{"id":9479,"date":"2024-01-01T16:22:32","date_gmt":"2024-01-01T10:52:32","guid":{"rendered":"https:\/\/envisionmathanswerkey.com\/?p=9479"},"modified":"2024-01-02T12:52:03","modified_gmt":"2024-01-02T07:22:03","slug":"envision-math-common-core-grade-4-answer-key-topic-4","status":"publish","type":"post","link":"https:\/\/envisionmathanswerkey.com\/envision-math-common-core-grade-4-answer-key-topic-4\/","title":{"rendered":"enVision Math Common Core Grade 4 Answer Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers"},"content":{"rendered":"

Go through the\u00a0enVision Math Common Core Grade 4 Answer Key<\/a>\u00a0Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers<\/strong>\u00a0regularly and improve your accuracy in solving questions.<\/p>\n

enVision Math Common Core 4th Grade Answers Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers<\/h2>\n

Essential Questions:<\/strong>
\nHow can you use a model to multiply? How can you use the Distributive Property to multiply? How can you use multiplication to solve problems?
\nAnswer:
\nMultiplication using\u00a0 the “Area model”:
\nWe know that,
\nAn “Area” is the amount of flat space occupied by a figure, You can find the area of a figure by filling it with unit squares and then counting how many squares entirely fill the shape
\nThe total area of the figure is the multiplication of the number of rows and columns
\nMultiplication using the Distributive Property:
\nLet the three numbers be a, b, c
\nNow,
\nAccording to the Distributive Property of Multiplication,
\na \u00d7 (b + c) = (a \u00d7 b) + (a \u00d7 c)
\n(a + b) \u00d7 c = (a \u00d7 c ) + (b \u00d7 c)
\nEx:
\nWe have to find the product of 12 \u00d7 4
\nNow,
\nThe steps to find the product of the given expression using the Distributive Property of Multiplication are:
\nStep 1:
\nSimplify the terms
\nSo,
\n12 \u00d7 4 = (10 + 2) \u00d7 4
\nStep 2:
\nUse the Distributive Property of Multiplication on the simplified expression that we obtained in Step 1
\nSo,
\n(10 + 2) \u00d7 4 = (10 \u00d7 4) + (2 \u00d7 4)
\n= 40 + 8
\nStep 3:
\nAdd the values that we obtained in the last step of Step 2
\nSo,
\n40 + 8 = 48
\nHence,
\nThe value of the given expression is: 48<\/p>\n

\"Envision<\/p>\n

enVision STEM Project: Renewable Energy and Multiplication<\/strong>
\nDo Research<\/strong>
\nUse the Internet or other sources to find information about different sources of renewable energy?
\nAnswer:
\nThe major types of renewable energy sources are:
\nA. Biomass:
\nObtained from wood and wood waste, municipal solid waste etc
\nB. Hydropower energy:
\nObtained from water
\nC. Geothermal energy:
\nObtained from land
\nD. Wind energy:
\nObtained from wind turbines
\nE. Solar energy:
\nObtained from the sun<\/p>\n

\"Use<\/p>\n

Journal: Write a Report that Includes what you found. Also in your report:
\n<\/strong>A)
\nA wind farm is an area of land with a large number of turbines. Draw an array with 15 rows to show the turbines in a wind farm. How many turbines are in your wind farm?
\nAnswer:
\nIt is given that
\nA wind farm is an area of land with a large number of turbines
\nNow,
\nThe representation of an array with 15 rows to show the turbines in a wind farm is:
\n\"\"
\nHence, from the above figure,
\nWe can conclude that there are 15 turbines in your wind farm<\/p>\n

B)
\nSome turbines produce 63 megawatt-hours of energy every week. Find how much energy one of these turbines would produce in one year. Remember, one year is 52 weeks?
\nAnswer:
\nIt is given that
\nSome turbines produce 63 megawatt-hours of energy every week.
\nSo,
\nThe amount of energy these turbines produce in 1 year = (The amount of energy produced by one of these turbines in a week) \u00d7 (The number of weeks in a year)
\n= 63 \u00d7 52
\n= 63 \u00d7 (50 + 2) [By using the Distributive Property of Multiplication]
\n= (63 \u00d7 50) + (63 \u00d7 2)
\n= 3,150 + 126
\n= 3,276 megawatt-hours
\nHence, from the above,
\nWe can conclude that the amount of energy produced by one of these turbines in one year is: 3,276 megawatt-hours<\/p>\n

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

Vocabulary<\/strong>
\nChoose the best term from the box. Write it on the blank.<\/p>\n

    \n
  • algorithm<\/li>\n
  • product<\/li>\n
  • variable<\/li>\n
  • array<\/li>\n<\/ul>\n

    Question 1.
    \nYou multiply numbers to find a(n) __________.
    \nAnswer:
    \nWe know that,
    \nYou multiply numbers to find a “Product”
    \nHence, from the above,
    \nWe can conclude that the best term that fits the given definition is a “Product”<\/p>\n

    Question 2.
    \nA(n) _________ shows the number of objects in rows and columns.
    \nAnswer:
    \nWe know that,
    \nAn “Array” shows the number of objects in rows and columns.
    \nHence, from the above,
    \nWe can conclude that the best term that fits the given definition is an “Array”<\/p>\n

    Question 3.
    \nA symbol or letter that stands for a number is called a(n) ______________.
    \nAnswer:
    \nWe know that,
    \nA symbol or letter that stands for a number is called a “Variable”
    \nHence, from the above,
    \nWe can conclude that the best term that fits the given definition is a “Variable”<\/p>\n

    Multiplication<\/strong><\/p>\n

    Find each product.
    \nQuestion 4.
    \n4 \u00d7 8
    \nAnswer:
    \nThe given expression is: 4 \u00d7 8
    \nNow,
    \nBy using the Distributive Property of Multiplication,
    \n4 \u00d7 8
    \n= 4 \u00d7 (4 + 4)
    \n= (4 \u00d7 4) + (4 \u00d7 4)
    \n= 16 + 16
    \n= 32
    \nHence, from the above,
    \nWe can conclude that the value of the given expression is: 32<\/p>\n

    Question 5.
    \n2 \u00d7 9
    \nAnswer:
    \nThe given expression is: 2 \u00d7 9
    \nNow,
    \nBy using the Distributive Property of Multiplication,
    \n2 \u00d7 9
    \n= 2 \u00d7 (6 + 3)
    \n= (2 \u00d7 6) + (2 \u00d7 3)
    \n= 12 + 6
    \n= 18
    \nHence, from the above,
    \nWe can conclude that the value of the given expression is: 18<\/p>\n

    Question 6.
    \n9 \u00d7 5
    \nAnswer:
    \nThe given expression is: 9 \u00d7 5
    \nNow,
    \nBy using the Distributive Property of Multiplication,
    \n9 \u00d7 5
    \n= 9 \u00d7 (4 + 1)
    \n= (9 \u00d7 4) + (9 \u00d7 1)
    \n= 36 + 9
    \n= 45
    \nHence, from the above,
    \nWe can conclude that the value of the given expression is: 45<\/p>\n

    Question 7.
    \n6 \u00d7 8
    \nAnswer:
    \nThe given expression is: 6 \u00d7 8
    \nNow,
    \nBy using the Distributive Property of Multiplication,
    \n6 \u00d7 8
    \n= 6 \u00d7 (4 + 4)
    \n= (6 \u00d7 4) + (6 \u00d7 4)
    \n= 24 + 24
    \n= 48
    \nHence, from the above,
    \nWe can conclude that the value of the given expression is: 48<\/p>\n

    Question 8.
    \n16 \u00d7 4
    \nAnswer:
    \nThe given expression is: 16 \u00d7 4
    \nNow,
    \nBy using the Distributive Property of Multiplication,
    \n16 \u00d7 4
    \n= (8 + 8) \u00d7 4
    \n= (4 \u00d7 8) + (4 \u00d7 8)
    \n= 32 + 32
    \n= 64
    \nHence, from the above,
    \nWe can conclude that the value of the given expression is: 64<\/p>\n

    Question 9.
    \n6 \u00d7 68
    \nAnswer:
    \nThe given expression is: 6 \u00d7 68
    \nNow,
    \nBy using the Distributive Property of Multiplication,
    \n6 \u00d7 68
    \n= (3 + 3) \u00d7 68
    \n= (3 \u00d7 68) + (3 \u00d7 68)
    \n= 204 + 204
    \n= 408
    \nHence, from the above,
    \nWe can conclude that the value of the given expression is: 408<\/p>\n

    Question 10.
    \n87 \u00d7 5
    \nAnswer:
    \nThe given expression is: 87 \u00d7 5
    \nNow,
    \nBy using the Distributive Property of Multiplication,
    \n87 \u00d7 5
    \n= 87 \u00d7 (3 + 2)
    \n= (87 \u00d7 3) + (87 \u00d7 2)
    \n= 261 + 174
    \n= 435
    \nHence, from the above,
    \nWe can conclude that the value of the given expression is: 435<\/p>\n

    Question 11.
    \n19 \u00d7 9
    \nAnswer:
    \nThe given expression is: 19 \u00d7 9
    \nNow,
    \nBy using the Distributive Property of Multiplication,
    \n19 \u00d7 9
    \n= 19 \u00d7 (3 + 6)
    \n= (19 \u00d7 3) + (19 \u00d7 6)
    \n= 57 + 114
    \n= 171
    \nHence, from the above,
    \nWe can conclude that the value of the given expression is: 171<\/p>\n

    Question 12.
    \n128 \u00d7 6
    \nAnswer:
    \nThe given expression is: 128 \u00d7 6
    \nNow,
    \nBy using the Distributive Property of Multiplication,
    \n128 \u00d7 6
    \n= (100 + 28) \u00d7 6
    \n= (100 \u00d7 6) + (28 \u00d7 6)
    \n= 600 + 168
    \n= 768
    \nHence, from the above,
    \nWe can conclude that the value of the given expression is: 768<\/p>\n

    Rounding<\/strong><\/p>\n

    Round each number to the nearest hundred.
    \n\"Envision
    \nQuestion 13.
    \n164
    \nAnswer:
    \nThe given number is: 164
    \nHence, from the above,
    \nWe can conclude that the number that is to the nearest hundred for the given number is: 200<\/p>\n

    Question 14.
    \n8,263
    \nAnswer:
    \nThe given number is: 8,263
    \nHence, from the above,
    \nWe can conclude that the number that is to the nearest hundred for the given number is: 8,300<\/p>\n

    Question 15.
    \n527
    \nAnswer:
    \nThe given number is: 527
    \nHence, from the above,
    \nWe can conclude that the number that is to the nearest hundred for the given number is: 500<\/p>\n

    Question 16.
    \n2,498
    \nAnswer:
    \nThe given number is: 2,498
    \nHence, from the above,
    \nWe can conclude that the number that is to the nearest hundred for the given number is: 2,500<\/p>\n

    Question 17.
    \n7,892
    \nAnswer:
    \nThe given number is: 7,892
    \nHence, from the above,
    \nWe can conclude that the number that is to the nearest hundred for the given number is: 7,900<\/p>\n

    \"Use<\/p>\n

    Question 18.
    \n472
    \nAnswer:
    \nThe given number is: 472
    \nHence, from the above,
    \nWe can conclude that the number that is to the nearest hundred for the given number is: 500<\/p>\n

    Round each number to the nearest thousand.
    \nQuestion 19.
    \n8,685
    \nAnswer:
    \nThe given number is: 8,685
    \nHence, from the above,
    \nWe can conclude that the number that is to the nearest thousand for the given number is: 9,000<\/p>\n

    Question 20.
    \n4,991
    \nAnswer:
    \nThe given number is: 4,991
    \nHence, from the above,
    \nWe can conclude that the number that is to the nearest thousand for the given number is: 5,000<\/p>\n

    Question 21.
    \n62,549
    \nAnswer:
    \nThe given number is: 62,549
    \nHence, from the above,
    \nWe can conclude that the number that is to the nearest thousand for the given number is: 62,000<\/p>\n

    Question 22.
    \n167,241
    \nAnswer:
    \nThe given number is: 167,241
    \nHence, from the above,
    \nWe can conclude that the number that is to the nearest thousand for the given number is: 167,200<\/p>\n

    Question 23.
    \n77,268
    \nAnswer:
    \nThe given number is: 77,268
    \nHence, from the above,
    \nWe can conclude that the number that is to the nearest thousand for the given number is: 77,000<\/p>\n

    Question 24.
    \n34,162
    \nAnswer:
    \nThe given number is: 34,162
    \nHence, from the above,
    \nWe can conclude that the number that is to the nearest thousand for the given number is: 34,000<\/p>\n

    Question 25.
    \n1,372
    \nAnswer:
    \nThe given number is: 1,372
    \nHence, from the above,
    \nWe can conclude that the number that is to the nearest thousand for the given number is: 1,000<\/p>\n

    Question 26.
    \n9,009
    \nAnswer:
    \nThe given number is: 9,009
    \nHence, from the above,
    \nWe can conclude that the number that is to the nearest thousand for the given number is: 9,000<\/p>\n

    Question 27.
    \n121,619
    \nAnswer:
    \nThe given number is: 121,619
    \nHence, from the above,
    \nWe can conclude that the number that is to the nearest thousand for the given number is: 122,000<\/p>\n

    Question 28.
    \nConstruct Arguments Explain how to round 608,149 to the nearest thousands place.
    \nAnswer:
    \nThe given number is: 608,149
    \nNow,
    \nFrom the given number,
    \nWe can observe that the total number is less than 608,500
    \nHence, from the above,
    \nWe can conclude that the number that is to the nearest thousand for the given number is: 608,000<\/p>\n

    Pick a Project<\/strong><\/p>\n

    PROJECT 4A
    \nHow high would a stack of sabal palms be?
    \nProject: Explain the Processes
    \n\"Envision<\/p>\n

    PROJECT 4B
    \nCan you estimate the weight of dozens of birds?
    \n“Project: Write a Report About the Northern Mockingbird
    \n\"Envision<\/p>\n

    PROJECT 4C
    \nHow many soccer players start in the Women’s World Cup ?
    \nProject: Create an Array Poster
    \n\"Envision<\/p>\n

    PROJECT 4D
    \nHow much weight can shotputters throw?
    \nProject: Compare Shot Masses
    \n\"Envision<\/p>\n

    Lesson 4.1 Multiply Multiples of 10<\/h3>\n

    Solve & Share<\/strong>
    \nThe principal of a school needs to order supplies for 20 new classrooms. Each classroom needs the following items: 20 desks, 30 chairs, and 40 pencils. How many of each item does the principal need to order? Solve these problems using any strategy you choose.
    \nI can … use place-value strategies or properties of operations to multiply by multiples of 10.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nThe principal of a school needs to order supplies for 20 new classrooms. Each classroom needs the following items: 20 desks, 30 chairs, and 40 pencils
    \nSo,
    \nThe total number of desks the principal needs to order = (The total number of classrooms) \u00d7 (The number of desks needed for each classroom)
    \n= 20 \u00d7 20
    \n= 2 tens \u00d7 2 tens [By using the Place-Value strategy]
    \n= 4 hundred
    \n= 400
    \nSo,
    \nThe total number of chairs the principal needs to order = (The total number of classrooms) \u00d7 (The number of chairs needed for each classroom)
    \n= 20 \u00d7 30
    \n= 2 tens \u00d7 3 tens [By using the Place-Value strategy]
    \n= 6 hundred
    \n= 600
    \nSo,
    \nThe total number of pencils the principal needs to order = (The total number of classrooms) \u00d7 (The number of pencils needed for each classroom)
    \n= 20 \u00d7 40
    \n= 2 tens \u00d7 4 tens [By using the Place-Value strategy]
    \n= 8 hundred
    \n= 800
    \nHence, from the above,
    \nWe can conclude that
    \nThe total number of desks the principal needs to order is: 400
    \nThe total number of chairs the principal needs to order is: 600
    \nThe total number of pencils the principal needs to order is: 800<\/p>\n

    Look Back! Look at the factors and products. What patterns do you notice?
    \nAnswer:
    \nThe patterns we noticed from the above factors and products are:
    \na. All multiples of 2 have a pattern of 2, 4, 6, 8, or 0 in the 1’s place.
    \nb. When multiplying Any number by 2, the result is even
    \nc. There are 5 even numbers and 5 odd numbers in each range of 10 numbers<\/p>\n

    Essential Question<\/strong>
    \nHow Can You Multiply on by Multiples of 10?
    \nAnswer:
    \nWhen we multiply any number by 10, the last digit will always be 0
    \nEx:
    \n2 \u00d7 20 = 2 \u00d7 2 \u00d7 10
    \n= 4 \u00d7 10
    \n= 40<\/p>\n

    Visual Learning Bridge
    \nThe number of visitors of each age group for the Sunny Day Amusement Park is shown below. How many children visit the park in 30 days?
    \n\"Envision<\/p>\n

    One Way<\/strong>
    \nFind 30 \u00d7 80.
    \nUse basic facts and place value.
    \n30 \u00d7 80 = 3 tens \u00d7 8 tens
    \n= 24 hundreds
    \n= 2,400
    \n2,400 children visit the park in 30 days.
    \n10 \u00d7 10 = 100<\/p>\n

    Another Way<\/strong>
    \nFind 30 \u00d7 80
    \nBreak apart numbers. Use the Commutative Property and the Associative Property of Multiplication.
    \n30 \u00d7 80 = (3 \u00d7 10) (8 \u00d7 10)
    \n= 3 \u00d7 8 \u00d7 10 \u00d7 10
    \n= (3 \u00d7 8) (10 \u00d7 10)
    \n= 24 \u00d7 100
    \n= 2,400
    \n2,400 children visit the park in 30 days.<\/p>\n

    Convince Me!<\/strong>
    \nLook for Relationships<\/strong> Use place value or properties of operations to determine how many adults age 65 and older visit the park in 30 days.
    \n\"Envision
    \nAnswer:
    \nFrom the given figure,
    \nWe can observe that
    \nThe number of visitors each day that are Adults 65 and over are: 40
    \nSo,
    \nThe number of visitors in 30 days that are Adults 65 and over = 30 \u00d7 (The number of visitors each day that are Adults 65 and over)
    \n= 30 \u00d7 40
    \n= 3 tens \u00d7 4 tens [By using the Place-Value strategy]
    \n= 12 hundred
    \n= 1,200
    \nHence, from the above,
    \nWe can conclude that the number of visitors in 30 days that are Adults 65 and over is: 1,200<\/p>\n

    \"Use<\/p>\n

    Another Example!<\/strong>
    \nUse the properties of operations to find 50 \u00d7 60.
    \n50 \u00d7 60 = 5 \u00d7 10 \u00d7 6 \u00d7 10
    \n= (5 \u00d7 6) \u00d7 (10 \u00d7 10)
    \n= 30 \u00d7 100
    \n= 3,000
    \nIf the product of the basic fact ends in zero, the product has one more zero than you see in the factors.<\/p>\n

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

    Do You Understand?<\/strong>
    \nQuestion 1.
    \nFind 50 \u00d7 20. How many zeros are in the product? Explain.
    \nAnswer:
    \nThe given expression is: 50 \u00d7 20
    \nNow,
    \nBy using the Place-Value Property,
    \n50 \u00d7 20
    \n= 5 tens \u00d7 2 tens
    \n= 10 hundred
    \n= 10 \u00d7 100
    \n= 1,000
    \nHence, from the above,
    \nWe can conclude that
    \nThe value of the product of the given expression is: 1,000
    \nThe number of zeroes in the obtained product is: 3<\/p>\n

    Question 2.
    \nHow many adults under 65 visit the park in 30 days?
    \n\"Envision
    \nAnswer:
    \nFrom the given figure,
    \nWe can observe that
    \nThe number of visitors each day that are adults under 65 are: 60
    \nSo,
    \nThe number of visitors in 30 days that are Adults under 65 = 30 \u00d7 (The number of visitors each day that are adults under 65)
    \n= 30 \u00d7 60
    \n= 3 \u00d7 10 \u00d7 6 \u00d7 10
    \n= (3 \u00d7 6) \u00d7 (10 \u00d7 10)
    \n= 18 \u00d7 100
    \n= 1,800
    \nHence, from the above,
    \nWe can conclude that
    \nThe number of visitors in 30 days that are Adults under 65 is: 1,800<\/p>\n

    Do You Know How?<\/strong>
    \nFor 3-8, use basic facts and place-value or properties of operations to find each product.
    \nQuestion 3.
    \n30 \u00d7 10
    \nAnswer:
    \nThe given expression is: 30 \u00d7 10
    \nNow,
    \nBy using the Properties of Operations,
    \n30 \u00d7 10
    \n= 3 \u00d7 10 \u00d7 1 \u00d7 10
    \n= (3 \u00d7 1) \u00d7 (10 \u00d7 10)
    \n= 3 \u00d7 100
    \n= 300
    \nHence, from the above,
    \nWe can conclude that the value of the given expression is: 300<\/p>\n

    Question 4.
    \n50 \u00d7 10
    \nAnswer:
    \nThe given expression is: 50 \u00d7 10
    \nNow,
    \nBy using the Properties of Operations,
    \n50 \u00d7 10
    \n= 5 \u00d7 10 \u00d7 1 \u00d7 10
    \n= (5 \u00d7 1) \u00d7 (10 \u00d7 10)
    \n= 5 \u00d7 100
    \n= 500
    \nHence, from the above,
    \nWe can conclude that the value of the given expression is: 500<\/p>\n

    Question 5.
    \n20 \u00d7 10
    \nAnswer:
    \nThe given expression is: 20 \u00d7 10
    \nNow,
    \nBy using the Properties of Operations,
    \n20 \u00d7 10
    \n= 2 \u00d7 10 \u00d7 1 \u00d7 10
    \n= (2 \u00d7 1) \u00d7 (10 \u00d7 10)
    \n= 2 \u00d7 100
    \n= 200
    \nHence, from the above,
    \nWe can conclude that the value of the given expression is: 200<\/p>\n

    Question 6.
    \n60 \u00d7 20
    \nAnswer:
    \nThe given expression is: 60 \u00d7 20
    \nNow,
    \nBy using the Properties of Operations,
    \n60 \u00d7 20
    \n= 6 \u00d7 10 \u00d7 2 \u00d7 10
    \n= (6 \u00d7 2) \u00d7 (10 \u00d7 10)
    \n= 12 \u00d7 100
    \n= 1,200
    \nHence, from the above,
    \nWe can conclude that the value of the given expression is: 1,200<\/p>\n

    Question 7.
    \n90 \u00d7 40
    \nAnswer:
    \nThe given expression is: 90 \u00d7 40
    \nNow,
    \nBy using the Properties of Operations,
    \n90 \u00d7 40
    \n= 9 \u00d7 10 \u00d7 4 \u00d7 10
    \n= (9 \u00d7 4) \u00d7 (10 \u00d7 10)
    \n= 36 \u00d7 100
    \n= 3,600
    \nHence, from the above,
    \nWe can conclude that the value of the given expression is: 3,600<\/p>\n

    \"Use<\/p>\n

    Question 8.
    \n80 \u00d7 50
    \nAnswer:
    \nThe given expression is: 80 \u00d7 50
    \nNow,
    \nBy using the Properties of Operations,
    \n80 \u00d7 50
    \n= 8 \u00d7 10 \u00d7 5 \u00d7 10
    \n= (8 \u00d7 5) \u00d7 (10 \u00d7 10)
    \n= 40 \u00d7 100
    \n= 4,000
    \nHence, from the above,
    \nWe can conclude that the value of the given expression is: 4,000<\/p>\n

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

    For 9-16, use basic facts and place-value or properties of operations to find each product.
    \nQuestion 9.
    \n20 \u00d7 70
    \nAnswer:
    \nThe given expression is: 20 \u00d7 70
    \nNow,
    \nBy using the Properties of Operations,
    \n20 \u00d7 70
    \n= 2 \u00d7 10 \u00d7 7 \u00d7 10
    \n= (2 \u00d7 7) \u00d7 (10 \u00d7 10)
    \n= 14 \u00d7 100
    \n= 1,400
    \nHence, from the above,
    \nWe can conclude that the value of the given expression is: 1,400<\/p>\n

    Question 10.
    \n70 \u00d7 90
    \nAnswer:
    \nThe given expression is: 70 \u00d7 90
    \nNow,
    \nBy using the Properties of Operations,
    \n70 \u00d7 90
    \n= 7 \u00d7 10 \u00d7 9 \u00d7 10
    \n= (7 \u00d7 9) \u00d7 (10 \u00d7 10)
    \n= 63 \u00d7 100
    \n= 6,300
    \nHence, from the above,
    \nWe can conclude that the value of the given expression is: 6,300<\/p>\n

    Question 11.
    \n40 \u00d7 20
    \nAnswer:
    \nThe given expression is: 40 \u00d7 20
    \nNow,
    \nBy using the Properties of Operations,
    \n40 \u00d7 20
    \n= 4 \u00d7 10 \u00d7 2 \u00d7 10
    \n= (4 \u00d7 2) \u00d7 (10 \u00d7 10)
    \n= 8 \u00d7 100
    \n= 800
    \nHence, from the above,
    \nWe can conclude that the value of the given expression is: 800<\/p>\n

    Question 12.
    \n40 \u00d7 30
    \nAnswer:
    \nThe given expression is: 40 \u00d7 30
    \nNow,
    \nBy using the Properties of Operations,
    \n40 \u00d7 30
    \n= 4 \u00d7 10 \u00d7 3 \u00d7 10
    \n= (4 \u00d7 3) \u00d7 (10 \u00d7 10)
    \n= 12 \u00d7 100
    \n= 1,200
    \nHence, from the above,
    \nWe can conclude that the value of the given expression is: 1,200<\/p>\n

    Question 13.
    \n70 \u00d7 40
    \nAnswer:
    \nThe given expression is: 70 \u00d7 40
    \nNow,
    \nBy using the Properties of Operations,
    \n70 \u00d7 40
    \n= 7 \u00d7 10 \u00d7 4 \u00d7 10
    \n= (7 \u00d7 4) \u00d7 (10 \u00d7 10)
    \n= 28 \u00d7 100
    \n= 2,800
    \nHence, from the above,
    \nWe can conclude that the value of the given expression is: 2,800<\/p>\n

    Question 14.
    \n20 \u00d7 30
    \nAnswer:
    \nThe given expression is: 20 \u00d7 30
    \nNow,
    \nBy using the Properties of Operations,
    \n20 \u00d7 30
    \n= 2 \u00d7 10 \u00d7 3 \u00d7 10
    \n= (2 \u00d7 3) \u00d7 (10 \u00d7 10)
    \n= 6 \u00d7 100
    \n= 600
    \nHence, from the above,
    \nWe can conclude that the value of the given expression is: 600<\/p>\n

    Question 15.
    \n60 \u00d7 40
    \nAnswer:
    \nThe given expression is: 60 \u00d7 40
    \nNow,
    \nBy using the Properties of Operations,
    \n60 \u00d7 40
    \n= 6 \u00d7 10 \u00d7 4 \u00d7 10
    \n= (6 \u00d7 4) \u00d7 (10 \u00d7 10)
    \n= 24 \u00d7 100
    \n= 2,400
    \nHence, from the above,
    \nWe can conclude that the value of the given expression is: 2,400<\/p>\n

    Question 16.
    \n60 \u00d7 90
    \nAnswer:
    \nThe given expression is: 60 \u00d7 90
    \nNow,
    \nBy using the Properties of Operations,
    \n60 \u00d7 90
    \n= 6 \u00d7 10 \u00d7 9 \u00d7 10
    \n= (6 \u00d7 9) \u00d7 (10 \u00d7 10)
    \n= 54 \u00d7 100
    \n= 5,400
    \nHence, from the above,
    \nWe can conclude that the value of the given expression is: 5,400<\/p>\n

    For 17-22, find the missing factor.
    \nQuestion 17.
    \n10 \u00d7 ______ = 100
    \nAnswer:
    \nThe given expression is: 10 \u00d7 _______ = 100
    \nNow,
    \nLet the missing factor be x
    \nSo,
    \n10 \u00d7 x = 100
    \nx = \\(\\frac{100}{10}\\)
    \nx = 10
    \nHence, from the above,
    \nWe can conclude that the missing factor is: 10<\/p>\n

    Question 18.
    \n_______ \u00d7 20 = 1,600
    \nAnswer:
    \nThe given expression is: ______ \u00d7 20 = 1,600
    \nNow,
    \nLet the missing factor be x
    \nSo,
    \nx \u00d7 20 = 1,600
    \nx = \\(\\frac{1,600}{20}\\)
    \nx = 80
    \nHence, from the above,
    \nWe can conclude that the missing factor is: 80<\/p>\n

    Question 19.
    \n_______ \u00d7 30 = 1,500
    \nAnswer:
    \nThe given expression is: ______ \u00d7 30 = 1,500
    \nNow,
    \nLet the missing factor be x
    \nSo,
    \nx \u00d7 30 = 1,500
    \nx = \\(\\frac{1,500}{30}\\)
    \nx = 50
    \nHence, from the above,
    \nWe can conclude that the missing factor is: 50<\/p>\n

    Question 20.
    \n20 \u00d7 ______ = 1,000
    \nAnswer:
    \nThe given expression is: 20 \u00d7 _______ = 1,000
    \nNow,
    \nLet the missing factor be x
    \nSo,
    \n20 \u00d7 x = 1,000
    \nx = \\(\\frac{1,000}{20}\\)
    \nx = 50
    \nHence, from the above,
    \nWe can conclude that the missing factor is: 50<\/p>\n

    Question 21.
    \n_______ \u00d7 90 = 8,100
    \nAnswer:
    \nThe given expression is: _______ \u00d7 90 = 8,100
    \nNow,
    \nLet the missing factor be x
    \nSo,
    \nx \u00d7 90 = 8,100
    \nx = \\(\\frac{8,100}{90}\\)
    \nx = 90
    \nHence, from the above,
    \nWe can conclude that the missing factor is: 90<\/p>\n

    Question 22.
    \n60 \u00d7 _______ = 4,200
    \nAnswer:
    \nThe given expression is: 60 \u00d7 _______ = 4,200
    \nNow,
    \nLet the missing factor be x
    \nSo,
    \n60 \u00d7 x = 4,200
    \nx = \\(\\frac{4,200}{60}\\)
    \nx = 70
    \nHence, from the above,
    \nWe can conclude that the missing factor is: 70<\/p>\n

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

    Question 23.
    \nReasoning<\/strong>
    \nThe product of two factors is 4,200. If one of the factors is 60, what is the other factor? Explain.
    \nAnswer:
    \nIt is given that
    \nThe product of two factors is 4,200 and one of the factors is 60
    \nNow,
    \nLet the other factor be z
    \nSo,
    \n60 \u00d7 z = 4,200
    \nz = \\(\\frac{4,200}{60}\\)
    \nz = 70
    \nHence, from the above,
    \nWe can conclude that the other factor for the given product is: 70<\/p>\n

    Question 24.
    \nAlgebra<\/strong> There is 30 players on each high school football team. Explain how you can find the total number of players if there are 40 teams. Write and solve an equation.
    \nAnswer:
    \nIt is given that
    \nThere is 30 players on each high school football team and there are 40 teams
    \nSo,
    \nThe total number of players = (The number of players present in each team) \u00d7 (The total number f teams)
    \n= 30 \u00d7 40
    \n= 3 \u00d7 10 \u00d7 4 \u00d7 10
    \n= (3 \u00d7 4) \u00d7 (10 \u00d7 10)
    \n= 12 \u00d7 100
    \n= 1,200
    \nHence, from the above,
    \nWe can conclude that the total number of players are: 1,200<\/p>\n

    Question 25.
    \nBob uses 2 gallons of water while brushing his teeth. He uses 10 gallons of water to wash clothes. How many more cups of water did Bob use while washing his clothes than brushing his teeth?
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nBob uses 2 gallons of water while brushing his teeth. He uses 10 gallons of water to wash clothes.
    \nNow,
    \nThe given figure is:
    \n\"Envision
    \nSo,
    \nThe number of cups Bob uses to wash his clothes = (The number of gallons Bob uses to wash his clothes) \u00d7 (The number of cups in each gallon)
    \n= 10 \u00d7 16
    \n= 10 \u00d7 (8 + 8) [By using the Distributive Property of Multiplication]
    \n= (10 \u00d7 8) + (10 \u00d7 8)
    \n= 80 + 80
    \n= 160 cups
    \nSo,
    \nThe number of cups Bob uses while brushing his teeth = (The number of gallons Bob uses to brush his teeth) \u00d7 (The number of cups in each gallon)
    \n= 2 \u00d7 16
    \n= 2 (8 + 8) [By using the Distributive Property of Multiplication]
    \n= (2 \u00d7 8) + (2 \u00d7 8)
    \n= 16 + 16
    \n= 32 cups
    \nSo,
    \nThe number of more cups that Bob uses to wash his clothes than while brushing his teeth
    \n= (The number of cups Bob uses to wash his clothes) – (the number of cups Bob uses while brushing his teeth)
    \n= 160 – 32
    \n= 128 cups
    \nHence, from the above,
    \nWe can conclude that
    \nThe number of more cups that Bob uses to wash his clothes than while brushing his teeth is: 128 cups<\/p>\n

    Question 26.
    \nJames walked 30 minutes each day for 90 days. Show how you can use place value or properties to find how many minutes James walked.
    \nAnswer:
    \nIt is given that James walked 30 minutes each day for 90 days
    \nSo,
    \nThe total number of minutes James walked = (The number of minutes James walks each day) \u00d7 (The total number of days James walked)
    \n= 30 \u00d7 90
    \n= 3 \u00d7 10 \u00d7 9 \u00d7 10
    \n= (3 \u00d7 9) \u00d7 (10 \u00d7 10)
    \n= 27 \u00d7 100
    \n= 2,700
    \nHence, from the above,
    \nWe can conclude that
    \nThe total number of minutes James walked is: 2,700 minutes<\/p>\n

    Question 27.
    \nHigher-Order Thinking<\/strong> What is one example of a product that will have the same number of zeros in the factors and the product? What is one example of a product that will NOT have the same number of zeros in the factors as the product?
    \nAnswer:
    \nThe example of a product that will have the same number of zeros in the factors and the product is:
    \n10 \u00d7 15
    \n= 10 \u00d7 (10 + 5) [By usingthe Distributive Property of Multiplication]
    \n= (10 \u00d7 10) + (10 \u00d7 5)
    \n= 100 + 50
    \n= 150
    \nThe example of a product that will not have the same number of zeros in the factors as the product is:
    \n40 \u00d7 5
    \n= 4 \u00d7 10 \u00d7 5
    \n= (4 \u00d7 5) \u00d7 10
    \n= 20 \u00d7 10
    \n= 2 \u00d7 10 \u00d7 10
    \n= 200<\/p>\n

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

    Question 28.
    \nSelect all of the expressions that have a product of 1,600.
    \n\u2610 20 \u00d7 80
    \n\u2610 20 \u00d7 60
    \n\u2610 40 \u00d7 40
    \n\u2610 60 \u00d7 30
    \n\u2610 90 \u00d7 20
    \nAnswer:
    \nIt is given that
    \nTo have a product of 1,600,
    \nThe expressions must be even numbers as factors and if we neglect zeroes, the product must be equal to 16
    \nHence, from the above,
    \nWe can conclude that
    \nAll the expressions that have a product of 1,600 are:
    \n\"\"<\/p>\n

    Question 29.
    \n<\/span>Which expression has 50 as the missing factor?
    \nA. 20 \u00d7 ? = 1,000
    \nB. 50 \u00d7 ? = 3,000
    \nC. 30 \u00d7 ? = 1,800
    \nD. 10 \u00d7 ? = 1,000
    \nAnswer:
    \nLet the missing factor be x
    \nNow,
    \nA.
    \n20 \u00d7 x = 1,000
    \nx = \\(\\frac{1,000}{20}\\)
    \n= 50
    \nB.
    \n50 \u00d7 x =3,000
    \nx = \\(\\frac{3,000}{50}\\)
    \n= 60
    \nC.
    \n30 \u00d7 x =1,800
    \nx = \\(\\frac{1,800}{30}\\)
    \n= 60
    \nD.
    \n10 \u00d7 x =1,000
    \nx = \\(\\frac{1,000}{10}\\)
    \n= 100
    \nHence, from the above,
    \nWe can conclude that the expression that has 50 as a missing factor is:
    \n\"\"<\/p>\n

    \"Use<\/p>\n

    Lesson 4.2 Use Models to Multiply 2-Digit Numbers by Multiples of 10<\/h3>\n

    Solve & Share<\/strong>
    \nThere are 10 teams in a baseball league. Each team has 25 players. How many players are in the league? Solve this problem using any strategy you choose.
    \nI can .. use models and properties of operations to help multiply.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nThere are 10 teams in a baseball league. Each team has 25 players
    \nNow,
    \nThe area-model that represents the number of teams and the number of players is:
    \n\"\"
    \nSo,
    \nFrom the above Area-Model,
    \nThe total number of players in the league = (The number of players in each team) \u00d7 (The total number of teams)
    \n= 25 \u00d7 10
    \n= 250 players
    \nHence, from the above,
    \nWe can conclude that the total number of players present in the league is: 250<\/p>\n

    Look Back!<\/strong>
    \nHow many players are in the league if there are 30 teams?
    \nAnswer:
    \nWe know that,
    \nThe total number of players in the league = (The number of players in each team) \u00d7 (The total number of teams)
    \nNow,
    \nIf there are 30 teams and each team consists of the same number of players as mentioned above, then
    \nThe total number of players present in the league
    \n= 25 \u00d7 30
    \n= 25 \u00d7 3 \u00d7 10
    \n= (25 \u00d7 3) \u00d7 10
    \n= 750 players
    \nHence, from the above,
    \nWe can conclude that the total number of players present in the league is: 750<\/p>\n

    Explain how you can use your answer above to help solve this problem.
    \nAnswer:
    \nMultiplication using\u00a0 the “Area model”:
    \nWe know that,
    \nAn “Area” is the amount of flat space occupied by a figure, You can find the area of a figure by filling it with unit squares and then counting how many squares entirely fill the shape
    \nThe total area of the figure is the multiplication of the number of rows and columns i.e., the value of the product
    \nHence, from the above process,
    \nWe can conclude that we used the above process to solve the above-given problems<\/p>\n

    Essentials Question<\/strong>
    \nHow Can You Use an Array or an Area Model to Multiply?
    \nAnswer:
    \nThe steps to find the product that uses an Array or an Area Model to multiply is:
    \na. First, write each factor in expanded form.
    \n<\/span>b. Then, draw your model.
    \n<\/span>c. Next, multiply to find the area of each smaller rectangle.
    \n<\/span>d. Finally, add those products to find the total area.<\/span><\/p>\n

    Visual Learning Bridge
    \nMax’s Moving Company has boxes for packing books. If each box holds 24 books, how many books would fit into 20 boxes?
    \n\"Envision<\/p>\n

    \"Envision<\/p>\n

    Convince Me!<\/strong>
    \nReasoning<\/strong> in the problem above, is the product, 480, reasonable? Explain.
    \nAnswer:
    \nFrom the above,
    \nIt is given that there are 24 books in each box
    \nSo,
    \nThe total number of books that would fit in 20 boxes = (The total number of boxes) \u00d7 (The number of books in each box)
    \n= 20 \u00d7 24
    \n= 2 \u00d7 10 \u00d7 24
    \n= 48 \u00d7 10
    \n= 480 books
    \nHence, from the above,
    \nWe can conclude that the product 480 is reasonable that we got in the above problem<\/p>\n

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

    Do You Understand?<\/strong>
    \nQuestion 1.
    \nDraw an area model to show 20 \u00d7 26. Then find the product.
    \nAnswer:
    \nThe given expression is: 20 \u00d7 26
    \nNow,
    \nBy using the Area-Model,
    \n20 \u00d7 26
    \n= (10 + 10) \u00d7 (20 + 6)
    \n= (10 \u00d7 20) + (10 \u00d7 6) + (10 \u00d7 20) + (10 \u00d7 6)
    \n= 200 + 60 + 200 + 60
    \n= 400 + 120
    \n= 520
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that the value of the given expression using the Area-Model is: 520<\/p>\n

    Do You Know How?<\/strong>
    \nQuestion 2.
    \nThe place-value block array shows 10 \u00d7 16. Find the product.
    \n\"Envision
    \nAnswer:
    \nThe given expression is: 10 \u00d7 16
    \nNow,
    \nBy using the Area-Model,
    \n10 \u00d7 16
    \n= 10 \u00d7 (10 + 6)
    \n= (10 \u00d7 10) + (10 \u00d7 6)
    \n= 100 + 60
    \n= 160
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that the value of the given expression by using the Area-Model is: 160<\/p>\n

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

    Leveled Practice For 3-8, use place-value blocks, area models, or arrays to find each product.
    \nQuestion 3.
    \n10 \u00d7 22
    \n\"Envision
    \nAnswer:
    \nThe given expression is: 10 \u00d7 22
    \nNow,
    \nBy using the Area-Model,
    \n10 \u00d7 22
    \n= 10 \u00d7 (20 + 2)
    \n= (10 \u00d7 20) + (10 \u00d7 2)
    \n= 200 + 20
    \n= 220
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that the value of the given expression by using the Area-Model is: 220<\/p>\n

    Question 4.
    \n10 \u00d7 13
    \n\"Envision
    \nAnswer:
    \nThe given expression is: 10 \u00d7 13
    \nNow,
    \nBy using the Area-Model,
    \n10 \u00d7 13
    \n= 10 \u00d7 (10 + 3)
    \n= (10 \u00d7 10) + (10 \u00d7 3)
    \n= 100 + 30
    \n= 130
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that the value of the given expression by using the Area-Model is: 130<\/p>\n

    Question 5.
    \n20 \u00d7 35
    \nAnswer:
    \nThe given expression is: 20 \u00d7 35
    \nNow,
    \nBy using the Area-Model,
    \n20 \u00d7 35
    \n= (10 + 10) \u00d7 (30 + 5)
    \n= (10 \u00d7 30) + (10 \u00d7 5) + (10 \u00d7 30) + (10 \u00d7 5)
    \n= 300 + 50 + 300 + 50
    \n= 600 + 100
    \n= 700
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that the value of the given expression by using the Area-Model is: 700<\/p>\n

    Question 6.
    \n20 \u00d7 41
    \nAnswer:
    \nThe given expression is: 20 \u00d7 41
    \nNow,
    \nBy using the Area-Model,
    \n20 \u00d7 41
    \n= (10 + 10) \u00d7 (40 + 1)
    \n= (10 \u00d7 40) + (10 \u00d7 1) + (10 \u00d7 40) + (10 \u00d7 1)
    \n= 400 + 10 + 400 + 10
    \n= 800 + 20
    \n= 820
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that the value of the given expression by using the Area-Model is: 820<\/p>\n

    Question 7.
    \n30 \u00d7 29
    \nAnswer:
    \nThe given expression is: 30 \u00d7 29
    \nNow,
    \nBy using the Area-Model,
    \n30 \u00d7 29
    \n= (20 + 10) \u00d7 (20 + 9)
    \n= (20 \u00d7 20) + (20 \u00d7 9) + (10 \u00d7 20) + (10 \u00d7 9)
    \n= 400 + 180 + 200 + 90
    \n= 600 + 270
    \n= 870
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that the value of the given expression by using the Area-Model is: 870<\/p>\n

    Question 8.
    \n40 \u00d7 37
    \nAnswer:
    \nThe given expression is: 40 \u00d7 37
    \nNow,
    \nBy using the Area-Model,
    \n40 \u00d7 37
    \n= (20 + 20) \u00d7 (30 + 7)
    \n= (20 \u00d7 30) + (20 \u00d7 7) + (20 \u00d7 30) + (20 \u00d7 7)
    \n= 600 + 140 + 600 + 140
    \n= 1,200 + 280
    \n= 1,480
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that the value of the given expression by using the Area-Model is: 1,480<\/p>\n

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

    Camera Sales
    \n\"Envision
    \nQuestion 9.
    \nAlgebra<\/strong> In the first 3 months of the year, an electronics store sold 1,446 cameras. How many cameras did the store sell in March? Write and solve an equation.
    \nAnswer:
    \nIt is given that
    \nIn the first 3 months of the year, an electronics store sold 1,446 cameras
    \nNow,
    \nThe given data is:
    \n\"Envision
    \nSo,
    \nThe number of cameras sold in the three months = (The number of cameras sold in January) + (The number of cameras sold in February) + (The number of cameras sold in March)
    \nNow,
    \nLet the number of cameras sold in March be x
    \nSo,
    \n1,446 = 486 + 385 + x
    \nx = 1,446 – (486 + 385)
    \nx = 1,446 – 871
    \nx = 575 cameras
    \nHence, from the above,
    \nWe can conclude that the number of cameras that the store sold in March\u00a0 is: 575<\/p>\n

    Question 10.
    \nFor every camera sold in February, the store donated $2 to a charity. How much did the store donate?
    \nAnswer:
    \nIt is given that
    \nFor every camera sold in February, the store donated $2 to a charity
    \nNow,
    \nThe given data is:
    \n\"Envision
    \nSo,
    \nThe total amount of money the store donated in February to a charity = (The number of cameras sold in February) \u00d7 (The amount of money the store donated to a charity for every camera)
    \n= 385 \u00d7 $2
    \n= (300 + 80 + 5) \u00d7 $2
    \n= (300 \u00d7 $2) + (80 \u00d7 $2) + (5 \u00d7 $2)
    \n= $600 + $160 + $10
    \n= $770
    \nHence, from the above,
    \nWe can conclude that
    \nThe total amount of money the store donated in Februaryto a charity is: $770<\/p>\n

    Question 11.
    \nModel with Math<\/strong>
    \nDuring a basketball game, 75 cups of fruit punch was sold. Each cup holds 20 fluid ounces. How many total fluid ounces of fruit punch were sold?
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nDuring a basketball game, 75 cups of fruit punch was sold. Each cup holds 20 fluid ounces.
    \nSo,
    \nThe amount of total fluid ounces of fruit punch that were sold = (The total number of cups) \u00d7 (The amount of fluid ounces that each cup holds)
    \n= 75 \u00d7 20
    \n= (70 + 5) \u00d7 (10 + 10)
    \n= (70 \u00d7 10) + (70 \u00d7 10) + (5 \u00d7 10) + (5 \u00d7 10)
    \n= 700 + 700 + 50 + 50
    \n= 1,400 + 100
    \n= 1,500 fluid ounces
    \nHence, from the above,
    \nWe can conclude that the amount of fluid ounces of fruit punch that were sold is: 1,500 fluid ounces<\/p>\n

    Question 12.
    \nHigher-Order Thinking<\/strong> Describe how to correct the partial product that is not correct in the worked problem below. What is the correct product?
    \n\"Envision
    \nAnswer:
    \nThe given expression is: 27 \u00d7 73
    \nNow,
    \nBy using the Area-Model,
    \n27 \u00d7 73
    \n= (20 + 7) \u00d7 (70 + 3)
    \n= (20 \u00d7 70) + (20 \u00d7 3) + (7 \u00d7 70) + (7 \u00d7 3)
    \n= 1,400 + 60 + 490 + 21
    \n= 1,971
    \nHence, from the above,
    \nWe can conclude that
    \nThe partial product that is not correct is: 490
    \nThe correct product for the given expression is: 1,971<\/p>\n

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

    Question 13.
    \nWhich is greater?
    \n15 \u00d7 30 or 22 \u00d7 20 Show your thinking.
    \n\"Envision
    \nAnswer:
    \nThe given expressions are:
    \na. 15 \u00d7 30\u00a0 \u00a0 \u00a0b. 22 \u00d7 20
    \nNow,
    \nBy using the Area-Model,
    \na.
    \n15 \u00d7 30
    \n= (10 + 5) \u00d7 (20 + 10)
    \n= (10 \u00d7 20) + (10 \u00d7 10) + (5 \u00d7 20) + (5 \u00d7 10)
    \n= 200 + 100 + 100 + 50
    \n= 400 + 50
    \n= 450
    \nb.
    \n22 \u00d7 20
    \n= (20 + 2) \u00d7 (10 + 10)
    \n= (20 \u00d7 10) + (20 \u00d7 10) + (2 \u00d7 10) + (2 \u00d7 10)
    \n= 200 + 200 + 20 + 20
    \n= 400 + 40
    \n= 440
    \nHence, from the above,
    \nWe can conclude that the value of the expression 15 \u00d7 30 is greater<\/p>\n

    Lesson 4.3 Estimate: Use Rounding or Compatible Numbers<\/h3>\n

    Solve & Share<\/strong>
    \nChoose two factors from the numbers below to find a product that is as close to 1,600 as possible. Solve this problem using any strategy you choose.
    \nI can .. use rounding or compatible numbers to estimate.
    \n\"Envision
    \nAnswer:
    \nThe given numbers are: 24, 32, 61, and 78
    \nSo,
    \nThe numbers that are closest to the given numbers are: 20, 30, 60, and 80
    \nNow,
    \nTo find the two factors from the given numbers to find a product that is as close to 1,600 as possible,
    \nWe have to find the product of the factors that are close to the given factors
    \nSo,
    \na.
    \n20 \u00d7 30
    \n= 2 \u00d7 10 \u00d7 3 \u00d7 10
    \n= (2 \u00d7 3) \u00d7 100
    \n= 6 \u00d7 100
    \n= 600
    \nb.
    \n20 \u00d7 60
    \n= 2 \u00d7 10 \u00d7 6 \u00d7 10
    \n= (2 \u00d7 6) \u00d7 100
    \n= 12 \u00d7 100
    \n= 1,200
    \nc.
    \n20 \u00d7 80
    \n= 2 \u00d7 10 \u00d7 8 \u00d7 10
    \n= (2 \u00d7 8) \u00d7 100
    \n= 16 \u00d7 100
    \n= 1,600
    \nd.
    \n30 \u00d7 60
    \n= 3 \u00d7 10 \u00d7 6 \u00d7 10
    \n= (3 \u00d7 6) \u00d7 100
    \n= 18 \u00d7 100
    \n= 1,800
    \ne.
    \n30 \u00d7 80
    \n= 3 \u00d7 10 \u00d7 8 \u00d7 10
    \n= (3 \u00d7 8) \u00d7 100
    \n= 24 \u00d7 100
    \n= 2,400
    \nf.
    \n60 \u00d7 80
    \n= 6 \u00d7 10 \u00d7 8 \u00d7 10
    \n= (6 \u00d7 8) \u00d7 100
    \n= 48 \u00d7 100
    \n= 4,800
    \nHence, from the above,
    \nWe can conclude that
    \nThe two factors that are closest to the product 1,600 are: 24, 78<\/p>\n

    Look Back!<\/strong>
    \nConstruct Arguments<\/strong> Why did you choose the two factors that you did? How do you know your factors will give the closest estimate of the product?
    \nAnswer:
    \nWe know that,
    \nThe difference between the product of the two exact numbers and the product of the numbers that are closest to the exact numbers must be small
    \nSo,
    \nWhen we observe the above products which we obtained in the above problem,
    \nWe can observe that there are two products that are closest to the given product
    \nBut,
    \nWhen we find the difference between the exact product and the estimated product,
    \nThe product of the factors 25 and 60 is close to the product of the exact factors 24, 78<\/p>\n

    Essential Question<\/strong>
    \nWhat Strategies Can I Use When Estimating?
    \nAnswer:
    \nThe front-end strategy\u00a0is a method of estimating computations by keeping the first digit in each of the numbers and changing all the other digits to zeros. This strategy can be used to estimate sums, differences, products and quotients.<\/p>\n

    Visual Learning Bridge
    \nThe workers picked 14 dozen apples at Ms. Piper’s apple grove and 12 dozen apples at Mr. Stuart’s apple grove. There are 12 apples in one dozen. About how many apples did the workers pick?
    \n14 + 12 = 26 dozen apples picked in total
    \n\"Envision<\/p>\n

    One Way<\/strong>
    \nUse rounding to estimate 26 \u00d7 12.
    \nRound 26 to the nearest ten.
    \nRound 12 to the nearest ten.
    \n\"Envision
    \nThe workers picked about 300 apples.
    \nSome problems do not need an exact answer.<\/p>\n

    Another Way<\/strong>
    \nUse compatible numbers to estimate 26 \u00d7 12.
    \nReplace the factors with numbers that are close and easy to multiply.
    \n26 is close to 25.
    \n12 is close to 10.
    \n25 \u00d7 10 = n
    \n250 = n
    \nThe workers picked about 250 apples.<\/p>\n

    Convince Me!<\/strong>
    \nReasoning<\/strong> Sue said she could find an estimate for 26 \u00d7 12 by rounding only 1 factor and multiplying 26 \u00d7 10. Do you agree? Explain.
    \nAnswer:
    \nSue said she could find an estimate for 26 \u00d7 12 by rounding only 1 factor and multiplying 26 \u00d7 10
    \nNow,
    \nWe know that,
    \nWhen we want to find an estimate of the factors, we have to estimate all the factors and it is possible when you do partial estimation i.e., rounding off 1 number and not rounding off another number
    \nHence, from the above,
    \nWe can conclude that we can agree with Sue<\/p>\n

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

    Do You Understand?<\/strong>
    \nQuestion 1.
    \nIn the example on the previous page, how do you know you only need an estimate and not an exact answer?
    \nAnswer:
    \nWe know that,
    \nThere are two types of estimations. They are:
    \na. Round-off method\u00a0 \u00a0 \u00a0 b. Closest estimated method
    \nNow,
    \nWhen we use the above two types of eatimations,
    \nThe answers we will get may differ
    \nSo,
    \nIn the example of previous page,
    \nWe got different answers when we use different types of estimation methods
    \nNow,
    \nFrom the example of previous page,
    \nWe can observe that the number of fruits had been asked in terms of estimations but not an exact number
    \nHence, from the above,
    \nWe can conclude that we only need an estimate and not an exact answer because of the mentioned quaestion<\/p>\n

    Question 2.
    \nIn the example on the previous page, what is another way you can multiply using compatible numbers by changing only one of the factors? Explain how you would find the estimated product.
    \nAnswer:
    \nThe given expression on the previous page is: 26 \u00d7 12
    \nNow,
    \nBy using the closest number estimation,
    \nThe factors will be changed into: 25 and 12; or 26 and 10
    \nSo,
    \nThe estimation of the product using compatible numbers will be:
    \n26 \u00d7 12
    \n= 25 \u00d7 12\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(or)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a026 \u00d7 10
    \n= 300\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(or)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0= 260
    \nHence, from the above,
    \nWe can coclude that the estimated product is approximatelu 300 (or) 260<\/p>\n

    Do You Know How?<\/strong>
    \nFor 3-6, estimate each product. Explain why you chose the strategy you used to estimate the product.
    \nQuestion 3.
    \n24 \u00d7 18
    \nAnswer:
    \nThe given expression is: 24 \u00d7 18
    \nNow,
    \nBy using the Compatible method estimation,
    \nThe given expression will become: 25 \u00d7 20
    \nSo,
    \n25 \u00d7 20
    \n= 25 \u00d7 2 \u00d7 10
    \n= (25 \u00d72) \u00d7 10
    \n= 50 \u00d710
    \n= 500
    \nHence, from the above,
    \nWe can conclude that the estimated product for the given expression is: 500<\/p>\n

    Question 4.
    \n33 \u00d7 31
    \nAnswer:
    \nThe given expression is: 33 \u00d7 31
    \nNow,
    \nBy using the Compatible method estimation,
    \nThe given expression will become: 35 \u00d7 30
    \nSo,
    \n35 \u00d7 30
    \n= 35 \u00d7 3 \u00d7 10
    \n= (35 \u00d7 3) \u00d7 10
    \n= 105 \u00d7 10
    \n= 1,050
    \nHence, from the above,
    \nWe can conclude that the estimated product for the given expression is: 1,050<\/p>\n

    Question 5.
    \n38 \u00d7 22
    \nAnswer:
    \nThe given expression is: 38 \u00d7 32
    \nNow,
    \nBy using the Compatible method estimation,
    \nThe given expression will become: 40 \u00d7 30
    \nSo,
    \n40 \u00d7 30
    \n= 4 \u00d7 10 \u00d7 3 \u00d7 10
    \n= (4 \u00d7 3) \u00d7 100
    \n= 12 \u00d7 100
    \n= 1,200
    \nHence, from the above,
    \nWe can conclude that the estimated product for the given expression is: 1,200<\/p>\n

    Question 6.
    \n45 \u00d7 48
    \nAnswer:
    \nThe given expression is: 45 \u00d7 48
    \nNow,
    \nBy using the Rounding-off method estimation,
    \nThe given expression will become: 50 \u00d7 50
    \nSo,
    \n50 \u00d7 50
    \n= 5 \u00d710 \u00d7 5 \u00d7 10
    \n= (5 \u00d7 5) \u00d7 100
    \n= 25 \u00d7 100
    \n= 2,500
    \nHence, from the above,
    \nWe can conclude that the estimated product for the given expression is: 2,500<\/p>\n

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

    For 7-12, estimate each product. Explain why you chose the strategy you used to estimate the product.
    \nQuestion 7.
    \n39 \u00d7 19
    \nAnswer:
    \nThe given expression is: 39 \u00d7 19
    \nNow,
    \nBy using the Rounding-off estimation method,
    \nThe given expression will become: 40 \u00d7 20
    \nSo,
    \n40 \u00d7 20
    \n= 4 \u00d7 10 \u00d7 2 \u00d7 10
    \n= (4 \u00d7 2) \u00d7100
    \n= 8 \u00d7 100
    \n= 800
    \nHence, from the above,
    \nWe can conclude that the estimated product for the given expression is: 800<\/p>\n

    Question 8.
    \n28 \u00d7 27
    \nAnswer:
    \nThe given expression is: 28 \u00d7 27
    \nNow,
    \nBy using the Rounding-off estimation method,
    \nThe given expression will become: 30 \u00d7 30
    \nSo,
    \n30 \u00d7 30
    \n= 3 \u00d7 10 \u00d7 3 \u00d7 10
    \n= (3 \u00d7 3) \u00d7100
    \n= 9 \u00d7 100
    \n= 900
    \nHence, from the above,
    \nWe can conclude that the estimated product for the given expression is: 900<\/p>\n

    Question 9.
    \n64 \u00d7 13
    \nAnswer:
    \nThe given expression is: 64 \u00d7 13
    \nNow,
    \nBy using the Compatible estimation method,
    \nThe given expression will become: 65 \u00d7 15
    \nSo,
    \n65 \u00d7 15
    \n= (60 + 5) \u00d7 (10 + 5)
    \n= (60 \u00d7 10) + (60 \u00d7 5) + (5 \u00d7 10) + (5 \u00d7 5)
    \n= 600 + 300 + 50 + 25
    \n= 975
    \nHence, from the above,
    \nWe can conclude that the estimated product for the given expression is: 975<\/p>\n

    Question 10.
    \n42 \u00d7 17
    \nAnswer:
    \nThe given expression is: 42 \u00d7 17
    \nNow,
    \nBy using the Rounding-off estimation method,
    \nThe given expression will become: 40 \u00d7 20
    \nSo,
    \n40 \u00d7 20
    \n= 4 \u00d7 10 \u00d7 2 \u00d7 10
    \n= (4 \u00d7 2) \u00d7100
    \n= 8 \u00d7 100
    \n= 800
    \nHence, from the above,
    \nWe can conclude that the estimated product for the given expression is: 800<\/p>\n

    Question 11.
    \n82 \u00d7 36
    \nAnswer:
    \nThe given expression is: 82 \u00d7 36
    \nNow,
    \nBy using the Compatible way estimation method,
    \nThe given expression will become: 80 \u00d7 35
    \nSo,
    \n80 \u00d7 35
    \n= (40 + 40) \u00d7 (30 + 5)
    \n= (40 \u00d7 30) + (40 \u00d7 5) + (40 \u00d7 30) + (40 \u00d7 5)
    \n= 1,200 + 200 + 1,200 + 200
    \n= 2,400 + 400
    \n= 2,800
    \nHence, from the above,
    \nWe can conclude that the estimated product for the given expression is: 2,800<\/p>\n

    Question 12.
    \n54 \u00d7 18
    \nAnswer:
    \nThe given expression is: 54 \u00d7 18
    \nNow,
    \nBy using the Compaible way estimation method,
    \nThe given expression will become: 55 \u00d7 20
    \nSo,
    \n55 \u00d720
    \n= 55 \u00d7 2 \u00d7 10
    \n= (55 \u00d7 2) \u00d7 10
    \n= 110 \u00d7 10
    \n= 1,100
    \nHence, from the above,
    \nWe can conclude that the estimated product for the given expression is: 1,100<\/p>\n

    For 13-14, estimate to check if the given answer is reasonable.
    \nCompare your estimate to the given answer to check if the answer is reasonable.
    \nQuestion 13.
    \n66 \u00d7 41 = 2,706
    \nRounds to ________ \u00d7 ________ = ________
    \nReasonable
    \nNot Reasonable
    \nAnswer:
    \nThe given expression is: 66 \u00d7 41
    \nNow,
    \nBy using the Rounding-off method,
    \n66 \u00d7 41
    \n= 70 \u00d7 40
    \n= 7 \u00d7 10 \u00d7 4 \u00d7 10
    \n= (7 \u00d7 4) \u00d7 100
    \n= 28 \u00d7 100
    \n= 2,800
    \nHence, from the above,
    \nWe can conclude that the Rounding is “Reasonable” since the difference between the estimated answer and the exact answer is small<\/p>\n

    Question 14.
    \n34 \u00d7 52 = 2,288
    \nRounds to ________ \u00d7 ________ = ________
    \nReasonable
    \nNot Reasonable
    \nAnswer:
    \nThe given expression is: 34 \u00d7 52
    \nNow,
    \nBy using the Rounding-off method,
    \n34 \u00d7 52
    \n= 30 \u00d7 50
    \n= 3 \u00d7 10 \u00d7 5 \u00d710
    \n= (3 \u00d7 5) \u00d7 100
    \n= 15 \u00d7 100
    \n= 1,500
    \nHence, from the above,
    \nWe can conclude that the Rounding is not “Reasonable” since the difference between the estimated answer and the exact answer is large<\/p>\n

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

    For 15-16, use the table at the right.
    \n\"Envision
    \nQuestion 15.
    \nAbout how many more Valencia orange trees than Temple orange trees does Mr. Gonzalez have? Explain.
    \nAnswer:
    \nThe given data is:
    \n\"Envision
    \nNow,
    \nFrom the given data,
    \nWe can find:
    \na.
    \nThe number of Orange trees in Valencia = (The number of trees in each row) \u00d7 (The number of rows)
    \n= 46 \u00d7 31
    \nNow,
    \nBy using the rounding-off method,
    \n46 \u00d7 31
    \n= 50 \u00d7 30
    \n= 5 \u00d7 10 \u00d7 3 \u00d7 10
    \n= (5 \u00d7 3) \u00d7 100
    \n= 1,500
    \nb.
    \nThe number of Orange trees in Temple = (The number of trees in each row) \u00d7 (The number of rows)
    \n= 38 \u00d7 28
    \nNow,
    \nBy using the rounding-off method,
    \n38 \u00d7 28
    \n= 40 \u00d7 30
    \n= 4 \u00d7 10 \u00d7 3 \u00d7 10
    \n= (4 \u00d7 3) \u00d7 100
    \n= 12 \u00d7 100
    \n= 1,200
    \nSo,
    \nThe number of Valencia orange trees more than Temple orange trees does Mr. Gonzalez have
    \n= (The number of Valencia Orange trees) – (The number of Temple Orange trees)
    \n= 1,500 – 1,200
    \n= 300
    \nHence, from the above,
    \nWe can conclude that Mr Gonzalez have about 300 Valencia Orange trees more than temple Orange trees<\/p>\n

    Question 16.
    \nAbout how many orange trees does Mr. Gonzalez have? Explain.
    \nAnswer:
    \nThe given data is:
    \n\"Envision
    \nNow,
    \nFrom the given data,
    \nWe can find that the total number of Mr. Gonzalez’s trees can be obtained by adding the given three types of Orange trees
    \nSo,
    \nThe number of Orange trees does Mr Gonzalez have = (The total number of trees in each row) \u00d7 (The total number of rows)
    \n= (38 + 28 + 46) \u00d7 (28 + 38 + 31)
    \n= 112 \u00d7 97
    \nNow,
    \nBy using the Rounding-off method,
    \n112 \u00d7 97
    \n= 110 \u00d7 100
    \n= 11 \u00d7 10 \u00d7 100
    \n= 11,000
    \nHence, from the above,
    \nWe can conclude that there are about 11,000 Orange trees does Mr. Gonzalez have<\/p>\n

    Question 17.
    \nHigher-Order Thinking<\/strong> How is using compatible numbers to estimate similar to using rounding? How is it different?
    \nAnswer:
    \nA)
    \nCompatible numbers:
    \nRound the whole number to the closest compatible number. “Compatible numbers” are numbers that are close in value to the real number that would make it easier to find an estimate calculation.
    \nB)
    \nRounding-off numbers:
    \nRounding off means a\u00a0number is made simpler by keeping its value intact but closer to the next number. It is done for whole numbers, and for decimals at various places of hundreds, tens, tenths, etc. Rounding off numbers is done to preserve the significant figures<\/p>\n

    Question 18.
    \nConstruct Arguments<\/strong> Explain how you would use estimation to decide which has the greater product, 39 \u00d7 21 or 32 \u00d7 32.
    \nAnswer:
    \nThe given expressions are: 39 \u00d7 21 and 32 \u00d7 32
    \nNow,
    \nA)
    \nBy using the rounding-off method:
    \na.
    \n39 \u00d7 21
    \n= 40 \u00d7 20
    \n= 4 \u00d7 10 \u00d7 2 \u00d7 10
    \n= (4 \u00d7 2) \u00d7 100
    \n= 8 \u00d7 100
    \n= 800
    \nb.
    \n32 \u00d7 32
    \n= 30 \u00d7 30
    \n= 3 \u00d7 10 \u00d7 3 \u00d7 10
    \n= (3 \u00d7 3) \u00d7 100
    \n= 9 \u00d7 100
    \n= 900
    \nB)
    \nBy using the compatible nembers method:
    \na.
    \n39 \u00d7 21
    \n= 40 \u00d7 20
    \n= 4 \u00d7 10 \u00d7 2 \u00d7 10
    \n= 8 \u00d7 100
    \n= 800
    \nb.
    \n32 \u00d7 32
    \n= 30 \u00d7 30
    \n= 3 \u00d7 10 \u00d7 3 \u00d7 10
    \n= (3 \u00d7 3) \u00d7 100
    \n= 9 \u00d7 100
    \n= 900
    \nHence, from the above,
    \nWe can conclude that the estimated product of 32 \u00d7 32 is greater<\/p>\n

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

    Question 19.
    \nSelect all of the expressions that could be used to estimate 32 \u00d7 14.
    \n\u2610 30 \u00d7 20
    \n\u2610 30 \u00d7 10
    \n\u2610 14 \u00d7 32
    \n\u2610 40 \u00d7 10
    \n\u2610 32 \u00d7 10
    \nAnswer:
    \nThe given expression is: 32 \u00d7 14
    \nNow,
    \nBy using the Rounding-off method,
    \nThe given expression will become: 30 \u00d7 10 and 32 \u00d7 10 and 30 \u00d7 14
    \nNow,
    \nBy using the Compatible numbers method,
    \nThe given expression will become: 30 \u00d7 15 and 30 \u00d7 14 and 32 \u00d7 15
    \nHence, from the above,
    \nWe can conclude that the expressions that can be used to estimate 32 \u00d7 14 are:
    \n\"\"<\/p>\n

    Question 20.
    \nSelect all of the equations that use compatible numbers to find an estimate for 28 \u00d7 24.
    \n\u2610 30 \u00d7 25 = 750
    \n\u2610 28 \u00d7 10 = 280
    \n\u2610 28 \u00d7 25 = 700
    \n\u2610 25 \u00d7 20 = 500
    \n\u2610 30 \u00d7 300 = 9,000
    \nAnswer:
    \nThe given expression is: 28 \u00d7 24
    \nNow,
    \nBy usin the compatible numbers method,
    \nThe given expression will become: 30 \u00d7 25 and 28 \u00d7 25 and 30 \u00d7 24
    \nHence, from the above,
    \nWe can conclude that
    \nThe equations that use compatible numbers to find an estimate for 28 \u00d7 24 are:
    \n\"\"<\/p>\n

    Lesson 4.4 Arrays and Partial Products<\/h3>\n

    Solve & Share<\/strong>
    \nA theater contains 14 rows of seats with 23 seats in each row. How many seats are in the theater? Solve this problem using any strategy you choose.
    \nI can … use place-value concepts and properties to multiply.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nA theater contains 14 rows of seats with 23 seats in each row.
    \nSo,
    \nThe total number of seats that are present in a theater = (The number of rows of seats a theater contains) \u00d7 (The number of seats in each row)
    \n= 14 \u00d7 23
    \nNow,
    \nBy using the Partial multiplication method,
    \n14 \u00d7 23
    \n= (10 + 4) \u00d7 (20 + 3)
    \n= (10 \u00d7 20) + (10 \u00d7 3) + (4 \u00d7 20) + (4 \u00d7 3)
    \n= 200 + 30 + 80 + 12
    \n= 322
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that
    \nThe total number of seats that are present in a theater is: 322 seats<\/p>\n

    Look Back!<\/strong>
    \nUse Structure<\/strong> Theater seating is an example of objects that are arranged in rows and columns, or arrays. How do the number of rows and the number of seats in each row relate to the total number of seats?
    \nAnswer:
    \nIt is given that
    \nTheater seating is an example of objects that are arranged in rows and columns or arrays
    \nNow,
    \nWe know that,
    \nThe total number of seats in a theater can be given by the multiplication of the number of rows present in a theater with the number of seats that are present in a row
    \nHence, from the above,
    \nWe can conclude that
    \nThe total number of seats in a theater = (The number of rows) \u00d7 (The number of seats in each row)<\/p>\n

    Essential Question<\/strong>
    \nHow Can You Multiply Using an Array?
    \nAnswer:
    \nThe steps that are to be followed when you multiply using an Array are:
    \nSTEP 1:
    \nLabel the height and width of a rectangle with the key numbers in the question.
    \nSTEP 2:
    \na. Break down the numbers into tens and ones.
    \nb Draw a separate column and row for each of these numbers.
    \nSTEP 3:
    \nMultiply each column and row with each other, until you get four numbers.<\/p>\n

    Visual Learning Bridge
    \nThere are 13 toy dogs in each row of a carnival booth. Twenty rows contain toy bulldogs and 4 rows contain toy huskies. How many toy dogs are there?
    \n\"Envision<\/p>\n

    \"Envision<\/p>\n

    Convince Me!<\/strong>
    \nModel with Math<\/strong> What 2-digit by 2-digit multiplication is shown by the model at the right? What is the product? Explain how you used the model to find the product.
    \n\"Envision
    \nAnswer:
    \nThe given model is:
    \n\"Envision
    \nNow,
    \nThe representation of the given model in the simplest form is:
    \n\"\"
    \nNow,
    \nFrom the above,
    \nWe can observe that
    \nThe two-digit numbers that are used for multiplication is:
    \n(10 + 8) \u00d7 (10 + 5)
    \n= 18 \u00d7 15
    \nNow,
    \nBy using the Partial Multiplication method,
    \n18 \u00d7 15
    \n= (10 + 8) \u00d7 (10 + 5)
    \n= (10 \u00d7 10) + (10 \u00d7 5) + (8 \u00d7 10) + (8 \u00d7 5)
    \n= 100 + 50 + 80 + 40
    \n= 270
    \nHence, from the above,
    \nWe can conclude that
    \nThe 2-digit by 2-digit multiplication that is shown in the given model is: 18 \u00d7 15
    \nThe product of the obtained 2-digit by 2-digit multiplication using the partial multiplication method is: 270<\/p>\n

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

    Do You Understand?<\/strong>
    \nQuestion 1.
    \nIn the example on the previous page, what four simpler multiplication problems were used to find 24 \u00d7 13?
    \nAnswer:
    \nThe given expression is: 24 \u00d7 13
    \nNow,
    \nBy using the Partial Multiplication method,
    \n24 \u00d7 13
    \n= (20 + 4) \u00d7 (10 + 3)
    \n= (20 \u00d7 10) + ( 20 \u00d7 3) + (4 \u00d7 10) + (4 \u00d7 3)
    \n= 200 + 60 + 40 + 12
    \n= 312
    \nHence, from the above,
    \nWe can conclude that
    \nThe four simpler multiplication problems that were used to find 24 \u00d7 13 that is the example on the previous page is:
    \na. 20 \u00d7 10 = 200
    \nb. 20 \u00d7 3 = 60
    \nc. 4 \u00d7 10 = 40
    \nd. 4 \u00d7 3 = 12<\/p>\n

    Question 2.
    \nHow can you use properties to help find the product of 24 \u00d7 13?
    \nAnswer:
    \nThe given expression is: 24 \u00d7 13
    \nNow,
    \nWe know that,
    \nDistributive Property of Multiplication:
    \na (b + c) = a.b + a.c
    \nNow,
    \nDivide 24 \u00d7 13 into parts
    \nSo,
    \n24 \u00d7 13
    \n= (20 + 4) \u00d7 (10 + 3)
    \nNow,
    \nBy using the Distributive Property of Multiplication,
    \n20 \u00d7 (10 + 3) = (20 \u00d7 10) + (20 \u00d7 3)
    \n= 200 + 60
    \n= 260
    \n4 \u00d7 (10 + 3) = (4 \u00d7 10) + (4 \u00d7 3)
    \n= 40 + 12
    \n= 52
    \nSo,
    \n24 \u00d7 13 = 260 + 52
    \n= 312
    \nHence, from the above,
    \nWe can conclude that the product of the given expression by using the properties is: 312<\/p>\n

    Do You Know How?<\/strong>
    \nFor 3, use the array drawn on a grid to find the product. Check if your answer is reasonable.
    \nQuestion 3.
    \n\"Envision
    \nAnswer:
    \nThe given expression is: 24 \u00d7 16
    \nNow,
    \nBy using the estimation of the Compatible numbers,
    \nThe given expression will become: 24 \u00d7 15
    \nNow,
    \nBy using the Partial Multiplication method,
    \n24 \u00d7 16\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 24 \u00d715
    \n= (20 + 4) \u00d7 (10 + 6)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0= (20 + 4) \u00d7 (10 + 5)
    \n= (20 \u00d7 10) + (20 \u00d7 6) + (4 \u00d7 10) + (4 \u00d7 6)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0= (20 \u00d7 10) + (20 \u00d7 5) + (4 \u00d7 10) + (4 \u00d7 5)
    \n= 200 + 120 + 40 + 24\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 = 200 + 100 + 40 + 20
    \n= 384\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0= 360
    \nHence, from the above,
    \nWe can conclude that
    \nThe product of the given expression is: 384
    \nThe product for the given expression is “Reasonable” since there is a small difference between the exact product and the estimated product<\/p>\n

    Independent Practice<\/strong>
    \nFor 4-7, use the array drawn on a grid to find each product.
    \nUsing the Commutative Property of Addition, you can add the partial products in any order.<\/p>\n

    Question 4.
    \n14 \u00d7 21
    \n\"Envision
    \nAnswer:
    \nThe given expression is: 14 \u00d7 21
    \nNow,
    \nBy using the Partial Multiplication method,
    \n14 \u00d7 21
    \n= (10 + 4) \u00d7 (20 + 1)
    \n= (10 \u00d7 20) + (10 \u00d7 1) + (4 \u00d7 20) + (4 \u00d7 1)
    \n= 200 + 10 + 80 + 4
    \n= 294
    \nHence, from the above,
    \nWe can conclude that the product for the given expression is: 294<\/p>\n

    Question 5.
    \n14 \u00d7 12
    \n\"Envision
    \nAnswer:
    \nThe given expression is: 14 \u00d7 12
    \nNow,
    \nBy using the Partial Multiplication Method,
    \n14 \u00d7 12
    \n= (10 + 4) \u00d7 (10 + 2)
    \n= (10 \u00d7 10) + (10 \u00d7 2) + (4 \u00d7 10) + (4 \u00d7 2)
    \n= 100 + 20 + 40 + 8
    \n= 168
    \nHence, from the above,
    \nWe can conclude that the product for the given expression is: 168<\/p>\n

    Question 6.
    \n18 \u00d7 18
    \n\"Envision
    \nAnswer:
    \nThe given expression is: 18 \u00d7 18
    \nNow,
    \nBy using the Partial Multiplication method,
    \n18 \u00d7 18
    \n= (10 + 8) \u00d7 (10 + 8)
    \n= (10 \u00d7 10) + (10 \u00d7 8) + (8 \u00d7 10) + (8 \u00d7 8)
    \n= 100 + 80 + 80 + 64
    \n= 324
    \nHence, from the above,
    \nWe can conclude that the product for the given expression is: 324<\/p>\n

    Question 7.
    \n15 \u00d7 13
    \n\"Envision
    \nAnswer:
    \nThe given expression is: 15 \u00d7 13
    \nNow,
    \nBy using the Partial Multiplication ethod,
    \n15 \u00d7 13
    \n= (10 + 5) \u00d7 (10 + 3)
    \n= (10 \u00d7 10) + (5 \u00d7 10) + (10 \u00d7 3) + (5 \u00d7 3)
    \n= 100 + 50 + 30 + 15
    \n= 195
    \nHence, from the above,
    \nWe can conclude that the product for the given expression is: 195<\/p>\n

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

    Question 8.
    \nFind the missing factor. 20 \u00d7 ? = 420 Explain how you solved.
    \nAnswer:
    \nThe given expression is: 20 \u00d7? = 420
    \nNow,
    \nLet the missing factor be x
    \nSo,
    \n20 \u00d7 x = 420
    \nx = \\(\\frac{420}{20}\\)
    \nx = 21
    \nHence, from the above,
    \nWe can conclude that the missing factor for the given expression is: 21<\/p>\n

    Question 9.
    \nThe flagpole in front of City Hall in Lou’s town is 35 feet tall. How many inches tall is the flagpole? Remember, there are 12 inches in 1 foot.
    \nAnswer:
    \nIt is given that
    \nThe flagpole in front of City Hall in Lou’s town is 35 feet tall
    \nNow,
    \nWe know that,
    \n1 foot = 12 inches
    \nSo,
    \nThe height of the flagpole in inches = (The height of the flagpole in feet) \u00d7 (The number of inches for 1 foot)
    \n= 35 \u00d7 12
    \nNow,
    \nBy using the Partial Multiplication method,
    \n35 \u00d7 12
    \n= (30 + 5) \u00d7 (10 + 2)
    \n= (30 \u00d7 10) + (30 \u00d7 2) + (5 \u00d7 10) + (5 \u00d7 2)
    \n= 300 + 60 + 50 + 10
    \n= 420
    \nHence, from the above,
    \nWe can conclude that the height of the flagpole in inches is: 420 inches<\/p>\n

    For 10-11, use the array at the right.
    \n\"Envision<\/p>\n

    Question 10.
    \nModel with Math<\/strong> Maggie is making a balloon game for the school fair. Students throw darts to pop the balloons. Draw lines on the array to separate each factor into tens and ones. How many balloons are used to set up the game?
    \nAnswer:
    \nIt is given that
    \nMaggie is making a balloon game for the school fair. Students throw darts to pop the balloons
    \nNow,
    \nThe given array is:
    \n\"Envision
    \nSo,
    \nThe total number of balloons used to set up the game = (The total number of rows of balloons) \u00d7 (The total number of columns of balloons)
    \n= 13 \u00d7 14
    \nNow,
    \nBy using the Partial Multiplication method,
    \n13 \u00d7 14
    \n= (10 + 3) \u00d7 (10 + 4)
    \n= (10 \u00d7 10) + (10 \u00d7 4) + (3 \u00d7 10) + (3 \u00d7 4)
    \n= 100 + 40 + 30 + 12
    \n= 182
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that the total number of balloons that are used to set up the game is: 182 balloons<\/p>\n

    Question 11.
    \nHigher-Order Thinking<\/strong> Maggie knows she will have to completely refill the balloon board about 15 times. Write an equation to show the number of balloons Maggie will need.
    \nAnswer:
    \nFrom Problem 10,
    \nWe know that,
    \nThe total number of balloons that are used to set up the game is: 182
    \nNow,
    \nIt is given that
    \nMaggie knows she will have to completely refill the balloon board about 15 times.
    \nSo,
    \nThe total number of balloons maggie will need to refill = (The total number of balloons that are used to set up the game) \u00d7 (The number of times the balloon board will have to refill)
    \n= 182 \u00d7 15
    \nNow,
    \nBy using the Partial Multiplication method,
    \n182 \u00d7 15
    \n= (100 + 80 + 2) \u00d7 15
    \n= (100 \u00d7 15) + (80 \u00d7 15) + (2 \u00d7 15)
    \n= 1,500 + 1,200 + 30
    \n= 2,730
    \nHence, from the above,
    \nWe can conclude that
    \nThe total number of balloons that Maggie will need to refill the balloon board 15 times is: 2,730 balloons<\/p>\n

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

    Question 12.
    \nInsert the missing partial products. Then add to find the product.
    \n\"Envision
    \nAnswer:
    \nThe given expressions are:
    \na. 13 \u00d7 23\u00a0 \u00a0 \u00a0b 33 \u00d7 42
    \nNow,
    \nBy using the Partial Multiplication Method,
    \na
    \n13 \u00d7 23
    \n= (10 + 3) \u00d7 (20 + 3)
    \n= (10 \u00d7 20) + (10 \u00d7 3) + (3 \u00d7 20) + (3 \u00d7 3)
    \n= 200 + 30 + 60 + 9
    \n= 299
    \nb
    \n33 \u00d7 42
    \n= (30 + 3) \u00d7 (40 + 2)
    \n= (30 \u00d7 40) + (30 \u00d7 2) + (3 \u00d7 40) + (3 \u00d7 2)
    \n= 1,200 + 60 + 120 + 6
    \n= 1,386
    \nHence, from the above,
    \nWe can conclude that
    \nThe missing partial products and the addition of partial products for the given expressions are:
    \n\"\"<\/p>\n

    Question 13.
    \nInsert the missing factor in each equation.
    \n\"Envision
    \nAnswer:
    \nLet the missing factor be x
    \nNow,
    \nThe given expressions are with the representation of missing factor are:
    \na. 19 \u00d7 x = 1,007
    \nb. 62 \u00d7 x = 1,736
    \nc. x\u00a0\u00d7 37 = 1,813
    \nd. 15 \u00d7 x = 1,260
    \nSo,
    \nThe missing factors for the given expressions will be:
    \na.
    \nx = \\(\\frac{1,007}{19}\\)
    \nx = 53
    \nb.
    \nx = \\(\\frac{1,736}{62}\\)
    \nx = 28
    \nc.
    \nx = \\(\\frac{1,813}{37}\\)
    \nx = 49
    \nd.
    \nx = \\(\\frac{1,260}{15}\\)
    \nx = 84
    \nHence, from the above,
    \nWe can conclude that
    \nThe missing factor in each equation is:
    \n\"\"<\/p>\n

    Lesson 4.5 Area Models and Partial Products<\/h3>\n

    Solve & Share<\/strong>
    \nA playground is divided into four sections as shown in the diagram below. Find the area of the playground. Explain how you found the answer. Solve this problem using any strategy you choose.
    \nI can … use area models and properties of operations to multiply two 2-digit numbers.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nA playground is divided into four sections as shown in the diagram below.
    \n\"Envision
    \nNow,
    \nFrom the given shape of the playground,
    \nWe can observe that the shape is like an “Area-Model” used for multiplication
    \nSo,
    \nBy using the Area-Model for multiplication,
    \nThe area of the playground = The sum of all the areas of the parts of the playground
    \nNow,
    \nBy using the partial Multiplication method,
    \nThe area of the playground = (10 \u00d7 20) + (10 \u00d7 4) + (8 \u00d7 20) + (8 \u00d7 4)
    \n= 200 + 40 + 160 + 32
    \n= 432 feet\u00b2
    \nHence, from the above,
    \nWe can conclude that the area of the given playground is: 432 feet\u00b2<\/p>\n

    Look Back!<\/strong> Explain how you found the area of the blue part of the playground?
    \nAnswer:
    \nThe given figure is:
    \n\"Envision
    \nNow,
    \nFrom the above problem,
    \nWe can observe that
    \nWe had found out the area of the playground by using the Partial Multiplication method
    \nNow,
    \nBy using the Partial Multiplication Method,
    \nThe area of the blue part of the playground = (20 \u00d7 10) + (8 \u00d7 20)
    \n= 200 + 160
    \n= 360 feet\u00b2
    \nHence, from the above,
    \nWe can conclude that the area of the blue part of the playground is: 360 feet\u00b2<\/p>\n

    Essential Question<\/strong>
    \nHow Can You Use the Distributive Property to Multiply?
    \nAnswer:
    \nThe steps to use the Distributive Property of Multiplication are:
    \na. Simplify the numbers.
    \nb. Split the problem into two easier problems.
    \nc Take the number outside the parentheses and multiply it by each number inside the parentheses, one at a time.
    \nd. Add the products.<\/p>\n

    Visual Learning Bridge
    \nThere are 15 players on each baseball team of the Strike Out Club. How many players are on all of the teams in the Strike Out Club?
    \n\"Envision<\/p>\n

    One Way<\/strong>
    \nUse an area model and the Distributive Property to find 25 \u00d7 15.
    \nBreak apart 15 into 10+ 5.
    \n25 \u00d7 (10+5)
    \n(25 \u00d7 10) + (25 \u00d7 5)
    \n\"Envision<\/p>\n

    Another Way<\/strong>
    \nUse an area model and the Distributive Property to find 25 \u00d7 15.
    \nBreak apart 25 into 20 + 5.
    \n(20 + 5) 15)
    \n(20 \u00d7 15) + (5 \u00d7 15)
    \n\"Envision<\/p>\n

    Another Way<\/strong>
    \nUse an area model and the Distributive Property to find 25 \u00d7 15.
    \nBreak apart both factors.
    \nBreak apart 25 into 20 + 5.
    \nBreak apart 15 into 10+ 5.
    \n\"Envision<\/p>\n

    Convince Me!<\/strong>
    \nUse Structure<\/strong> How does an area model illustrate the Distributive Property?
    \nAnswer:
    \nThe distributive property says that multiplication distributes over addition.
    \nThe representation of the area model will be in two ways:
    \nA) a(b + c)
    \nB) ab + ac
    \nThe equality of the expressions,
    \n<\/strong>a(b + c) = ab + ac
    \nis the distributive property.<\/p>\n

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

    Do You Understand?<\/strong>
    \nQuestion 1.
    \nWhat four simpler multiplication problems are used to find 24 \u00d7 23?
    \n\"Envision
    \nAnswer:
    \nThe given expression is: 24 \u00d7 23
    \nNow,
    \nBy using the Partial multiplication method,
    \n24 \u00d7 23
    \n= (20 + 4) \u00d7 (20 + 3)
    \n= (20 \u00d7 20) + (20 \u00d7 3) + (4 \u00d7 20) + (4 \u00d7 3)
    \n= 400 + 60 + 80 + 12
    \n= 552
    \nHence, from the above,
    \nWe can conclude that
    \nThe four simpler multiplication problems used to find 24 \u00d7 23 are:
    \na. 20 \u00d7 20 = 400
    \nb. 20 \u00d7 3 = 60
    \nc. 4 \u00d7 20 = 80
    \nd. 4 \u00d7 3 = 12<\/p>\n

    Question 2.
    \nHow can an area model and the Distributive Property help you multiply? Use 12 \u00d7 16 to explain.
    \nAnswer:
    \nWe know that,
    \nThe distributive property says that multiplication distributes over addition.
    \nThe representation of the area model will be in two ways:
    \nA) a(b + c)
    \nB) ab + ac
    \nThe equality of the expressions,
    \n<\/strong>a(b + c) = ab + ac
    \nis the distributive property.
    \nNow,
    \nBy using the Distributive Property of Multiplication,
    \n12 \u00d7 16
    \n= 12 \u00d7 (10 + 6)
    \n= (12 \u00d7 10) + (12 \u00d7 6)
    \n= 120 + 72
    \n= 192
    \nHence, from the above,
    \nWe can conclude that the product for the expression 12 \u00d7 16 is: 192<\/p>\n

    Do You Know How?<\/strong>
    \nQuestion 3.
    \nUse the area model and the Distributive Property to find 35 \u00d7 12.
    \n\"Envision
    \n35 \u00d7 12 =
    \n30 \u00d7 10 =
    \n5 \u00d7 10 =
    \n30 \u00d7 2 =
    \n5 \u00d7 2 =
    \n______ + _____ + ______ + _______ = _________
    \nAnswer:
    \nThe given expression is: 35 \u00d7 12
    \nNow,
    \nBy using the area model and the Distributive Property of Multiplication,
    \n35 \u00d7 12
    \n= (30 + 5) \u00d7 (10 + 2)
    \n= (30 \u00d7 10) + (30 \u00d7 2) + (5 \u00d7 10) + (5 \u00d7 2)
    \n= 300 + 60 + 50 + 10
    \n= 420
    \nHence, from the above,
    \nWe can conclude that the value of the product for the given expression is: 420<\/p>\n

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

    For 4-10, draw an area model to find each product.
    \nQuestion 4.
    \n18 \u00d7 25
    \nAnswer:
    \nThe given expression is: 18 \u00d7 25
    \nNow,
    \nBy using the area model,
    \n18 \u00d7 25
    \n= (10 + 8) \u00d7 (20 + 5)
    \n= (10 \u00d7 20) + (10 \u00d7 5) + (8 \u00d7 20) + (8 \u00d7 5)
    \n= 200 + 50 + 160 + 40
    \n= 450
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that the product value for the given expression is: 450<\/p>\n

    Question 5.
    \n14 \u00d7 25
    \nAnswer:
    \nThe given expression is: 14 \u00d7 25
    \nNow,
    \nBy using the area model,
    \n14 \u00d7 25
    \n= (10 + 4) \u00d7 (20 + 5)
    \n= (10 \u00d7 20) + (10 \u00d7 5) + (4 \u00d7 20) + (4 \u00d7 5)
    \n= 200 + 50 + 80 + 20
    \n= 350
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that the product value for the given expression is: 450<\/p>\n

    Question 6.
    \n22 \u00d7 88
    \nAnswer:
    \nThe given expression is: 22 \u00d7 88
    \nNow,
    \nBy using the area model,
    \n22 \u00d7 88
    \n= (20 + 2) \u00d7 (80 + 8)
    \n= (20 \u00d7 80) + (20 \u00d7 8) + (2 \u00d7 80) + (2 \u00d7 8)
    \n= 1,600 + 160 + 160 + 16
    \n= 1,936
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that the product value for the given expression is: 1,936<\/p>\n

    Question 7.
    \n\"Envision
    \nAnswer:
    \nThe given expression is: 41 \u00d712
    \nNow,
    \nBy using the area model,
    \n41 \u00d7 12
    \n= (40 + 1) \u00d7 (10 + 2)
    \n= (40 \u00d7 10) + (40 \u00d7 2) + (1 \u00d7 10) + (1 \u00d7 2)
    \n= 400 + 80 + 10 + 2
    \n= 492
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that the product value for the given expression is: 492<\/p>\n

    Question 8.
    \n\"Envision
    \nAnswer:
    \nThe given expression is: 30 \u00d7 27
    \nNow,
    \nBy using the area model,
    \n30 \u00d7 27
    \n= 30 \u00d7 (20 + 7)
    \n= (30 \u00d7 20) + (30 \u00d7 7)
    \n= 600 + 210
    \n= 810
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that the product value for the given expression is: 810<\/p>\n

    Question 9.
    \n\"Envision
    \nAnswer:
    \nThe given expression is: 58 \u00d7 19
    \nNow,
    \nBy using the area model,
    \n58 \u00d7 19
    \n= (50 + 8) \u00d7 (10 + 9)
    \n= (50 \u00d7 10) + (50 \u00d7 9) + (8 \u00d7 10) + (8 \u00d7 9)
    \n= 500 + 450 + 80 + 72
    \n= 1,102
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that the product value for the given expression is: 1,102<\/p>\n

    Question 10.
    \n\"Envision
    \nAnswer:
    \nThe given expression is: 29 \u00d7 50
    \nNow,
    \nBy using the area model,
    \n29 \u00d7 50
    \n= (20 + 9) \u00d7 50
    \n= (20 \u00d7 50) + (9 \u00d7 50)
    \n= 1,000 + 450
    \n= 1,450
    \n\"\"
    \nHence, from the above,
    \nWe can conclude that the product value for the given expression is: 1,450<\/p>\n

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

    Question 11.
    \nWrite 652,079 using number names and in expanded form.
    \nAnswer:
    \nThe given number is: 652,079
    \nNow,
    \nWe know that,
    \n“Expanded form” or the “Expanded notation” is a way of writing numbers to see the math value of individual digits. When numbers are separated into individual place values and decimal places they can also form a mathematical expression
    \nHence,
    \nThe representation of the given number in the word form is:
    \nSix hundred fifty-two thousand, and Seventy-nine
    \nThe representation of the given number in the expanded form is:
    \n600,000 + 50,000 + 2,000 + 0 + 70 + 9<\/p>\n

    Question 12.
    \nNumber Sense Sara estimated 23 \u00d7 43 using 20 \u00d7 40. Sam estimated 23 \u00d7 43 using 25 \u00d7 40. Will Sara’s or Sam’s method give an estimate closer to the exact answer? Explain.
    \nAnswer:
    \nIt is given that
    \nSara estimated 23 \u00d7 43 using 20 \u00d7 40. Sam estimated 23 \u00d7 43 using 25 \u00d7 40
    \nNow,
    \nThe given expression is: 23 \u00d7 43
    \nNow,
    \nBy using the Rounding method,
    \nThe given expression will become: 20 \u00d7 40
    \nBy using the Compatible numbers method,
    \nThe given expression will become: 25 \u00d7 40 (or) 25 \u00d7 45
    \nNow,
    \nThe exact answer is:
    \n23 \u00d7 43
    \n= (20 + 3) \u00d7 (40 + 3)
    \n= (20 \u00d7 40) + (20 \u00d7 3) + (3 \u00d7 40) + (3 \u00d7 3)
    \n= 800 + 60 + 120 + 9
    \n= 989
    \nNow,
    \nThe estimation of Sara is:
    \n20 \u00d7 40
    \n= 2 \u00d7 10 \u00d7 4 \u00d7 10
    \n= (2 \u00d7 4) \u00d7 100
    \n= 8 \u00d7 100
    \n= 800
    \nNow,
    \nThe estimation of Sam is:
    \n25 \u00d7 40
    \n= (20 + 5) \u00d7 40
    \n= (20 \u00d7 40) + (5 \u00d7 40)
    \n= 800 + 200
    \n= 1,000
    \nHence, from the above,
    \nWe can conclude that
    \nSam’s method gives an estimate closer to the exact answer<\/p>\n

    Question 13.
    \nUse Structure<\/strong> Each family of Florida Scrub-Jays inhabits 25 acres of land. No other Scrub-Jay families live within this area. How many acres of land are needed for 24 families of Florida Scrub-Jays? Show how you can use the Distributive Property to solve this problem.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nEach family of Florida Scrub-Jays inhabits 25 acres of land. No other Scrub-Jay families live within this area.
    \nSo,
    \nThe acres of land that is needed for 24 families of Florida Scrub-jays = (The amount of land needed by each family of Florida Scrub-Jays) \u00d7 (The total number of families)
    \n= 25 \u00d7 24
    \nNow,
    \nBy using the Distributive Property of Multiplication,
    \n25 \u00d7 24
    \n= (20 + 5) \u00d7 (20 + 4)
    \n= (20 \u00d7 20) + (20 \u00d7 4) + (5 \u00d7 20) + (5 \u00d7 4)
    \n= 400 + 80 +100 + 20
    \n= 600 acres of land
    \nHence, from the above,
    \nWe can conclude that
    \nThe total amount of acres of land needed for 24 families of Florida Scrub-Jays is: 600 acres of land<\/p>\n

    Question 14.
    \nMarla wants to buy a new tablet that costs $565, including tax. She saved $15 per week for 30 weeks. Does Marla have enough money saved to buy the tablet? Explain.
    \nAnswer:
    \nIt is given that
    \nMarla wants to buy a new tablet that costs $565, including tax. She saved $15 per week for 30 weeks.
    \nNow,
    \nThe amount of money that Marla saved = (The amount of money she saved per week) \u00d7 (Total number of weeks)
    \n= $15 \u00d7 30
    \n= (10 + 5) \u00d7 30
    \n= (10 \u00d730) + (5 \u00d7 30)
    \n= 300 + 150
    \n= $450
    \nSo,
    \nThe amount of money saved by Marla < The cost of a new tablet
    \n$450 < $565
    \nHence, from the above,
    \nWe can conclude that Marla does not have enough money saved to buy the tablet<\/p>\n

    Question 15.
    \nHigher-Order Thinking<\/strong> Which costs less: 13 oranges that cost 29 cents each or 17 apples that cost 25 cents each? How much less?
    \nAnswer:
    \nIt is given that
    \n13 oranges cost 29 cents each and 17 apples cost 25 cents each
    \nNow,
    \nThe total cost of Oranges = (The number of Oranges) \u00d7 (The cost of each Orange)
    \n= 13 \u00d7 29
    \n= (10 + 3) \u00d7 (20 + 9)
    \n= (10 \u00d7 20) + (10 \u00d7 9) + (3 \u00d7 20) + (3 \u00d7 9)
    \n= 200 + 90 + 60 + 27
    \n= $377
    \nNow,
    \nThe total cost of apples = (The number of apples) \u00d7 (The cost of each apple)
    \n= 17 \u00d7 25
    \n= (10 + 7) \u00d7 (20 + 5)
    \n= (10 \u00d7 20) + (10 \u00d7 5) + (7 \u00d7 20) + (7 \u00d7 5)
    \n= 200 + 50 + 140 + 35
    \n= $425
    \nSo,
    \nThe difference between the cost of Oranges and the cost of apples = $425 – $377
    \n= $48
    \nHence, from the above,
    \nWe can conclude that
    \nThe cost of Oranges is less
    \nThe cost of Oranges is less by $48 than the cost of apples<\/p>\n

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

    Question 16.
    \nSelect all of the partial products which would be used to find 19 \u00d7 26.
    \n\u2610 45; 60; 180; 200
    \n\u2610 200; 180; 60; 54
    \n\u2610 6; 45; 180; 200
    \n\u2610 200; 60; 180; 54
    \n\u2610 0; 54; 180; 200
    \nAnswer:
    \nThe given expression is: 19 \u00d7 26
    \nNow,
    \nBy using the area model,
    \n19 \u00d7 26
    \n= (10 + 9) \u00d7 (20 + 6)
    \n= (10 \u00d7 20) + (10 \u00d7 6) + (9 \u00d7 20) + (9 \u00d7 6)
    \n= 200 + 60 + 180 + 54
    \nHence, from the above,
    \nWe can conclude that all the partial products that are used to find 19 \u00d7 26 are:
    \n\"\"<\/p>\n

    Question 17.
    \n<\/span>Select all of the ways you can use breaking apart and the Distributive Property to find the product of 35 \u00d7 12.
    \n\u2610 35 \u00d7 (10 + 2)
    \n\u2610 (12 \u00d7 30) + (12 \u00d7 5)
    \n\u2610 (12 \u00d7 5) + (35 \u00d7 5)
    \n\u2610 (30 \u00d7 5) + (10 \u00d7 2)
    \n\u2610 (30 \u00d7 10) + (5 \u00d7 10) + (30 \u00d7 2) + (5 \u00d7 2)
    \nAnswer:
    \nThe given expression is: 35 \u00d7 12
    \nNow,
    \nBy using the Distributive Property of Multiplication,
    \n35 \u00d7 12
    \n= 35 \u00d7 (10 + 2) —- (1)
    \n= (30 + 5) \u00d7 12
    \n= (30 \u00d7 12) + (5 \u00d7 12) —- (2)
    \n= (30 + 5) \u00d7 (10 + 2)
    \n= (30 \u00d7 10) + (30 \u00d7 2) + (5 \u00d7 10) + (5 \u00d7 2) —– (3)
    \nHence, from the above,
    \nWe can conclude that
    \nAll of the ways you can use breaking apart and the Distributive Property to find the product of 35 \u00d7 12 are:
    \n\"\"<\/p>\n

    Lesson 4.6 Use Partial Products to Multiply by 2-Digit Numbers<\/h3>\n

    Solve & Share<\/strong>
    \nThere are 11 regular players and 5 substitute players on a soccer team. How many players are on 15 soccer teams? Solve this problem using any strategy you choose.
    \nI can … use place value and partial products to multiply.
    \n\"Envision
    \nAnswer:
    \nIt is given that
    \nThere are 11 regular players and 5 substitute players on a soccer team.
    \nSo,
    \nThe total number of players in 15 teams = (The number of players) \u00d7 (The number of teams)
    \n= (Regular players + Substitute players) \u00d7 (The number of teams)
    \n= (11 + 5) \u00d7 15
    \n= (11 \u00d7 15) + (5 \u00d7 15)
    \n= 165 + 75
    \n= 240 players
    \nHence, from the above,
    \nWe can conclude that there are a total of 240 players in 15 teams<\/p>\n

    Look Back!<\/strong> How could you use an array and rounding or an array and compatible numbers to estimate the product for the problem above?
    \nAnswer:
    \nFrom the above problem,
    \nWe can observe that
    \nWe have to use the area model or the Distributive property to find the total number of players because there are different types of players i.e., Regular players, Substitute players, etc.<\/p>\n

    Essential Question<\/strong>
    \nHow Can You Record Multiplication?
    \nAnswer:
    \nThe ways to record Multiplication are:
    \na. Grid method.
    \nb. Long multiplication.
    \nc. Lattice multiplication.
    \nd. Binary or Peasant multiplication.
    \ne. Binary multiplication in computers.
    \nf. Shift and add.
    \ng Quarter square multiplication.
    \nh. Fast multiplication algorithms for large inputs<\/p>\n

    Visual Learning Bridge
    \nMarcia put 7 oranges and 8 apples into each of 12 bags. How many pieces of fruit did Marcia put into all of the bags?
    \n\"Envision<\/p>\n

    Find 12 \u00d7 15.
    \nEstimate: 12 \u00d7 15 is about 10 \u00d7 15 = 150.
    \nFirst, multiply the ones.
    \n\"Envision<\/p>\n

    Then, multiply the tens.
    \n\"Envision<\/p>\n

    Convince Me!<\/strong>
    \nReasoning<\/strong> Which partial products are incorrect? What is the correct final product?
    \n\"Envision
    \nAnswer:
    \nThe given expression is: 26 \u00d7 12
    \nNow,
    \nBy using the Partial multiplication method,
    \n26 \u00d7 12
    \n= (20 + 6) \u00d7 (10 + 2)
    \n= (20 \u00d7 10) + (20 \u00d7 2) + (6 \u00d7 10) + (6 \u00d7 2)
    \n= 200 + 40 + 60 + 12
    \n= 312
    \nHence, from the above,
    \nWe can conclude that
    \nThe given incorrect partial products are: 200, 60
    \nThe correct final product is: 312<\/p>\n

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

    Do You Understand?<\/strong>
    \nQuestion 1.
    \nIn the example on the previous page, why do you find 2 \u00d7 1 ten, rather than 2 \u00d7 1?
    \nAnswer:
    \nWe know that,
    \n1 ten = 10 —–> 10’s place-value
    \n1 = 1 ——> Unit’s place-value
    \nNow,
    \nFrom the above,
    \nWe can observe that there is a lot of difference 10’s position and unit’s position
    \nHence, from the above,
    \nWe can conclude that
    \nWe would write 2 \u00d7 1 ten rather than 2 \u00d7 1 as we observed in the example on the previous page<\/p>\n

    Question 2.
    \nIn the example on the previous page, could you record the 4 partial products in a different order? Explain.
    \nAnswer:
    \nThe given expression in the example on the previous page is: 15 \u00d7 12
    \nNow,
    \nBy using the partial Products method,
    \n15 \u00d7 13
    \n= (10 + 5) \u00d7 (10 + 2)
    \n= (10 \u00d7 10) + (10 \u00d7 2) + (5 \u00d7 10) + (5 \u00d7 2)
    \n= 100 + 20 + 50 + 10
    \n= 180
    \nHence, from the above,
    \nWe can conclude that
    \nWe can record the 4 partial products in a different order for the example that is present in the previous page<\/p>\n

    Do You Know How?<\/strong>
    \nFor 3-4, find all the partial products. Then add to find the final product. Draw area models as needed.
    \nQuestion 3.
    \n\"Envision
    \nAnswer:
    \nThe given expression is: 23 \u00d7 14
    \nNow,
    \nBy using the Partial Products method,
    \n23 \u00d7 14
    \n= (20 + 3) \u00d7 (10 + 4)
    \n= (20 \u00d7 10) + (20 \u00d7 4) + (3 \u00d7 10) + (3 \u00d7 4)
    \n= 200 + 80 + 30 + 12
    \n= 322
    \nHence, from the above,
    \nWe can conclude that the final product for the given expression is: 322<\/p>\n

    Question 4.
    \n\"Envision
    \nAnswer:
    \nThe given expression is: 41 \u00d7 25
    \nNow,
    \nBy using the Partial Products method,
    \n41 \u00d7 25
    \n= (40 + 1) \u00d7 (20 + 5)
    \n= (40 \u00d7 20) + (40 \u00d7 5) + (1 \u00d7 20) + (1 \u00d7 5)
    \n= 800 + 200 + 20 + 5
    \n= 1,025
    \nHence, from the above,
    \nWe can conclude that the final product for the given expression is: 1,025<\/p>\n

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

    For 5-12, estimate. Find all the partial products. Then add to find the final product. Draw area models as needed.
    \nRemember to check if your final product is reasonable.
    \nQuestion 5.
    \n\"Envision
    \nAnswer:
    \nThe given expression is: 34 \u00d7 51
    \nNow,
    \nBy using the Partial Products method,
    \n34 \u00d7 51
    \n= (30 + 4) \u00d7 (50 + 1)
    \n= (30 \u00d7 50) + (30 \u00d7 1) + (4 \u00d7 50) + (4 \u00d7 1)
    \n= 1,500 + 30 + 200 + 4
    \n= 1,734
    \nEstimate:
    \n35 \u00d7 50
    \n= (30 + 5) \u00d7 50
    \n= (30 \u00d7 50) + (5 \u00d7 50)
    \n= 1,500 + 250
    \n= 1,750
    \nHence, from the above,
    \nWe can conclude that
    \nThe final product for the given expression is: 1,750
    \nThe final product is reasonalble when we compare to its estimate<\/p>\n

    Question 6.
    \n\"Envision
    \nAnswer:
    \nThe given expression is: 73 \u00d7 81
    \nNow,
    \nBy using the Partial Products method,
    \n73 \u00d7 81
    \n= (70 + 3) \u00d7 (80 + 1)
    \n= (70 \u00d7 80) + (70 \u00d7 1) + (3 \u00d7 80) + (3 \u00d7 1)
    \n= 5,600 + 70 + 240 + 3
    \n= 5,913
    \nEstimate:
    \n75 \u00d7 80
    \n= (70 + 5) \u00d7 80
    \n= (70 \u00d7 80) + (5 \u00d7 80)
    \n= 5,600 + 400
    \n= 6,000
    \nHence, from the above,
    \nWe can conclude that
    \nThe final product for the given expression is: 5,913
    \nThe final product is reasonalble when we compare to its estimate<\/p>\n

    Question 7.
    \n\"Envision
    \nAnswer:
    \nThe given expression is: 64 \u00d7 32
    \nNow,
    \nBy using the Partial products method,
    \n64 \u00d7 32
    \n= (60 + 4) \u00d7 (30 + 2)
    \n= (60 \u00d7 30) + (60 \u00d7 2) + (4 \u00d7 30) + (4 \u00d7 2)
    \n= 1,800 + 120 + 120 + 8
    \n= 2,048
    \nEstimate:
    \n65 \u00d7 30
    \n= (60 + 5) \u00d7 30
    \n= (60 \u00d7 30) + (5 \u00d7 30)
    \n= 1,800 + 150
    \n= 1,950
    \nHence, from the above,
    \nWe can conclude that
    \nThe final product for the given expression is: 2,048
    \nThe final product is reasonalble when we compare to its estimate<\/p>\n

    Question 8.
    \n\"Envision
    \nAnswer:
    \nThe given expression is: 26 \u00d7 53
    \nNow,
    \nBy using the Partial Products method,
    \n26 \u00d7 53
    \n= (20 + 6) \u00d7 (50 + 3)
    \n= (20 \u00d7 50) + (20 \u00d7 3) + (6 \u00d7 50) + (6 \u00d7 3)
    \n= 1,000 + 60 + 300 + 18
    \n= 1,378
    \nEstimate:
    \n25 \u00d7 50
    \n= (20 + 5) \u00d7 50
    \n= (20 \u00d7 50) + (5 \u00d7 50)
    \n= 1,000 + 250
    \n= 1,250
    \nHence, from the above,
    \nWe can conclude that
    \nThe final product for the given expression is: 1,378
    \nThe final product is not reasonalble when we compare to its estimate<\/p>\n

    Question 9.
    \n\"Envision
    \nAnswer:
    \nThe given expression is: 38 \u00d7 17
    \nNow,
    \nBy using the partial products method,
    \n38 \u00d7 17
    \n= (30\u00a0 +8) \u00d7 (10 + 7)
    \n= (30 \u00d7 10) + (30 \u00d7 7) + (8 \u00d7 10) + (8 \u00d7 7)
    \n= 300 + 210 + 80 + 56
    \n= 646
    \nEstimate:
    \n40 \u00d7 15
    \n= 40 \u00d7 (10 + 5)
    \n= (40 \u00d7 10) + (40 \u00d7 5)
    \n= 400 + 200
    \n= 600
    \nHence, from the above,
    \nWe can conclude that
    \nThe final product for the given expression is: 646
    \nThe final product is reasonalble when we compare to its estimate<\/p>\n

    Question 10.
    \n\"Envision
    \nAnswer:
    \nThe given expression is: 34 \u00d7 30
    \nNow,
    \nBy using the Partial Products method,
    \n34 \u00d7 30
    \n= (30 + 4) \u00d7 30
    \n= (30 \u00d7 30) + (4 \u00d7 30)
    \n= 900 + 120
    \n= 1,020
    \nEstimate:
    \n35 \u00d7 30
    \n= (30 + 5) \u00d7 30
    \n= (30 \u00d7 30) + (5 \u00d7 30)
    \n= 900 + 150
    \n= 1,050
    \nHence, from the above,
    \nWe can conclude that
    \nThe final product for the given expression is: 1,020
    \nThe final product is reasonalble when we compare to its estimate<\/p>\n

    Question 11.
    \n\"Envision
    \nAnswer:
    \nThe given expression is: 19 \u00d7 43
    \nNow,
    \nBy using the Partial products method,
    \n19 \u00d7 43
    \n= (10 + 9) \u00d7 (40 + 3)
    \n= (10 \u00d7 40) + (10 \u00d7 3) + (9 \u00d7 40) + (9 \u00d7 3)
    \n= 400 + 30 + 360 + 27
    \n= 817
    \nEstimate:
    \n20 \u00d7 45
    \n= 20 \u00d7 (40 + 5)
    \n= (20 \u00d7 40) + (20 \u00d7 5)
    \n= 800 + 100
    \n= 900
    \nHence, from the above,
    \nWe can conclude that
    \nThe final product for the given expression is: 817
    \nThe final product is reasonalble when we compare to its estimate<\/p>\n

    Question 12.
    \n\"Envision
    \nAnswer:
    \nThe given expression is: 19 \u00d7 52
    \nNow,
    \nBy using the Partial Products method,
    \n19 \u00d7 52
    \n= (10 + 9) \u00d7 (50 + 2)
    \n= (10 \u00d7 50) + (10 \u00d7 2) + (9 \u00d7 50) + (9 \u00d7 2)
    \n= 500 + 20 + 450 + 18
    \n= 988
    \nEstimate:
    \n20 \u00d750
    \n= 2 \u00d7 10 \u00d7 5 \u00d7 10
    \n= (2 \u00d7 5) \u00d7 100
    \n= 10 \u00d7 100
    \n= 1,000
    \nHence, from the above,
    \nWe can conclude that
    \nThe final product for the given expression is: 988
    \nThe final product is reasonalble when we compare to its estimate<\/p>\n

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

    Question 13.
    \nThe Castillo de San Marcos is a Spanish fortress that was built between 1672 and 1695.
    \n\"Envision
    \na. Rounded to the nearest ten thousand, how many pesos did it cost to build the fortress?
    \nAnswer:
    \nIt is given that
    \nThe Castillo de San Marcos is a Spanish fortress that was built between 1672 and 1695.
    \nNow,
    \nThe given figure is:
    \n\"Envision
    \nNow,
    \nFrom the given figure,
    \nWe can observe that
    \nIt is given that it costs 138,375 pesos to build the given fortress
    \nNow,
    \nFrom the given number,
    \nWe can observe that
    \nFrom the left of the digits’ positions, the 5th position value is “Ten-Thousand”
    \nSo,
    \nThe given number when it is rounded to the nearest ten thousand is: 140,000
    \nHence, from the above,
    \nWe can conclude that
    \nThe number we can obtain when we rounded the given number to the nearest ten thousand is: 140,000
    \nIt costs about 140,000 pesos to build the fortress<\/p>\n

    b. How many years did it take to build the fortress?
    \nAnswer:
    \nIt is given that
    \nThe Castillo de San Marcos is a Spanish fortress that was built between 1672 and 1695.
    \nSo,
    \nThe number of years did it take to build the fortress = 1695 – 1672
    \n= 23 years
    \nHence, from the above,
    \nWe can conclude that it took 23 years to build a fortress<\/p>\n

    Question 14.
    \nConstruct Arguments<\/strong> A school has 2 large patios. One is rectangular and is 24 feet long by 18 feet wide. The other is square and each side is 21 feet long. Which patio has a greater area? Explain.
    \nAnswer:
    \nIt is given that
    \nA school has 2 large patios. One is rectangular and is 24 feet long by 18 feet wide. The other is square and each side is 21 feet long
    \nNow,
    \nWe know that,
    \nThe area of a rectangle = Length \u00d7 Width
    \nThe area of a square = Side \u00d7 Side
    \nNow,
    \nThe area of a rectangular patio = 24 \u00d7 18
    \n= (20 + 4) \u00d7 (10 + 8)
    \n= (20 \u00d7 10) + (20 \u00d7 8) + (4 \u00d7 10) + (4 \u00d7 8)
    \n= 200 + 160 + 40 + 32
    \n= 432 feet\u00b2
    \nNow,
    \nThe area of a square patio = 21 \u00d7 21
    \n= (20 + 1) \u00d7 (20 + 1)
    \n= (20 \u00d720) + (20 \u00d7 1) + (1 \u00d7 20) + (1 \u00d7 1)
    \n= 400 + 20 + 20 + 1
    \n= 441 feet\u00b2
    \nSo,
    \nThe area of a square patio > The area of a rectangular patio
    \nHence, from the above,
    \nWe can conclude that square patio has a greater area<\/p>\n

    Question 15.
    \nHigher Order Thinking<\/strong> Show two ways! to break apart the factors, and then find the product of 14 \u00d7 22.
    \nAnswer:
    \nThe given expression is: 14 \u00d7 22
    \nNow,
    \nBy using the Distributive Property of Multiplication,
    \nWay 1:
    \n14 \u00d7 22
    \n= (10 + 4) \u00d7 (20 + 2)
    \n= (10 \u00d7 20) \u00d7 (10 \u00d7 2) + (4 \u00d7 20) + (4 \u00d7 2)
    \n= 200 + 20 + 80 + 8
    \n= 308
    \nWay 2:
    \n14 \u00d7 22
    \n= 14 \u00d7 (20 + 2)
    \n= (14 \u00d7 20) + (14 \u00d7 2)
    \n= 280 + 28
    \n= 308
    \nHence, from the above,
    \nWe can conclude that the product value for the given expression is: 308<\/p>\n

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

    Question 16.
    \nWhich set of numbers has the missing partial product and the final product?
    \n\"Envision
    \nA. 210; 836
    \nB. 18; 806
    \nC. 180; 806
    \nD. 18; 644
    \nAnswer:
    \nThe given expression is: 13 \u00d7 62
    \nNow,
    \nBy using the Partial Products method,
    \n13 \u00d7 62
    \n= (10 + 3) \u00d7 (60 + 2)
    \n= (10 \u00d7 60) + (10 \u00d7 2) + (3 \u00d7 60) + (3 \u00d7 2)
    \n= 600 + 20 + 180 + 6
    \n= 806
    \nHence, from the above,
    \nWe can conclude that
    \nThe missing partial product and the final product are:
    \n\"\"<\/p>\n

    Question 17.
    \nSelect all the equations in which 12 is the missing factor
    \n\u2610 b \u00d7 14 = 336
    \n\u2610 b \u00d7 36 = 432
    \n\u2610 18 \u00d7 b = 216
    \n\u2610 39 \u00d7 b = 468
    \n\u2610 21 \u00d7 b = 231
    \nAnswer:
    \nThe given equations are:
    \na. b \u00d7 14 = 336\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0b. b \u00d736 = 432\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0c. 18 \u00d7 b = 216
    \nd. 39 \u00d7 b = 468\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0e. 21 \u00d7 b = 231
    \nNow,
    \na.
    \nb \u00d7 14 = 336
    \nb = \\(\\frac{336}{14}\\)
    \nb = 24
    \nb.
    \nb \u00d7 36 = 432
    \nb = \\(\\frac{432}{36}\\)
    \nb = 12
    \nc.
    \n18 \u00d7 b = 216
    \nb = \\(\\frac{216}{18}\\)
    \nb = 12
    \nd.
    \n39 \u00d7 b = 468
    \nb = \\(\\frac{468}{39}\\)
    \nb = 12
    \ne.
    \n21 \u00d7 b = 231
    \nb = \\(\\frac{231}{21}\\)
    \nb = 11
    \nHence, from the above,
    \nWe can conclude that the expressions that have the missing factor 12 are:
    \n\"\"<\/p>\n

    Lesson 4.7 Problem Solving<\/h3>\n

    Make Sense and Persevere<\/strong>
    \nSolve & Share<\/strong>
    \nFive students set a goal to raise $500 from their charity walk. Sponsors donated $25 for each mile walked. By how much did these students exceed or miss their goals? Solve this problem using any strategy you choose.
    \nI can … make sense of problems and keep working if I get stuck.
    \n\"Envision<\/p>\n

    Thinking Habits
    \nBe a good thinker! These questions can help you.
    \n1. What do I need to find?
    \n2. What do I know?
    \n3. What’s my plan for solving the problem?
    \n4. What else can I try if I get stuck?
    \n5. How can I check that my solution makes sense?
    \nAnswer:
    \nIt is given that
    \nFive students set a goal to raise $500 from their charity walk. Sponsors donated $25 for each mile walked
    \nNow,
    \nThe given table is:
    \n\"Envision
    \nNow,
    \nFrom the given table,
    \nThe total number of miles walked by all the students = 5 + 5 + 3 + 4 + 4
    \n= 21 miles
    \nSo,
    \nThe amount of money raised by the given students = (The total number of miles walked by all the students) \u00d7 (The amount of money donated by the sponsor for each mile)
    \n= 21 \u00d7 25
    \n= (20 + 1) \u00d7 (20 + 5)
    \n= (20 \u00d7 20) + (20 \u00d7 5) + (1 \u00d7 20) + (1 \u00d7 5)
    \n= 400 + 100 + 20 + 5
    \n= $525
    \nSo,
    \n$500 < $525
    \nNow,
    \nThe amount of money exceeded = (The amount of money raised by all the students) – (The amount of money that have to be raised by the students)
    \n= $525 – $500
    \n= $25
    \nHence, from the above,
    \nWe can conclude that the amount of money exceeded is: $25<\/p>\n

    Look Back!<\/strong>
    \nMake Sense and Persevere<\/strong> Is there more than one way to solve the problem? Explain.
    \nAnswer:
    \nNo, there is no other way to solve the above problem because we must find the total number of miles traveled by all the students and the amount of money raised by them by walking the number of miles<\/p>\n

    Essential Question<\/strong>
    \nHow Can You Make Sense of and Persevere in Solving Problems with More Than One Step?
    \nAnswer:
    \nMathematically, proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals<\/p>\n

    Visual Learning Bridge
    \nThe park has a large garden with a walkway around it. The park’s ground crew are going to paint the walkway. What is the area of the walkway?
    \n\"Envision
    \nWhat is a good plan for solving this problem?
    \nI need to find the area of the garden and walkway together. Then I need to subtract to find only the area of the walkway.<\/p>\n

    How can I make sense of and solve this problem?
    \nI can<\/p>\n

      \n
    • identify the quantities given.<\/li>\n
    • understand how the quantities are related.<\/li>\n
    • choose and implement an appropriate strategy.<\/li>\n
    • check to be sure my work and answer make sense.<\/li>\n<\/ul>\n

      First find the area of the whole park.
      \n\"Envision
      \n52 \u00d7 85 = p
      \n4,420 = p
      \nNext, find the area of the of the garden.
      \n32 \u00d7 65 = g
      \n2,080 = g
      \nSubtract to find the area of the walkway. 4,420 \u2013 2,080 = w
      \nThe area of the walkway is 2,340 square feet.<\/p>\n

      Convince Me!<\/strong>
      \nMake Sense and Persevere<\/strong> Can you solve the problem a different way and still calculate the same answer? Explain.
      \nAnswer:
      \nThere is no other way to solve the above problem because we must calculate the area of the park and the area of the garden to find the area of the walkway since the walkway is enclosed between the park and the garden<\/p>\n

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

      Make Sense and Persevere<\/strong> On her vacation, Julia filled 3 memory cards like the one shown. She printed 2 copies of each picture. How many pictures did Julia print?
      \n\"Envision
      \nQuestion 1.
      \nWhat do you know and what do you need to find?
      \nAnswer:
      \nIt is given that
      \nOn her vacation, Julia filled 3 memory cards like the one shown. She printed 2 copies of each picture.
      \nNow,
      \nWe know that,
      \na. The number of filled memory cards
      \nb. The number of copies of each picture
      \nc. The number of pictures in each memory card
      \nWe need to find:
      \na. The number of pictures in all the filled memory cards
      \nb. The total number of copies of pictures<\/p>\n

      Question 2.
      \nWhat steps might you take to solve the problem?
      \nAnswer:
      \nThe steps you have to take to solve the given problem are:
      \nStep 1:
      \nFind the number of pictures in all the filled cards by multiplying the number of filled memory cards and the number of pictures in each memory card
      \nStep 2:
      \nAfter finding the number of pictures in all the filled memory cards, find the number of copies of pictures by multiplying the number of pictures in all the filled memory cards with 2<\/p>\n

      Question 3.
      \nHow many pictures did Julia print? Explain.
      \nAnswer:
      \nIt is given that
      \nOn her vacation, Julia filled 3 memory cards like the one shown. She printed 2 copies of each picture.
      \nNow,
      \nThe given figure is:
      \n\"Envision
      \nNow,
      \nFrom the given figure,
      \nWe can observe that
      \nA memory card holds 28 pictures
      \nSo,
      \nThe number of pictures present in 3 memory cards = (The number of filled memory cards) \u00d7 (The number of pictures in each memory card)
      \n= 28 \u00d7 3
      \n= (20 + 8) \u00d7 3
      \n= (20 \u00d7 3) + (8 \u00d7 3)
      \n= 60 + 24
      \n= 84 pictures
      \nNow,
      \nThe\u00a0 total number of copies of pictures = (The number of pictures present in 3 memory cards) \u00d7 2
      \n= 84 \u00d7 2
      \n= (80 + 4) \u00d7 2
      \n= (80 \u00d7 2) + (4 \u00d7 2)
      \n= 160 + 8
      \n= 168 pictures
      \nHence, from the above,
      \nWe can conclude that
      \nThe total number of pictures did Julia print are: 168<\/p>\n

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

      Make Sense and Persevere<\/strong> Jarrod delivers 63 newspapers each Monday through Saturday and 78 newspapers each Sunday. Last month consisted of 4 Sundays and 26 other days. How many newspapers did Jarrod deliver last month? Use Exercises 4-6 to solve this problem.
      \nQuestion 4.
      \nWhat strategies can you use to find how many newspapers Jarrod delivered last month?
      \nAnswer:
      \nIt is given that
      \nJarrod delivers 63 newspapers each Monday through Saturday and 78 newspapers each Sunday. Last month consisted of 4 Sundays and 26 other days
      \nNow,
      \nThe strategy can you use to find the number of newspapers Jarrod delivered last month is:
      \nStep 1:
      \nFind the total number of newspapers Jarrod delivered on other days
      \nStep 2:
      \nFind the total number of newspapers Jarrod delivered only on Sunday
      \nStep 3:
      \nAdd the results we obtained in Step 1 and Step 2<\/p>\n

      Question 5.
      \nHow are the quantities related?
      \nAnswer:
      \nThe given quantities are related in a way such as:
      \nThe total number of newspapers Jarrod delivered last month = (The total amount of newspapers Jarrod delivered on other days) + (The total number of newspapers Jarrod delivered on Sunday)<\/p>\n

      Question 6.
      \nExplain how to solve the problem.
      \nAnswer:
      \nThe steps to solve the given problem are:
      \nStep 1:
      \nFind the total number of newspapers Jarrod delivered on other days
      \nStep 2:
      \nFind the total number of newspapers Jarrod delivered only on Sunday
      \nStep 3:
      \nAdd the results we obtained in Step 1 and Step 2<\/p>\n

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

      Performance Task<\/strong>
      \nUniforms<\/strong>
      \nThe Stillwater Storm soccer team consists of 16 players. Each player needs a uniform set. The uniform set includes two jerseys, a pair of shorts, and a pair of socks. The price for each separate item is shown. Nine of the players need a medium size and the others need a small size. If the team buys more than 10 sets, the team price for each set is $56. How much money is saved if the team buys all the uniform sets together rather than separately?
      \n\"Envision
      \nQuestion 7.
      \nReasoning<\/strong> What are the quantities in the problem and how are they related?
      \nAnswer:
      \nIt is given that
      \nThe Stillwater Storm soccer team consists of 16 players. Each player needs a uniform set. The uniform set includes two jerseys, a pair of shorts, and a pair of socks. The price for each separate item is shown. Nine of the players need a medium size and the others need a small size. If the team buys more than 10 sets, the team price for each set is $56
      \nSo,
      \nThe quantities related to the given problem are;
      \na. The price for 2 pairs of jerseys
      \nb. The price for a pair of shorts
      \nc. The price for a pair of socks
      \nHence,
      \nThe relation between the quantities are:
      \nThe total cost of a set = (The cost of 2 pairs of jerseys) + (The cost of a pair of shorts) + (The cost of a pair of socks)<\/span><\/p>\n

      Question 8.
      \n<\/span>Model with Math<\/strong>
      \nUse the bar diagram to write an addition equation and find the total cost of the uniform sets when purchased separately. Then find the total cost if 10 sets or more are purchased.
      \nWhen you make sense and persevere, you think about the amounts given.
      \n\"Envision
      \nAnswer:<\/p>\n

      Question 9.
      \nBe Precise What is the difference in cost if the team bought all the uniform sets together or if they bought them separately?
      \nAnswer:<\/p>\n

      Topic 4 Fluency Practice Activity<\/h3>\n

      Find a Match<\/strong>
      \nWork with a partner. Point to a clue. Read the clue. Look below the clues to find a match. Write the clue letter in the box next to the match. Find a match for every clue.
      \nI can … add and subtract multi-digit whole numbers.<\/p>\n

      Clues
      \nA. The difference is between 950 and 1,000.
      \nB. The difference is exactly 913.
      \nC. The sum is between 600 and 700.
      \nD. The sum is exactly 500.
      \nE. The difference is between 700 and 800.
      \nF. The sum is greater than 300 but less than 400.
      \nG. The sum is exactly 753.
      \nH. The difference is exactly 413.
      \n\"\"<\/p>\n

      Topic 4 Vocabulary Review<\/h3>\n

      Understand Vocabulary
      \nWord List<\/p>\n

        \n
      • array<\/li>\n
      • compatible numbers<\/li>\n
      • estimate<\/li>\n
      • factors<\/li>\n
      • partial product<\/li>\n
      • product<\/li>\n
      • rounding<\/li>\n
      • variable<\/li>\n<\/ul>\n

        Question 1.
        \nCross out the numbers that are NOT factors of 12.
        \n1
        \n3
        \n5
        \n6
        \n8
        \nAnswer:
        \nWe know that,
        \nA “factor” is defined as a number or algebraic expression that divides another number or expression evenly i.e., with no remainder
        \nHence, from the above,
        \nWe can conclude that
        \nThe numbers that are not factors of 12 are:
        \n\"\"<\/p>\n

        Question 2.
        \nCross out the numbers that are NOT good estimates for the product of 17 \u00d7 23.
        \n600
        \n400
        \n300
        \n200
        \n100
        \nAnswer:
        \nThe given expression is: 17 \u00d7 23
        \nNow,
        \nThe estimate of the given expression is: 20 \u00d7 25 (or) 15 \u00d7 25 (or) 20 \u00d7 20
        \nHence, from the above,
        \nWe can conclude that
        \nThe numbers that are not good estimates for the product 17 \u00d7 23 are:
        \n\"\"<\/p>\n

        Question 3.
        \nCross out the numbers that are NOT partial products for 12 \u00d7 41.
        \n2
        \n10
        \n18
        \n80
        \n400
        \nAnswer:
        \nThe given expression is: 12 \u00d7 41
        \nNow,
        \nBy using the Partial products method,
        \n12 \u00d7 41
        \n= (10 + 2) \u00d7 (40 +1)
        \n= (10 \u00d7 40) + (10 \u00d7 1) + (2 \u00d7 40) + (2 \u00d7 1)
        \n= 400 + 10 + 80 + 2
        \n= 492
        \nHence, from the above,
        \nWe can conclude that
        \nThe numbers that are not the partial products for the given expression are:
        \n\"\"<\/p>\n

        Label each example with a term from the Word List.
        \n\"\"
        \nQuestion 4.
        \n\"Envision _________
        \nAnswer:
        \nThe given figure is:
        \n\"Envision
        \nNow,
        \nWe know that,
        \nAn “Array” is a collection of rows and columns
        \nHence, from the above,
        \nWe can conclude that the term that is sutable for the given figure is: Array<\/p>\n

        Question 5.
        \nn. __________
        \nAnswer:
        \nWe know that,
        \nA “Variable”, in algebra, is a symbol (usually a letter) standing in for an unknown numerical value in an equation
        \nHence, from the above,
        \nWe can conclude that the term that is best suitable for the given problem is: Variable<\/p>\n

        Question 6.
        \n2,318 to the nearest thousand is 2,000 _________
        \nAnswer:
        \nWe know that,
        \nThe “Rounding-off” \u00a0is\u00a0a mathematical technique of adjusting the number’s digits to make the number easier to use during calculations
        \nHence, from the above,
        \nWe can conclude that the term that is suitable for the given sentence is: Rounding<\/p>\n

        Question 7.
        \n3 \u00d7 4 = 12 _________
        \nAnswer:
        \nWe know that,
        \nA “Product” is a final value when we multiply the number of factors
        \nHence, from the above,
        \nWe can conclude that the term that is suitable for the given problem is: product<\/p>\n

        Use Vocabulary in Writing<\/strong>
        \nQuestion 8.
        \nAlicia needs to find 23 \u00d7 47. Use at least 3 terms from the Word List to explain how Alicia might find 23 \u00d7 47.
        \nAnswer:
        \nIt is given that
        \nAlicia needs to find 23 \u00d7 47
        \nHence, from the above,
        \nWe can conclude that
        \nThe three words from the given word list to explain how Alicia might find the product for the given expression is:
        \na. Compatible numbers
        \nb. Estimate
        \nc. partial Product<\/p>\n

        Topic 4 Reteaching<\/h3>\n

        Set A pages 129-132<\/strong><\/p>\n

        Use place-value strategies to find 20 \u00d7 80.
        \nThink about the pattern.
        \n20 \u00d7 80 = 2 tens \u00d7 8 tens
        \n= 16 hundreds
        \n= 1,600<\/p>\n

        50 \u00d7 40 = (5 \u00d7 10) \u00d7 (4 \u00d7 10)
        \n= (5 \u00d7 4) \u00d7 (10 \u00d7 10)
        \n= 20 \u00d7 100
        \n= 2,000<\/p>\n

        Remember<\/strong> when the product of a basic fact has a zero, there is one more zero in the answer.<\/p>\n

        Use a pattern to find each product.
        \nQuestion 1.
        \n40 \u00d7 10
        \nAnswer:
        \nThe given expression is: 40 \u00d7 10
        \nNow,
        \nBy using the Place-value method,
        \n40 \u00d7 10
        \n= 4 \u00d7 10 \u00d7 1 \u00d7 10
        \n= (4 \u00d7 1) \u00d7 100
        \n= 400
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is: 400<\/p>\n

        Question 2.
        \n60 \u00d7 20
        \nAnswer:
        \nThe given expression is: 60 \u00d7 20
        \nNow,
        \nBy using the Place-value method,
        \n60 \u00d7 20
        \n= 6 \u00d7 10 \u00d7 2 \u00d7 10
        \n= (6 \u00d7 2) \u00d7 100
        \n= 1,200
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is: 1,200<\/p>\n

        Question 3.
        \n80 \u00d7 50
        \nAnswer:
        \nThe given expression is: 80 \u00d7 50
        \nNow,
        \nBy using the Place-value method,
        \n80 \u00d7 50
        \n= 8 \u00d7 10 \u00d7 5 \u00d7 10
        \n= (8 \u00d7 5) \u00d7 100
        \n= 4,000
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is: 4,000<\/p>\n

        Question 4.
        \n30 \u00d7 90
        \nAnswer:
        \nThe given expression is: 30 \u00d7 90
        \nNow,
        \nBy using the Place-value method,
        \n30 \u00d7 90
        \n= 3 \u00d7 10 \u00d7 9 \u00d7 10
        \n= (3 \u00d7 9) \u00d7 100
        \n= 2,700
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is: 2,700<\/p>\n

        Question 5.
        \n80 \u00d7 70
        \nAnswer:
        \nThe given expression is: 80 \u00d7 70
        \nNow,
        \nBy using the Place-value method,
        \n80 \u00d7 70
        \n= 8 \u00d7 10 \u00d7 7 \u00d7 10
        \n= (8 \u00d7 7) \u00d7 100
        \n= 5,600
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is: 5,600<\/p>\n

        Question 6.
        \n60 \u00d7 60
        \nAnswer:
        \nThe given expression is: 60 \u00d7 60
        \nNow,
        \nBy using the Place-value method,
        \n60 \u00d7 60
        \n= 6 \u00d7 10 \u00d7 6 \u00d7 10
        \n= (6 \u00d7 6) \u00d7 100
        \n= 3,600
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is: 3,600<\/p>\n

        Question 7.
        \n80 \u00d7 30
        \nAnswer:
        \nThe given expression is: 80 \u00d7 30
        \nNow,
        \nBy using the Place-value method,
        \n80 \u00d7 30
        \n= 8 \u00d7 10 \u00d7 3 \u00d7 10
        \n= (8 \u00d7 3) \u00d7 100
        \n= 2,400
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is: 2,400<\/p>\n

        Question 8.
        \n20 \u00d7 50
        \nAnswer:
        \nThe given expression is: 20 \u00d7 50
        \nNow,
        \nBy using the Place-value method,
        \n20 \u00d7 50
        \n= 2 \u00d7 10 \u00d7 5 \u00d7 10
        \n= (2 \u00d7 5) \u00d7 100
        \n= 1,000
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is: 1,000<\/p>\n

        Set B pages 133-136<\/strong><\/p>\n

        Use an array or an area model to multiply 20 \u00d7 14.
        \n\"Envision
        \n200 + 80 = 280
        \nSo, 20 \u00d7 14 = 280.<\/p>\n

        Remember<\/strong> you can draw arrays or area models to represent multiplication problems.<\/p>\n

        Draw a model to find each product.
        \nQuestion 1.
        \n10 \u00d7 23
        \nAnswer:
        \nThe given expression is: 10 \u00d7 23
        \nNow,
        \nBy using the Area-model,
        \n10 \u00d7 23
        \n= 10 \u00d7 (20 + 3)
        \n= (10 \u00d7 20) + (10 \u00d7 3)
        \n= 200 + 30
        \n= 230
        \n\"\"
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is: 230<\/p>\n

        Question 2.
        \n16 \u00d7 20
        \nAnswer:
        \nThe given expression is: 16 \u00d7 20
        \nNow,
        \nBy using the Area-model,
        \n16 \u00d7 20
        \n= (10 + 6) \u00d7 20
        \n= (10 \u00d7 20) + (6 \u00d7 20)
        \n= 200 + 120
        \n= 320
        \n\"\"
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is: 320<\/p>\n

        Set C pages 137-140<\/strong><\/p>\n

        Use compatible numbers to estimate 28 \u00d7 19.
        \n28 is close to 25.
        \n19 is close to 20.
        \nIf 25 \u00d7 2 = 50, then
        \n25 \u00d7 20 = 500.
        \nSo, 28 \u00d7 19 is about 500.<\/p>\n

        Remember<\/strong> that compatible numbers are numbers that are easy to compute mentally.<\/p>\n

        Estimate each product.
        \nQuestion 1.
        \n29 \u00d7 31
        \nAnswer:
        \nThe given expression is: 29 \u00d7 31
        \nNow,
        \nBy using the estimation of the Compatible number,
        \nThe given expression will become: 30 \u00d7 30
        \nSo,
        \n30 \u00d7 30
        \n= 3 \u00d7 10 \u00d7 3 \u00d7 10
        \n= (3 \u00d7 3) \u00d7 100
        \n= 9 \u00d7 100
        \n= 900
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is about 900<\/p>\n

        Question 2.
        \n42 \u00d7 49
        \nAnswer:
        \nThe given expression is: 42 \u00d7 49
        \nNow,
        \nBy using the estimation of the Compatible number,
        \nThe given expression will become: 40 \u00d7 50
        \nSo,
        \n40 \u00d7 50
        \n= 4 \u00d7 10 \u00d7 5 \u00d7 10
        \n= (4 \u00d7 5) \u00d7 100
        \n= 20 \u00d7 100
        \n= 2,000
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is about 2,000<\/p>\n

        Question 3.
        \n73 \u00d7 18
        \nAnswer:
        \nThe given expression is: 73 \u00d7 18
        \nNow,
        \nBy using the estimation of the Compatible number,
        \nThe given expression will become: 70 \u00d7 20
        \nSo,
        \n70 \u00d7 20
        \n= 7 \u00d7 10 \u00d7 2 \u00d7 10
        \n= (7 \u00d7 2) \u00d7 100
        \n= 14 \u00d7 100
        \n= 1,400
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is about 1,400<\/p>\n

        Question 4.
        \n24 \u00d7 38
        \nAnswer:
        \nThe given expression is: 24 \u00d7 38
        \nNow,
        \nBy using the estimation of the Compatible number,
        \nThe given expression will become: 25 \u00d7 40
        \nSo,
        \n25 \u00d7 40
        \n= (20\u00a0 +5) \u00d7 40
        \n= (20 \u00d7 40) + (5 \u00d7 40)
        \n= 800 + 200
        \n= 1,000
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is about 1,000<\/p>\n

        Question 5.
        \n19 \u00d7 31
        \nAnswer:
        \nThe given expression is: 19 \u00d7 31
        \nNow,
        \nBy using the estimation of the Compatible number,
        \nThe given expression will become: 20 \u00d7 30
        \nSo,
        \n20 \u00d7 30
        \n= 2 \u00d7 10 \u00d7 3 \u00d7 10
        \n= (2 \u00d7 3) \u00d7 100
        \n= 6 \u00d7 100
        \n= 600
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is about 600<\/p>\n

        Question 6.
        \n63 \u00d7 87
        \nAnswer:
        \nThe given expression is: 63 \u00d7 87
        \nNow,
        \nBy using the estimation of the Compatible number,
        \nThe given expression will become: 60 \u00d7 90
        \nSo,
        \n60 \u00d7 90
        \n= 6 \u00d710 \u00d79 \u00d7 10
        \n= (6 \u00d7 9) \u00d7 100
        \n= 54 \u00d7 100
        \n= 5,400
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is about 5,400<\/p>\n

        Set D pages 141-144<\/strong><\/p>\n

        Find 14 \u00d7 12. Draw a 14 \u00d7 12 array.
        \nSeparate each factor into tens and ones. Color each section a different color. Add each part to find the product.
        \n\"Envision<\/p>\n

        Remember<\/strong> when you break apart a multiplication problem, you can solve the simpler problems in any order and the answer will remain the same.<\/p>\n

        Question 1.
        \n14 \u00d7 32
        \nAnswer:
        \nThe given expression is: 14 \u00d7 32
        \nNow,
        \nBy using the Partial Products method,
        \n14 \u00d7 32
        \n= (10 + 4) \u00d7 (30 + 2)
        \n= (10 \u00d7 30) + (10 \u00d7 2) + (4 \u00d7 30) + (4 \u00d7 2)
        \n= 300 + 20 + 120 + 8
        \n= 448
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is: 448<\/p>\n

        Question 2.
        \n64 \u00d7 12
        \nAnswer:
        \nThe given expression is: 64 \u00d7 12
        \nNow,
        \nBy using the Partial Products method,
        \n64 \u00d7 12
        \n= (60 + 4) \u00d7 (10 + 2)
        \n= (60 \u00d7 10) + (60 \u00d7 2) + (4 \u00d7 10) + (4 \u00d7 2)
        \n= 600 + 120 + 40 + 8
        \n= 768
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is: 768<\/p>\n

        Question 3.
        \n56 \u00d7 17
        \nAnswer:
        \nThe given expression is: 56 \u00d7 17
        \nNow,
        \nBy using the Partial Products method,
        \n56 \u00d7 17
        \n= (50 + 6) \u00d7 (10 + 7)
        \n= (50 \u00d7 10) + (50 \u00d7 7) + (6 \u00d7 10) + (6 \u00d7 7)
        \n= 500 + 350 + 60 + 42
        \n= 952
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is: 952<\/p>\n

        Question 4.
        \n72 \u00d7 15
        \nAnswer:
        \nThe given expression is: 72 \u00d7 15
        \nNow,
        \nBy using the Partial Products method,
        \n72 \u00d7 15
        \n= (70 + 2) \u00d7 (10 + 5)
        \n= (70 \u00d7 10) + (70 \u00d7 5) + (2 \u00d7 10) + (2 \u00d7 5)
        \n= 700 + 350 + 20 + 10
        \n= 1,080
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is: 1,080<\/p>\n

        Question 5.
        \n26 \u00d7 63
        \nAnswer:
        \nThe given expression is: 26 \u00d7 63
        \nNow,
        \nBy using the Partial Products method,
        \n26 \u00d7 63
        \n= (20 + 6) \u00d7 (60 + 3)
        \n= (20 \u00d7 60) + (20 \u00d7 3) + (6 \u00d7 60) + (6 \u00d7 3)
        \n= 1,200 + 60 + 360 + 18
        \n= 1,638
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is: 1,638<\/p>\n

        Question 6.
        \n47 \u00d7 27
        \nAnswer:
        \nThe given expression is: 47 \u00d7 37
        \nNow,
        \nBy using the Partial Products method,
        \n47 \u00d7 37
        \n= (40 + 7) \u00d7 (30 + 7)
        \n= (40 \u00d7 30) + (40 \u00d7 7) + (7 \u00d7 30) + (7 \u00d7 7)
        \n= 1,200 + 280 + 210 + 49
        \n= 1,739
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is: 1,739<\/p>\n

        Question 7.
        \n19 \u00d7 51
        \nAnswer:
        \nThe given expression is: 19 \u00d7 51
        \nNow,
        \nBy using the Partial Products method,
        \n19 \u00d7 51
        \n= (10 + 9) \u00d7 (50 + 1)
        \n= (10 \u00d7 50) + (10 \u00d7 1) + (9 \u00d7 50) + (9 \u00d7 1)
        \n= 500 + 10 + 450 + 9
        \n= 969
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is: 969<\/p>\n

        Question 8.
        \n12 \u00d7 56
        \nAnswer:
        \nThe given expression is: 12 \u00d7 56
        \nNow,
        \nBy using the Partial Products method,
        \n12 \u00d7 56
        \n= (10 + 2) \u00d7 (50 + 6)
        \n= (10 \u00d7 50) + (10 \u00d7 6) + (2 \u00d7 50) + (2 \u00d7 6)
        \n= 500 + 60 + 100 + 12
        \n= 662
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is: 662<\/p>\n

        Question 9.
        \n76 \u00d7 23
        \nAnswer:
        \nThe given expression is: 76 \u00d7 23
        \nNow,
        \nBy using the Partial Products method,
        \n76 \u00d7 23
        \n= (70 + 6) \u00d7 (20 + 3)
        \n= (70 \u00d7 20) + (70 \u00d7 3) + (6 \u00d7 20) + (6 \u00d7 3)
        \n= 1,400 + 210 + 120 + 18
        \n= 1,748
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is: 1,748<\/p>\n

        Question 10.
        \n84 \u00d7 37
        \nAnswer:
        \nThe given expression is: 84 \u00d7 37
        \nNow,
        \nBy using the Partial Products method,
        \n84 \u00d7 37
        \n= (80 + 4) \u00d7 (30 + 7)
        \n= (80 \u00d7 30) + (80 \u00d7 7) + (4 \u00d7 30) + (4 \u00d7 7)
        \n= 2,400 + 560 + 120 + 28
        \n= 3,108
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is: 3,108<\/p>\n

        Question 11.
        \n14 \u00d7 72
        \nAnswer:
        \nThe given expression is: 14 \u00d7 72
        \nNow,
        \nBy using the Partial Products method,
        \n14 \u00d7 72
        \n= (10 + 4) \u00d7 (70 + 2)
        \n= (10 \u00d7 70) + (10 \u00d7 2) + (4 \u00d7 70) + (4 \u00d7 2)
        \n= 700 + 20 + 280 + 8
        \n= 1,008
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is: 1,008<\/p>\n

        Question 12.
        \n21 \u00d7 51
        \nAnswer:
        \nThe given expression is: 21 \u00d7 51
        \nNow,
        \nBy using the Partial Products method,
        \n21 \u00d7 51
        \n= (20 + 1) \u00d7 (50 + 1)
        \n= (20 \u00d7 50) + (20 \u00d7 1) + (1 \u00d7 50) + (1 \u00d7 1)
        \n= 1,000 + 20 + 50 + 1
        \n= 1,071
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is: 1,071<\/p>\n

        Set E pages 145-152<\/strong><\/p>\n

        Use the Distributive Property to find 13 \u00d7 55.
        \n13 \u00d7 55
        \n= (10 + 3) (50 + 5)
        \n= (10 + 3) 50 + (10 + 3) \u00d7 5
        \n= (10 \u00d7 50) + (3 \u00d7 50) + (10 \u00d7 5) + (3 \u00d7 5)
        \n= 500 + 150 + 50 + 15
        \n= 715<\/p>\n

        Remember<\/strong> you can break numbers apart in more than one way when using the Distributive Property to solve multiplication problems.<\/p>\n

        Question 1.
        \n12 \u00d7 19
        \nAnswer:
        \nThe given expression is: 12 \u00d7 19
        \nNow,
        \nBy using the Distributive Property of Multiplication,
        \n12 \u00d7 19
        \n= (10 + 2) \u00d7 (10 + 9)
        \n= (10 \u00d7 10) + (10 \u00d7 9) + (2 \u00d7 10) + (2 \u00d7 9)
        \n= 100 + 90 + 20 + 18
        \n= 228
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is: 228<\/p>\n

        Question 2.
        \n38 \u00d7 12
        \nAnswer:
        \nThe given expression is: 38 \u00d7 12
        \nNow,
        \nBy using the Distributive Property of Multiplication,
        \n38 \u00d7 12
        \n= (30 + 8) \u00d7 (10 + 2)
        \n= (30 \u00d7 10) + (30 \u00d7 2) + (8 \u00d7 10) + (8 \u00d7 2)
        \n= 300 + 60 + 80 + 16
        \n= 456
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is: 456<\/p>\n

        Question 3.
        \n19 \u00d7 25
        \nAnswer:
        \nThe given expression is: 19 \u00d7 25
        \nNow,
        \nBy using the Distributive Property of Multiplication,
        \n19 \u00d7 25
        \n= (10 + 9) \u00d7 (20 + 5)
        \n= (10 \u00d7 20) + (10 \u00d7 5) + (9 \u00d7 20) + (9 \u00d7 5)
        \n= 200 + 50 + 180 + 45
        \n= 475
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is: 475<\/p>\n

        Question 4.
        \n45 \u00d7 23
        \nAnswer:
        \nThe given expression is: 45 \u00d7 23
        \nNow,
        \nBy using the Distributive Property of Multiplication,
        \n45 \u00d7 23
        \n= (40 + 5) \u00d7 (20 + 3)
        \n= (40 \u00d7 20) + (40 \u00d7 3) + (5 \u00d7 20) + (5 \u00d7 3)
        \n= 800 + 120 + 100 + 15
        \n= 1,035
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is: 1,035<\/p>\n

        Question 5.
        \n62 \u00d7 11
        \nAnswer:
        \nThe given expression is: 62 \u00d7 11
        \nNow,
        \nBy using the Distributive Property of Multiplication,
        \n62 \u00d7 11
        \n= (60 + 2) \u00d7 (10 + 1)
        \n= (60 \u00d7 10) + (60 \u00d7 1) + (2 \u00d7 10) + (2 \u00d7 1)
        \n= 600 + 60 + 20 + 2
        \n= 682
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is: 682<\/p>\n

        Question 6.
        \n46 \u00d7 26
        \nAnswer:
        \nThe given expression is: 46 \u00d7 26
        \nNow,
        \nBy using the Distributive Property of Multiplication,
        \n46 \u00d7 26
        \n= (40 + 6) \u00d7 (20 + 6)
        \n= (40 \u00d7 20) + (40 \u00d7 6) + (6 \u00d7 20) + (6 \u00d7 6)
        \n= 800 + 240 + 120 + 36
        \n= 1,196
        \nHence, from the above,
        \nWe can conclude that the product value for the given expression is: 1,196<\/p>\n

        Set F pages 153-156<\/strong><\/p>\n

        Think about these questions to help you make sense and persevere in solving problems.
        \nThinking Habits<\/p>\n

          \n
        • What do I need to find?<\/li>\n
        • What do I know?<\/li>\n
        • What’s my plan for solving the problem?<\/li>\n
        • What else can I try if I get stuck?<\/li>\n
        • How can I check that my solution makes sense?
          \n\"Envision<\/li>\n<\/ul>\n

          Remember<\/strong> to use the information given to solve the problem.<\/p>\n

          Rose visited 14 cities on her vacation. She bought 8 souvenirs in each city to send to her friends. Rose paid $2 in postage for each souvenir to be sent.
          \nQuestion 1.
          \nWhat hidden question do you need to answer to find how much it cost Rose to send all of the souvenirs?
          \nAnswer:
          \nIt is given that
          \nRose visited 14 cities on her vacation. She bought 8 souvenirs in each city to send to her friends. Rose paid $2 in postage for each souvenir to be sent.
          \nSo,
          \nThe hidden question you need to answer to find how much it cost Rose to send all of the souvenirs is:
          \na. What is the total number of souvenirs Rose bought from all the cities?<\/p>\n

          Question 2.
          \nHow much did Rose spend on the souvenirs and postage?
          \nAnswer:
          \nIt is given that
          \nRose visited 14 cities on her vacation. She bought 8 souvenirs in each city to send to her friends. Rose paid $2 in postage for each souvenir to be sent.
          \nSo,
          \nThe total number of souvenirs Rose bought from all the cities = (The total number of cities) \u00d7 (The number of souvenirs bought by Rose in each city)
          \n= 14 \u00d7 8
          \n= (10\u00a0 +4) \u00d7 8
          \n= (10 \u00d7 8) + (4 \u00d7 8)
          \n= 80 + 32
          \n= 112 souvenirs
          \nSo,
          \nThe total cost to send all the souvenirs in postage by Rose = (The total number of souvenirs Rose bought) \u00d7 (The cost of each souvenir to be sent by Rose in postage)
          \n= 112 \u00d7 $2
          \n= (100 + 2) \u00d7 2
          \n= (100 \u00d7 2) + (2 \u00d7 2)
          \n= 200 + 4
          \n= $204
          \nHence, from the above,
          \nWe can conclude that the total money spent by Rose on souvenirs and postage is: $204<\/p>\n

          Topic 4 Assessment Practice<\/h3>\n

          Question 1.
          \nDon works 18 hours a week. Which expression shows a good way to use rounding to estimate how many hours Don will work in 52 weeks?
          \nA. 10 \u00d7 50
          \nB. 10 \u00d7 600
          \nC. 20 \u00d7 50
          \nD. 18 \u00d7 60
          \nAnswer:
          \nIt is given that Don works 18 hours a week
          \nSo,
          \nThe number of hours Don will work in 52 weeks = (The number of hours Don works in a week) \u00d7 (The number of weeks)
          \n= 18 \u00d7 52
          \nNow,
          \nBy using the rounding-off estimation,
          \nThe number of hours Don will work in 52 weeks is about 20 \u00d7 50
          \nHence,
          \nThe number of hours Don will work in 52 weeks is about:
          \n\"\"<\/p>\n

          Question 2.
          \nFind the product 21 \u00d7 40. Show your work.
          \nAnswer:
          \nThe given expression is: 21 \u00d7 40
          \nNow,
          \nBy using the Distributive Property of Multiplication,
          \n21 \u00d7 40
          \n= (20 + 1) \u00d7 40
          \n= (20 \u00d7 40) + (1 \u00d7 40)
          \n= 800 + 40
          \n= 840
          \nHence, from the above,
          \nWe can conclude that the product value for the given expression is: 840<\/p>\n

          Question 3.
          \nThe product of two factors is 2,000. One of the factors is 50. What is the other factor?
          \nA. 20
          \nB. 40
          \nC. 200
          \nD. 400
          \nAnswer:
          \nIt is given that
          \nThe product of two factors is 2,000. One of the factors is 50
          \nNow,
          \nLet the other factor be x
          \nSo,
          \n50 \u00d7 x = 2,000
          \nx = \\(\\frac{2,000}{50}\\)
          \nx = 40
          \nHence, from the above,
          \nWe can conclude that the other factor is:
          \n\"\"<\/p>\n

          Question 4.
          \nSelect all the expressions that are equal to 260.
          \n\u2610 23 \u00d7 10
          \n\u2610 20 \u00d7 10 + 3 \u00d7 10
          \n\u2610 13 \u00d7 20
          \n\u2610 1 \u00d7 20 + 3 \u00d7 20
          \n\u2610 10 \u00d7 20 + 3 \u00d7 20
          \nAnswer:
          \nThe given expressions are:
          \na. 23 \u00d7 10
          \nb. (20 \u00d7 10) + (3 \u00d7 10)
          \nc. 13 \u00d7 20
          \nd. (1 \u00d7 20) + (3 \u00d7 20)
          \ne. (10 \u00d7 20) + (3 \u00d7 20)
          \nSo,
          \nThe product values for the given expressions are:
          \na.
          \n23 \u00d7 10
          \n= (20 + 3) \u00d7 10
          \n= (20 \u00d7 10) + (3 \u00d7 10)
          \n= 200 + 30
          \n= 230
          \nb.
          \n(20 \u00d7 10) + (3 \u00d7 10)
          \n= 230
          \nc.
          \n13 \u00d7 20
          \n= (10 + 3) \u00d7 20)
          \n= (10 \u00d7 20) + (3 \u00d7 20)
          \n= 200 + 60
          \n= 260
          \nd.
          \n(1 \u00d7 20) + (3 \u00d7 20)
          \n= 70
          \ne.
          \n(10 \u00d7 20) + (3 \u00d7 20)
          \n= 200 + 60
          \n= 260
          \nHence, from the above,
          \nWe can conclude that all of the expressions that are equal to 260 are:
          \n\"\"<\/p>\n

          Question 5.
          \nA florist makes centerpieces. He puts 18 roses in each centerpiece. Which is the best way to use compatible numbers to estimate the number of roses the florist needs for 24 centerpieces? What is the exact number of roses?
          \nA. 10 \u00d7 25 = 250; 360
          \nB. 20 \u00d7 25 = 500; 432
          \nC. 25 \u00d7 30 = 750; 672
          \nD. 30 \u00d7 30 = 900; 784
          \nAnswer:
          \nIt is given that
          \nA florist makes centerpieces. He puts 18 roses in each centerpiece
          \nSo,
          \nThe number of roses the florist needs for 24 centerpieces = (The number of centerpieces) \u00d7 (The number of roses in each centerpiece)
          \n= 18 \u00d7 24
          \nNow,
          \nBy using the partial Products method,
          \n18 \u00d7 24
          \n= (10 + 8) \u00d7 (20 + 4)
          \n= (10 \u00d7 20) + (10 \u00d7 4) + (8 \u00d7 20) + (8 \u00d7 4)
          \n= 200 + 40 + 160 + 32
          \n= 432
          \nNow,
          \nEstimate:
          \n18 \u00d7 24
          \n= 20 \u00d7 25 [Estimate of the total number of roses]
          \nNow,
          \nBy using the Partial Products method,
          \n20 \u00d7 25
          \n= 20 \u00d7 (20 + 5)
          \n= (20 \u00d7 20) + (20 \u00d7 5)
          \n= 400 + 100
          \n= 500
          \nHence, from the above,
          \nWe can conclude that
          \nThe estimated and the exact number of the total number of roses in 24 centerpieces are:
          \n\"\"<\/p>\n

          Question 6.
          \nMargo hiked 12 miles 13 times last month. She hiked 14 miles 12 times this month.
          \nA. Draw arrays or area models to find the number of miles Margo hiked during the past two months.
          \nAnswer:
          \nIt is given that
          \nMargo hiked 12 miles 13 times last month. She hiked 14 miles 12 times this month.
          \nSo,
          \nThe number of miles Margo hiked during the past month = (The number of miles margo hiked past month) \u00d7 (The number of times Margo hiked during the past month)
          \n= 12 \u00d7 13
          \n= (10 + 2) \u00d7 (10 + 3)
          \nThe number of miles Margo hiked during this month = (The number of miles margo hiked this month) \u00d7 (The number of times Margo hiked during this month)
          \n= 14 \u00d7 12
          \n= (10 + 4) \u00d7 (10 + 2)
          \nHence, from the above,
          \nThe representations of the area models for the number of miles Margo hiked this month and the past month is:
          \n\"\"<\/p>\n

          B. Write and solve equations to represent your arrays or area models.
          \nAnswer:
          \nFrom part (A),
          \nWe know that,
          \nThe number of miles Margo hiked during the past month = (The number of miles margo hiked past month) \u00d7 (The number of times Margo hiked during the past month)
          \n= 12 \u00d7 13
          \nThe number of miles Margo hiked during this month = (The number of miles margo hiked this month) \u00d7 (The number of times Margo hiked during this month)
          \n= 14 \u00d7 12
          \nSo,
          \nBy using the Partial Products method,
          \n12 \u00d7 13
          \n= (10 + 2) \u00d7 (10 + 3)
          \n= (10 \u00d7 10) + (10 \u00d7 3) + (2 \u00d7 10) + (2 \u00d7 3)
          \n= 100 + 30 + 20 + 6
          \n= 156 miles
          \nSo,
          \nBy using the Partial products method,
          \n14 \u00d7 12
          \n= (10 + 4) \u00d7 (10 + 2)
          \n= (10 \u00d7 10) + (10 \u00d7 2) + (4 \u00d7 10) + (4 \u00d7 2)
          \n= 100 + 20 + 40 + 8
          \n= 168 miles
          \nHence, from the above,
          \nWe can conclude that
          \nThe number of miles Margo hiked the past month is: 156 miles
          \nThe number of miles Margo hiked this month is: 168 miles<\/p>\n

          Question 7.
          \nLiz makes necklaces. Each has 16 beads. Use the numbers in the box to complete the table.
          \n\"Envision
          \nAnswer:
          \nIt is given that
          \nLiz makes necklaces. Each has 16 beads
          \nSo,
          \nThe total number of beads = (The number of necklaces) \u00d7 (The number of beads in each necklace)
          \nThe number of beads in each necklace = \\(\\frac{The total number of beads}{The number of encklaces}\\)
          \nHence from the above,
          \nWe can conclude that
          \nThe completed table is:
          \n\"\"<\/p>\n

          Question 8.
          \nJustine’s plant shop has 7 shelves. Each shelf holds 18 plants. Suppose she adds 2 more shelves to the shop. Write and solve equations to find the number of plants all of the shelves can hold. Check if your answer is reasonable.
          \nAnswer:
          \nIt is given that
          \nJustine’s plant shop has 7 shelves. Each shelf holds 18 plants. Suppose she adds 2 more shelves to the shop.
          \nSo,
          \nThe total number of shelves in the shop = 7 + 2
          \n= 9 shelves
          \nNow,
          \nThe total number of plants all of the shelves can hold = (The total number of shelves) \u00d7 (The number of plants each shelf can hold)
          \n= 9 \u00d7 18
          \n= 9 \u00d7 (10 + 8)
          \n= (9 \u00d7 10) + (9 \u00d7 8)
          \n= 90 + 72
          \n= 162 plants
          \nEstimate:
          \n10 \u00d7 20
          \n= 1\u00d7 10 \u00d7 2 \u00d7 10
          \n= (1 \u00d7 2) \u00d7 100
          \n= 2 \u00d7 100
          \n= 200 plants
          \nHence, from the above,
          \nWe can conclude that the total number of plants all of the shelves can hold is exactly: 162 plants
          \nAs the estimated and the exact answer does not have much difference, the answer is reasonable<\/p>\n

          Question 9.
          \nJack’s landscape service charges $78 to plant a tree. What is the total cost to plant 18 trees on Tuesday and 23 trees on Wednesday? Write and solve equations.
          \nAnswer:
          \nIt is given that
          \nJack’s landscape service charges $78 to plant a tree
          \nSo,
          \nThe total cost to plant the trees on Tuesday = (The total number of trees planted on Tuesday) \u00d7 (The cost to plant each tree)
          \n= 18 \u00d7 78
          \n= (10 + 8) \u00d7 (70 + 8)
          \n= (10 \u00d7 70) + (10 \u00d7 8) + (8 \u00d7 70) + (8 \u00d7 8)
          \n= 700 + 80 +560 + 64
          \n= $1,404
          \nSo,
          \nThe total cost to plant the trees on Wednesday = (The total number of trees planted on Wednesday) \u00d7 (The cost to plant each tree)
          \n= 23 \u00d7 78
          \n= (20 + 3) \u00d7 (70 + 8)
          \n= (20 \u00d7 70) + (20 \u00d7 8) + (3 \u00d7 70) + (3 \u00d7 8)
          \n= 1,400 + 160 +210 + 24
          \n= $1,794
          \nHence, from the above,
          \nWe can conclude that
          \nThe total cost to plant trees on Tuesday is: $1,404
          \nThe total cost to plant trees on Wednesday is: $1,794<\/p>\n

          Question 10.
          \nWhich of the following uses properties of operations to help find 27 \u00d7 14?
          \nA. 20 + 7 + 10 + 4
          \nB. 20 \u00d7 7 \u00d7 10 \u00d7 4
          \nC. 27 + 14
          \nD. (20 + 7) (10 + 4)
          \nAnswer:
          \nThe given expression is: 27 \u00d7 14
          \nNow,
          \nBy using the Distributive Property of Multiplication,
          \n27 \u00d7 14
          \n= (20 + 7) \u00d7 (10 + 4)
          \nHence, from the above,
          \nWe can conclude that the properties of operations used to find 27 \u00d7 14 are:
          \n\"\"<\/p>\n

          Question 11.
          \nLorin drew an area model to find 19 \u00d7 15. Write the partial product for each rectangle in the area model.
          \n\"Envision
          \nAnswer:
          \nThe given expression is: 19 \u00d7 15
          \nNow,
          \nBy using the Partial products method,
          \n19 \u00d7 15
          \n= (10 + 9) \u00d7 (10 + 5)
          \n= (10 \u00d7 10) + (10 \u00d7 5) + (9 \u00d7 10) + (9 \u00d7 5)
          \n= 100 + 50 + 90 + 45
          \n= 285
          \nHence, from the above,
          \nWe can conclude that the partial products for the given expression are:
          \n\"\"<\/p>\n

          Question 12.
          \nThe biggest painting in an art show is 31 inches long and 28 inches wide. The smallest painting is 14 inches long and 11 inches wide. What is the difference between the areas of the two paintings? Use equations to show your work.
          \n________ square inches
          \nAnswer:
          \nIt is given that
          \nThe biggest painting in an art show is 31 inches long and 28 inches wide. The smallest painting is 14 inches long and 11 inches wide.
          \nNow,
          \nThe area of the biggest painting = Length \u00d7 Width
          \n= 31 \u00d7 28
          \n= (30 + 1) \u00d7 (20 + 8)
          \n= (30 \u00d7 20) + (30 \u00d7 8) + (1 \u00d7 20) + (1 \u00d7 8)
          \n= 600 +240 + 20 + 8
          \n= 868 inches\u00b2
          \nNow,
          \nThe area of the samllest painting = Length \u00d7 Width
          \n= 14 \u00d7 11
          \n= (10 + 4) \u00d7 (10 + 1)
          \n= (10 \u00d7 10) + (10 \u00d7 1) + (4 \u00d7 10) + (4 \u00d7 1)
          \n= 100 + 10 + 40 + 4
          \n= 154 inches\u00b2
          \nSo,
          \nThe difference between the areas of the biggest painting and the smallest painting = 868 – 154
          \n= 714 square inches
          \nHence, from the above,
          \nWe can conclude that the difference between the areas between the two paintings is: 714 square inches<\/p>\n

          Topic 4 Performance Task<\/h3>\n

          Raising Money<\/strong>
          \nThe fourth-grade students at Skyline School sold candles to raise money. The money raised will be used to buy toys for children who live in a group home. The Candle Fundraiser table shows how many candles each class sold.<\/p>\n

          \"Envision Beehive candles sell for $20 each. The class raised $12 for each candle sold.
          \n\"Envision Soybean candles sell for $22 each. The class raised $14 for each candle sold.
          \n\"Envision<\/p>\n

          Question 1.
          \nAbout how much money did the fourth-grade students raise? Explain.
          \nAnswer:
          \nIt is given that
          \nBeehive candles sell for $20 each. The class raised $12 for each candle sold.
          \nSoybean candles sell for $22 each. The class raised $14 for each candle sold.
          \nSo,
          \nThe total cost of each Beehive candle = $20 + $12
          \n= $32
          \nThe total cost of each Soybean candle = $22 + $14
          \n= $36
          \nNow,
          \nThe given table is:
          \n\"Envision
          \nSo,
          \nThe money raised by the fourth-grade students = (The total number of candles sold) + (The amount raised for each candle)
          \n= (32 \u00d7 32) + (25 \u00d7 36) + (18 \u00d7 36) + (47 \u00d7 32)
          \n= 1,024 + 900 + 648 + 1,504
          \n= $4,076
          \nHence, from the above,
          \nWe can conclude that
          \nThe total money raised by the fourth-grade students is: $4,076<\/p>\n

          Question 2.
          \nThe school secretary needs to know exactly how much money was raised.
          \nPart A
          \nHow much money did Ms. Schmidt’s class raise? Use an area model and partial products to find the product.
          \nAnswer:
          \nThe given data is:
          \n\"Envision
          \nNow,
          \nFrom the given table,
          \nWe can observe that
          \nThe amount of money raised by Mr. Schmidt’s class = (The number of candles sold by Mr.Schmidt’s class) \u00d7 (The total amount raised for each candle sold)
          \n= 25 \u00d7 36
          \nNow,
          \nBy using the area-Model,
          \n25 \u00d7 36
          \n= (20 + 5) \u00d7 (30 + 6)
          \n= (20 \u00d7 30) + (20 \u00d7 6) + (5 \u00d7 30) + (5 \u00d7 6)
          \n= 600 + 120 + 150 + 30
          \n= 900
          \n\"\"
          \nHence, from the above,
          \nWe can conclude that the money raised by Mr. Schmidt’s class is: $900<\/p>\n

          Part B
          \nHow much money did Ms. Picard’s class raise? Use the Distributive Property to find the product.
          \nAnswer:
          \nThe given data is:
          \n\"Envision
          \nNow,
          \nFrom the given table,
          \nWe can observe that
          \nThe amount of money raised by Ms. Picard’s class = (The number of candles sold by Ms.Picord’s class) \u00d7 (The total amount raised for each candle sold)
          \n= 18 \u00d7 36
          \nNow,
          \nBy using the Distributive Property of Multiplication,
          \n18 \u00d7 36
          \n= (10 + 8) \u00d7 (30 + 6)
          \n= (10 \u00d7 30) + (10 \u00d7 6) + (8 \u00d7 30) + (8 \u00d7 6)
          \n= 300 + 60 + 240 + 48
          \n= 648
          \nHence, from the above,
          \nWe can conclude that the amount of money raised by Ms. Picard’s class is: $648<\/p>\n

          Part C
          \nHow much money did the fourth-grade students raise? Explain.
          \nAnswer:
          \nIt is given that
          \nBeehive candles sell for $20 each. The class raised $12 for each candle sold.
          \nSoybean candles sell for $22 each. The class raised $14 for each candle sold.
          \nSo,
          \nThe total cost of each Beehive candle = $20 + $12
          \n= $32
          \nThe total cost of each Soybean candle = $22 + $14
          \n= $36
          \nNow,
          \nThe given table is:
          \n\"Envision
          \nSo,
          \nThe money raised by the fourth-grade students = (The total number of candles sold) + (The amount raised for each candle)
          \n= (32 \u00d7 32) + (25 \u00d7 36) + (18 \u00d7 36) + (47 \u00d7 32)
          \n= 1,024 + 900 + 648 + 1,504
          \n= $4,076
          \nHence, from the above,
          \nWe can conclude that
          \nThe total money raised by the fourth-grade students is: $4,076<\/p>\n

          Part D
          \nIs the total amount of money raised from Part C reasonable based on the estimate you found in Exercise 1? Explain.
          \nAnswer:
          \nFrom part (C) and Exercise 1,
          \nWe can observe that
          \nIn Exercise 1,
          \nWe found out the total amount of money raised by the fourth-grade students
          \nIn part (C),
          \nWe found out the total amount of money raised by the fourth-grade students
          \nSo,
          \nFrom the above,
          \nWe can observe that
          \nWe found the same value in both
          \nHence, from the above,
          \nWe can conclude that
          \nThe total amount of money raised from Part C is reasonable based on the estimate you found in Exercise 1<\/p>\n

          The fourth-grade students at Skyline School decide to buy the toys shown.
          \n\"Envision
          \nBuying Toys<\/p>\n

            \n
          • They buy 68 toys in all.<\/li>\n
          • They buy at least 25 of each type.<\/li>\n
          • They cannot spend more money than was raised.<\/li>\n<\/ul>\n

            Question 3.
            \nHow many of each toy can be bought following the directions from the Buying Toys list? Find the total cost for the numbers you choose.
            \nAnswer:
            \nIt is given that
            \nThe cost of each teddy bear is: $17
            \nThe cost of each scooter is: $28
            \nThe total number of toys are: 68
            \nNow,
            \nLet there be an equal number of teddy bears and scooters
            \nSo,
            \nThe number of each toy = \\(\\frac{68}{2}\\)
            \n= 34
            \nSo,
            \nThe total cost of the toys = (34 \u00d7 17) + (34 \u00d7 28)
            \n= $1,530
            \nHence, from the above,
            \nWe can conclude that
            \nThe total cost for the number of toys you choose is: $1,530<\/p>\n","protected":false},"excerpt":{"rendered":"

            Go through the\u00a0enVision Math Common Core Grade 4 Answer Key\u00a0Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers\u00a0regularly and improve your accuracy in solving questions. enVision Math Common Core 4th Grade Answers Key Topic 4 Use Strategies and Properties to Multiply by 2-Digit Numbers Essential Questions: How can you use a model to …<\/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\/9479"}],"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=9479"}],"version-history":[{"count":4,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/posts\/9479\/revisions"}],"predecessor-version":[{"id":28178,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/posts\/9479\/revisions\/28178"}],"wp:attachment":[{"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/media?parent=9479"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/categories?post=9479"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/tags?post=9479"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}