is:
\n-7\u00b0 < -5\u00b0 < 4\u00b0 < 7\u00b0
\nThe coldest day from the given days is: Wednesday
\nHence, from the above,
\nWe can conclude that
\nThe temperatures in order from the last to the greatest is:
\n-7\u00b0 < -5\u00b0 < 4\u00b0 < 7\u00b0
\nThe coldest day from the given days is: Wednesday<\/p>\n

Question 40.
\nIn a bank account, a paid-out expense is called a debit, and a deposit is called a credit. Would you use positive or negative integers to represent credits? Debits? Explain.
\nAnswer:
\nIt is given that
\nIn a bank account, a paid-out expense is called a debit, and a deposit is called a credit.
\nNow,
\nWe know that,
\nA “Paid-out expense” or “Debit” is a negative value
\nA “Deposit” or “Credit” is a positive value
\nHence, from the above,
\nWe can conclude that
\nWe will use “Positive integers” to represent “Credits”
\nWe will use “Negative integers” to represent “Debits”<\/p>\n

Question 41.
\nHigher-Order Thinking<\/strong> Atoms have negatively charged particles called electrons and positively charged particles called protons. If an atom loses an electron, it has a positive electric charge. If it gains an electron, it has a negative electric charge. Which integer would represent the electric charge of an atom that has an equal number of electrons and protons?
\nAnswer:
\nIt is given that
\nAtoms have negatively charged particles called electrons and positively charged particles called protons. If an atom loses an electron, it has a positive electric charge. If it gains an electron, it has a negative electric charge.
\nNow,
\nWe know that,
\n“Protons” and “Electrons” have the same values but opposite signs
\nSo,
\nThe value of the integer that has an equal number of electrons and protons =\u00a0 (The number of Protons) – (The number of Electrons)
\n= 0 (Since both have the same values but with opposite signs)
\nHence, from the above,
\nWe can conclude that
\nThe value of the integer that has an equal number of electrons and protons is: 0<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 42.
\nMarco goes on a recreational scuba diving expedition. What is a possible diving depth for his expedition?
\nA. 0 meters
\nB. 40 meters
\nC. 400 meters
\nD. -40 meters
\nAnswer:
\nIt is given that
\nMarco goes on a recreational scuba diving expedition.
\nNow,
\nIn terms of sea level,
\nThe “Height” represents the positive values
\nThe “Depth” represents the negative values
\nHence, from the above,
\nWe can conclude that
\nThe possible diving depth for Marco’s expedition is:
\n\"\"<\/p>\n

Question 43.
\nFill in the bubbles to match each integer with its opposite.
\n\"Envision
\nAnswer:
\nThe bubbles that matched with each integer and its opposite are:
\n\"\"<\/p>\n

Lesson 2.2 Represent Rational Numbers on the Number Line<\/h3>\n

ACTIVITY<\/strong><\/p>\n

Explore It!<\/strong>
\nThe locations of four animals relative to sea level are shown.
\n\"Envision<\/p>\n

A. What can you say about the animals and their positions relative to sea level?
\nAnswer: It is given that
\nThe locations of four animals relative to sea level are shown.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nWe know that,
\nThe sea level is considered as zero
\nThe values above the sea level are positive
\nThe values below the sea level are negative
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe animals that are below the sea level are: Dolphin, Sea Turtle, and Shark
\nThe animals that are above sea level are: Seagull
\nHence, from the above,
\nWe can conclude that
\nThe animals that are below the sea level are: Dolphin, Sea Turtle, and Shark
\nThe animals that are above sea level are: Seagull<\/p>\n

B. How can you use a number line to represent the locations of the animals?<\/p>\n

\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nIn terms of a number line,
\nThe sea level is considered zero
\nThe values above zero (Sea level) are positive
\nThe values below zero (Sea level) are negative
\nSo,
\nThe representations of the locations of the given animals in a number line are:
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe representation of the locations of the animals in a number line is:
\n\"\"<\/p>\n

Focus on math practices<\/strong>
\nGeneralize<\/strong> How is representing the locations of negative fractions and decimals like representing the locations of positive fractions and decimals? How is it different?
\nAnswer:
\nWe know that,
\nIn a number line,
\nThe values above zero are “Positive numbers”
\nThe values below zero are “Negative numbers”
\nHence,
\nIf the fractions and decimals are positive, then they are above zero in a number line
\nIf the fractions and decimals are negative, then they are below zero in a number line<\/p>\n

Visual Learning<\/strong><\/p>\n

? Essential Question<\/strong> How can you plot, compare, and order rational numbers using a number line?
\nAnswer:
\nJust as positive and negative integers can be represented on a number line, so can positive and negative rational numbers. You can use a number line to help you compare and order negative rational numbers.
\nWhen comparing the values of two numbers, you can use a number line to determine\u00a0which number is greater. The number on the right is always greater than the number on the left<\/p>\n

Try It!<\/strong>
\nHow can you find and position –\\(\\frac{5}{4}\\) and -1.75 on the number lines? Write –\\(\\frac{5}{4}\\) and -1.75 as mixed numbers, then plot the points on the number lines.
\n\"Envision<\/p>\n

\"Envision
\nAnswer:
\nThe given numbers are: –\\(\\frac{5}{4}\\) and -1.75
\nNow,
\nThe representation of –\\(\\frac{5}{4}\\) into a mixed number is:
\n\"\"
\nThe representation of -1.75 into a mixed number is:
\n\"\"
\nSo,
\nThe representation of –\\(\\frac{5}{4}\\) on a number line is:
\n\"\"
\nSo,
\nThe representation of -1.75 on a number line is:
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given numbers as mixed numbers are:
\n\"\"
\nThe representation of the given numbers on a number line is:
\n\"\"\"\"<\/p>\n

Convince Me!<\/strong> Why is it helpful to rename –\\(\\frac{5}{4}\\) and -1.75 as mixed numbers when plotting these points on number lines?
\nAnswer:
\nThe given numbers are: –\\(\\frac{5}{4}\\) and -1.75
\nNow,
\nThe representations of the given numbers into mixed numbers are:
\n\"\"
\nNow,
\nWhen we observe the fraction part in the mixed numbers,
\nWe can draw a conclusion as to how much nearer to the whole numbers the fraction part is
\nNow,
\nWe know that,
\nBetween 2 numbers,
\n\\(\\frac{1}{4}\\) is near to the previous number
\n\\(\\frac{1}{2}\\) is between 2 numbers
\n\\(\\frac{3}{4}\\) is near to the next number
\nHence,
\nWe will rewrite the numbers into mixed numbers to facilitate the marking of the numbers on the number line<\/p>\n

Try It!<\/strong>
\nIf \\(\\frac{1}{4}\\) is ordered within the list of numbers in the example above, between which two numbers would it be placed?
\nAnswer:
\nThe given numbers in Example 2 are:
\n\\(\\frac{2}{3}\\), 1.75 and -0.75
\nNow,
\nThe given numbers from the given information will be:
\n\\(\\frac{2}{3}\\), \\(\\frac{1}{4}\\), 1.75 and -0.75
\nNow,
\nConvert all the given numbers into decimal numbers to compare the numbers from the least to the greatest
\nSo,
\nThe representation of the given fractions into decimal numbers are:
\n\\(\\frac{2}{3}\\) = 0.66
\n\\(\\frac{1}{4}\\) = 0.25
\nSo,
\nThe order of the given numbers from the least t the greatest is:
\n-0.75 < \\(\\frac{1}{4}\\) < \\(\\frac{2}{3}\\) < 1.75
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{1}{4}\\) would be placed between -0.75 and \\(\\frac{2}{3}\\)<\/p>\n

Try It!<\/strong>
\nAt 10:00 P.M. one winter night, the temperature was -3\u00b0C. At midnight, the temperature was -7\u00b0C. Use <, >, or = to compare the two temperatures and explain their relationship.
\nAnswer:
\nIt is given that
\nAt 10:00 P.M. one winter night, the temperature was -3\u00b0C. At midnight, the temperature was -7\u00b0C.
\nNow,
\nWe know that,
\nThe values before zero are called “Negative Numbers”
\nThe values before zero decrease as we move from right to left
\nThe farthest value at the leftmost side of a number line is the “least value”
\nSo,
\n-3\u00b0 C > -7\u00b0 C
\nHence, from the above,
\nWe can conclude that
\nThe temperature at 10 P.M is greater than the midnight<\/p>\n

KEY CONCEPT<\/strong>
\nA rational number can be expressed as a fraction in the form \\(\\frac{a}{b}\\) and –\\(\\frac{a}{b}\\) are integers and b is not 0.<\/p>\n

\"Envision
\nThe numbers, in order from least to greatest, are: -1.75, \\(\\frac{3}{5}\\), 1.25<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n? Essential Question<\/strong> How can you plot, compare, and order rational numbers using a number line?
\nAnswer:
\nJust as positive and negative integers can be represented on a number line, so can positive and negative rational numbers. You can use a number line to help you compare and order negative rational numbers.
\nWhen comparing the values of two numbers, you can use a number line to determine\u00a0which number is greater. The number on the right is always greater than the number on the left<\/p>\n

Question 2.
\nGeneralize<\/strong> Why are whole numbers rational numbers? Use 15 as an example.
\nAnswer:
\nRational numbers are the numbers that can be written in the form of \\(\\frac{p}{q}\\) where q \u2260 0.\u00a0 “Whole numbers” are the numbers that start from 0 to \u221e. Whole numbers can be written in the form of \\(\\frac{0}{1}\\), \\(\\frac{1}{1}\\), \\(\\frac{2}{1}\\), etc. Thus, every whole number is a rational number but every rational number is not a whole number.
\nExample:
\n15 = \\(\\frac{15}{1}\\)
\nWe know that,
\n15 is a “Whole number”
\nSo,
\nWe can write 15 in the form of \\(\\frac{p}{q}\\) i.e., a “Rational number”<\/p>\n

Question 3.
\nVocabulary<\/strong> Why are integers rational numbers? Give an example.
\nAnswer:
\nAn “Integer” can be written as a fraction by giving it a denominator of one, So, any integer is a rational number
\nExamples: 17 and -34 are integers as well as rational numbers<\/p>\n

Question 4.
\nReasoning<\/strong> Explain how the inequality -4\u00b0C > -9\u00b0C describes how the temperatures are related.
\nAnswer:
\nThe given inequality is: -4\u00b0C > -9\u00b0C
\nNow,
\nWe know that,
\nIn a number line,
\nThe values before zero are “Negative values”
\nThe negative values on a number line decrease from right to left
\nSo,
\nIn the negative values,
\nThe large number will be a small number and vice-versa
\nHence, from the above,
\nWe can conclude that
\n-4\u00b0 C is greater than -9\u00b0 C<\/p>\n

Do You Know How?<\/strong><\/p>\n

In 5-7, write the number positioned at each point.<\/strong>
\n\"Envision<\/p>\n

Question 5.
\nA
\nAnswer:
\nThe given number line is:
\n\"Envision
\nHence,
\nThe representation of the value of A on the given number line is:
\n\"\"<\/p>\n

Question 6.
\nB
\nAnswer:
\nThe given number line is:
\n\"Envision
\nHence,
\nThe representation of the value of B on the given number line is:
\n\"\"<\/p>\n

Question 7.
\nC
\nAnswer:
\nThe given number line is:
\n\"Envision
\nHence,
\nThe representation of the value of C on the given number line is:
\n\"\"<\/p>\n

In 8-11, plot the points on the number line below.<\/strong><\/p>\n

\"Envision<\/p>\n

Question 8.
\nP at -1\\(\\frac{1}{4}\\)
\nAnswer:
\nThe given point is: P (-1\\(\\frac{1}{4}\\))
\nNow,
\nThe representation of -1\\(\\frac{1}{4}\\) into a decimal number is:
\n\"\"
\n-1\\(\\frac{1}{4}\\) = -1.25
\nHence,
\nThe representation of P on the given number line is:
\n\"\"<\/p>\n

Question 9.
\nQ at 0.25
\nAnswer:
\nThe given point is: Q (0.25)
\nHence,
\nThe representation of Q on the given number line is:
\n\"\"<\/p>\n

Question 10.
\nR at -0.75
\nAnswer:
\nThe given point is: R (-0.75)
\nHence,
\nThe representation of R on the given number line is:
\n\"\"<\/p>\n

Question 11.
\nS at –\\(\\frac{1}{4}\\)
\nAnswer:
\nThe given point is: S (-\\(\\frac{1}{4}\\))
\nNow,
\nThe representation of –\\(\\frac{1}{4}\\) into a decimal number is:
\n\"\"<\/p>\n

In 12-14, use the number line to help order the numbers from least to greatest.<\/strong><\/p>\n

\"Envision<\/p>\n

Question 12.
\n1.25, –\\(\\frac{3}{2}\\), -1.25, 1\\(\\frac{1}{2}\\)
\nAnswer:
\nThe given numbers are:
\n1.25, –\\(\\frac{3}{2}\\), -1.25, 1\\(\\frac{1}{2}\\)
\nNow,
\nThe representation of the given numbers into decimal numbers are:
\n1.25, -1.5, -1.25, 1.5
\nHence, from the above,
\nWe can conclude that
\nThe numbers that are from the least to the greatest are: -1.5, -1.25, 1.25, 1.5<\/p>\n

Question 13.
\n\u2013 0.5, \\(\\frac{1}{2}\\), -0.75, \\(\\frac{3}{4}\\)
\nAnswer:
\nThe given numbers are:
\n\u2013 0.5, \\(\\frac{1}{2}\\), -0.75, \\(\\frac{3}{4}\\)
\nNow,
\nThe representation of the given numbers into decimal numbers are:
\n-0.5, 0.5, -0.75, 0.75
\nHence, from the above,
\nWe can conclude that
\nThe numbers that are from the least to the greatest are: -0.75, -0.5, 0.5, 0.75<\/p>\n

Question 14.
\n-1.5, -0.75, -1,2
\nAnswer:
\nThe given numbers are:
\n-1.5, -0.75, -1,2
\nHence, from the above,
\nWe can conclude that
\nThe numbers that are from the least to the greatest are: -1.5, -1.2, -0.75<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Scan for Multimedia<\/strong><\/p>\n

In 15-20, write the number positioned at each point.<\/strong><\/p>\n

\"Envision<\/p>\n

Question 15.
\nA
\nAnswer:
\nThe given number line is:
\n\"Envision
\nNow,
\nWe know that,
\nEach line in the given number line represents 0.25 units
\nHence, from the above,
\nWe can conclude that
\nThe value of A in the given number line is: -3.25<\/p>\n

Question 16.
\nB
\nAnswer:
\nThe given number line is:
\n\"Envision
\nNow,
\nWe know that,
\nEach line in the given number line represents 0.25 units
\nHence, from the above,
\nWe can conclude that
\nThe value of B in the given number line is: -4.5<\/p>\n

Question 17.
\nC
\nAnswer:
\nThe given number line is:
\n\"Envision
\nNow,
\nWe know that,
\nEach line in the given number line represents 0.25 units
\nHence, from the above,
\nWe can conclude that
\nThe value of C in the given number line is: 1.25<\/p>\n

Question 18.
\nD
\nAnswer:
\nThe given number line is:
\n\"Envision
\nNow,
\nWe know that,
\nEach line in the given number line represents 0.25 units
\nHence, from the above,
\nWe can conclude that
\nThe value of D in the given number line is: -5.75<\/p>\n

Question 19.
\nE
\nAnswer:
\nThe given number line is:
\n\"Envision
\nNow,
\nWe know that,
\nEach line in the given number line represents 0.25 units
\nHence, from the above,
\nWe can conclude that
\nThe value of E in the given number line is: 0.5<\/p>\n

Question 20.
\nF
\nAnswer:
\nThe given number line is:
\n\"Envision
\nNow,
\nWe know that,
\nEach line in the given number line represents 0.25 units
\nHence, from the above,
\nWe can conclude that
\nThe value of F in the given number line is: -2.5<\/p>\n

Question 21.
\nPlot the numbers on the number line below.
\n\"Envision<\/p>\n

A. -5\\(\\frac{1}{2}\\)
\nAnswer:
\nThe given fraction is: -5\\(\\frac{1}{2}\\)
\nNow,
\nThe given number line is:
\n\"Envision
\nNow,
\nThe representation of the given fraction into a decimal number is: -5.5
\nFrom the given number line,
\nWe can observe that
\nEach line in the number line represents 0.1 units
\nHence,
\nThe representation of the given fraction in the given number line is:\"\"<\/p>\n

B. -6.3
\nAnswer:
\nThe given number is: -6.3
\nNow,
\nThe given number line is:
\n\"Envision
\nNow,
\nFrom the given number line,
\nWe can observe that
\nEach line in the number line represents 0.1 units
\nHence,
\nThe representation of the given decimal number in the given number line is:\"\"<\/p>\n

C. -5.8
\nAnswer:
\nThe given number is: -5.8
\nNow,
\nThe given number line is:
\n\"Envision
\nNow,
\nFrom the given number line,
\nWe can observe that
\nEach line in the number line represents 0.1 units
\nHence,
\nThe representation of the given decimal number in the given number line is:
\n\"\"<\/p>\n

D. -6\\(\\frac{7}{10}\\)
\nAnswer:
\nThe given fraction is: -6\\(\\frac{7}{10}\\)
\nNow,
\nThe given number line is:
\n\"Envision
\nNow,
\nThe representation of the given fraction into a decimal number is: -6.7
\nFrom the given number line,
\nWe can observe that
\nEach line in the number line represents 0.1 units
\nHence,
\nThe representation of the given fraction in the given number line is:\"\"<\/p>\n

E. -4.9
\nAnswer:
\nThe given number is: -4.9
\nNow,
\nThe given number line is:
\n\"Envision
\nNow,
\nFrom the given number line,
\nWe can observe that
\nEach line in the number line represents 0.1 units
\nHence,
\nThe representation of the given decimal number in the given number line is:
\n\"\"
\nF. -6\\(\\frac{9}{10}\\)
\nAnswer:
\nThe given fraction is: -6\\(\\frac{9}{10}\\)
\nNow,
\nThe given number line is:
\n\"Envision
\nNow,
\nThe representation of the given fraction into a decimal number is: -6.9
\nFrom the given number line,
\nWe can observe that
\nEach line in the number line represents 0.1 units
\nHence,
\nThe representation of the given fraction in the given number line is:
\n\"\"<\/p>\n

Question 22.
\nUse <, >, or = to compare.<\/strong>
\nA. \\(\\frac{1}{10}\\) \"Envision 0.09
\nAnswer:
\nThe given numbers are: \\(\\frac{1}{10}\\), 0.09
\nNow,
\nConvert –\\(\\frac{1}{10}\\) into a decimal number
\nSo,
\n\\(\\frac{1}{10}\\) = 0.10
\nHence, from the above,
\nWe can conclude that
\n\"\"
\nB. \u20131.44 \"Envision -1\\(\\frac{1}{4}\\)
\nAnswer:
\nThe given numbers are: -1.44, -1\\(\\frac{1}{4}\\)
\nNow,
\nConvert -1\\(\\frac{1}{4}\\) into a decimal number
\nSo,
\n1\\(\\frac{1}{4}\\) = 1.25
\nHence, from the above,
\nWe can conclude that
\n\"\"
\nC. –\\(\\frac{2}{3}\\) \"Envision -0.8
\nAnswer:
\nThe given numbers are: -0.8, –\\(\\frac{2}{3}\\)
\nNow,
\nConvert –\\(\\frac{2}{3}\\) into a decimal number
\nSo,
\n\\(\\frac{2}{3}\\) = 0.66
\nHence, from the above,
\nWe can conclude that
\n\"\"
\nD. 0.5 \"Envision \\(\\frac{2}{4}\\)
\nAnswer:
\nThe given numbers are: 0.5, \\(\\frac{2}{4}\\)
\nNow,
\nConvert \\(\\frac{2}{4}\\) into a decimal number
\nSo,
\n\\(\\frac{2}{4}\\) = 0.5
\nHence, from the above,
\nWe can conclude that
\n\"\"
\nE. -2\\(\\frac{3}{4}\\) \"Envision -2.25
\nAnswer:
\nThe given numbers are: -2\\(\\frac{3}{4}\\), -2.25
\nNow,
\nConvert 2\\(\\frac{3}{4}\\) into a decimal number
\nSo,
\n2\\(\\frac{3}{4}\\) = 2.75
\nHence, from the above,
\nWe can conclude that
\n\"\"
\nF. –\\(\\frac{3}{5}\\) \"Envision -0.35
\nAnswer:
\nThe given numbers are: –\\(\\frac{3}{5}\\), -0.35
\nNow,
\nConvert \\(\\frac{3}{5}\\) into a decimal number
\nSo,
\n\\(\\frac{3}{5}\\) = 0.60
\nHence, from the above,
\nWe can conclude that
\n\"\"<\/p>\n

Question 23.
\nOrder the numbers from least to greatest.
\nA. -6, 8, -9, 13
\nAnswer:
\nThe given numbers are: -6, 8, -9, 13
\nNow,
\nWe know that,
\nIn a number line,
\nThe positive values are greater than negative values
\nThe positive values increase from left to right
\nThe negative values decrease from right to left
\nHence, from the above,
\nWe can conclude that
\nThe order of the given numbers from the least to the greatest is: -9, -6, 8, 13<\/p>\n

B. –\\(\\frac{4}{5}\\), –\\(\\frac{1}{2}\\), 0.25, -0.2
\nAnswer:
\nThe given numbers are: –\\(\\frac{4}{5}\\), –\\(\\frac{1}{2}\\), 0.25, -0.2
\nNow,
\nWe know that,
\nIn a number line,
\nThe positive values are greater than negative values
\nThe positive values increase from left to right
\nThe negative values decrease from right to left
\nNow,
\nThe representation of the given numbers into a decimal number is: -0.8, -0.5, 0.25, -0.2
\nHence, from the above,
\nWe can conclude that
\nThe order of the given numbers from the least to the greatest is: -0.8, -0.5, -0.2, 0.25<\/p>\n

C. 4.75, -2\\(\\frac{1}{2}\\), –\\(\\frac{8}{3}\\), –\\(\\frac{9}{2}\\)
\nAnswer:
\nThe given numbers are: 4.75, -2\\(\\frac{1}{2}\\), –\\(\\frac{8}{3}\\), –\\(\\frac{9}{2}\\)
\nNow,
\nWe know that,
\nIn a number line,
\nThe positive values are greater than negative values
\nThe positive values increase from left to right
\nThe negative values decrease from right to left
\nNow,
\nThe representation of the given numbers into a decimal number is: 4.75, -2.5, -2.66, -4.5
\nHence, from the above,
\nWe can conclude that
\nThe order of the given numbers from the least to the greatest is: -4.5, -2.66, -2.5, 4.75<\/p>\n

D. 4, -3, -8, -1
\nAnswer:
\nThe given numbers are 4, -3, -8, -1
\nNow,
\nWe know that,
\nIn a number line,
\nThe positive values are greater than negative values
\nThe positive values increase from left to right
\nThe negative values decrease from right to left
\nHence, from the above,
\nWe can conclude that
\nThe order of the given numbers from the least to the greatest is: -8, -3, -1, 4<\/p>\n

E. –\\(\\frac{1}{4}\\), 0.5, \\(\\frac{3}{4}\\), –\\(\\frac{1}{2}\\)
\nAnswer:
\nThe given numbers are: –\\(\\frac{1}{4}\\), 0.5, \\(\\frac{3}{4}\\), –\\(\\frac{1}{2}\\)
\nNow,
\nWe know that,
\nIn a number line,
\nThe positive values are greater than negative values
\nThe positive values increase from left to right
\nThe negative values decrease from right to left
\nNow,
\nThe representation of the given numbers into a decimal number is: -0.25, 0.5, 0.75, -0.5
\nHence, from the above,
\nWe can conclude that
\nThe order of the given numbers from the least to the greatest is: -0.5, -0.25, 0.5, 0.75<\/p>\n

F. –\\(\\frac{4}{5}\\), –\\(\\frac{5}{4}\\), –\\(\\frac{3}{2}\\), 1.5
\nAnswer:
\nThe given numbers are: –\\(\\frac{4}{5}\\), –\\(\\frac{5}{4}\\), –\\(\\frac{3}{2}\\), 1.5
\nNow,
\nWe know that,
\nIn a number line,
\nThe positive values are greater than negative values
\nThe positive values increase from left to right
\nThe negative values decrease from right to left
\nNow,
\nThe representation of the given numbers into a decimal number is -0.8, -1.25, -1.5, 1.5
\nHence, from the above,
\nWe can conclude that
\nThe order of the given numbers from the least to the greatest is: -1.5, -1.25, -0.8, 1.5<\/p>\n

Question 24.
\nMake Sense and Persevere<\/strong> What is the least number of points you must plot to have examples of all four sets of numbers, including at least one positive integer and one negative integer? Explain.
\n\"Envision
\nAnswer:
\nLet’s define the sets:
\nIntegers: The set of all whole numbers.
\nRational numbers: Numbers that can be written as the quotient of two integer numbers.
\nNatural numbers: The set of positive integers.
\nWhole numbers: All the numbers that can be made by adding (or subtracting) 1 a given number of times.
\nHence, from the above,
\nWe can conclude that
\nWe can use only one example for all four sets of numbers, including at least one positive integer and one negative integer<\/p>\n

Question 25.
\nReasoning<\/strong> Suppose you plot the locations of the animals on a number line. Which animal would be represented by the point farthest from 0 on the number line? Explain.
\n\"Envision
\nAnswer:
\nIt is given that
\nYou plot the locations of the animals on a number line
\nNow,
\nThe given table is:
\n\"Envision
\nNow,
\nThe representation of the given data present in the table into a decimal number is:
\nBloodbelly comb jelly\u00a0 \u00a0 \u00a0 \u00a0 \u00a0–\u00a0 \u00a0 \u00a0 \u00a0 -0.8 km
\nDeep-sea anglerfish\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0–\u00a0 \u00a0 \u00a0 \u00a0 -0.66 km
\nFanfin anglerfish\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0–\u00a0 \u00a0 \u00a0 \u00a0 -2.25 km
\nGulper eel\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 –\u00a0 \u00a0 \u00a0 \u00a0 \u00a0-1.1 km
\nPacific blackdragon\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 –\u00a0 \u00a0 \u00a0 \u00a0 \u00a0-0.3 km
\nSlender snipe eel\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 –\u00a0 \u00a0 \u00a0 \u00a0 \u00a0-0.6 km
\nNow,
\nWe know that,
\nIn a number line,
\nThe negative decrease from right to left
\nThe negative number that is at the end of the number line on the left side is the “Least number”
\nHence, from the above,
\nWe can conclude that
\nThe animal which is farthest from 0 at the left side of the number line is: Fanfin anglerfish
\nThe animal which is farthest from 0 at the right side of the number line is: Pacific blackdragon<\/p>\n

Question 26.
\nWhich animal is closest to a depth of -0.7 km?
\nAnswer:
\nFrom Question 25,
\nThe given information is:
\n\"Envision
\nNow,
\nThe representation of the given data present in the table into a decimal number is:
\nBloodbelly comb jelly\u00a0 \u00a0 \u00a0 \u00a0 \u00a0–\u00a0 \u00a0 \u00a0 \u00a0 -0.8 km
\nDeep-sea anglerfish\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0–\u00a0 \u00a0 \u00a0 \u00a0 -0.66 km
\nFanfin anglerfish\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0–\u00a0 \u00a0 \u00a0 \u00a0 -2.25 km
\nGulper eel\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 –\u00a0 \u00a0 \u00a0 \u00a0 \u00a0-1.1 km
\nPacific blackdragon\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 –\u00a0 \u00a0 \u00a0 \u00a0 \u00a0-0.3 km
\nSlender snipe eel\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 –\u00a0 \u00a0 \u00a0 \u00a0 \u00a0-0.6 km
\nHence, from the above,
\nWe can conclude that
\nThe animal that is closest to a depth of -0.7 km is: Deep-sea anglerfish<\/p>\n

Question 27.
\nThe change in the value of a stock is represented by the rational number -5.90. Describe, in words, what this means.
\nAnswer:
\nIt is given that
\nThe change in the value of a stock is represented by the rational number -5.90.
\nNow,
\nFrom the given information,
\nWe can observe that
\nThe given number is a negative number.
\nNow,
\nWe know that,
\nWhen a stock is given by a negative number, it represents the decrease in stock and when a stock is given by a positive number, it represents the increase in stock.
\nAs we are given number is a negative number,
\nWe can just say that
\nThe stock value is being decreased by 5.90.
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given information in words is:
\nThe stock value is being decreased by 5.90.<\/p>\n

Question 28.
\nConstruct Arguments<\/strong> A classmate ordered these numbers from greatest to least. Is he correct? Construct an argument to justify your answer.
\n4.4, 4.2, -4.42, -4.24
\nAnswer:
\nIt is given that
\nA classmate ordered these numbers from greatest to least and the given order of these numbers is:
\n4.4, 4.2, -4.42, -4.24
\nNow,
\nWe know that,
\nThe positive values are greater than the negative values
\nIn a number line,
\nThe negative values decrease from right to left
\nThe positive values increase from left to right
\nSo,
\nThe order of the numbers ordered by a classmate from the greatest to the least is:
\n4.4, 4.2, -4.24, -4.42
\nHence, from the above,
\nWe can conclude that
\nThe classmate is not correct<\/p>\n

Question 29.
\nMake Sense and Persevere<\/strong> Order -3.25, -3\\(\\frac{1}{8}\\), -3\\(\\frac{3}{4}\\), and -3.1 from least to greatest. Explain.
\nAnswer:
\nThe given numbers are: -3.25, -3\\(\\frac{1}{8}\\), -3\\(\\frac{3}{4}\\), and -3.1
\nNow,
\nThe representation of the given numbers into a decimal number is:
\n-3.25, -3.12, -3.75, and -3.1
\nHence, from the above,
\nWe can conclude that
\nThe order of the given numbers from the least to the greatest is:
\n-3.75, -3.25, -3.12, -3.1<\/p>\n

Question 30.
\nHigher Order Thinking<\/strong> Suppose \\(\\frac{a}{b}\\), \\(\\frac{c}{d}\\), and \\(\\frac{e}{f}\\) represent three rational numbers. If \\(\\frac{a}{b}\\) is less than \\(\\frac{c}{d}\\), and \\(\\frac{c}{d}\\) is less than \\(\\frac{e}{f}\\), compare \\(\\frac{a}{b}\\) and \\(\\frac{e}{f}\\). Explain.
\nAnswer:
\nIt is given that
\nSuppose \\(\\frac{a}{b}\\), \\(\\frac{c}{d}\\), and \\(\\frac{e}{f}\\) represent three rational numbers. If \\(\\frac{a}{b}\\) is less than \\(\\frac{c}{d}\\), and \\(\\frac{c}{d}\\) is less than \\(\\frac{e}{f}\\)
\nNow,
\nAccording to the given information,
\n\\(\\frac{a}{b}\\) < \\(\\frac{c}{d}\\) and \\(\\frac{c}{d}\\) < \\(\\frac{e}{f}\\)
\nSo,
\n\\(\\frac{a}{b}\\) < \\(\\frac{c}{d}\\) < \\(\\frac{e}{f}\\)
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{a}{b}\\) < \\(\\frac{e}{f}\\)<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 31.
\nWhich could be a value for n?
\n\"Envision
\nA. –\\(\\frac{1}{2}\\)
\nB. –\\(\\frac{1}{3}\\)
\nC. –\\(\\frac{1}{4}\\)
\nD. –\\(\\frac{1}{6}\\)
\nAnswer:
\nThe given number line is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nEach line in the given number line is: 0.1 units
\nSo,
\nThe representation of the value of n on the given number line is:
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of n on the given number line is:
\n\"\"<\/p>\n

Question 32.
\nWhich inequality does NOT represent the correct position of two numbers on a number line?
\nA. 4\\(\\frac{1}{2}\\) > \\(\\frac{25}{4}\\)
\nB. -4\\(\\frac{1}{2}\\) > –\\(\\frac{25}{5}\\)
\nC. -6 < -5
\nD. –\\(\\frac{1}{2}\\) < \\(\\frac{1}{2}\\)
\nAnswer:
\nThe given inequalities are:
\nA. 4\\(\\frac{1}{2}\\) > \\(\\frac{25}{4}\\)
\nB. -4\\(\\frac{1}{2}\\) > –\\(\\frac{25}{5}\\)
\nC. -6 < -5
\nD. –\\(\\frac{1}{2}\\) < \\(\\frac{1}{2}\\)
\nSo,
\nThe representation of the given inequalities in the form of a decimal number is:
\nA. 4.5 > 6.25
\nB. -9.5 > -5
\nC. -6 < -5
\nD. -0.5 < 0.5
\nHence, from the above,
\nWe can conclude that
\nThe inequality that does not represent the correct position of two numbers on a number line is:
\n\"\"<\/p>\n

Lesson 2.3 Absolute Values of Rational Numbers<\/h3>\n

Solve & Discuss It!<\/strong>
\nA portion of a bank account statement is shown below How would you interpret the value of the ending balance? Explain.
\n\"Envision
\nAnswer:
\nIt is given that
\nA portion of a bank account statement is shown
\nNow,
\nThe given figure is:
\n\"\"
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe Ending Balance is: -$30
\nNow,
\nWe know that,
\nA negative balance in your Debit Account means you owe money to the bank. It probably means you have used more than what you had in your account
\nHence, from the above,
\nWe can conclude that
\nThe Ending Balance of -$30 in your account statement represents a negative balance in your Debit Account which means you owe money to the bank. It probably means you have used more than what you had in your account<\/p>\n

Focus on math practices<\/strong>
\nReasoning<\/strong> What is an example of a bank account balance that represents an amount owed greater than $40?
\nAnswer:
\nWe know that,
\nThe money owed should be represented as “Negative”
\nNow,
\nLet the bank account balance that represents an amount be x
\nSo,
\nAccording to the given information,
\nx > -$40
\nHence, from the above,
\nWe can conclude that
\nThe representation of the bank account balance that represents an amount greater than 40$ is:
\nx > -$40<\/p>\n

Visual learning<\/strong><\/p>\n

? Essential Question<\/strong>
\nHow are absolute values used to describe quantities?
\nAnswer:
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive<\/p>\n

Try It!<\/strong>
\nThe students in a science class recorded the change in the water level of a local river. During which week did the water level change by the greatest amount?
\nUse absolute values to represent the change in the water level.
\n\"Envision
\nThe water level changed by the greatest amount in Week\"Envision.
\nAnswer:
\nIt is given that
\nThe students in a science class recorded the change in the water level of a local river
\nNow,
\nThe given table is:
\n\"\"
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nSo,
\nThe representation of the changes in the water level is:
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\"\"<\/p>\n

Convince Me!<\/strong> Can a lesser number represent a greater change in water level than a greater number? Explain.
\nAnswer:
\nThe given table is:
\n\"\"
\nNow,
\nFrom the given table,
\nWe can observe that
\nThe greatest number in the given table represents the greatest change in the water level
\nHence, from the above,
\nWe can conclude that
\nA lesser number can not represent a greater change in water level than a greater number<\/p>\n

Try It!<\/strong>
\nA bank has two customers with overdrawn accounts. Which balance is the greater number? Which balance is the lesser amount owed?
\n\"Envision
\nAnswer:
\nIt is given that
\nA bank has two customers with overdrawn accounts
\nNow,
\nThe given information is:
\n\"Envision
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nSo,
\nThe absolute value of V.Wong’s Account’s balance is: $19.45
\nThe absolute value of J.Olson’s Account’s balance is: $23.76
\nNow,
\nWe know that,
\nIn a number line,
\nThe value of a negative number decrease from right to left and the negative number that is farthest to zero is the least number
\nWe represent the amount of money owed as “Negative”
\nSo,
\nThe greater number is: $23.76
\nThe balance that is the lesser amount owed is: $-19.45
\nHence, from the above,
\nWe can conclude that
\nThe greater number is: $23.76
\nThe balance that is the lesser amount owed is: $-19.45<\/p>\n

KEY CONCEPT<\/strong>
\nThe absolute value of a number is its distance from 0 on a number line. Distance is always positive. The absolute value of any number, n, is written |n|.
\n\"Envision
\n-4 and 4 are opposites as they are the same distance from 0.<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n? Essential Question<\/strong> How are absolute values used to describe quantities?
\nAnswer:
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive<\/p>\n

Question 2.
\nConstruct Arguments<\/strong> Explain why – 7 has a greater absolute value than the absolute value of 6.
\nAnswer:
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nSo,
\nThe absolute value of -7 is:
\n|-7| = 7
\nSo,
\nIn the number line,
\nWe know that,
\nThe positive values increase from left to right
\nHence, from the above,
\nWe can conclude that
\nAccording to the Property of representation of Integers in a number line,
\n7 > 6<\/p>\n

Question 3.
\nReasoning<\/strong> Give an example of a balance that has a greater integer value than a balance of -$12 but represents a debt of less than $5.
\nAnswer:
\nIt is given that
\nGive an example of a balance that has a greater integer value than a balance of -$12 but represents a debt of less than $5.
\nNow,
\nLet x be the balance that has a greater integer value than a balance of -$12, but represents a debt of less than $5.
\nNow,
\nWe are given that a balance that has a greater integer value than a balance of -$12
\nSo,
\nx >-12
\nNow,
\nWe are given that balance represents a debt of less than $5.
\nSo,
\nDebt < -5
\nSo,
\nx < -5
\nSo,
\n-5 >x >-12
\nSo,
\nx \u2208 [-6,-11]
\nHence, from the above,
\nWe can conclude that
\nThe balance that has a greater integer value than a balance of -$12, but represents a debt of less than $5 is:
\n-5 > x > -12<\/p>\n

Question 4.
\nOf the three elevations, -2 feet, -12 feet, and 30 feet, which represents the least number? Which represents the farthest distance from sea level?
\nAnswer:
\nIt is given that
\nThere are three elevations namely: -2 feet, -12 feet, and 30 feet
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nSo,
\nThe absolute values of the three elevations are: 2 feet, 12 feet, and 30 feet
\nNow,
\nWe know that,
\nIn a number line,
\nThe positive values increase from left to right
\nSo,
\nThe least number is: 2 feet
\nThe farthest distance from the sea level is: 30 feet
\nHence, from the above,
\nWe can conclude that
\nThe least number is: 2 feet
\nThe farthest distance from the sea level is: 30 feet<\/p>\n

Do You Know How?<\/strong><\/p>\n

In 5-14, find each absolute value.<\/strong><\/p>\n

Question 5.
\n|-9|
\nAnswer:
\nThe given number is |-9|
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nHence, from the above,
\nWe can conclude that
\nThe absolute value of |-9| is: 9<\/p>\n

Question 6.
\n|5\\(\\frac{3}{4}\\)|
\nAnswer:
\nThe given number is |5\\(\\frac{3}{4}\\)|
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nHence, from the above,
\nWe can conclude that
\nThe absolute value of |5\\(\\frac{3}{4}\\)| is: 5\\(\\frac{3}{4}\\)<\/p>\n

Question 7.
\n|-5.5|
\nAnswer:
\nThe given number is |-5.5|
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nHence, from the above,
\nWe can conclude that
\nThe absolute value of |-5.5| is: 5.5<\/p>\n

Question 8.
\n|82.5|
\nAnswer:
\nThe given number is |82.5|
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nHence, from the above,
\nWe can conclude that
\nThe absolute value of |82.5| is: 82.5<\/p>\n

Question 9.
\n|-14\\(\\frac{1}{3}\\)|
\nAnswer:
\nThe given number is |-14\\(\\frac{1}{3}\\)|
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nHence, from the above,
\nWe can conclude that
\nThe absolute value of |-14\\(\\frac{1}{3}\\)| is: 14\\(\\frac{1}{3}\\)<\/p>\n

Question 10.
\n|-7.75|
\nAnswer:
\nThe given number is |-7.75|
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nHence, from the above,
\nWe can conclude that
\nThe absolute value of |-7.75| is: 7.75<\/p>\n

Question 11.
\n|\u201319|
\nAnswer:
\nThe given number is |-19|
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nHence, from the above,
\nWe can conclude that
\nThe absolute value of |-19| is: 19<\/p>\n

Question 12.
\n|-2\\(\\frac{1}{2}\\)|
\nAnswer:
\nThe given number is |-2\\(\\frac{1}{2}\\)|
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nHence, from the above,
\nWe can conclude that
\nThe absolute value of |-2\\(\\frac{1}{2}\\)| is: 2\\(\\frac{1}{2}\\)<\/p>\n

Question 13.
\n|24|
\nAnswer:
\nThe given number is |24|
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nHence, from the above,
\nWe can conclude that
\nThe absolute value of |24| is: 24<\/p>\n

Question 14.
\n|35.4|
\nAnswer:
\nThe given number is |35.4|
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nHence, from the above,
\nWe can conclude that
\nThe absolute value of |35.4| is: 35.4<\/p>\n

In 15-17, use the absolute value of each account balance to determine which account has the greater overdrawn amount.<\/strong><\/p>\n

Question 15.
\nAccount A: -$5.42
\nAccount B: -$35.76
\nAnswer:
\nThe given information is:
\nAccount A: -$5.42
\nAccount B: -$35.76
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nSo,
\nThe absolute value of each account balance is:
\nAccount A: $5.42
\nAccount B: $35.76
\nNow,
\nWe know that,
\nIn a number line,
\nThe positive values increase from left to right
\nSo,
\nThe account that has the greater overdrawn amount is: Account B
\nHence, from the above,
\nWe can conclude that
\nThe account that has the greater overdrawn amount is: Account B<\/p>\n

Question 16.
\nAccount A: – $6.47
\nAccount B: -$2.56
\nAnswer:
\nThe given information is:
\nAccount A: -$6.47
\nAccount B: -$2.56
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nSo,
\nThe absolute value of each account balance is:
\nAccount A: $6.47
\nAccount B: $2.56
\nNow,
\nWe know that,
\nIn a number line,
\nThe positive values increase from left to right
\nSo,
\nThe account that has the greater overdrawn amount is: Account A
\nHence, from the above,
\nWe can conclude that
\nThe account that has the greater overdrawn amount is: Account A<\/p>\n

Question 17.
\nAccount A: -$32.56
\nAccount B: -$29.12
\nAnswer:
\nThe given information is:
\nAccount A: -$32.56
\nAccount B: -$29.12
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nSo,
\nThe absolute value of each account balance is:
\nAccount A: $32.56
\nAccount B: $29.12
\nNow,
\nWe know that,
\nIn a number line,
\nThe positive values increase from left to right
\nSo,
\nThe account that has the greater overdrawn amount is: Account A
\nHence, from the above,
\nWe can conclude that
\nThe account that has the greater overdrawn amount is: Account A<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

In 18-33, find each absolute value.<\/strong><\/p>\n

Question 18.
\n|-46|
\nAnswer:
\nThe given number is |-46|
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nHence, from the above,
\nWe can conclude that
\nThe absolute value of |-46| is: 46<\/p>\n

Question 19.
\n|0.7|
\nAnswer:
\nThe given number is |0.7|
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nHence, from the above,
\nWe can conclude that
\nThe absolute value of |0.7| is: 0.7<\/p>\n

Question 20.
\n|-\\(\\frac{2}{3}\\)|
\nAnswer:
\nThe given number is |-\\(\\frac{2}{3}\\)|
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nHence, from the above,
\nWe can conclude that
\nThe absolute value of |-\\(\\frac{2}{3}\\)| is: \\(\\frac{2}{3}\\)<\/p>\n

Question 21.
\n|-7.35|
\nAnswer:
\nThe given number is |-7.35|
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nHence, from the above,
\nWe can conclude that
\nThe absolute value of |-7.35| is: 7.35<\/p>\n

Question 22.
\n|-4\\(\\frac{3}{4}\\)|
\nAnswer:
\nThe given number is |-4\\(\\frac{3}{4}\\)|
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nHence, from the above,
\nWe can conclude that
\nThe absolute value of |-4\\(\\frac{3}{4}\\)| is: 4\\(\\frac{3}{4}\\)<\/p>\n

Question 23.
\n|-54.5|
\nAnswer:
\nThe given number is |-54.5|
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nHence, from the above,
\nWe can conclude that
\nThe absolute value of |-54.5| is: 54.5<\/p>\n

Question 24.
\n|27\\(\\frac{1}{4}\\)|
\nAnswer:
\nThe given number is |27\\(\\frac{1}{4}\\)|
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nHence, from the above,
\nWe can conclude that
\nThe absolute value of |27\\(\\frac{1}{4}\\)| is: 27\\(\\frac{1}{4}\\)<\/p>\n

Question 25.
\n|\u201313.35|
\nAnswer:
\nThe given number is |-13.35|
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nHence, from the above,
\nWe can conclude that
\nThe absolute value of |-13.35| is: 13.35<\/p>\n

Question 26.
\n|14|
\nAnswer:
\nThe given number is |14|
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nHence, from the above,
\nWe can conclude that
\nThe absolute value of |14| is: 14<\/p>\n

Question 27.
\n|-11.5|
\nAnswer:
\nThe given number is |-11.5|
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nHence, from the above,
\nWe can conclude that
\nThe absolute value of |-11.5| is: 11.5<\/p>\n

Question 28.
\n|-6.3|
\nAnswer:
\nThe given number is |-6.3|
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nHence, from the above,
\nWe can conclude that
\nThe absolute value of |-6.3| is: 6.3<\/p>\n

Question 29.
\n|3.75|
\nAnswer:
\nThe given number is |3.75|
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nHence, from the above,
\nWe can conclude that
\nThe absolute value of |3.75| is: 3.75<\/p>\n

Question 30.
\n|-8.5|
\nAnswer:
\nThe given number is |-8.5|
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nHence, from the above,
\nWe can conclude that
\nThe absolute value of |-8.5| is: 8.5<\/p>\n

Question 31.
\n|15|
\nAnswer:
\nThe given number is |15|
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nHence, from the above,
\nWe can conclude that
\nThe absolute value of |15| is: 15<\/p>\n

Question 32.
\n|-6\\(\\frac{3}{4}\\)|
\nAnswer:
\nThe given number is |-6\\(\\frac{3}{4}\\)|
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nHence, from the above,
\nWe can conclude that
\nThe absolute value of |-6\\(\\frac{3}{4}\\)| is: 6\\(\\frac{3}{4}\\)<\/p>\n

Question 33.
\n|-5.3|
\nAnswer:
\nThe given number is |-5.3|
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nHence, from the above,
\nWe can conclude that
\nThe absolute value of |-5.3| is: 5.3<\/p>\n

In 34-37, order the numbers from least to greatest.<\/strong><\/p>\n

Question 34.
\n|-12|, |11\\(\\frac{3}{4}\\)|, |-20.5|, |2|
\nAnswer:
\nThe given numbers are:
\n|-12|, |11\\(\\frac{3}{4}\\)|, |-20.5|, |2|
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nSo,
\nThe absolute values of the given numbers are:
\n12, 11\\(\\frac{3}{4}\\), 20.5, 2
\nSo,
\nThe absolute values of the given numbers in the decimal form are:
\n12, 11.75, 20.5, 2
\nNow,
\nWe know that,
\nIn a number line,
\nThe positive values increase from left to right
\nHence, from the above,
\nWe can conclude that
\nThe order of the numbers from the least to the greatest is:
\n2, 11.75, 12, 20.5<\/p>\n

Question 35.
\n|10|, |-3|, |0|, |-5.25|
\nAnswer:
\nThe given numbers are:
\n|10|, |-3|, |0|, |-5.25|
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nSo,
\nThe absolute values of the given numbers are:
\n10, 3, 0, 5.25
\nNow,
\nWe know that,
\nIn a number line,
\nThe positive values increase from left to right
\nHence, from the above,
\nWe can conclude that
\nThe order of the numbers from the least to the greatest is:
\n0, 3, 5.25, 10<\/p>\n

Question 36.
\n|-6|, |-4|, |11|, |0|
\nAnswer:
\nThe given numbers are:
\n|-6|, |-4|, |11|, |0|
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nSo,
\nThe absolute values of the given numbers are:
\n6, 4, 11, 0
\nNow,
\nWe know that,
\nIn a number line,
\nThe positive values increase from left to right
\nHence, from the above,
\nWe can conclude that
\nThe order of the numbers from the least to the greatest is:
\n0, 4, 6, 11<\/p>\n

Question 37.
\n|4|, |-3|, |\u201318|, |-3.18|
\nAnswer:
\nThe given numbers are:
\n|4|, |-3|, |\u201318|, |-3.18|
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nSo,
\nThe absolute values of the given numbers are:
\n4, 3, 18, 3.18
\nNow,
\nWe know that,
\nIn a number line,
\nThe positive values increase from left to right
\nHence, from the above,
\nWe can conclude that
\nThe order of the numbers from the least to the greatest is:
\n3, 3.18, 4, 18<\/p>\n

Alberto and Rebecca toss horseshoes at a stake. Whoever’s horseshoe is closer to the stake wins a point.<\/strong><\/p>\n

Question 38.
\nReasoning<\/strong> What integer best describes the location of Alberto’s horseshoe in relation to the stake? What integer best describes the location of Rebecca’s horseshoe?
\n\"Envision
\nAnswer:
\nIt is given that
\nAlberto and Rebecca toss horseshoes at a stake. Whoever’s horseshoe is closer to the stake wins a point.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe stake is at zero position
\nThe location of Alberto’s horseshoe is at: -3
\nThe location of Rebecca’s horseshoe is at: 2
\nSo,
\nThe location of Alberto’s horseshoe is: 3 feet
\nThe location of Reecca’s horseshoe is: 2 feet
\nHence, from the above,
\nWe can conclude that
\nThe integer that best describes the location of Alberto’s horseshoe in relation to the stake is: 3 feet
\nThe integer that best describes the location of Rebecca’s horseshoe is: 2 feet<\/p>\n

Question 39.
\nCritique Reasoning<\/strong> Alberto says that -3 is less than 2, so he wins a point. Is Alberto correct? Explain.
\nAnswer:
\nIt is given that
\nAlberto says that -3 is less than 2, so he wins a point
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nThe number of points between Alberto’s horseshoe and Rebecca’s horseshoe = |(The location of Alberto’s horseshoe) + (The location of Rebecca’s horseshoe)|
\n= |-3 + 2|
\n= |-1|
\n= 1
\nHence, from the above,
\nWe can conclude that Alberto is correct<\/p>\n

Question 40.
\nA model with Math<\/strong>
\nFind the distance from Alberto’s horseshoe to Rebecca’s horseshoe. Explain.
\nAnswer:
\nThe given figure is:
\n\"Envision
\nSo,
\nThe distance between Alberto’s horseshoe and Rebecca’s horseshoe = |(The location of Alberto’s horseshoe) + (The location of Rebecca’s horseshoe)|
\n= |-3 + 2|
\n= |-1|
\n= 1 feet
\nHence, from the above,
\nWe can conclude that
\nThe distance from Alberto’s horseshoe to Rebecca’s horseshoe is: 1 feet<\/p>\n

Question 41.
\nHigher-Order Thinking<\/strong> Let a = any rational number. Is the absolute value of a difference if a is a positive number or a negative number? Explain.
\nAnswer:
\nIt is given that
\n“a” is any rational number
\nNow,
\nLet
\na = 6 (or) a = -5
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nSo,
\n|a| = |6| (or) |a| = |-5|
\n|a| = 6 (or) |a| = 5
\nSo,
\n|6 – 5| = |1| =1
\nHence, from the above,
\nWe can conclude that
\nThe absolute value of a difference is positive even if “a” is a positive number or a negative number<\/p>\n

Question 42.
\nConstruct Arguments<\/strong> Samuel and Leticia are playing a game. After the first round of the game, Samuel’s score was -19, and Leticia’s score was 21. The score with the greater absolute value wins each round. Who won the first round? Explain.
\nAnswer:
\nIt is given that
\nSamuel and Leticia are playing a game. After the first round of the game, Samuel’s score was -19, and Leticia’s score was 21. The score with the greater absolute value wins each round
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nSo,
\nThe abso;ute value of Samuel’s score was:
\n|-19| = 19
\nThe absolute value of Leticia’s score was:
\n|21| = 21
\nNow,
\nWe know that,
\nThe positive values increase from left to right
\nSo,
\nLeticia won the first round
\nHence, from the above,
\nWe can conclude that
\nLeticia won the first round<\/p>\n

Question 43.
\nUse Structure<\/strong> Ana and Chuyen are exploring underwater sea life while on a helmet diving adventure. Ana’s location is -30 feet below sea level, and Chuyen’s location is -12 feet below sea level. Which girl is located farther from sea level?
\nAnswer:
\nIt is given that
\nAna and Chuyen are exploring underwater sea life while on a helmet diving adventure. Ana’s location is -30 feet below sea level, and Chuyen’s location is -12 feet below sea level
\nNow,
\nWe know that,
\nIn a number line,
\nThe negative values decrease from right to left
\nThe negative value that is farthest to the number line is the least value
\nHence, from the above,
\nWe can conclude that
\nAna is located farther from sea level<\/p>\n

Question 44.
\nMarie’s account balance is – $45.62. Tom’s account balance is $42.55. Which balance represents the greater number? Which balance represents the lesser amount owed?
\nAnswer:
\nIt is given that
\nMarie’s account balance is – $45.62. Tom’s account balance is $42.55
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nSo,
\nThe absolute value of Marie’s account balance is: $45.62
\nThe absolute value of Tom’s account balance is: $42.55
\nSo,
\nThe balance that represents the greater number is: Marie’s account balance
\nThe balance that represents the lesser amount owed is: Tom’s account balance
\nHence, from the above,
\nWe can conclude that
\nThe balance that represents the greater number is: Marie’s account balance
\nThe balance that represents the lesser amount owed is: Tom’s account balance<\/p>\n

Question 45.
\nIn New York, the Federal Reserve gold vault is located at a depth of |-80| feet below ground. The treasure at Oak Island is believed to be at a depth of |-134| feet. Which is farther below ground, the gold vault or the Oak Island treasure?
\nAnswer:
\nIt is given that
\nIn New York, the Federal Reserve gold vault is located at a depth of |-80| feet below ground. The treasure at Oak Island is believed to be at a depth of |-134| feet
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nSo,
\nThe absolute value of the depth of the Federal reserve gold vault is: 80 feet
\nThe absolute value of the depth of the treasure at Oak Island is: 134 feet
\nSo,
\nThe treasure at oak Island is farther below ground
\nHence, from the above,
\nWe can conclude that
\nThe treasure at Oak Island is farther below ground<\/p>\n

Question 46.
\nTwo scuba divers are swimming below sea level. The locations of the divers can be represented by -30 feet and -42 feet. Which measure represents the location that is closest to sea level?
\nAnswer:
\nIt is given that
\nTwo scuba divers are swimming below sea level. The locations of the divers can be represented by -30 feet and -42 feet
\nNow,
\nWe know that,
\nIn a number line,
\nThe negative values decrease from left to right
\nHence, from the above,
\nWe can conclude that
\nThe measure that represents the location that is closest to sea level is: -30 feet<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 47.
\nThe table at the right shows the scores at the end of the first round of a golf tournament. The scores are relative to par.
\n\"Envision
\nPART A<\/strong>
\nPar is represented as 0. Using absolute value, show the distance each score is from par.
\nAnswer:
\nIt is given that
\nThe table at the right shows the scores at the end of the first round of a golf tournament. The scores are relative to par.
\nNow,
\nThe given table is:
\n\"Envision
\nHence,
\nThe representation of the distance each score is from Par in a number line is:
\n\"\"
\nPART B<\/strong>
\nThe golfer with the least score wins the round. Who won the first round of the tournament? Explain.
\nAnswer:
\nIt is given that
\nThe golfer with the least score wins the round
\nThe given table is:
\n\"Envision
\nNow,
\nFrom part (a),
\nThe representation of the scores on a number line is:
\n\"\"
\nNow,
\nFrom the given table,
\nWe can observe that
\nThe golfer who has the least score is: Kate
\nHence, from the above,
\nWe can conclude that
\nKate won the first round of the tournament<\/p>\n

Topic 2 MID-TOPIC CHECKPOINT<\/h3>\n

Question 1.
\nVocabulary<\/strong> Describe the relative locations of the rational numbers -(-\\(\\frac{a}{b}\\)) and \\(\\frac{a}{b}\\) on a number line. Lessons 2.1 and 2.2
\nAnswer:
\nThe given rational numbers are: \\(\\frac{a}{b}\\) and -(-\\(\\frac{a}{b}\\))
\nNow,
\nWe know that,
\n– \u00d7 – = +
\nSo,
\nThe given rational numbers are: \\(\\frac{a}{b}\\) and \\(\\frac{a}{b}\\)
\nNow,
\nFrom the given fractions,
\nWe can observe that both are positive rational numbers
\nNow,
\nWe know that,
\nIn a number line,
\nThe values after zero are the “positive Values”
\nHence, from the above,
\nWe can conclude that
\nThe relative locations of the given rational numbers on a number line are: At the right side of the number line<\/p>\n

Question 2.
\nMarc deposited $175 in a new bank account. After buying some furniture, he was overdrawn by $55. Select all the true statements about Marc’s account. Lesson 2.1
\n\"Envision To start, Marc had a negative balance.
\n\"Envision In this situation, 0 represents an empty bank account.
\n\"Envision When Marc was overdrawn, he had a negative balance.
\n\"Envision After buying furniture, Marc had a positive balance.
\n\"Envision The lowest balance in the account was -$55.
\nAnswer:
\nIt is given that
\nMarc deposited $175 in a new bank account. After buying some furniture, he was overdrawn by $55.
\nHence, from the above,
\nWe can conclude that
\nAll the true statements about Marc’s account are:
\n\"\"<\/p>\n

Question 3.
\nWhat number is represented on the number line? Give your answer as a decimal and as a fraction. Lesson 2.2
\n\"Envision
\nAnswer:
\nThe given number line is:
\n\"Envision
\nNow,
\nFrom the given number line,
\nWe can observe that
\nEach line in the given number line represents 0.25 units
\nSo,
\nThe number that is represented on the number line is: -2.25
\nHence, from the above,
\nWe can conclude that
\nThe number that is represented on the number line as a decimal number is: -2.25
\nThe number that is represented on the number line as a fraction is: -2\\(\\frac{1}{4}\\)<\/p>\n

Question 4.
\nThe absolute value of a number is 52. Select all the integers that this number could be. Lesson 2.3
\n\"Envision -52
\n\"Envision -25
\n\"Envision 25
\n\"Envision –\\(\\frac{1}{52}\\)
\n\"Envision 52
\nAnswer:
\nIt is given that
\nThe absolute value of a number is 52
\nHence, from the above,
\nWe can conclude that
\nAll the integers that the number could be:
\n\"\"<\/p>\n

Question 5.
\nThe table shows the location of four treasure chests relative to sea level. How can you use the number line to find the treasure chest that is farthest from sea level? Lesson 2.2
\n\"Envision
\n\"Envision
\nAnswer:
\nIt is given that
\nThe table shows the location of four treasure chests relative to sea level
\nNow,
\nThe given table is:
\n\"Envision
\nNow,
\nThe representation of \\(\\frac{5}{4}\\) feet in a decimal form is: 1.25 feet
\nNow,
\nThe given number line is:
\n\"Envision
\nNow,
\nFrom the given number line,
\nWe can observe that
\nEach unit represents 0.25 on the given number line
\nHence,
\nThe representation of the locations of the treasure chests on the given number line is:
\n\"\"<\/p>\n

Question 6.
\nThree customers have accounts owing money. The table shows the account balances that represent what the customers owe. Which customer owes the least amount of money? Lesson 2.3
\n\"Envision
\nAnswer:
\nIt is given that
\nThree customers have accounts owing money. The table shows the account balances that represent what the customers owe.
\nNow,
\nThe given table is:
\n\"Envision
\nNow,
\nWe know that,
\nIn a number line,
\nThe negative value that is farthest from zero is the least value
\nThe negative values decrease from right to left
\nHence, from the above,
\nWe can conclude that
\nThe customet that owes the least amount of money is: B Barker<\/p>\n

How well did you do on the mid-topic checkpoint? Fill in the stars. \"Envision<\/p>\n

Topic 2 MID-TOPIC PERFORMANCE TASK<\/strong><\/p>\n

Warren and Natasha started a dog-walking business. During their first week, they paid $10 to make their business cards and $6 for a 4.5-pound box of doggie treats. Warren walked a dog for 15 minutes, and Natasha walked a dog for 30 minutes.<\/p>\n

PART A<\/strong>
\nWhich integers represent the dollar amounts either spent or earned during the first week Warren and Natasha were in business? Select all that apply.
\n\"Envision $5
\n\"Envision -$5
\n\"Envision $10
\n\"Envision -$10
\n\"Envision -$6
\n\"Envision
\nAnswer:
\nIt is given that
\nWarren and Natasha started a dog-walking business. During their first week, they paid $10 to make their business cards and $6 for a 4.5-pound box of doggie treats. Warren walked a dog for 15 minutes, and Natasha walked a dog for 30 minutes.
\nNow,
\nThe given table is:
\n\"Envision
\nSo,
\nThe amount of money Warren and Natasha spent during the first week = (The amount spent on business cards) + 9the amount spent on a 4.5-pound box of doggie treats)
\n= $10 + $6
\n= $16
\nSo,
\nThe amount earned by Warren = $5
\nThe amount earned by Natasha = $10
\nHence, from the above,
\nWe can conclude that
\nThe integers that represent the dollar amounts either spent or earned during the first week Warren and Natasha were in business are:
\n\"\"<\/p>\n

PART B<\/strong>
\nAt the end of each week, Warren records the weight in pounds of doggie treats eaten as a negative rational number. Plot the numbers of pounds eaten each week on the number line. Order the numbers from most pounds eaten to fewest pounds eaten.
\n\"Envision
\n\"Envision
\nAnswer:
\nIt is given that
\nAt the end of each week, Warren records the weight in pounds of doggie treats eaten as a negative rational number
\nNow,
\nThe given table is:
\n\"Envision
\nNow,
\nThe representation of the number of pounds eaten into a decimal number are:
\n–\\(\\frac{3}{2}\\) = -1.5
\n–\\(\\frac{2}{3}\\) = -0.6
\n–\\(\\frac{5}{4}\\) = -1.25
\nNow,
\nWe know that,
\nIn a number line,
\nThe negative numbers decrease from right to left
\nSo,
\nThe representation of the number of pounds eaten on a number line is:
\n\"\"
\nSo,
\nThe order of the numbers from most pounds eaten to fewest pounds eaten is:
\n-1.5 < -1.25 < -0.66 < -0.5
\n–\\(\\frac{3}{2}\\) < –\\(\\frac{5}{4}\\) < –\\(\\frac{2}{3}\\) < -0.5
\nHence, from the above,
\nWe can conclude that
\nThe representation of the number of pounds eaten on a number line is:
\n\"\"
\nThe order of the numbers from most pounds eaten to fewest pounds eaten is:
\n–\\(\\frac{3}{2}\\) < –\\(\\frac{5}{4}\\) < –\\(\\frac{2}{3}\\) < -0.5<\/p>\n

PART C<\/strong>
\nFind the absolute value for the number of pounds of doggie treats eaten each week. Which two weeks had the greatest number of pounds eaten?
\nAnswer:
\nThe given table from part B is:
\n\"Envision
\nNow,
\nWe know that,
\n“Absolute value” refers to a point’s distance from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive
\nSo,
\nThe absolute values for the number of pounds eaten are:
\n|-\\(\\frac{3}{2}\\)| = \\(\\frac{3}{2}\\)
\n|-\\(\\frac{2}{3}\\)| = \\(\\frac{2}{3}\\)
\n|-0.5| = 0.5
\n|-\\(\\frac{5}{4}\\)| = \\(\\frac{5}{4}\\)
\nSo,
\nThe representation of the absolute values for the number of pounds eaten into a decimal number is:
\n1.5, 0.66, 0.5, 1.25
\nNow,
\nFrom the given table,
\nWe can observe that
\nWeek 1 and Week 4 had the greatest number of pounds eaten
\nHence, from the above,
\nWe can conclude that
\na. The absolute values for the number of pounds eaten are:
\n|-\\(\\frac{3}{2}\\)| = \\(\\frac{3}{2}\\)
\n|-\\(\\frac{2}{3}\\)| = \\(\\frac{2}{3}\\)
\n|-0.5| = 0.5
\n|-\\(\\frac{5}{4}\\)| = \\(\\frac{5}{4}\\)
\nb. Week 1 and Week 4 had the greatest number of pounds eaten<\/p>\n

Lesson 2.4 Represent Rational Numbers on the Coordinate Plane<\/h3>\n

Solve & Discuss It!<\/strong><\/p>\n

Point B has the same x-coordinate as point A, but its y-coordinate is the opposite of the y-coordinate of point A. Plot point B and write its coordinates.
\n\"Envision
\nAnswer:
\nIt is given that
\nPoint B has the same x-coordinate as point A, but its y-coordinate is the opposite of the y-coordinate of point A.
\nNow,
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nWe can observe that A (3, 5)
\nNow,
\nWe know that,
\nThe opposite of y-coordinate in the coordinate plane is: -y
\nSo,
\nThe coordinates for point B is: B (3, -5)
\nHence, from the above,
\nWe can conclude that
\nThe coordinates for point B is: B (3, -5)
\nThe representation of B (3, -5) in the coordinate plane is:
\n\"\"<\/p>\n

Make Sense and Persevere<\/strong> How can you use what you know about integers and graphing points on a coordinate plane to plot point B?
\nAnswer:
\nFrom the above problem,
\nWe know that,
\nThe coordinates of point B are: B (3, -5)
\nNow,
\nWe know that,
\nA coordinate plane consists of 2 number lines i.e., 1 horizontal number line and 1 vertical number line, and both intersect at zero
\nNow,
\nWe know that,
\nIn a number line,
\nThe right side of zero and the upside of zero will be positive numbers
\nThe left side of zero and the downside of zero will be negative numbers
\nHence, from the above,
\nWe can conclude that
\nB (3, -5) will be located on the right side of the horizontal number line and the downside of the vertical number line<\/p>\n

Focus on math practices<\/strong>
\nGeneralize<\/strong> Two points that have the same x-coordinate but opposite y-coordinates. Across which axis do they form mirror images of each other?
\nAnswer:
\nIt is given that
\nTwo points that have the same x-coordinate but opposite y-coordinates.
\nSo,
\nThe representation of the point and its mirror image respectively are:
\n(x, y) ——–> (x, -y)
\nHence, from the above,
\nWe can conclude that
\nThe mirror images of the given points will form across the x-axis<\/p>\n

Visual Learning<\/strong><\/p>\n

? Essential Question<\/strong> How can you graph a point with rational coordinates on a coordinate plane?
\nAnswer:
\nGraphing Points with Rational Number Coordinates:
\na. Locate “a” on the x-axis.
\nb. From “a” on the x-axis, if “b” is positive, move up “b” units, and if “b” is negative, move down “b” units.
\nc. The location should now line up to “a” on the x-axis and to “b” on the y-axis, so we draw a point at the location. This is the point (a, b).<\/p>\n

Try It!<\/strong>
\nGraph point P(-2, -3) on the coordinate plane shown.
\nStart at the origin\"Envision.
\nThe x-coordinate is negative, so move\"Envision units to the left.
\nThen use y-coordinate to move\"Envision units down.
\nDraw and label the point.
\n\"Envision
\nAnswer:
\nThe given point is: P (-2, -3)
\nNow,
\n\"\"
\nHence,
\nThe representation of the given point in the coordinate plane is:
\n\"\"<\/p>\n

Convince Me!<\/strong> How do the signs of the coordinates relate to the quadrant in which a point is located? Explain for each of the four quadrants.
\nAnswer:
\na. If both x and y are positive, then the point lies in the “First quadrant”
\nb. If x is negative and y is positive, then the point lies in the “Second quadrant”
\nc. If both x and y are negative, then the point lies in the “Third quadrant”
\nd. If x is positive and y is negative, then the point lies in the “Fourth quadrant”<\/p>\n

Try It!<\/strong>
\nWhat landmark is located on the map at (2, \\(\\frac{1}{4}\\))?
\n\"\"
\nAnswer:
\nThe given point is: (2, \\(\\frac{1}{4}\\)
\nNow,
\nThe given figure is:
\n\"\"
\nNow,
\nThe representation of the given point in the decimal form is: (2, 0.25)
\nNow,
\nFrom the given map,
\nLocate the given point
\nSo,
\nThe location of the given point in the map is:
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe landmark that is located on the map at (2, \\(\\frac{1}{4}\\)) is: FBI Building<\/p>\n

Try It!<\/strong>
\nThe coordinates of point A are (-3, 5). What are the coordinates of point B, which is a reflection of point A across the x-axis?
\nAnswer:
\nIt is given that
\nThe coordinates of point A are (-3, 5)
\nNow,
\nIt is given that
\nPoint B is the reflection of point A across the x-axis
\nNow,
\nWe know that,
\nWhen (x, y) reflects across the x-axis, the point will become (x, -y)
\nHence, from the above,
\nWe can conclude that
\nThe coordinates of point B, which is a reflection of point A across the x-axis is: (-3, -5)<\/p>\n

KEY CONCEPT<\/strong>
\nA coordinate plane is a grid that contains number lines that intersect at right angles and divide the plane into four quadrants. The horizontal number line is called the x-axis, and the vertical number line is called the y-axis.
\nThe location of a point on a coordinate plane is written as an ordered pair (x, y).
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n? Essential Question<\/strong> How can you graph a point with rational coordinates on a coordinate plane?
\nAnswer:
\nGraphing Points with Rational Number Coordinates:
\na. Locate “a” on the x-axis.
\nb. From “a” on the x-axis, if “b” is positive, move up “b” units, and if “b” is negative, move down “b” units.
\nc. The location should now line up to “a” on the x-axis and to “b” on the y-axis, so we draw a point at the location. This is the point (a, b).<\/p>\n

Question 2.
\nWhat is the y-coordinate of any point that lies on the x-axis?
\nAnswer:
\nEvery point on the x-axis has no distance (zero distance) from the x-axis, therefore, the y – coordinate of every point lying on the x-axis is always “Zero”<\/p>\n

Question 3.
\nLook for Relationships<\/strong> How are the points (4, 5) and (-4, 5) related?
\nAnswer:
\nThe given points are: (4, 5), and (-4, 5)
\nNow,
\nWe know that,
\nThe reflection of A (x, y) across the x-axis is: B(x, -y)
\nThe reflection of A (x, y) across the y-axis is: B(-x, y)
\nNow,
\nFrom the given points,
\nWe can observe that
\n(-4, 5) is the reflection of (4, 5) across the y-axis
\nHence, from the above,
\nWe can conclude that
\n(-4, 5) is the reflection of (4, 5) across the y-axis<\/p>\n

Question 4.
\nConstruct Arguments<\/strong> On a larger map, the coordinates for the location of another Washington, D.C. landmark are (8, -10). In which quadrant of the map is this landmark located? Explain.
\nAnswer:
\nIt is given that
\nOn a larger map, the coordinates for the location of another Washington, D.C. landmark are (8, -10)
\nNow,
\nWe know that,
\na. If both x and y are positive, then the point lies in the “First quadrant”
\nb. If x is negative and y is positive, then the point lies in the “Second quadrant”
\nc. If both x and y are negative, then the point lies in the “Third quadrant”
\nd. If x is positive and y is negative, then the point lies in the “Fourth quadrant”
\nHence, from the above,
\nWe can conclude that
\nThe location of another Washington D. C of the map is located in the fourth quadrant<\/p>\n

In 5-7, graph and label each point on the coordinate plane.
\n\"Envision
\n<\/strong><\/p>\n

Question 5.
\nA(-4, 1)
\nAnswer:
\nThe given point is: A (-4, 1)
\nHence,
\nThe representation of the given point in the coordinate plane is:
\n\"\"<\/p>\n

Question 6.
\nB(4, 3)
\nAnswer:
\nThe given point is: B (4, 3)
\nHence,
\nThe representation of the given point in the coordinate plane is:
\n\"\"<\/p>\n

Question 7.
\nC(0, -2)
\nAnswer:
\nThe given point is: C (0, -2)
\nHence,
\nThe representation of the given point in the coordinate plane is:
\n\"\"<\/p>\n

Question 8.
\nWhat ordered pair gives the coordinates of point P above?
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision<\/strong>
\nNow,
\nFrom the given coordinate plane,
\nWe can observe that
\nPoint P is in the 4th quadrant
\nThe coordinates of point P is: P (3, -2)
\nHence, from the above,
\nWe can conclude that
\nThe ordered pair that gives the coordinates of Point P is: P (3, -2)<\/p>\n

In 9 and 10, use the map in Example 2 and write the ordered pair of each location.
\n\"\"
\n<\/strong><\/p>\n

Question 9.
\nWhite House
\nAnswer:
\nThe given map is:
\n\"\"<\/strong>
\nNow,
\nFrom the given map,
\nWe can observe that
\nThe coordinates of the White House are: (1, 1)
\nHence, from the above,
\nWe can conclude that
\nThe coordinates of the White House are: (1, 1)<\/p>\n

Question 10.
\nLincoln Memorial
\nAnswer:
\nThe given map is:
\n\"\"
\n<\/strong>Now,
\nFrom the given map,
\nWe can observe that
\nThe coordinates of the Lincoln Memorial are: (-1, -0.5)
\nHence, from the above,
\nWe can conclude that
\nThe coordinates of the White House are: (-1, -0.5)<\/p>\n

In 11 and 12, use the map in Example 2 and write the landmark located at each ordered pair.<\/strong><\/p>\n

Question 11.
\n(0.5, 0)
\nAnswer:
\nThe given point is: (0.5, 0)
\nNow,
\nThe given map is:
\n\"\"
\n<\/strong>Now,
\nFrom the given map,
\nWe can observe that
\nThe landmark that is located at (0.5, 0) is: Ellipse
\nHence, from the above,
\nWe can conclude that
\nThe landmark that is located at (0.5, 0) is: Ellipse<\/p>\n

Question 12.
\n(\\(\\frac{3}{4}\\), –\\(\\frac{1}{2}\\))
\nAnswer:
\nThe given point is: (\\(\\frac{3}{4}\\), –\\(\\frac{1}{2}\\))
\nNow,
\nThe given map is:
\n\"\"
\n<\/strong>Now,
\nFrom the given map,
\nWe can observe that
\nThe landmark that is located at (\\(\\frac{3}{4}\\), –\\(\\frac{1}{2}\\)) is: Washington Monument
\nHence, from the above,
\nWe can conclude that
\nThe landmark that is located at (\\(\\frac{3}{4}\\), –\\(\\frac{1}{2}\\)) is: Washington Monument<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

In 13-20, graph and label each point.<\/strong><\/p>\n

Question 13.
\nA(1, -1)
\nAnswer:
\nThe given point is: A (1, -1)
\nHence,
\nThe representation of the given point in the coordinate plane is:
\n\"\"<\/p>\n

Question 14.
\nB(4, 3)
\nAnswer:
\nThe given point is: B (4, 3)
\nHence,
\nThe representation of the given point in the coordinate plane is:
\n\"\"<\/p>\n

Question 15.
\nC(-4, 3)
\nAnswer:
\nThe given point is: C (-4, 3)
\nHence,
\nThe representation of the given point in the coordinate plane is:
\n\"\"<\/p>\n

Question 16.
\nD(5, -2)
\nAnswer:
\nThe given point is: D (5, -2)
\nHence,
\nThe representation of the given point in the coordinate plane is:
\n\"\"<\/p>\n

Question 17.
\nE(-2.5, 1.5)
\nAnswer:
\nThe given point is: E (-2.5, 1.5)
\nHence,
\nThe representation of the given point in the coordinate plane is:
\n\"\"<\/p>\n

Question 18.
\nF(2, 1.5)
\nAnswer:
\nThe given point is: F (2, 1.5)
\nHence,
\nThe representation of the given point in the coordinate plane is:
\n\"\"<\/p>\n

Question 19.
\nG(-2, -1\\(\\frac{1}{2}\\))
\nAnswer:
\nThe given point is: G (-2, -1\\(\\frac{1}{2}\\))
\nHence,
\nThe representation of the given point in the coordinate plane is:
\n\"\"<\/p>\n

Question 20.
\nH(1\\(\\frac{1}{2}\\), -1)
\nAnswer:
\nThe given point is: H (1\\(\\frac{1}{2}\\), -1)
\nHence,
\nThe representation of the given point in the coordinate plane is:
\n\"\"<\/p>\n

In 21-26, write the ordered pair for each point.<\/strong>
\n\"Envision<\/p>\n

Question 21.
\nP
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nWe can observe that
\nThe coordinates of point P from the given coordinate plane is: P (0, -8)
\nHence, from the above,
\nWe can conclude that
\nThe coordinates of point P from the given coordinate plane is: P (0, -8)<\/p>\n

Question 22.
\nQ
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nWe can observe that
\nThe coordinates of point Q from the given coordinate plane is: Q (5, -3)
\nHence, from the above,
\nWe can conclude that
\nThe coordinates of point Q from the given coordinate plane is: Q (5, -3)<\/p>\n

Question 23.
\nR
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nWe can observe that
\nThe coordinates of point R from the given coordinate plane is: R (-8, 0)
\nHence, from the above,
\nWe can conclude that
\nThe coordinates of point R from the given coordinate plane is: R (-8, 0)<\/p>\n

Question 24.
\nS
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nWe can observe that
\nThe coordinates of point S from the given coordinate plane is: S (-2.5, -0.5)
\nHence, from the above,
\nWe can conclude that
\nThe coordinates of point S from the given coordinate plane is: S (-2.5, -0.5)<\/p>\n

Question 25.
\nT
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nWe can observe that
\nThe coordinates of point T from the given coordinate plane is: T (1.5, 2.5)
\nHence, from the above,
\nWe can conclude that
\nThe coordinates of point T from the given coordinate plane is: T (1.5, 2.5)<\/p>\n

Question 26.
\nU
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nWe can observe that
\nThe coordinates of point U from the given coordinate plane is: U (-1, -0.5)
\nHence, from the above,
\nWe can conclude that
\nThe coordinates of point U from the given coordinate plane is: U (-1, -0.5)<\/p>\n

In 27-30, use the map at the right.
\n\"Envision
\n<\/strong><\/p>\n

Question 27.
\nWhich building is located in Quadrant III?
\nAnswer:
\nThe given map is:
\n\"Envision
\nNow,
\nWe know that,
\na. If both x and y are positive, then the point lies in the “First quadrant”
\nb. If x is negative and y is positive, then the point lies in the “Second quadrant”
\nc. If both x and y are negative, then the point lies in the “Third quadrant”
\nd. If x is positive and y is negative, then the point lies in the “Fourth quadrant”
\nSo,
\nFrom the given map,
\nWe can observe that
\nThe building that is located in Quadrant III is: Fire House
\nHence, from the above,
\nWe can conclude that
\nThe building that is located in Quadrant III is: Fire House<\/p>\n

Question 28.
\nWhich two places have the same x-coordinate?
\nAnswer:
\nThe given map is:
\n\"Envision
\nNow,
\nWe know that,
\na. If both x and y are positive, then the point lies in the “First quadrant”
\nb. If x is negative and y is positive, then the point lies in the “Second quadrant”
\nc. If both x and y are negative, then the point lies in the “Third quadrant”
\nd. If x is positive and y is negative, then the point lies in the “Fourth quadrant”
\nSo,
\nFrom the given map,
\nWe can observe that
\nThe two buildings that have the same x-coordinate are: Swimming pool and clubhouse
\nHence, from the above,
\nWe can conclude that
\nThe two buildings that have the same x-coordinate are: Swimming pool and clubhouse<\/p>\n

Question 29.
\nUse Structure<\/strong> The city council wants the location of the entrance to a new city park to be determined by the reflection of the school entrance across the y-axis. What are the coordinates of the entrance to the new city park on this map?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe city council wants the location of the entrance to a new city park to be determined by the reflection of the school entrance across the y-axis
\nNow,
\nThe given map is:
\n\"Envision
\nNow,
\nFrom the given map,
\nWe can observe that
\nThe coordinates of the school entrance are: (4, 6)
\nNow,
\nWe know that,
\nThe reflection of (x, y) across the y-axis is: (-x, y)
\nSo,
\nThe coordinates of the entrance to a new city park are: (-4, 6)
\nHence, from the above,
\nWe can conclude that
\nThe coordinates of the entrance to a new city park are: (-4, 6)<\/p>\n

Question 30.
\nHigher-Order Thinking<\/strong> You are at the market square (0, 0) and want to get to the doctor’s office. Following the gridlines, what is the shortest route?
\n\"Envision
\nAnswer:
\nIt is given that
\nYou are at the market square (0, 0) and want to get to the doctor’s office.
\nNow,
\nThe given map is:
\n\"Envision
\nNow,
\nThe representation of the distance of the Market and doctor’s office in the map are:
\n\"\"
\nNow,
\nFrom the given map,
\nWe can observe that
\nThe shortest route can be determined by using the Pythagoras Theorem since a right triangle is formed
\nSo,
\nThe shortest route between the market and doctor’s office = \\(\\sqrt{4\u00b2 + 2\u00b2}\\)
\n= \\(\\sqrt{20}\\)
\n= 4.47 units
\nHence, from the above,
\nWe can conclude that
\nThe shortest route between the market and the doctor’s office is: 4.47 units<\/p>\n

In 31-36, use the coordinate plane at the right.
\n\"Envision
\n<\/strong><\/p>\n

Question 31.
\nWhat is located at (-0.7,-0.2)?
\nAnswer:
\nThe given point is: (-0.7, -0.2)
\nNow,
\nThe given coordinate plane is:<\/p>\n

\"Envision
\nNow,
\nFrom the given coordinate plane,
\nWe can observe that the landmark that is located at (-0.7, -0.2) is: Pond
\nHence, from the above,
\nWe can conclude that
\nThe pond is located at (-0.7, -0.2)<\/p>\n

Question 32.
\nWhat is located at (\\(\\frac{3}{10}\\), –\\(\\frac{1}{5}\\))?
\nAnswer:
\nThe given point is: (\\(\\frac{3}{10}\\), –\\(\\frac{1}{5}\\))
\nNow,
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nThe representation of the given point in the form of a decimal number is: (0.3, -0.2)
\nNow,
\nFrom the given coordinate plane,
\nWe can observe that the landmark that is located at (\\(\\frac{3}{10}\\), –\\(\\frac{1}{5}\\)) is: Start of Hiking Trail
\nHence, from the above,
\nWe can conclude that
\nThe start of Hiking Trail is located at (\\(\\frac{3}{10}\\), –\\(\\frac{1}{5}\\))<\/p>\n

Question 33.
\nBe Precise<\/strong> Write the ordered pair to locate the end of the hiking trail in two different ways.
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nWe can observe that
\nThe coordinates to locate the end of the hiking trail is: (0.2, -0.8)
\nHence, from the above,
\nWe can conclude that
\nThe coordinates to locate the end of the hiking trail is: (0.2, -0.8)<\/p>\n

Question 34.
\nWhat are the coordinates of the information center? Explain.
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nWe can observe that
\nThe coordinates of the Information center are: (-0.2, 0.7)
\nHence, from the above,
\nWe can conclude that
\nThe coordinates of the Information center are: (-0.2, 0.7)<\/p>\n

Question 35.
\nWhat are the coordinates of the point that is a reflection across the x-axis of the pond?
\nAnswer:
\nFrom Question 31,
\nWe can observe that,
\nThe location of the pond in the coordinate plane is: (-0.7, -0.2)
\nNow,
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nWe know that,
\nThe reflection of (x, y) across the x-axis is: (x, -y)
\nSo,
\nThe coordinates of the point that is a reflection across the x-axis of the pond are: (-0.7, 0.2)
\nHence, from the above,
\nWe can conclude that
\nThe coordinates of the point that is a reflection across the x-axis of the pond are: (-0.7, 0.2)<\/p>\n

Question 36.
\nUse Structure<\/strong> Which picnic areas are located at points that are reflections of each other across one of the axes of the coordinate plane?
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nWe can observe that
\nThe coordinates of Picnic area 1 are: (-0.8, 0.6)
\nThe coordinates of Picnic area 2 are: (0.6, 0.9)
\nThe coordinates of Picnic area 3 are: (0.6, 0.3)
\nThe coordinates of Picnic area 4 are: (-0.8, -0.6)
\nNow,
\nWe know that,
\nThe reflection of (x, y) across the x-axis is: (x, -y)
\nThe reflection of (x, y) across the y-axis is: (-x, y)
\nHence, from the above,
\nWe can conclude that
\nPicnic area 4 is the reflection of Picnic area 1 across the x-axis<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 37.
\nGraph and label each point on the coordinate plane at the right.
\nA. (\\(\\frac{3}{4}\\), -1\\(\\frac{1}{2}\\))
\nB. (-2.75, -2.25)
\nC. (0, 2\\(\\frac{1}{4}\\))
\nD. (-1.75, 2)
\n\"Envision
\nAnswer:
\nThe given points are:
\nA. (\\(\\frac{3}{4}\\), -1\\(\\frac{1}{2}\\))
\nB. (-2.75, -2.25)
\nC. (0, 2\\(\\frac{1}{4}\\))
\nD. (-1.75, 2)
\nHence,
\nThe representation of the given points in the coordinate plane are:
\n\"\"<\/p>\n

3-Act Mathematical Modeling: The Ultimate Throw<\/strong><\/p>\n

3-ACT MATH<\/strong><\/p>\n

The ULTIMATE THROW<\/strong><\/p>\n

ACT 1<\/strong><\/p>\n

Question 1.
\nAfter watching the video, what is the first question that comes to mind?
\nAnswer:<\/p>\n

Question 2.
\nWrite the Main Question you will answer.
\nAnswer:<\/p>\n

Question 3.
\nMake a prediction to answer this Main Question.
\nThe person who threw the flying disc farther is \"Envision
\nAnswer:<\/p>\n

Question 4.
\nConstruct Arguments<\/strong> Explain how you arrived at your prediction.
\n\"Envision
\nAnswer:<\/p>\n

ACT 2<\/strong><\/p>\n

Question 5.
\nWhat information in this situation would be helpful to know? How would you use that information?
\nAnswer:<\/p>\n

Question 6.
\nUse Appropriate Tools<\/strong> What tools can you use to solve the problem? Explain how you would use them strategically.
\nAnswer:<\/p>\n

Question 7.
\nA Model with Math<\/strong>
\nRepresent the situation using mathematics. Use your representation to answer the Main Question.
\nAnswer:<\/p>\n

Question 8.
\nWhat is your answer to the Main Question? Does it differ from your prediction? Explain.
\n\"Envision
\nAnswer:<\/p>\n

\u0410\u0421\u0422 3<\/strong><\/p>\n

Question 9.
\nWrite the answer you saw in the video.
\nAnswer:<\/p>\n

Question 10.
\nReasoning<\/strong> Does your answer match the answer in the video? If not, what are some reasons that would explain the difference?
\n\"Envision
\nAnswer:<\/p>\n

Question 11.
\nMake Sense and Persevere<\/strong> Would you change your model now that you know the answer? Explain.
\nAnswer:<\/p>\n

Reflect<\/strong><\/p>\n

Question 12.
\nModel with Math<\/strong>
\nExplain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?
\nAnswer:<\/p>\n

Question 13.
\nMake Sense and Persevere<\/strong> When did you struggle most while solving the problem? How did you overcome that obstacle?
\n\"Envision
\nAnswer:<\/p>\n

SEQUEL<\/strong><\/p>\n

Question 14.
\nReasoning<\/strong> Suppose each person walks to the other person’s disc. They throw each other’s discs toward the starting point. Where do you think each disc will land?
\n\"Envision
\nAnswer:<\/p>\n

Lesson 2.5 Find Distances on the Coordinate Plane<\/h3>\n

Solve & Discuss It!<\/strong><\/p>\n

Graph the points on the coordinate plane below. What picture do you make when you connect the points in order?
\n(3, 3), (0, 0), (-4,-4), (-9,0), (-4, 4), (0, 0), (3, -3), (3, 3)
\nName a pair of points that are the same distance from the x-axis. Explain your choice.
\n\"Envision
\nAnswer:
\nThe given points are:
\n(3, 3), (0, 0), (-4,-4), (-9,0), (-4, 4), (0, 0), (3, -3), (3, 3)
\nNow,
\nRepresent the given points in the coordinate plane and connect the points in the given order
\nSo,
\nThe representation of the given points in the coordinate plane is:
\n\"\"
\nNow,
\nFrom the given coordinate plane,
\nWe can observe that
\nWhen we connect the given points in order,
\nThe picture you made looks like a composite figure of a rhombus and a triangle
\nNow,
\nThe pair of points that are the same distance from the x-axis is:
\n(3, 3), (3, -3); and (-4, 4), (-4, -4)
\nHence, from the above,
\nWe can conclude that
\na. When we connect the given points in order,
\nThe picture you made looks like a composite figure of a rhombus and a triangle
\nb. The pair of points that are the same distance from the x-axis is:
\n(3, 3), (3, -3); and (-4, 4), (-4, -4)<\/p>\n

Use Structure<\/strong> How can you use the structure of the grid to find a pair of points that are the same distance from the x-axis?
\nAnswer:
\nDerived from the Pythagorean Theorem, the distance formula is used to find the distance between two points in the plane. The Pythagorean Theorem,\u00a0a\u00b2+b\u00b2=c\u00b2\u00a0is based on a right triangle where a and b are the lengths of the legs adjacent to the right angle, and c is the length of the hypotenuse.<\/p>\n

Focus on math practices<\/strong>
\nUse Structure<\/strong> How can you use the coordinate plane to find the total length of the picture you graphed?
\nAnswer:
\nWe know that,
\nThe “Linear distance” between the two points is the square root of the sum of the squared values of the x-axis distance and the y-axis distance
\nEx:
\nThe distance between (3,2) and (7,8) is \\(\\sqrt{52}\\), or approximately 7.21 units.
\n\"\"<\/p>\n

? Essential Question<\/strong> How can you find the distance between two points on a coordinate plane?
\nAnswer:
\nWe know that,
\nThe distance between two points by using the distance formula, which is an application of the Pythagorean theorem. We can rewrite the Pythagorean theorem as
\nd=\u221a(x2<\/sub>-x1<\/sub>)\u00b2+(y2<\/sub>-y1<\/sub>)\u00b2
\nto find the distance between any two points.<\/p>\n

Try It!<\/strong>
\nWhat is the distance from the school to the playground? Explain how you used absolute values to find the distance?
\n\"\"
\nAnswer:
\nThe given map is:
\n\"\"
\nNow,
\nFrom the given coordinate plane,
\nWe can observe that
\nThe coordinates at which the school is located are: (-4, 2)
\nThe coordinates at which the playground is located are: (-4, 5)
\nNow,
\nCompare the given points with (x, a), and (x, b) respectively
\nWe know that,
\nLet (x, a), and (y, b) are the two points
\nNow,
\nThe absolute distance of (x, a) = | x + a|
\nThe absolute distance of (y, b) = | y + b|
\nSo,
\nThe absolute distance between (x, a), and (y, b) = |x + a| + |y + b|
\nSo,
\nThe absolute distance between (-4, 2), and (-4, 5) = | -4 + 2| + |-4 +5|
\n= |-2| + |1|
\n= 2 + 1
\n= 3 units
\nHence, from the above,
\nWe can conclude that
\nThe distance from the school to the playground is: 3 units<\/p>\n

Convince Me!<\/strong> To find the distance from the school to the playground, do you add or subtract the absolute values of the y-coordinates? Explain.
\nAnswer:
\nLet (x, a), and (y, b) are the two points
\nNow,
\nThe absolute distance of (x, a) = | x + a|
\nThe absolute distance of (y, b) = | y + b|
\nSo,
\nThe absolute distance between (x, a), and (y, b) = |x + a| + |y + b|
\nHence, from the above,
\nWe can conclude that
\nTo find the distance from the school to the playground, we added the absolute values of the y-coordinates<\/p>\n

Try It!<\/strong>
\nWhat is the total distance of the Coulters’ return trip after their day at the water park?
\n\"\"
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"\"
\nNow,
\nFrom the given figure,
\nwe can observe that
\nThe total distance of the Coulters’ return trip after their day at the water park = (The absolute distance of Water park) + (The absolute distance of Coulters’ home)
\nNow,
\nWe know that,
\nLet (x, a), and (y, b) are the two points
\nNow,
\nThe absolute distance of (x, a) = | x + a|
\nThe absolute distance of (y, b) = | y + b|
\nSo,
\nThe absolute distance between (x, a), and (y, b) = |x + a| + |y + b|
\nNow,
\nFrom the given points,
\nWe can observe that the x-coordinates are the same for the Water park and the Coulters’ home
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe total distance of the Coulters’ return trip after their day at the Water Park is: 151 miles<\/p>\n

Try It!<\/strong>
\nPoint D is in Quadrant IV and is the same distance from point B as point A. What are the coordinates of point D?
\n\"\"
\nAnswer:<\/p>\n

KEY CONCEPT<\/strong>
\nYou can use absolute values to find distances between points on a coordinate plane.
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n?Essential Question<\/strong> How can you find the distance between two points on a coordinate plane?
\nAnswer:
\nWe know that,
\nThe distance between two points by using the distance formula, which is an application of the Pythagorean theorem. We can rewrite the Pythagorean theorem as
\nd=\u221a(x2<\/sub>-x1<\/sub>)\u00b2+(y2<\/sub>-y1<\/sub>)\u00b2
\nto find the distance between any two points.<\/p>\n

Question 2.
\nLook for Relationships<\/strong> To find the distance between two points using their coordinates, when do you add their absolute values and when do you subtract them?
\nAnswer:
\nCase 1:
\nLet the coordinates of points be (x, y), and (a, b)
\nSo,
\nThe absolute value of (x, y) is: |x + y|
\nThe absolute value of (a, b) is: |a + b|
\nHence, from the above,
\nWe can conclude that
\nWe will add the absolute values when both the coordinates of the given points are positive
\nCase 2:
\nLet the coordinates of points be (x, -y), and (-a, b)
\nSo,
\nThe absolute value of (x, -y) is: |x – y|
\nThe absolute value of (-a, b) is: |b – a|
\nHence, from the above,
\nWe can conclude that
\nWe will subtract the absolute values when any one of the coordinates of the given points are negative<\/p>\n

Question 3.
\nReasoning<\/strong> Can you use absolute value to find the distance between Li’s house and Tammy’s house in
\nExample 1? Explain.
\n\"\"
\nAnswer:
\nThe given figure is:
\n\"\"
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe coordinates at which Li’s House is located = (-4, -3)
\nThe coordinates at which Tammy’s house is located = (2, 0)
\nNow,
\nWe know that,
\nThe absolute distance between two points is: |x + a| + |y + b|
\nHence, from the above,
\nWe can conclude that
\nYou can use absolute value to find the distance between Li’s house and Tammy’s house that is present in Example 1<\/p>\n

Do You Know How?<\/strong><\/p>\n

In 4-9, find the distance between each pair of points.<\/strong><\/p>\n

Question 4.
\n(-5, 2) and (-5, 6)
\nAnswer:
\nThe given points are: (-5, 2) , and (-5, 6)
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe absolute distance between two points = |x + a| + |y + b|
\nSo,
\nThe distance between (-5, 2), and (-5, 6) = |-5 + 2| + |-5 + 6|
\n= 3 + 1
\n= 4 units
\nHence, from the above,
\nWe can conclude that
\nThe distance between the given pair of points is: 4 units<\/p>\n

Question 5.
\n(4.5, -3.3) and (4.5, 5.5)
\nAnswer:
\nThe given points are: (4.5, -3.3), and (4.5, 5.5)
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe absolute distance between two points = |x + a| – |y + b|
\nSo,
\nThe distance between (4.5, -3.3), and (4.5, 5.5) = |4.5 – 3.3| – |4.5 + 5.5|
\n= |1.2 – 10|
\n= 8.8 units
\nHence, from the above,
\nWe can conclude that
\nThe distance between the given pair of points is: 11.2 units<\/p>\n

Question 6.
\n(5\\(\\frac{1}{2}\\), -7\\(\\frac{1}{2}\\)) and (5\\(\\frac{1}{2}\\), -1\\(\\frac{1}{2}\\))
\nAnswer:
\nThe given points are: (5\\(\\frac{1}{2}\\), -7\\(\\frac{1}{2}\\)) and (5\\(\\frac{1}{2}\\), -1\\(\\frac{1}{2}\\))
\nNow,
\nConvert the coordinates of the given points into decimal numbers
\nSo,
\nThe given points are: (5.5, -3.5), and (5.5, -1.5)
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe absolute distance between two points = |x + a| – |y + b|
\nSo,
\nThe distance between (5.5, -3.5), and (5.5, -1.5) = |5.5 – 3.5| – |5.5 – 1.5|
\n= |2 – 4|
\n= 2 units
\nHence, from the above,
\nWe can conclude that
\nThe distance between the given pair of points is: 2 units<\/p>\n

Question 7.
\n(-2\\(\\frac{1}{4}\\), -8 ) and (7\\(\\frac{3}{4}\\), -8)
\nAnswer:
\nThe given points are: (-2\\(\\frac{1}{4}\\), -8 ) and (7\\(\\frac{3}{4}\\), -8)
\nNow,
\nConvert the coordinates of the given points into a decimal number
\nSo,
\nThe given points are: (-2.25, -8), and (7.75, -8)
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe absolute distance between two points = |x + a| – |y + b|
\nSo,
\nThe distance between (-2.25, -8), and (7.75, -8) = |-2.25 – 8| – |7.75 – 8|
\n= |10.25 – 0.25|
\n= 10 units
\nHence, from the above,
\nWe can conclude that
\nThe distance between the given pair of points is: 10 units<\/p>\n

Question 8.
\n(5\\(\\frac{1}{4}\\), -3\\(\\frac{1}{4}\\)) and (5\\(\\frac{1}{4}\\), -6\\(\\frac{1}{4}\\))
\nAnswer:
\nThe given points are: (5\\(\\frac{1}{4}\\), -3\\(\\frac{1}{4}\\)) and (5\\(\\frac{1}{4}\\), -6\\(\\frac{1}{4}\\))
\nNow,
\nConvert the coordinates of the given points into a decimal number
\nSo,
\nThe given points are: (5.25, -3.25), and (5.25, -6.25)
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe absolute distance between two points = |x + a| + |y + b|
\nSo,
\nThe distance between (5.25, -3.25), and (5.25, -6.25) = |5.25 – 3.25| + |5.25 – 6.25|
\n= 2 + 1
\n= 3 units
\nHence, from the above,
\nWe can conclude that
\nThe distance between the given pair of points is: 3 units<\/p>\n

Question 9.
\n(-1\\(\\frac{1}{2}\\), \u20136\\(\\frac{1}{2}\\)) and (-2\\(\\frac{1}{2}\\), -6\\(\\frac{1}{2}\\))
\nAnswer:
\nThe given points are: (-1\\(\\frac{1}{2}\\), \u20136\\(\\frac{1}{2}\\)) and (-2\\(\\frac{1}{2}\\), -6\\(\\frac{1}{2}\\))
\nNow,
\nConvert the coordinates of the given points into a decimal number
\nSo,
\nThe given points are: (-1.5, -6.5), and (-2.5, -6.5)
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe absolute distance between two points = |x + a| + |y + b|
\nSo,
\nThe distance between (-1.5, -6.5), and (-2.5, -6.5) = |-1.5 – 6.5| – |-2.5 – 6.5|
\n= |8 – 9|
\n= 1 units
\nHence, from the above,
\nWe can conclude that
\nThe distance between the given pair of points is: 1 units<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Scan for Multimedia<\/strong><\/p>\n

Leveled Practice<\/strong> In 10-15, find the distance between each pair of points.<\/strong><\/p>\n

Question 10.
\n\"Envision
\nAnswer:
\nThe given points are: (-2, 8), and (7, 8)
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe distance between two points = |x + a| + |y + b|
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe distance between the given points is: 21 units<\/p>\n

Question 11.
\n\"Envision
\nAnswer:
\nThe given points are: (-6.1, -8.4), and (-6.1, -4.2)
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe distance between two points = |x + a| – |y + b|
\nNow,
\nSince the x-coordinates are the same, subtract the y-coordinates to find the distance between the given points
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe distance between the given points is: 4.2 units<\/p>\n

Question 12.
\n\"Envision
\nAnswer:
\nThe given points are: (12\\(\\frac{1}{2}\\), 3\\(\\frac{3}{4}\\)), and (-4\\(\\frac{1}{2}\\), 3\\(\\frac{3}{4}\\))
\nNow,
\nConvert the coordinates of the given points into a decimal number
\nSo,
\nThe given points are: (12.5, 3.75), and (-4.5, 3.75)
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe distance between two points = |x + a| + |y + b|
\nNow,
\nSince y-coordinates are the same, add the x-coordinates to find the distance between the given points
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe distance between the given points is: 17 units<\/p>\n

Question 13.
\n(-5, -3) and (-5, -6)
\nAnswer:
\nThe given points are: (-5, -3) , and (-5, -6)
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe absolute distance between two points = |x + a| – |y + b|
\nSo,
\nThe distance between (-5, -3), and (-5, -6) = |-5 – 3| – |-5 – 6|
\n= |8 – 11|
\n= 3 units
\nHence, from the above,
\nWe can conclude that
\nThe distance between the given pair of points is: 3 units<\/p>\n

Question 14.
\n(-5.4, 4.7) and (0.6, 4.7)
\nAnswer:
\nThe given points are: (-5.4, 4.7), and (0.6, 4.7)
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe absolute distance between two points = |x + a| – |y + b|
\nSo,
\nThe distance between (-5.4, 4.7), and (0.6, 4.7) = |-5.4 + 4.7| – |0.6 + 4.7|
\n= |-0.7 – 5.3|
\n= 6 units
\nHence, from the above,
\nWe can conclude that
\nThe distance between the given pair of points is: 6 units<\/p>\n

Question 15.
\n(7\\(\\frac{1}{2}\\), -5\\(\\frac{3}{4}\\)) and (7\\(\\frac{1}{2}\\), -1\\(\\frac{1}{4}\\))
\nAnswer:
\nThe given points are: (7\\(\\frac{1}{2}\\), -5\\(\\frac{3}{4}\\)) and (7\\(\\frac{1}{2}\\), -1\\(\\frac{1}{4}\\))
\nNow,
\nConvert the coordinates of the given points into a decimal number
\nSo,
\nThe given points are: (7.5, -5.75), and (7.5, -1.25)
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe absolute distance between two points = |x + a| – |y + b|
\nSo,
\nThe distance between (7.5, -5.75), and (7.5, -1.25) = |7.5 – 5.75| – |7.5 – 1.25|
\n= |1.75 – 6.25|
\n= 4.5 units
\nHence, from the above,
\nWe can conclude that
\nThe distance between the given pair of points is: 4.5 units<\/p>\n

In 16-19, use the map at the right.
\n\"Envision
\n<\/strong><\/p>\n

Question 16.
\nFind the distance from roller coaster 1 to the swings.
\nAnswer:
\nThe given map is:<\/p>\n

\"Envision
\nNow,
\nFrom the given map,
\nWe can observe that
\nThe coordinates of roller coaster 1 are: (-6, 7)
\nThe coordinates of the swings are: (1, 7)
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe distance between the two points = |x + a| – |y – b|
\nSo,
\nThe distance from roller coaster 1 to the swings = |-6 + 7| – |1 + 7|
\n= |1 – 8|
\n= 7 units
\nHence, from the above,
\nWe can conclude that
\nThe distance from roller coaster 1 to the swings is: 7 units<\/p>\n

Question 17.
\nFind the distance from the Ferris wheel to the roller coaster 3.
\nAnswer:
\nThe given map is:
\n\"Envision
\nNow,
\nFrom the given map,
\nWe can observe that
\nThe coordinates of the Ferris Wheel are: (-6, 2)
\nThe coordinates of roller coaster 3 are: (-6, -8)
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe distance between the two points = |x + a| + |y – b|
\nSo,
\nThe distance from Ferris Wheel to the roller coaster 3 = |-6 + 2| – |-6 – 8|
\n= |-4 + 14|
\n= 10 units
\nHence, from the above,
\nWe can conclude that
\nThe distance from Ferris Wheel to the roller coaster 3 is: 10 units<\/p>\n

Question 18.
\nFind the total distance from roller coaster 2 to roller coaster 3 and then to the water slide.
\nAnswer:
\nThe given map is:
\n\"Envision
\nNow,
\nFrom the given map,
\nWe can observe that
\nThe coordinates of roller coaster 2 are: (-6, -3)
\nThe coordinates of roller coaster 3 are: (-6, -8)
\nThe coordinates of the Water slide are: (8, -8)
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe distance between the two points = |x + a| + |y – b|
\nSo,
\nThe distance from Ferris Wheel to the roller coaster 3 = |-6 + 2| – |-6 – 8|
\n= |-4 + 14|
\n= 10 units
\nHence, from the above,
\nWe can conclude that
\nThe distance from Ferris Wheel to the roller coaster 3 is: 10 units<\/p>\n

Question 19.
\nHigher Order Thinking<\/strong> Is the distance from the merry-go-round to the water slide the same as the distance from the water slide to the merry-go-round? Explain.
\nAnswer:
\nThe given map is:
\n\"Envision
\nNow,
\nFrom the given map,
\nWe can observe that
\nThe coordinates of merry-go-round are: (8, 2)
\nThe coordinates of the Water slide are: (8, -8)
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe distance between the two points = |x + a| + |y – b|
\nSo,
\nThe distance from merry-go-round to the Water slide\u00a0 = |8 + 2| – |8 – 8|
\n= |0 + 10|
\n= 10 units
\nSo,
\nThe distance from the Water slide to merry-go-round = |8 – 8| – |8 + 2|
\n= |0 – 10|
\n= 10 units
\nHence, from the above,
\nWe can conclude that
\nThe distance from merry-go-round to the Water slide and the distance from the Water slide to merry-go-round are the same<\/p>\n

In 20 and 21, use the coordinate plane at the right.<\/strong><\/p>\n

The graph shows the locations of point G and point H. Point J is graphed at (n, -3). The distance from point H to point J is equal to the distance from point H to point G.<\/p>\n

\"Envision<\/p>\n

Question 20.
\nWhat is the distance from point H to point J?
\nAnswer:
\nIt is given that
\nThe graph shows the locations of point G and point H. Point J is graphed at (n, -3). The distance from point H to point J is equal to the distance from point H to point G.
\nNow,
\nThe given coordinate plane is:<\/p>\n

\"Envision
\nNow,
\nFrom the given coordinate plane,
\nWe can observe that
\nThe coordinates of G are: (0, -y)
\nIt is given that
\nThe coordinates of point J are: (n, -3)
\nNow,
\nCompare points H and J with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe distance between 2 points = | x + a| – |y + b|
\nSo,
\nThe distance between points H and J = |n – 6| – |n – 3|
\n= |n – 6 – n + 3|
\n= 3 units
\nHence, from the above,
\nWe can conclude that
\nThe distance from point G to point H is: 3 units<\/p>\n

Question 21.
\nWhat is the value of n?
\nAnswer:
\nIt is given that
\nThe graph shows the locations of point G and point H. Point J is graphed at (n, -3). The distance from point H to point J is equal to the distance from point H to point G.
\nNow,
\nThe given coordinate plane is:<\/p>\n

\"Envision
\nNow,
\nFrom Exercise 20,
\nWe know that,
\nThe distance from point G to point H is: 3 units
\nNow,
\n|n – 6 + n – 3| = 3
\n|2n – 9| = 3
\n2n = 3 + 9
\n2n = 12
\nn = \\(\\frac{12}{2}\\)
\nn = 6 units
\nHence, from the above,
\nWe can conclude that
\nThe value of n is: 6 units<\/p>\n

Question 22.
\nUse Structure<\/strong> Suppose a, b, and c are all negative numbers. How do you find the distance between points (a, b) and (a, c)?
\nAnswer:
\nIt is given that
\na, b, and c are all negative numbers.
\nNow,
\nThe given points are: (a, b), and (a, c)
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe distance between 2 points = |x + a| – |y + b|
\nSo,
\nThe distance between the given points = |-a + (-b)| – |-a + (-c)|
\n= |-a – b| – |-a – c|
\n= |-a – b + a + c|
\n= |c – b| units
\nHence, from the above,
\nWe can conclude that
\nThe distance between points (a, b), and (a, c) is: |c – b| units<\/p>\n

Question 23.
\nA scientist graphed the locations of the epicenter of an earthquake and all of the places where people reported feeling the earthquake. She positioned the epicenter at (-1, 8) and the farthest location reported to have felt the quake was positioned at (85, 8). If each unit on the graph represents 1 mile, how far from its epicenter was the earthquake felt?
\nAnswer:
\nIt is given that
\nA scientist graphed the locations of the epicenter of an earthquake and all of the places where people reported feeling the earthquake. She positioned the epicenter at (-1, 8) and the farthest location reported to have felt the quake was positioned at (85, 8) and each unit on the graph represents 1 mile
\nNow,
\nThe given points are: (-1, 8), (85, 8)
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe distance between 2 points = |x + a| – |y + b|
\nSo,
\nThe distance between the given points = |-1 + 8| – |85 + 8|
\n= |7 – 93|
\n= 86 miles
\nHence, from the above,
\nWe can conclude that
\nThe earthquake felt 86 miles away from its epiccenter<\/p>\n

Question 24.
\nThe rectangle ABCD shown on the coordinate plane represents an overhead view of a piece of land. Each unit represents 1,000 feet. What are the dimensions of the rectangular piece of land, in feet?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe rectangle ABCD shown on the coordinate plane represents an overhead view of a piece of land. Each unit represents 1,000 feet
\nNow,
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nWe can observe that
\nThe dimensions of the rectangle present in the given coordinate plane are:
\nA (0, 0), B (0, 4), C (5, 5), and D (5, 0)
\nSo,
\nThe dimensions of the rectangular piece of land, in feet are:
\nA (0, 0), B (0, 4,000), C (5,000, 5,000), and D (5,000, 0)
\nHence, from the above,
\nWe can conclude that
\nThe dimensions of the rectangular piece of land, in feet are:
\nA (0, 0), B (0, 4,000), C (5,000, 5,000), and D (5,000, 0)<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 25.
\nYou are given the following ordered pairs.
\n(3.5, -1) (-1.5, 3) (-3, 3) (3.5, 2.5) (-1.5, -1.5)
\nPART A<\/strong>
\nGraph the ordered pairs on the coordinate plane.
\nAnswer:
\nThe given ordered pairs are:
\n(3.5, -1), (-1.5, 3), (-3, 3), (3.5, 2.5), (-1.5, -1.5)
\nHence,
\nThe representation of the given ordered pairs in the coordinate plane is:
\n\"\"<\/p>\n

PART B<\/strong>
\nFind the two ordered pairs on the coordinate plane that are 4.5 units apart.
\n\"Envision
\nAnswer:
\nThe given ordered pairs are:
\nA (3.5, -1), B(-1.5, 3), C(-3, 3), D(3.5, 2.5), E(-1.5, -1.5)
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe distance between 2 points = |x – a| + |y + b|
\nSo,
\nThe distance of AB = 6.41 units
\nThe distance of AC = 7.63 units
\nThe dsitance of AD = 3.5 units
\nThe distance of AE = 5.02 units
\nThe distance of BC = 4.5 units
\nThe distance of BD = 5.02 units
\nThe distance of BE = 4.5 units
\nHence, from the above,
\nWe can conclude that
\nThe two ordered pairs on the coordinate plane that are 4.5 units apart is:
\nB(-1.5, 3), and C (-3, 3); B(-1.5, 3), and E (-1.5, -1.5)<\/p>\n

Lesson 2.6 Represent Polygons on the Coordinate Plane<\/h3>\n

Solve & Discuss It!<\/strong><\/p>\n

ACTIVITY<\/strong><\/p>\n

Draw a polygon with vertices at A(-1, 6), B(-7, 6), C(-7, -3), and D(-1, -3). Then find the perimeter of the polygon.
\n\"Envision
\nAnswer:
\nThe vertices of a given polygon are:
\nA(-1, 6), B(-7, 6), C(-7, -3), and D(-1, -3)
\nSo,
\nThe representtaion of the given vertices of a polygon in the coordinate plane are:
\n\"\"
\nNow,
\nWe know that,
\nThe “Perimeter” of a figure is the sum of all of the side lengths of a given figure
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe distance betwee 2points = |x + a| + |y + b|
\nSo,
\nThe distace of AB = |-1 + 6| + |-7 + 6|
\n= |5 + 1|
\n= 6 units
\nThe distance of BC = |-7 + 6| + |-7 – 3|
\n= |1 – 10|
\n= 9 units
\nThe distace of CD = |-7 – 3| + |-1 – 3|
\n= |10 – 4|
\n= 6 units
\nThe distace of DA = |-1 – 3| + |-1 + 6|
\n= |5 + 4|
\n= 9 units
\nSo,
\nThe perimeter of the given polygon = 6 + 6 + 9 + 9
\n= 30 units
\nHence, from the above,
\nWe can conclude that
\nThe perimeter of the given polygon is: 30 units<\/p>\n

Use Structure<\/strong> How can you use the coordinate plane to draw the polygon and find its perimeter?
\nAnswer:
\nYou\u00a0can use ordered pairs to represent vertices of polygons. To draw a polygon in a coordinate plane, plot and connect the ordered pairs
\nTo find the perimeter of a regular polygon, we\u00a0take the length of each side, , and multiply it by the number of sides<\/p>\n

Focus on math practices<\/strong>
\nConstruct Arguments<\/strong> What type of polygon did you draw? Use a definition to justify your answer.
\nAnswer:
\nThe representation of the vertices of the given polyon is:
\n\"\"
\nNow,
\nFrom the given coordinate plane,
\nWe can observe that
\nThe opposite side lengths are equal and all the angle measures are 90\u00b0
\nNow,
\nWe know that,
\nThe polygon that has the same opposite side lengths is called as “Rectangle”
\nHence, from the above,
\nWe can conclude that
\nThe type of polyon did you draw is: Rectangle<\/p>\n

Visual Learning<\/strong><\/p>\n

? Essential Question<\/strong> How is distance used to solve problems about polygons in a coordinate plane?
\nAnswer:
\nTo find the distance between two points with\u00a0the same x-coordinates, subtract their y-coordinates. To find the distance between two points with the same y-coordinates, subtract their x-coordinates
\nTo find the distance between the different x and y-coordinates, subtract the x-coordinates of the 2 points and y-coordinates of the 2 points and add both the values<\/p>\n

Try It!<\/strong>
\nThe archaeologist later decides to extend the roped-off area so that the new perimeter goes from A to B to the food tent to the working tent and then back to A. How much rope does she need now?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe archaeologist later decides to extend the roped-off area so that the new perimeter goes from A to B to the food tent to the working tent and then back to A
\nNow,
\nThe given figure is:
\n\"\"
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe coordinates of A are: (-4, 6)
\nThe coordinates of B are: (2, 6)
\nThe coordinates of Food Tent are: (2, -2)
\nThe coordinates of Working Tent are: (-4, -2)
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nSo,
\n\"\"
\nHence, from the above,
\nWe can concldue that
\nThe length of rope the archaeologist needs now is: 28 meters<\/p>\n

Convince Me!<\/strong> How could you use the formula for the perimeter of a rectangle to find the perimeter of the larger rectangle using two of the distances?
\nAnswer:
\nWe know that,
\nThe perimeter of a rectangle = 2(l + w)
\nNow,
\nIf the perimeter and the length of a rectangle are known, then
\nWidth = P\/2 – l,
\nwhere,
\nl = length, w = width, and P = perimeter of the rectangle.
\nNow,
\nIf the perimeter and the width are known, then
\nLength(L) = P\/2 – w.<\/p>\n

Try It!<\/strong>
\nThe rancher needs to replace the fence for the holding pen for the horses. How much fencing does he need?
\nThe rancher needs \"Envision yards of fencing.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe rancher needs to replace the fence for the holding pen for the horses
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nThe coordinates of the fence that is used for the holding pen for the houses is:
\nA (4.5, 14), B (15, 14), C (15, 8), D (10, 8), E (10, 3), F (15, 3), G (15, -2.25), and H (4.5, -2.25)
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe distance between 2 points = |x + a| + |y + b|
\nNow,
\nThe distance of AB = |4.5 – 15| = 11.5 units
\nThe distance of BC = |14 – 8| = 6 units
\nThe distance of CD = |15 – 10| = 5 units
\nThe distance of DE = |8 – 3| = 5 units
\nThe distance of EF = |10 – 15| = 5 units
\nThe distance of FG = | 3 + 2.25| = 5.25 units
\nThe distance of GH = |15 – 4.5| = 11.5 units
\nThe distance of HA = |14 + 2.25| = 16.25 units
\nNow,
\nWe know that,
\nTo find the length of fencing, we have to find the perimeter of the given polyon
\nSo,
\nThe length of the fencing that Rancher needed = 11.5 + 6 + 5 + 5 + 5 + 5.25 + 11.5 + 16.25
\n= 65.5 yards
\nHence, from the above,
\nWe can conclude that
\nThe rancher needs 65.5 yards of fencing<\/p>\n

Try It!<\/strong>
\nJoaquin says that quadrilateral ADEF is a square. Is he correct? Explain.
\nAnswer:
\nIt is given that
\nJoaquin says that quadrilateral ADEF is a square
\nNow,
\nThe given figure is:
\n\"\"
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe coordinates of A are: (-5, 2)
\nThe coordinates of D are: (7, 2)
\nThe coordinates of E are: (7, -8)
\nThe coordinates of F are: (-5, -8)
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe distance between 2 points = |x + a| + |y + b|
\nNow,
\nThe distance of AD = |-5 + 2| + |7 + 2|
\n= 12 units
\nThe distance of DE = |7 + 2| + |7 – 8|
\n= 10 units
\nThe distance of EF = |7 – 8| + |-5 – 8|
\n= 12 units
\nThe disatnce of FD = |-5 – 8| – |-5 + 2|
\n= 10 units
\nNow,
\nFrom the given side lengths,
\nWe can observe that
\nThe opposite side lengths are equal
\nNow,
\nwe know that,
\nThe rectangle has the opposite side lengths
\nHence, from the above,
\nWe can conclude that
\nJoaquin is not correct<\/p>\n

KEY CONCEPT<\/strong>
\nYou can represent polygons on a coordinate plane and solve problems by using absolute values to find side lengths.
\nAdd or subtract absolute values to find the length of each side.
\n\"Envision
\nAB: |-3| + |2| = 3 + 2 = 5 units
\nBC: |4| – |2| = 4 – 2 = 2 units
\nCD: |-3| + |2| = 3 + 2 = 5 units
\nDA: |4| – |2| = 4 – 2 = 2 units<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n?Essential Question<\/strong> How is distance used to solve problems about polygons in a coordinate plane?
\nAnswer:
\nTo find the distance between two points with\u00a0the same x-coordinates, subtract their y-coordinates. To find the distance between two points with the same y-coordinates, subtract their x-coordinates
\nTo find the distance between the different x and y-coordinates, subtract the x-coordinates of the 2 points and y-coordinates of the 2 points and add both the values<\/p>\n

Question 2.
\nReasoning<\/strong> In Example 1, why do you add absolute values to find the distance from A to B but subtract absolute values to find the distance from B to C?
\nAnswer:
\nFrom Example 1,
\nWe know that,
\nThe coordinates of A, B, and C are:
\nA (-4, 6), B (2, 6), and C (2, 1)
\nNow,
\nWe know that,
\nThe absolute distance between given points = |x + a| + |y + b|
\nNow,
\nWhen the y-coordinates are the same,
\nThe absolute distance between 2 points = |x + y|
\nSo,
\nThe absolute distance from A and B is positive because the x-coordinates move from left to right (or) vice-versa
\nWhen the x-coordinates are the same,
\nThe absolute distance between 2 points = |a – b|
\nSo,
\nThe absolute distance from B and C is negative because the y-coordinates move from top to bottom (or) vice-versa<\/p>\n

Question 3.
\nConstruct Arguments<\/strong> Could you add or subtract the absolute values of coordinates to find the length of the diagonal AC of rectangle ABCD in Example 1? Explain.
\nAnswer:
\nFrom Example 1,
\nWe know that,
\nA (-4, 6), B (2, 6), C (2, 1), and D (-4, 1)
\nNow,
\nWe know that,
\nThe distance between 2 points = |x + a| + |y + b|
\nSo,
\nThe distance of the diagonal AC of rectangle ABCD is solved by the addition of the absolute values of the coordinates of A and C
\nHence, from the above,
\nWe can conclude that
\nWe add the absolute values of coordinates to find the length of the diagonal AC of rectangle ABCD in Example 1<\/p>\n

Do You Know How?<\/strong><\/p>\n

Question 4.
\nFind the perimeter of rectangle MNOP with vertices M(-2,5), N(-2, -4), 0(3,-4), and P(3, 5).
\nAnswer:
\nThe given vertices of rectangle MNOP are:
\nM(-2,5), N(-2, -4), 0(3,-4), and P(3, 5)
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe distance between 2 points = |x + a| + |y + b|
\nNow,
\nThe disatnce of MN = |-2 + 5| + |-2 – 4|
\n= |3 + 6|
\n= 9 units
\nThe distance of NO = |-2 – 4| + |3 – 4|
\n= |6 – 1|
\n= 5 units
\nThe distance of OP = |3 – 4| + |3 + 5|
\n= |1 + 8|
\n= 9 units
\nThe distance of PM = |3 + 5| + |-2 + 5|
\n= |8 – 3|
\n= 5 units
\nNow,
\nThe perimeter of the rectangle MNOP = MN + NO + OP + PM
\n= 9 + 5 + 9 + 5
\n= 28 units
\nHence, from the above,
\nWe can conclude that
\nThe perimeter of the given rectangle MNOP is: 28 units<\/p>\n

Question 5.
\nJen draws a polygon with vertices E(-2, 3.5), F(3, 3.5), G(3, -1.5), and H(-2, -1.5). Is EFGH a square? Justify your answer.
\nAnswer:
\nIt is given that
\nJen draws a polygon with vertices E(-2, 3.5), F(3, 3.5), G(3, -1.5), and H(-2, -1.5)
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe distance between 2 points = |x + a| + |y + b|
\nNow,
\nThe distance of EF = |3 + 2|
\n= 5 units
\nThe distance of FG = |3.5 + 1.5|
\n= 5 units
\nThe distance of GH = |3 + 2|
\n= 5 units
\nThe distance of HE = |3.5 + 1.5|
\n= 5 units
\nNow,
\nWe know that,
\nThe “Square” has all the same side lengths
\nHence, from the above,
\nWe can conclude that
\nEFGH is a square<\/p>\n

Question 6.
\nSquare ABCD has vertices A(-4.5, 4), B(3.5, 4), C(3.5, -4), and D(-4.5, -4). What is the area of square ABCD?
\nAnswer:
\nIt is given that
\nSquare ABCD has vertices A(-4.5, 4), B(3.5, 4), C(3.5, -4), and D(-4.5, -4)
\nNow,
\nWe know that,
\nThe square has the same side lengths
\nSo,
\nThe distance between any 2 points is sufficient to find the area of square ABCD
\nNow,
\nWe know that,
\nThe area of a square = Side\u00b2
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe distance between 2 points = |x + a| + |y + b|
\nNow,
\nThe distance of AB =|3.5 + 4.5|
\n= 8 units
\nSo,
\nThe area of square ABCD = Side\u00b2
\n= 8\u00b2
\n= 64 units\u00b2
\nHence, from the above,
\nWe can conclude that
\nThe area of square ABCD is: 64 units\u00b2<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Scan for Multimedia<\/strong><\/p>\n

Leveled Practice<\/strong>
\nIn 7 and 8, find the perimeter of each rectangle.<\/strong><\/p>\n

Question 7.
\nRectangle JKLM: J(-3, 8), K(-3,-1), L(4, -1), M(4,8)
\nJK = |8| + |-1| = \"Envision
\nKL = |\u20133| + |4| = \"Envision
\nPerimeter = \"Envision units
\nAnswer:
\nThe given vertices of Rectangle JKLM are: J(-3, 8), K(-3,-1), L(4, -1), M(4,8)
\nNow,
\nWe know that,
\nThe side lengths of the parallel sides are the same
\nSo,
\nIn Rectangle JKLM,
\nJK = LM and KL = MJ
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe distance between 2 points = |x + a| + |y + b|
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe perimeter of Rectangle JKLM is: 16 units<\/p>\n

Question 8.
\nRectangle WXYZ: W(-3, -2), X(4, -2), Y(4, -5), Z(-3, -5)
\nWX = |-3| + |4| = \"Envision
\nXY = |\u20135| – |-2| = \"Envision
\nPerimeter = \"Envision units
\nAnswer:
\nThe given vertices of Rectangle WXYZ are: W(-3, -2), X(4, -2), Y(4, -5), Z(-3, -5)
\nNow,
\nWe know that,
\nThe side lengths of the parallel sides are the same
\nSo,
\nIn Rectangle WXYZ,
\nWX = YZ and XY = ZW
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe distance between 2 points = |x + a| + |y + b|
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe perimeter of Rectangle JKLM is: 10 units<\/p>\n

Question 9.
\nTriangle JKL has vertices J(0, 0), K(5, 0), and L(0, -3). Is triangle JKL equilateral? Justify your answer.
\nAnswer:
\nIt is given that
\nTriangle JKL has vertices J(0, 0), K(5, 0), and L(0, -3)
\nNow,
\nWe know that,
\nThe “Equilateral Triangle” is a triangle that has the same side lengths
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe distance between 2 points = |x + a| + |y + b|
\nSo,
\nThe distance of JK = |0 + 0| + |5 + 0|
\n= 5 units
\nThe distance of KL = |5 + 0| + |0 – 3|
\n= 8 units
\nThe distance of LJ = |0 – 3| + |0 + 0|
\n= 3 units
\nSo,
\nFrom the given side lengths,
\nJK \u2260 L \u2260 LJ
\nHence, from the above,
\nHence, from the above,
\nWe can conclude that
\nTriangle JKL is not an equilateral triangle<\/p>\n

Question 10.
\nPolygon WXYZ has vertices W(-1.5, 1.5), X(6, 1.5), Y(6, -4.5), and Z(-1.5, -4.5). Is WXYZ a rectangle? Justify your answer.
\nAnswer:
\nIt is given that
\nPolygon WXYZ has vertices W(-1.5, 1.5), X(6, 1.5), Y(6, -4.5), and Z(-1.5, -4.5)
\nNow,
\nThe representation of the given polygon WXYZ with its side lengths is:
\n\"\"
\nNow,
\nFrom the above figure,
\nWe can observe that
\nThe side lengths of the opposite sides are equal
\nNow,
\nWe know that,
\nA “Rectangle” has the same opposite side lengths
\nHence, from the above,
\nWe can conclude that
\nPolygon WXYZ is a rectangle<\/p>\n

Question 11.
\nWhat are the perimeter and area of rectangle ABCD?
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe coordinates of the rectangle ABCD are:
\nA (-3, -7), B (4, -7), C (4, 6), and D (-3, 6)
\nNow,
\nWe know that,
\nIn a Rectangle,
\nThe side lengths of the opposite sides are the same
\nSo,
\nAB = CD and BC = DA
\nNow,
\nWe know that,
\nThe perimeter of a rectangle = 2 (l + w)
\nThe area of a rectangle = l \u00d7 w
\nNow,
\nThe distance of AB = |-3 – 7| + |4 – 7|
\n= |-10 + 3|
\n= 7 units
\nThe disatnce of BC = |4 – 3| + |6 + 6|
\n= |1 + 12|
\n=13 units
\nSo,
\nThe perimeter of Rectangle ABCD = 2 (AB + BC)
\n= 2 (7 + 13)
\n= 2 (20)
\n= 40 units
\nThe area of Rectangle ABCD = AB \u00d7 CD
\n= 7 \u00d7 13
\n= 91 units\u00b2
\nHence, from the above,
\nWe can conclude that
\nThe perimeter of Rectangle BACD is: 40 units
\nThe area of Rectangle BACD is: 91 units\u00b2<\/p>\n

Question 12.
\nMike used a coordinate plane to design the patio shown at the right. Each unit on the grid represents 1 yard. To buy materials to build the patio, Mike needs to know its perimeter. What is the perimeter of the patio?
\n\"Envision
\nAnswer:
\nIt is given that
\nMike used a coordinate plane to design the patio shown at the right. Each unit on the grid represents 1 yard.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe coordinates of the given patio are:
\nA (3, 2), B (3, 4), C (7, 4), D (7, 7), E (9, 7), and F (9, 2)
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe distance between 2 points = |x + a| + |y + b|
\nNow,
\nThe distance of AB = |4 – 2| = 2 units
\nThe disatnce of BC = |7 – 3| = 4 units
\nThe distance of CD = |7 – 4| = 3 units
\nThe distance of DE = |9 – 7| = 2 units
\nThe distance of EF = |7 – 2| = 5 units
\nThe distance of FA = |9 – 3| = 6 units
\nNow,
\nThe perimeter of the patio = 2 + 4 + 3 + 2 + 5 + 6
\n= 22 units
\nHence, from the above,
\nWe can conclude that
\nThe perimeter of the patio is: 22 units<\/p>\n

Question 13.
\nJordan started at her home at point H. She ran to the bank (B), the library (1), the post office (P), the caf\u00e9 (C), her school (S), and then back to her home, as shown. The coordinates represent the position, in miles, of each of these locations with respect to the center of town, which is located at the origin. What is the total distance that Jordan ran?
\n\"Envision
\nAnswer:
\nIt is given that
\nJordan started at her home at point H. She ran to the bank (B), the library (1), the post office (P), the caf\u00e9 (C), her school (S), and then back to her home, as shown. The coordinates represent the position, in miles, of each of these locations with respect to the center of town, which is located at the origin
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe coordinates are:
\nH (\\(\\frac{-3}{4}\\), \\(\\frac{1}{2}\\)), B (1, \\(\\frac{1}{2}\\)), L (1, –\\(\\frac{1}{4}\\)), P (\\(\\frac{1}{2}\\), –\\(\\frac{1}{4}\\)), C (\\(\\frac{1}{2}\\), –\\(\\frac{3}{4}\\)), S (-\\(\\frac{3}{4}\\), –\\(\\frac{3}{4}\\))
\nNow,
\nThe representation of the given coordinates with its side lengths is:
\n\"\"
\nSo,
\nThe total distance that Jordan ran = 1.75 + 0.75 + 0.70 + 1.25 + 1.25
\n= 5.7 miles
\nHence, from the above,
\nWe can conclude that
\nThe total distance that Jordan ran is: 5.7 miles<\/p>\n

Question 14.
\nUse Structure<\/strong> Ana drew a plan for a rectangular piece of material that she will use for a quilt. The vertices are (-1.2, -3.5), (-1.2, 4.4), and (5.5, 4.4). What are the coordinates of the fourth vertex?
\nAnswer:
\nIt is given that
\nAna drew a plan for a rectangular piece of material that she will use for a quilt. The vertices are (-1.2, -3.5), (-1.2, 4.4), and (5.5, 4.4)
\nNow,
\nLet the vertices be A (-1.2, -3.5), B (-1.2, 4.4), C (5.5, 4.4), and D (x, y)
\nNow,
\nWe know that,
\nIn a Rectangle,
\nThe opposite side lengths are the same
\nSo,
\nFor the given quilt,
\nAB = CD and BC = DA
\nNow,
\nIf you observe carefully the vertices given, you will see that there are two vertex with same coordinates in x axis: (-1.2, -3.5), and (-1.2, 4.4),
\nAnd one vertex has other coordinate in x axis, it is: (5.5, 4.4)
\nSo,
\nTo form a rectangle, you need a fourth vertex that will have same coordinate in x than (5.5, 4.4) and same coordinate in y axis as (-1.2, -3.5),
\nSo,
\nThe fourth vertex coordinate is (5.5, -3.5)
\nHence, from the above,
\nWe can conclude that
\nThe coordinates of the fourth vertex are: (5.5, -3.5)<\/p>\n

Question 15.
\nMr. Janas is building a pool in his backyard. He sketches the rectangular pool on a coordinate plane. The vertices of the pool are A(-5, 7), B(1,7), C(1, -1), and D(-5, -1). If each unit represents 1 yard, how much area of the backyard is needed for the pool?
\nAnswer:
\nIt is given that
\nMr. Janas is building a pool in his backyard. He sketches the rectangular pool on a coordinate plane. The vertices of the pool are A(-5, 7), B(1,7), C(1, -1), and D(-5, -1)
\nNow,
\nThe representation of the given vertices of a rectangular pool and its side lengths in the coordinate plane is:
\n\"\"
\nNow,
\nWe know that,
\nIn a Rectangle,
\nThe opposite side lengths have the same value
\nNow,
\nWe know that,
\nThe area of a Rectangle = Length \u00d7 Width
\nSo,
\nThe area of the rectangular swimming pool = AB \u00d7 BC
\n= 6 \u00d7 8
\n= 48 yard\u00b2
\nHence, from the above,
\nWe can conclude that
\nThe area of the given rectangular swimming pool is: 48 yard\u00b2<\/p>\n

Question 16.
\nVocabulary<\/strong> Why is absolute value used to find distances on a coordinate plane?
\nAnswer:
\nWe know that the absolute value of a point on the number line (or the absolute value of the coordinate of a tree in your backyard) tells you the distance between that point (or tree) and the number zero at origin of your coordinate system<\/p>\n

Question 17.
\nHigher Order Thinking<\/strong> A square on a coordinate plane has one vertex at (-0.5, -2) and a perimeter of 10 units. If all of the vertices are located in Quadrant III, what are the coordinates of the other three vertices?
\nAnswer:
\nIt is given that
\nA square on a coordinate plane has one vertex at (-0.5, -2) and a perimeter of 10 units and all of the vertices are located in Quadrant III
\nNow,
\nWe know that,
\nThe perimeter of square is:
\n
\nwhere,
\nb is the length side of the square
\nNow,
\nAccording to the given information,
\n
\nNow,
\nDivide the above equation by 4 both sides
\nSo,
\n
\nNow,
\nA (-0.5, -2) —> The given coordinates of one vertex
\nNow,
\nwe know that
\nAll of the vertices are located in Quadrant III
\nSo,
\nThe other three vertices are located at the left and down of vertex A
\nNow,
\nThe coordinate of vertex B located at 2.5 units at left of vertex A
\nThe coordinate of vertex C located at 2.5 units at left and 2.5 units down of vertex A
\nThe coordinate of vertex D located at 2.5 units down of vertex A
\nSo,
\nB(-0.5-2.5,-2) —–> B(-3, -2)
\nC(-0.5-2.5,-2-2.5) —–> C(-3, -4.5)
\nD(-0.5,-2-2.5) —–> D(-0.5, -4.5)
\nHence, from the above,
\nWe can conclude that
\nThe coordinates of the other three vertices are:
\nB (-3, -2), C (-3, -4.5), and D (-0.5, -4.5)<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 18.
\nYou are given the following points on a coordinate plane: A(-1\\(\\frac{1}{2}\\), –\\(\\frac{1}{2}\\)), B(-1\\(\\frac{1}{2}\\), -3), and C (4, -3).
\nPART A<\/strong>
\nUsing absolute value, find the distance (number of units) between points A and B.
\nAnswer:
\nThe given points are:
\nA(-1\\(\\frac{1}{2}\\), –\\(\\frac{1}{2}\\)), B(-1\\(\\frac{1}{2}\\), -3), and C (4, -3)
\nNow,
\nThe representation of the given coordinates along with its side lengths in the coordinate plane is:
\n\"\"
\nSo,
\nFrom the above figure,
\nWe can observe that
\nThe distance between points A and B is: 2.5 units
\nHence, from the above,
\nWe can conclude that
\nThe distance between points A and B is: 2.5 units
\nPART B<\/strong>
\nSelect all the coordinates that are 8 units from point C.
\n\"Envision (12, -3)
\n\"Envision (12, -11)
\n\"Envision (4, -3)
\n\"Envision (-4, -3)
\n\"Envision (4,-11)
\nAnswer:
\nIt is given that
\nThe coordinates of Point C are: C (4, -3)
\nNow,
\nWhen the coordiantes of point C moves 8 units towards right,
\nC (4 + 8, -3) = C (12, -3)
\nWhen the coordiantes of point C moves 8 units towards left,
\nC (4 – 8, -3) = C (-4, -3)
\nHence, from the above,
\nWe can conclude that
\nAll the coordinates that are 8 units from point C are:
\n\"\"<\/p>\n

Topic 2 REVIEW<\/h3>\n

? Topic Essential Question<\/strong>
\nWhat are integers and rational numbers? How are points graphed on a coordinate plane?
\nAnswer:
\nAn “Integer” can be written as a fraction by giving it a denominator of one. So, any integer is a rational number
\n“Rational numbers” are those numbers that are integers and can be expressed in the form of \\(\\frac{x}{y}\\) where both numerator and denominator are integers
\nTo graph or plot points, we use two perpendicular lines called the x-axis and the y-axis. The horizontal number line is the x-axis and the vertical line is the y-axis.\u00a0 Every point in the coordinate plane is represented by an ordered pair of x and y coordinates.<\/p>\n

Vocabulary Review<\/strong><\/p>\n

Complete each definition and then provide an example of each vocabulary word.<\/strong><\/p>\n

Vocabulary<\/strong><\/p>\n

absolute value
\nopposite
\nordered pair
\nrational number<\/p>\n

\"Envision
\nAnswer:
\n\"\"<\/p>\n

Use Vocabulary in Writing<\/strong><\/p>\n

Explain how the points A(9, –\\(\\frac{2}{5}\\)) and B(9, \\(\\frac{2}{5}\\)) are related. Use vocabulary words in your explanation.
\nAnswer:
\nThe given points are:
\nA(9, –\\(\\frac{2}{5}\\)) and B(9, \\(\\frac{2}{5}\\))
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nFrom the given points,
\nWe can observe that
\na. The x-coordinates are the same
\nb. The y-coordinates are opposite to each other
\nc. The y-coordinates and x-coordinates are rational numbers since they can be written in the form of \\(\\frac{x}{y}\\)<\/p>\n

Concepts and Skills Review<\/strong><\/p>\n

LESSON 2.1 Understand Integers<\/strong><\/p>\n

Quick Review <\/strong>
\nIntegers are all of the counting numbers, their opposites, and 0. Opposites are integers located on opposite sides of 0 and the same distance from 0 on a number line.<\/p>\n

Example<\/strong>
\nFor each point on the number line, write the integer and its opposite.
\n\"Envision
\nA: 4,-4 B: 0,0 C: -6,6
\nThe opposite of the opposite of a number is the number itself.<\/p>\n

Practice <\/strong>
\nFor each point on the number line, write the integer and its opposite.
\n\"Envision
\nQuestion 1.
\nA
\nAnswer:
\nThe given number line is:
\n\"Envision
\nNow,
\nFrom the given number line,
\nWe can observe that
\nEach line in the given number line represents 1 unit
\nNow,
\nWe know that,
\nThe opposite value of a is: -a
\nThe opposite value of -a is: a
\nHence, from the above,
\nWe can conclude that
\nThe value of A from the given number line is: 3
\nThe opposite value of A is: -3<\/p>\n

Question 2.
\nB
\nAnswer:
\nThe given number line is:
\n\"Envision
\nNow,
\nFrom the given number line,
\nWe can observe that
\nEach line in the given number line represents 1 unit
\nNow,
\nWe know that,
\nThe opposite value of a is: -a
\nThe opposite value of -a is: a
\nHence, from the above,
\nWe can conclude that
\nThe value of B from the given number line is: -1
\nThe opposite value of B is: 1<\/p>\n

Question 3.
\nC
\nAnswer:
\nThe given number line is:
\n\"Envision
\nNow,
\nFrom the given number line,
\nWe can observe that
\nEach line in the given number line represents 1 unit
\nNow,
\nWe know that,
\nThe opposite value of a is: -a
\nThe opposite value of -a is: a
\nHence, from the above,
\nWe can conclude that
\nThe value of C from the given number line is: 6
\nThe opposite value of C is: -6<\/p>\n

Question 4.
\nD
\nAnswer:
\nThe given number line is:
\n\"Envision
\nNow,
\nFrom the given number line,
\nWe can observe that
\nEach line in the given number line represents 1 unit
\nNow,
\nWe know that,
\nThe opposite value of a is: -a
\nThe opposite value of -a is: a
\nHence, from the above,
\nWe can conclude that
\nThe value of D from the given number line is: -7
\nThe opposite value of D is: 7<\/p>\n

Question 5.
\nE
\nAnswer:
\nThe given number line is:
\n\"Envision
\nNow,
\nFrom the given number line,
\nWe can observe that
\nEach line in the given number line represents 1 unit
\nNow,
\nWe know that,
\nThe opposite value of a is: -a
\nThe opposite value of -a is: a
\nHence, from the above,
\nWe can conclude that
\nThe value of E from the given number line is: -5
\nThe opposite value of E is: 5<\/p>\n

Question 6.
\nF
\nAnswer:
\nThe given number line is:
\n\"Envision
\nNow,
\nFrom the given number line,
\nWe can observe that
\nEach line in the given number line represents 1 unit
\nNow,
\nWe know that,
\nThe opposite value of a is: -a
\nThe opposite value of -a is: a
\nHence, from the above,
\nWe can conclude that
\nThe value of F from the given number line is: 1
\nThe opposite value of F is: -1<\/p>\n

LESSON 2.2 Represent Rational Numbers on the Number Line<\/strong><\/p>\n

Quick Review<\/strong>
\nRational numbers are numbers that can be written as a quotient \\(\\frac{a}{b}\\), where a and b are integers and b does not equal 0. You can use number lines to represent, compare, and order rational numbers.<\/p>\n

Example <\/strong>
\nCompare and order -0.1, 0.75, and –\\(\\frac{1}{4}\\) from least to greatest.
\nPlot the numbers on a number line.
\n\"Envision
\nso –\\(\\frac{1}{4}\\) < -0.1 < 0.75, and their order from least to greatest is –\\(\\frac{1}{4}\\), -0.1, 0.75.<\/p>\n

Practice <\/strong>
\nIn 1-3, plot each rational number on the number line.<\/strong><\/p>\n

\"Envision<\/p>\n

Question 1.
\n\\(\\frac{3}{4}\\)
\nAnswer:
\nThe given point is: \\(\\frac{3}{4}\\)
\nNow,
\nThe given number line is:
\n\"Envision
\nNow,
\nFrom the given number line,
\nWe can observe that
\nEach line in the given number line represents 0.1 unit
\nNow,
\nThe representation of \\(\\frac{3}{4}\\) in the decimal form is: 0.75
\nHence,
\nThe representation of the given point on the given number line is:
\n\"\"<\/p>\n

Question 2.
\n–\\(\\frac{2}{5}\\)
\nAnswer:
\nThe given point is: –\\(\\frac{2}{5}\\)
\nNow,
\nThe given number line is:
\n\"Envision
\nNow,
\nFrom the given number line,
\nWe can observe that
\nEach line in the given number line represents 0.1 unit
\nNow,
\nThe representation of –\\(\\frac{2}{5}\\) in the decimal form is: -0.4
\nHence,
\nThe representation of the given point on the given number line is:
\n\"\"<\/p>\n

Question 3.
\n0.5
\nAnswer:
\nThe given point is: 0.5
\nNow,
\nThe given number line is:
\n\"Envision
\nNow,
\nFrom the given number line,
\nWe can observe that
\nEach line in the given number line represents 0.1 unit
\nHence,
\nThe representation of the given point on the given number line is:
\n\"\"<\/p>\n

In 4-7, use <, >, or = to compare.<\/strong><\/p>\n

Question 4.
\n0.25 \"Envision \\(\\frac{1}{4}\\)
\nAnswer:
\nThe given numbers are: 0.25 and \\(\\frac{1}{4}\\)
\nNow,
\nThe conversion of \\(\\frac{1}{4}\\) into a decimal number is: 0.25
\nHence, from the above,
\nWe can conclude that
\n\"\"<\/p>\n

Question 5.
\n1\\(\\frac{5}{8}\\) \"Envision 1.6
\nAnswer:
\nThe given numbers are: 1\\(\\frac{5}{8}\\) and 1.6
\nNow,
\nThe conversion of 1\\(\\frac{5}{8}\\) into a decimal number is: 1.625
\nHence, from the above,
\nWe can conclude that
\n\"\"<\/p>\n

Question 6.
\n3.65 \"Envision 3\\(\\frac{3}{4}\\)
\nAnswer:
\nThe given numbers are: 3.65 and 3\\(\\frac{3}{4}\\)
\nNow,
\nThe conversion of 3\\(\\frac{3}{4}\\) into a decimal number is: 3.75
\nHence, from the above,
\nWe can conclude that
\n\"\"<\/p>\n

Question 7.
\n–\\(\\frac{2}{3}\\) \"Envision \\(\\frac{3}{4}\\)
\nAnswer:
\nThe given numbers are: –\\(\\frac{2}{3}\\) and \\(\\frac{3}{4}\\)
\nNow,
\nThe conversion of \\(\\frac{3}{4}\\) into a decimal number is: 0.75
\nThe conversion of –\\(\\frac{2}{3}\\) into a decimal number is: -0.66
\nHence, from the above,
\nWe can conclude that
\n\"\"<\/p>\n

LESSON 2.3 Absolute Values of Rational Numbers<\/strong><\/p>\n

Quick Review<\/strong>
\nThe absolute value of a number is its distance from 0 on the number line. Distance is always positive. Absolute values are never negative.<\/p>\n

Example<\/strong>
\nFind the absolute values and order |3|, |4|, |-2|, |\u20135| from least to greatest.
\n\"Envision<\/p>\n

Practice<\/strong>
\nIn 1-4, find each value.<\/strong><\/p>\n

Question 1.
\n|-9|
\nAnswer:
\nThe given absolute number is: |-9|
\nNow,
\nWe know that,
\nThe absolute value of any number is positive
\nHence, from the above,
\nWe can conclude that
\nThe value of |-9| is: 9<\/p>\n

Question 2.
\n|-2|
\nAnswer:
\nThe given absolute number is: |-2|
\nNow,
\nWe know that,
\nThe absolute value of any number is positive
\nHence, from the above,
\nWe can conclude that
\nThe value of |-2| is: 2<\/p>\n

Question 3.
\n|4|
\nAnswer:
\nThe given absolute number is: |4|
\nNow,
\nWe know that,
\nThe absolute value of any number is positive
\nHence, from the above,
\nWe can conclude that
\nThe value of |4| is: 4<\/p>\n

Question 4.
\n-|-10|
\nAnswer:
\nThe given absolute number is: -|-10|
\nNow,
\nWe know that,
\nThe absolute value of any number is positive
\nHence, from the above,
\nWe can conclude that
\nThe value of -|-10| is: -10<\/p>\n

In 5-8, order the values from least to greatest.<\/strong><\/p>\n

Question 5.
\n|-3|, |-2|, |10|
\nAnswer:
\nThe given absolute numbers are: |-3|, |-2|, and |10|
\nNow,
\nWe know that,
\nThe absolute value of any number is positive
\nSo,
\nThe values of the given absolute numbers are: 3, 2, and 10
\nSo,
\nThe order of the given absolute numbers from the least to the greatest is: 2, 3, and 10
\nHence, from the above,
\nWe can conclude that
\nThe order of the given absolute numbers from the least to the greatest is: 2, 3, and 10<\/p>\n

Question 6.
\n|-7|, |0|, |-5|
\nAnswer:
\nThe given absolute numbers are: |-7|, |0|, and |-5|
\nNow,
\nWe know that,
\nThe absolute value of any number is positive
\nSo,
\nThe values of the given absolute numbers are: 7, 0, and 5
\nSo,
\nThe order of the given absolute numbers from the least to the greatest is: 0, 5, and 7
\nHence, from the above,
\nWe can conclude that
\nThe order of the given absolute numbers from the least to the greatest is: 0, 5, and 7<\/p>\n

Question 7.
\n|-18.5|, |18|, |-12.5|
\nAnswer:
\nThe given absolute numbers are: |-18.5|, |18|, and |-12.5|
\nNow,
\nWe know that,
\nThe absolute value of any number is positive
\nSo,
\nThe values of the given absolute numbers are: 18.5, 18, and 12.5
\nSo,
\nThe order of the given absolute numbers from the least to the greatest is: 12.5, 18, and 18.5
\nHence, from the above,
\nWe can conclude that
\nThe order of the given absolute numbers from the least to the greatest is: 12.5, 18, and 18.5<\/p>\n

Question 8.
\n|26|, |-20|, |-24.5|
\nAnswer:
\nThe given absolute numbers are: |26|, |-20|, and |-24.5|
\nNow,
\nWe know that,
\nThe absolute value of any number is positive
\nSo,
\nThe values of the given absolute numbers are: 26, 20, and 24.5
\nSo,
\nThe order of the given absolute numbers from the least to the greatest is: 20, 24.5, and 26
\nHence, from the above,
\nWe can conclude that
\nThe order of the given absolute numbers from the least to the greatest is: 20, 24.5, and 26<\/p>\n

LESSON 2.4 Represent Rational Numbers on the Coordinate Plane<\/strong><\/p>\n

Quick Review<\/strong>
\nAn ordered pair (x, y) of numbers gives the coordinates that locate a point on a coordinate plane. Coordinates can be whole numbers, fractions, mixed numbers, or decimals.<\/p>\n

Example<\/strong>
\nExplain how to plot any point with coordinates (x, y).
\n\u2022 Start at the origin, (0, 0).
\n\u2022 Use the x-coordinate to move right (if positive) or left (if negative) along the x-axis.
\n\u2022 Then use the y-coordinate of the point to move up (if positive) or down (if negative) following the y-axis.
\n\u2022 Draw and label the point on the coordinate plane. Explain how to name the location of a point on a coordinate plane. Follow the grid line from the point to the x-axis to name the x-coordinate, and follow the grid line from the point to the y-axis to name the y-coordinate.<\/p>\n

Practice<\/strong>
\nIn 1-6, give the ordered pair for each point.<\/strong>
\n\"Envision<\/p>\n

Question 1.
\nU
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nWe can observe that
\nThe coordinates of point U are: (0, 2.5)
\nHence, from the above,
\nWe can conclude that
\nThe ordered pair for point U is: (0, 2.5)<\/p>\n

Question 2.
\nV
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nWe can observe that
\nThe coordinates of point V are: (-2, 1.5)
\nHence, from the above,
\nWe can conclude that
\nThe ordered pair for point V is: (-2, 1.5)<\/p>\n

Question 3.
\nW
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nWe can observe that
\nThe coordinates of point W are: (-4, -1)
\nHence, from the above,
\nWe can conclude that
\nThe ordered pair for point W is: (-4, -1)<\/p>\n

Question 4.
\nX
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nWe can observe that
\nThe coordinates of point X are: (2.5, 0)
\nHence, from the above,
\nWe can conclude that
\nThe ordered pair for point X is: (2.5, 0)<\/p>\n

Question 5.
\nY
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nWe can observe that
\nThe coordinates of point Y are: (2, -1.5)
\nHence, from the above,
\nWe can conclude that
\nThe ordered pair for point Y is: (2, -1.5)<\/p>\n

Question 6.
\nZ
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nWe can observe that
\nThe coordinates of point Z are: (-1.5, -3)
\nHence, from the above,
\nWe can conclude that
\nThe ordered pair for point Z is: (-1.5, -3)<\/p>\n

LESSONS 2.5 AND 2.6 Find Distances and Represent Polygons on the Coordinate Plane<\/strong><\/p>\n

Quick Review<\/strong>
\nYou can use absolute value to find the distance between two points that share the same X- or y-coordinate. When the y-coordinates are the same, use the x-coordinates to find the distance. When the x-coordinates are the same, use the y-coordinates. If the points are in different quadrants, add their absolute values. If the points are in the same quadrant, subtract their absolute values.
\nYou can use what you know about finding the distance between two points to find the lengths of the sides of a polygon on a coordinate plane.<\/p>\n

Example<\/strong>
\nFind the length of side AB.
\n\"Envision
\nThe ordered pairs for points A and B are A(-3, 2) and B(-1, 2). The points are in the same quadrant, so subtract the absolute values of the x-coordinates.
\n|-3| – |-1| = 3 – 1 = 2 units
\nThe length of side AB is 2 units.<\/p>\n

Practice<\/strong><\/p>\n

In 1-6, find the remaining side lengths of polygon ABCDEF. Then find the polygon’s perimeter.
\n\"Envision
\n<\/strong>Answer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nThe coordinates of the vertices are:
\nA (-3, 2) B (-1, 2), C (-1, 1), D (1, 1), E (1, -2), F (-3, -2)<\/p>\n

Question 1.
\nLength of BC
\nAnswer:
\nThe given coordinates of points B and C are: B (-1, 2), and C (-1, 1)
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe distance between 2 points = |x + a| + |y + b|
\nSo,
\nThe length of BC = |2 – 1|
\n= 1 unit
\nHence, from the above,
\nWe can conclude that
\nThe length of BC is: 1 unit<\/p>\n

Question 2.
\nLength of CD
\nAnswer:
\nThe given coordinates of points C and D are: C (-1, 1), and D (1, 1)
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe distance between 2 points = |x + a| + |y + b|
\nSo,
\nThe length of CD = |1 + 1|
\n= 2 units
\nHence, from the above,
\nWe can conclude that
\nThe length of CD is: 2 units<\/p>\n

Question 3.
\nLength of DE
\nAnswer:
\nThe given coordinates of points D and E are: D (1, 1), and E (1, -2)
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe distance between 2 points = |x + a| + |y + b|
\nSo,
\nThe length of DE = |2 + 1|
\n= 3 units
\nHence, from the above,
\nWe can conclude that
\nThe length of DE is: 3 units<\/p>\n

Question 4.
\nLength of EF
\nAnswer:
\nThe given coordinates of points E and F are: E (1, -2), and F (-3, -2)
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe distance between 2 points = |x + a| + |y + b|
\nSo,
\nThe length of EF = |3 + 1|
\n= 4 units
\nHence, from the above,
\nWe can conclude that
\nThe length of EF is: 4 units<\/p>\n

Question 5.
\nLength of FA
\nAnswer:
\nThe given coordinates of points F and A are: F (-3, -2), and A (-3, 2)
\nNow,
\nCompare the given points with (x, a), and (y, b)
\nNow,
\nWe know that,
\nThe distance between 2 points = |x + a| + |y + b|
\nSo,
\nThe length of FA = |2 + 2|
\n= 4 units
\nHence, from the above,
\nWe can conclude that
\nThe length of FA is: 4 units<\/p>\n

Question 6.
\nPerimeter of ABCDEF
\nAnswer:
\nWe know that,
\nThe perimeter of any polygon is the sum of its side lengths
\nSo,
\nThe perimeter of polygon ABCDEF = AB + BC + CD+ DE+ EF+ FA
\n= 2 + 1 + 2 + 3 + 4 + 4
\n= 16 units
\nHence, from the above,
\nWe can conclude that
\nThe perimeter of the given polygon ABCDEF is: 16 units<\/p>\n

In 7 and 8, polygon QRST has vertices Q(-4, -1), R(-4,5), S(2,5), and T(2, -1)<\/strong>.<\/p>\n

Question 7.
\nDraw and label polygon QRST on the coordinate plane.
\n\"Envision
\nAnswer:
\nThe given vertices of the polygon QRST are:
\nQ (-4, -1), R (-4,5), S (2,5), and T (2, -1)
\nHence,
\nThe representation of the given vertices along with its side lengths in the given coordinate plane is:
\n\"\"<\/p>\n

Question 8.
\nConstruct an argument to justify whether or not polygon QRST is a square.
\nAnswer:
\nFrom Exercise 7,
\nThe representation of polygon QRST along with its side lengths in the given coordinate plane is:
\n\"\"
\nNow,
\nFrom the given coordinate plane,
\nWe can observe that
\nThe sidelengths of all the sides are the same
\nNow,
\nWe know that,
\nA “Square” has the same side lengths on all the four sides
\nHence, from the above,
\nWe can conclude that
\nPolygon QRST is a square<\/p>\n

Topic 2 Fluency Practice<\/h3>\n

Hidden Clue<\/strong>
\nFor each ordered pair, simplify the two coordinates. Then locate and label the corresponding point on the graph. Draw line segments to connect the points in alphabetical order. Use the completed picture to help answer the riddle below.
\n\"Envision
\n\"Envision<\/p>\n","protected":false},"excerpt":{"rendered":"

Go through the\u00a0enVision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers\u00a0regularly and improve your accuracy in solving questions. Envision Math Common Core 6th Grade Answers Key Topic 2 Integers and Rational Numbers ?Topic Essential Question\u00a0What are integers and rational numbers? How are points graphed on a coordinate plane? Answer: An …<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/posts\/16571"}],"collection":[{"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/comments?post=16571"}],"version-history":[{"count":4,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/posts\/16571\/revisions"}],"predecessor-version":[{"id":28271,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/posts\/16571\/revisions\/28271"}],"wp:attachment":[{"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/media?parent=16571"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/categories?post=16571"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/tags?post=16571"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}