Category: Envision Math

Envision Math 5th Grade Textbook Answer Key Topic 19.9 Problem Solving

Problem Solving

Solve a Simpler Problem
Owen needs to read 2 books from a list of 6 books. How many different combinations of books are possible?

Answer:
Follow these steps to solve a simpler problem.

1. Break apart or change the problem into one that is simpler.
2. Solve the simpler problem.
3. Use the answers to the simpler problem to solve the original problem.

Guided Practice

Do you know HOW?
Question 1.
Draw a picture to show the number of combinations of pairs for 5 books.
Answer:

Question 2.
Think of extending the table at the top to find the number of pairs with 7 books. What number would you add to 15 to find the number of pairs? How many pairs would there be for 7 books?
Answer:

Do you UNDERSTAND?
Question 3.
Writing to Explain Is it easier to use the table or to draw a picture as the number of books increases?
Answer:

Question 4.
Write a Problem Write a real-world problem that can be solved by solving a simpler problem.
Answer:

Independent Practice

Solve each problem.
Question 5.
Continue the pattern in the book problem above. How many pairs of books would there be for 8 books? 9 books? 10 books?
Answer:

Question 6.
Find the number of degrees in a hexagon. HINT: Divide the hexagon into triangles.

a. How many triangles are formed?
b. How many degrees are in each triangle?
c. What is the total number of degrees in the hexagon?
Answer:

Question 7.
Using the same strategy in Problem 6, what is the total number of degrees in a pentagon? An octagon?
Answer:

Use letters to represent the books.

Look for a pattern.
Find the pattern. Continue the pattern to 6 books.

There are 15 different possible pairs.

Question 8.
Juanita tossed 3 number cubes, and these digits came up.

If each of the numbers 2, 1 and 4 is used only once to form a 3-digit number, which shows all the possible 3-digit numbers?
A. 214, 421, 142
B. 214, 124, 412, 421
C. 214, 241, 142, 412, 124
D. 214, 241, 142, 124, 412, 421
Answer:

Question 9.
Jill has 3 colored vases to arrange on a shelf

Let B stand for blue, O for orange, and G for green. Which list shows all the possible arrangements of the vases?
A. BOG, OBG, GOB
B. BOG, OBG, GOB, GBO
C. BOG, OBG, GOB, GBO, OGB
D. BOG, OBG, GOB, GBO, OGB, BGO
Answer:

Question 10.
George was choosing his clothes for the next day. In his closet he had 2 ties, 5 shirts, 3 trousers, and 2 belts. How many different ways could George choose a tie, shirt, trousers, and belt?
Answer:

Question 11.
The McMillan family wanted to buy a new vehicle. They could choose a van or a car; a black, silver, or white exterior; and a tan or black interior. How many different vehicles can they buy?
Answer:

Question 12.
After seeing a movie, 2 friends stopped for frozen yogurt. Three flavors were available in small, medium, and large sizes. How many different combinations of flavors and sizes are possible?
Answer:

Question 13.
Algebra Draw a picture and write an equation to solve.
Niko had 17\(\frac{1}{3}\) ft of fencing. He uses 5\(\frac{2}{3}\) ft to finish a job. How many feet of fencing does Niko have now? Let f = feet of fencing left?
Answer:

Go through the enVision Math Common Core Kindergarten Answer Key Topic 1 Numbers 0 to 5 regularly and improve your accuracy in solving questions.

Envision Math Common Core Kindergarten Answers Key Topic 1 Numbers 0 to 5

Essential Question:
How can numbers from 0 to 5 be counted, read, and written?

Directions Read the character speech bubbles to students. Find Out! Have students pay attention to the daily weather changes. Say: The weather changes from day to day. Talk to friends and relatives about the weather. Ask them to help you record the number of sunny days and rainy days from Monday to Friday. Journal: Make a Poster Have students make a poster of the weather information they collected. Have them draw suns for the number of sunny days and clouds with raindrops for the number of rainy days. Then have students write the numbers to tell how many.

Review What You Know

Question 1.

Answer:

Explanation:
I circled the animal to the right.

Question 2.

Answer:

Explanation:
I circled the animal to the left.

Question 3.

Answer:

Explanation:
I circled the animal that is green in color.

Question 4.

Answer:

Explanation:
I drew lines from each object in the top row to bottom row.

Question 5.

Answer:

Explanation:
I drew lines from each object in the top row to bottom row.

Question 6.

Answer:

Explanation:
I drew lines from each object in the top row to bottom row.

Directions Have students: 1 draw a circle around the animal that is on the right; 2 draw a circle around the animal that is on the left; 3 draw a circle around the animal that is green; 4-6 draw a line from each object in the top row to an object in the bottom row.

Pick a Project

Directions Say: You will choose one of these projects. Look at picture A. Think about this question: What would animals say if they could talk? If you choose Project A, you will create funny animal characters and tell a story. Look at picture B. Think about this question: Does your favorite sport or game use a ball? If you choose Project B, you will make a poster of different balls that are used in sports and games. Look at picture C. Think about this question: What would you want to show on a license plate? If you choose Project C, you will create a license plate for your bike.

3-ACT MATH PREVIEW

Math Modeling

Set the Table

Directions Read the robot’s speech bubble to students. Generate Interest Ask students if they have ever set the table. Say: Who sets the table before meals? What chores are you responsible for at home? Have them share stories about responsibilities and rules around the house.

Lesson 1.1 Count 1, 2 and 3

Solve & Share

Answer:

Explanation:
I drew 2 counters in the nest and circled the colored box that shows the number 2 as 2 boxes are colored.

Directions Have students place 2 counters in the nest on the workmat. Say: Peeps the bird found these worms for her babies. Draw a circle around the colored box that shows how many worms Peeps found. Tell how you know you are correct.

Visual Learning Bridge

Guided Practice

Question 1.

Answer:

Explanation:
I colored a box as i counted each worm.There are 2 worms.

Question 2.

Answer:

Explanation:
I colored a box as i counted each worm.There is 1 worm.

Directions 1 and 2 Have students color a box as they count each worm to show how many.

Question 3.

Answer:

Explanation:
I colored a box as i counted each worm.There are 3 worms.

Question 4.

Answer:

Explanation:
I colored a box as i counted each worm.There are 2 worms.

Question 5.

Answer:

Explanation:
I colored a box as i counted each worm.There are 3 worms.

Directions 3 and 4 Have students color a box as they count each worm to show how many. 5 Vocabulary Have students count the worms, and color a box as they count each worm aloud.

Independent Practice

Question 6.

Answer:

Explanation:
I colored a box as i counted each nest.There are 2 nests.

Question 7.

Answer:

Explanation:
I colored a box as i counted each nest.There is 1 nest.

Question 8.

Answer:

Explanation:
I colored a box as i counted each nest.There are 3 nests.

Question 9.

Answer:

Explanation:
I drew 2 nests and colored 2 boxes as i counted the number of nests.

Directions 6-8 Have students color a box as they count each nest to show how many. 9 Higher Order Thinking Have students draw 1, 2, or 3 nests, and then color a box as they draw each nest to show how many.

Lesson 1.2 Recognize 1, 2 and 3 in Different Arrangements

Solve & Share

Directions Say: Redbird and Bluebird each have 2 babies. Redbird and Blue bird get worms for their babies and put them in their nests. Bluebird’s worms moved around in the nest. Show and count how many worms with your counters. Color the boxes to show the worms Th each nest. Tell how you know you are correct.

Visual Learning Bridge

Guided Practice

Question 1.

Answer:

Explanation:
I counted each bird, there are 2 birds and i colored 2 boxes.

Question 2.

Answer:

Explanation:
I counted each bird, there are 3 birds and i colored 3 boxes.

Directions 1 and 2 Have students count each bird, and then color the boxes to show how many.

Question 3.

Answer:

Explanation:
I counted each bird, there are 3 birds and i colored 3 boxes.

Question 4.

Answer:

Explanation:
I counted each bird, there are 2 birds and i colored 2 boxes.

Question 5.

Answer:

Explanation:
I counted each bird, there is 1 bird and i colored 1 box.

Question 6.

Answer:

Explanation:
I counted each bird, there are 3 birds and i colored 3 boxes.

Question 7.

Answer:

Explanation:
I counted each bird, there is 1 bird and i colored 1 box.

Question 8.

Answer:

Explanation:
I counted each bird, there are 2 birds and i colored 2 boxes.

Directions 3-8 Have students count the dots, and then color the boxes to show how many.

Independent Practice

Question 9.

Answer:

Explanation:
I counted each dot, there is 1 dot and i colored 1 box.

Question 10.

Answer:

Explanation:
I counted each dot, there are 3 dots and i colored 3 boxes.

Question 11.

Answer:

Explanation:
I counted each dot, there are 2 dots and i colored 2 boxes.

Question 12.

Answer:

Explanation:
I counted each dot, there are 3 dots and i colored 3 boxes.

Question 13.

Answer:

Explanation:
I drew 2 counters in two different ways.

Directions 9-12 Have students count the dots, and then color the boxes to show how many. 13 Higher Order Thinking Have students draw 2 counters, and then draw 2 counters in a different way.

Lesson 1.3 Read, Make, and Write 1, 2 and 3

Answer:

Explanation:
I drew 2 counters in the big cloud.I drew 2 circles to show that Alex saw 2 stars.

Directions Say: Alex sees 2 stars in the sky. He draws 2 stars in a cloud. Place counters in the large cloud on the workmat to show how many stars. How can you show how many stars Alex sees in other ways? Draw or use objects to show other ways in the small, empty cloud.

Visual Learning Bridge

Guided Practice

Question 1.

Answer:

Explanation:
I counted the stars.There is 1 star.SO, i wrote number 1 in the blank.

Question 2.

Answer:

Explanation:
I counted the stars.They are 2 in number.So, i wrote the number 2 in the blank.

Question 3.

Answer:

Explanation:
I counted the stars.They are 3 in number.So, i wrote the number 3 in the blank.

Directions 1-3 Have students count the stars, and then write the number to tell how many.

Question 4.

Answer:

Explanation:
I counted the smilies.There is 1 smilie.So, i practiced writing the number 1..

Question 5.

Answer:

Explanation:
I counted the smilies.There are 2 smilies.So, i  practiced writing the number 2.

Question 6.

Answer:

Explanation:
I counted the smilies.There are 3 smilies.So, i practiced writing the number 3.

Question 7.

Answer:

I counted the smilies.There is 1 smilie.So, i practiced writing the number 1.

Directions 4-7 Have students count the objects, and then practice writing the number that tells how many.

Independent Practice

Question 8.

Answer:

Explanation:
I used 2 counters to make the number 2, drew circles to represent it.

Question 9.

Answer:

Explanation:
I used 2 counters to make the number 3, drew circles to represent it.

Question 10.

Answer:

Explanation:
I used 1 counter to make the number 1, drew circles to represent it.

Question 11.

Answer:

Explanation:
I drew 3 stars and practiced writing the number 3.

Directions 8-10 Have students use counters to make the number. Then have them draw circles to represent the number. 11 Higher Order Thinking Have students draw 1, 2, or 3 stars, and then practice writing the number that tells how many.

Lesson 1.4 Count 4 and 5

Solve & Share

Answer:

Directions Have students place 5 counters in the tree on the workmat. Then say: Chips the chipmunk found these nuts. Draw a circle around the colored box that shows how many nuts Chips found. Tell how you know you are correct.

Visual Learning Bridge

Guided Practice

Question 1.

Answer:

Explanation:
I colored a box as i counted each orange.There are 4 oranges.

Question 2.

Answer:

Explanation:
I colored a box as i counted each orange.There are 5 oranges.

Directions 1 and 2 Have students color a box as they count each orange to show how many.

Question 3.

Answer:

Explanation:
I colored a box as i counted each orange.There are 4 oranges.

Question 4.

Answer:

Explanation:
I colored a box as i counted each fruit.There are 4 fruits.

Question 5.

Answer:

Explanation:
I colored a box as i counted each fruit.There are 5 fruits.

Question 6.

Answer:

Explanation:
I colored a box as i counted each fruit.There are 4 fruits.

Directions 3-6 Have students color a box as they count each piece of fruit to show how many.

Independent Practice

Question 7.

Answer:

Explanation:
I colored a box as i counted each fruit.There are 5 fruits.

Question 8.

Answer:

Explanation:
I colored a box as i counted each fruit.There are 4 fruits.

Question 9.

Answer:

Explanation:
I colored a box as i counted each fruit.There are 5 fruits.

Question 10.

Answer:

Explanation:
I colored a box as i drew each orange.I drew 4 oranges.

Directions 7-9 Have students color a box as they count each piece of fruit to show how many. 10 Higher Order Thinking Have students draw 4 or 5 oranges, and then color a box as they draw each orange to show how many.

Lesson 1.5 Recognize 4 and 5 in Different Arrangements

Directions Say: Some bees are flying around before landing on their beehives. Then 4 bees land on one hive and 4 bees land on the other hive. Use cubes to show different ways the bees can land on the hives. Count the bees and color the boxes to show how many landed on each hive. Tell how you know you are correct.

Visual Learning Bridge

Guided Practice

Question 1.

Answer:

Explanation:
I counted the bees, there are 4 bees.So, i colored 4 boxes.

Question 2.

Answer:

Explanation:
I counted the cats.There are 5 cats.SO, i colored 5 boxes.

Directions 1 and 2 Have students count the insects or animals in each group, and then color the boxes to show how many.

Question 3.

Answer:

Explanation:
I counted the number of birds.There are 4 birds.So, i colored 4 boxes.

Question 4.

Answer:

Explanation:
I counted the number of birds.There are 5 birds.So, i colored 5 boxes.

Question 5.

Answer:

Explanation:
I counted the number of birds.There are 5 birds.So, i colored 5 boxes.

Question 6.

Answer:

Explanation:
I counted the number of birds.There are 4 birds.So, i colored 4 boxes.

Question 7.

Answer:

Explanation:
I counted the number of birds.There are 5 birds.So, i colored 5 boxes.

Question 8.

Answer:

Explanation:
I counted the number of birds.There are 4 birds.So, i colored 4 boxes.

Directions 3-8 Have students count the birds, and then color the boxes to show how many.

Independent Practice

Question 9.

Answer:

Explanation:
I counted the number of dots.There are 4 dotss.So, i colored 4 boxes.

Question 10.

Answer:

Explanation:
I counted the number of dots.There are 5 dots.So, i colored 5 boxes.

Question 11.

Answer:

Explanation:
I counted and circled those groups that are 4 in number.

Question 12.

Answer:

Explanation:
I counted and circled those groups that are 5 in number.

Question 13.

Answer:

Explanation:
I drew 5 counters in the first space and drew 5 counters in two different ways in the next two spaces.

Directions Have students: 9 and 10 count the dots, and then color the boxes to show how many; 11 count the groups, and then draw a circle around the groups that show 4; 12 count the groups, and then draw a circle around the groups that show 5. 13 Higher Order Thinking Have students draw 5 counters in the first space, and then draw 5 counters in two different ways in the other two spaces.

Lesson 1.6 Read, Make, and Write 4 and 5

Answer:

Explanation:
I drew 5 counters and 5 circles to show the numbe rof forgs.

Directions Say: Alex sees some frogs sitting on a lily pod. Show how many frogs he sees by placing counters on the large lily pad. Count and tell how many. Then draw or use objects to show how many frogs ¡n other ways on the small, empty lily pod on the workmat.

Visual Learning Bridge

Guided Practice

Question 1.

Answer:

Explanation:
I counted the butterflies.They are 4 in number.I practiced writing the number 4.

Question 2.

Answer:

Explanation:
I counted the butterflies.They are 5 in number.I practiced writing the number 5.

Directions 1 and 2 Have students count the butterflies, and then practice writing the number that tells how many.

Question 3.

Answer:

Explanation:
I counted the frogs.They are 4 in number.I practiced writing the number 4.

Question 4.

Answer:

Explanation:
I counted the frogs.They are 5 in number.I practiced writing the number 5.

Question 5.

Answer:

Explanation:
I counted the frogs.They are 4 in number.I practiced writing the number 4.

Question 6.

Answer:

Explanation:
I counted the frogs.They are 5 in number.I practiced writing the number 5.

Directions 3-6 Have students count the frogs, and then practice writing the number that tells how many.

Independent Practice

Question 7.

Answer:

Explanation:
I drew 4 counters and 4 squares to represent the number 4.

Question 8.

Answer:

Explanation:
I drew 5 counters and 5 squares to represent the number 5.

Question 9.

Answer:

Explanation:
I counted the birds in the above picture.There are 5 blue birds and 4 yellow birds.So, i colored 5 for blue bird and 4 boxes for yellow bird and wrote the numbers 5 and 4 in the blanks.

Directions 7 and 8 Have students use counters to make the number. Then have them draw squares to represent the number. 9 Higher Order Thinking Have students count the blue birds and the yellow birds, color a box for each bird, and then write the numbers to tell how many.

Lesson 1.7 Identify the Number 0

Solve & Share

Directions Say: Alex has a vegetable garden. Toss a number cube to see how many potatoes he has in his basket. Place counters on the basket to show how many. What does it mean if the cube face you roll has no dots? How can Alex use the boxes to show that there are no potatoes in the basket?

Visual Learning Bridge

Guided Practice

Question 1.

Answer:

Explanation:
I colored a box as i counted each apple.There are 3apples.

Question 2.

Answer:

Explanation:
I dinot color any box as the plate is empty.

Directions 1 and 2 Have students color a box as they count each apple to show how many.

Question 3.

Answer:

Explanation:
I colored a box as i counted each banana.There are 2 bananas.

Question 4.

Answer:

Explanation:
I colored a box as i counted each strawberry.There are 5 strawberries.

Question 5.

Answer:

Explanation:
I dinot color any box as the plate is empty.

Question 6.

Answer:

Explanation:
I colored a box as i counted each strawberry.There are 4 strawberries.

Question 7.

Answer:

Explanation:
I colored a box as i counted each banana.There are 3 bananas.

Question 8.

Answer:

Explanation:
I dinot color any box as the plate is empty.

Directions 3-8 Have students color a box as they count each piece of fruit to show how many.

Independent Practice

Question 9.

Answer:

Explanation:
I dinot color any box as the vase is empty.

Question 10.

Answer:

Explanation:
I counted and colored 1 box as there is 1 flower in the box.

Question 11.

Answer:

Explanation:
I counted and colored 3 boxes as there are 3 flowers in the vase.

Question 12.

Answer:

Explanation:
I counted and colored 3 boxes as there are 5 flowers in the vase.

Question 13.

Answer:

Explanation:
I dinot color any box as the vase is empty.

Question 14.

Answer:

Explanation:
I counted and colored 3 boxes as there are 2 flowers in the vase.

Question 15.

Answer:

Explanation:
I drew 3 flowers and colored 3 boxes to show the number of flowers.

Directions 9-14 Have students color a box as they count each flower in the vase to show how many. 15 Higher Order Thinking Have students pick a number between 0 and 5, draw that many flowers, and then color the boxes to show how many.

Lesson 1.8 Read and Write 0

Solve & Share

Answer:

Explanation:
I counted the number of colored boxes next to each crayon on the workmat and colored crayons inside the bag.There are 2 blue crayons, 3 red crayons, 1 yellow crayon in the bag.

Directions Say: Count the number of colored boxes next to each crayon on the workmat. Then color the crayons inside the bag to show how many of each type Alex has and tell how you know. How many green crayons does Alex have? Tell how you know.

Visual Learning Bridge

Guided Practice

Question 1.

Answer:

Explanation:
I counted the pencils in each pencil holder, and then practiced writing the number3.There are 3 pencils.

Question 2.

Answer:

Explanation:
I counted the pencils in each pencil holder, and then practiced writing the number 0.There are no pencils.

Directions 1 and 2 Have students count the pencils in each pencil holder, and then practice writing the number that tells how many.

Question 3.

Answer:

Explanation:
I counted the pencils in each pencil holder, and then practiced writing the number 2.There are 2 pencils.

Question 4.

Answer:

Explanation:
I counted the pencils in each pencil holder, and then practiced writing the number 3.There are 3 pencils.

Question 5.

Answer:

Explanation:
I counted the pencils in each pencil holder, and then practiced writing the number 0.There are no pencils.

Question 6.

Answer:

Explanation:
I counted the pencils in each pencil holder, and then practiced writing the number 1.There is 1 pencil.

Question 7.

Answer:

Explanation:
I counted the pencils in each pencil holder, and then practiced writing the number 2.There are 2 pencils.

Question 8.

Answer:

Explanation:
I counted the pencils in each pencil holder, and then practiced writing the number 2.There are 2 pencils.

Directions 3-8 Have students count the balls in each box, and then practice writing the number that tells how many.

Independent Practice

Question 9.

Answer:

Explanation:
I counted the balls in each box, and then practiced writing the number 0.There are no balls.

Question 10.

Answer:

Explanation:
I counted the balls in each box, and then practiced writing the number 2.There are 2 balls.

Question 11.

Answer:

Explanation:
I counted the balls in each box, and then practiced writing the number 1.There is 1 ball.

Question 12.

Answer:

Explanation:
I counted the balls in each box, and then practiced writing the number 0.There are no balls.

Question 13.

Answer:

Explanation:
I counted and wrote the numbers from 0 to 5.

Question 14.

Answer:

Explanation:
I drew 0 counters and wrote the number 0 in the blank and drew 4 counters and wrote number 4 in the box.

Directions Have students: 9-12 count the balls in each box, and then practice writing the number that tells how many; 13 practice writing the numbers 0 to 5. 14 Higher Order Thinking Have students draw zero counters and write the number to tell how many, and then draw 1 to 5 counters and write the number to tell how many.

Lesson 1.9 Numbers to 5

Answer:

Explanation:
The numbe before 4 is 3 and after 4 is 5.So, i wrote the numbers 3 and 5 and drew counters to show the numbers3, 4 and 5.

Directions Say: Marta is thinking of two numbers-one is the number that comes just before 4 when counting, and the other is the number that comes just after 4 when counting. Write the two numbers Marta is thinking of. Show how you know you are correct.

Visual Learning Bridge

Guided Practice

Question 1.

Answer:

Explanation:
The number that comes just before 1 is 0 and after 1 is 3
The number that comes just before 4 is 3 and after 4 is 5
So, i wrote the numbers 0, 2, 3 and 5 in the blanks.

Directions 1 Have students write the number that comes just before 1 and the number that comes just after 1. Then have them write the number that comes just before 4 when counting, and the number that comes just after 4 when counting. Have them say the numbers in order from 0 to 5.

Question 2.

Answer:

Explanation:
I colored the cubes as i counted the numbers from 0 to 5 and wrote the numbers 0, 1, 2, 3, 4 and 5.

Question 3.

Answer:

Explanation:
I counted the flowers in each vase and wrote the numbers 0, 1, 2, 3, 4 and 5 in the blanks that tell how many.The number that comes just after 1 is 2 so, i circled 2.

Directions Have students: 2 color the cubes to show each number, write the numbers in order, and then draw a circle around the number that comes just after 1 when counting; 3 count the flowers in each vase, write the numbers, and then say the numbers in order from 0 to 5.

Independent Practice

Question 4.

Answer:

Explanation:
I counted the balls in each boxes and wrote the numbers 0, 1, 2, 3, 4 and 5 in the blanks that tell how many.The number that comes just after 4 is 5 so, i circled 5.

Question 5.

Answer:

Explanation:
I counted backwards from 5 to 0 and colored the cubes to match with the number.I wrote the numbers 5, 4 , 3, 2, 1 and 0.

Directions 4 Have students count the toys in each box, write the numbers, and then draw a circle around the number that comes just after 4 when counting how many. 5 Higher Order Thinking Have students color 5 cubes, and then write the number. Have them color cubes to show the number that comes next when counting backwards from 5, and then write the number of cubes they colored in the tower. Repeat for the remaining towers.

Lesson 1.10 Construct Arguments

Solve & Share

Answer:

Directions Say: Alex needs to count the group of shapes. How can you count these shapes? Use objects or words to help. Write the number to tell how many shapes. Tell why your number is correct.

Visual Learning Bridge

Guided Practice

Question 1.

Answer:

Explanation:
I counted the number of birds and wrote 4 in the blank.There are 4 birds and i drew 4 circles to show the number 4.

Question 2.

Answer:

Explanation:
I counted the number of birds and wrote 3 in the blank.There are 3 birds and i wrote the number 1, 2 and 3 circles to show the number 3.

Directions 1 and 2 Have students make a math argument about how many birds are in each row, and then write the number. Have them use objects, words, or a method of their choice to explain their arguments and tell why they are correct.

Independent Practice

Question 3.

Answer:

Explanation:
I counted the number of leaves and wrote 5 in the blank.There are 5 leaves and i drew 5 circles to show the number 5.

Question 4.

Answer:

Explanation:
I counted the number of leaves and wrote 2 in the blank.There are 2 leaves and i wrote the number 1 and 2 circles to show the number 2.

Question 5.

Answer:

Explanation:
I counted the number of leaves and wrote 4 in the blank.There are 4 leaves and i colored 4 boxes to show the number 4.

Question 6.

Answer:

Explanation:
I counted the number of leaves and wrote 3 in the blank.There are 4 leaves and i colored 3 boxes to show the number 3.

Directions 3-5 Have students make a math argument about how many leaves are in each row, and then write the number. Have them use objects, words, or a method of their choice to explain their arguments and tell why they are correct. 6 enVision’ STEM Say: Chlorophyll makes leaves green. There is less sunlight in the winter, so trees save their chlorophyll This turns leaves brown, orange, red, and yellow. Have students make a math argument about how many orange leaves are in the row, and then write the number. Have them use objects, words, or a method of their choice to explain their arguments and tell why they are correct.

Problem Solving

Performance Task

Question 7,8,9.

Answer:

Explanation:
There are 3 rabbits in the picture.I used the tool of coloring a box as i count each rabbit.So, i colored 3 boxes to show the number of rabbits.

Directions Read the problem to students. Then have them use multiple problem-solving methods to solve the problem. Say: Brooke sees some rabbits. How many rabbits does she see? 7 Reasoning How can you find the number of rabbits Brooke sees? 8 Use Tools What tool can you use to help solve the problem? 9 Explain How can you use math to explain why your work is correct?

Topic 1 Vocabulary Review

Question 1.

Answer:

Explanation:
I circled the number 2.

Question 2.

Answer:

Explanation:
I wrote the number 0 as it means none.

Question 3.

Answer:

Explanation:
I circled the number 4.

Question 4.

Answer:

Explanation:
I marked a X on red cube and circled all the 5 cubes.

Directions Understand Vocabulary Have students: 1 draw a circle around the number; 2 write the number that means none; 3 draw a circle around the number four; 4 mark an X on the one red cube, and draw a circle around all five cubes in the group.

Question 5.

Answer:

Explanation:
I circled the number 1.

Question 6.

Answer:

Explanation:
I wrote the number 3.

Question 7.

Answer:

Explanation:
I counted the number of cubes and wrote the number 2 as there are 2 cubes.

Question 8.

Answer:

Explanation:
I wrote the numbers from 0 to 5 in order, and then drew counters to show the the numbers 0, 1, 2, 3, 4 and 5.

Directions Understand Vocabulary Have students: 5 draw a circle around the number one; 6 write the number three; 7 count the number of cubes, and then write the number to tell how many; 8 write the numbers 0 to 5 in order, and then draw counters to show that many of each number.

Topic 1 Reteaching

Set A

Question 1.

Answer:

Explanation:
I colored a box as i count each ball.There are 3 balls in the picture.

Question 2.

Answer:

Explanation:
I colored a box as i counted each ball.There ia 1 ball in the picture.

Set B

Question 3.

Answer:

Explanation:
I counted and practiced writing the number 2.There are 2 flowers in the vase.

Question 4.

Answer:

Explanation:
I counted and practiced writing the number 1.There are 1 flower in the vase.

Directions Have students: 1 and 2 color a box as they count each ball to show how many; 3 and 4 count the flowers in the vase, and then practice writing the number that tells how many.

Set C

Question 5.

Answer:

Explanation:
I counted the number of snakes and colored 5 boxes as there are 5 snakes.

Question 6.

Answer:

Explanation:
I counted the number of birds and colored 4 boxes as there are 4 birds

Directions 5 and 6 Have students color a box as they count each animal to show how many.

Set D

Question 7.

Answer:

Explanation:
The number that comes just before 3 is 2 and just after 3 is 4.So, i wrote the number 2 and 4.

Set E

Question 8.

Answer:

Explanation:
I counted the number of flowers in the vase and practiced writing the number 3.There are 3 flowers in the vase.

Question 9.

Answer:

Explanation:
I counted the number of flowers in the vase and practiced writing the number 0.There are no flowers in the vase.

Directions Have students: 7 write the numbers that are 1 less than 3 and 1 more than 3 when counting; 8 and 9 Count the flowers in the vase, and then practice writing the number that tells how many.

Set F

Question 10.

Answer:

Explanation:
I counted the number of octopuses and practiced writing number 3 as there are 3 octopuses.

Question 11.

Answer:

Explanation:
I drew 5 circles to show the number 5.

Directions 10 and 11 Have students count the octopuses or read the number, and then practice writing the number or draw circles to tell how many.

Topic 1 Assessment Practice

Question 1.

Answer:

Explanation:
I marked option c as it shows 3 flowers.

Question 2.

Answer:

Explanation:
I marked option D as it shows the number of counters 4.

Question 3.

Answer:

Explanation:
I marked the option D as it shows the number of pears in the picture.There are 5 pears.

Question 4.

Answer:

Explanation:
I marked the option A as it shows that the box has no toys in it.

Directions Have students mark the best answer. 1 Which shows 3 flowers? 2 How many counters are there? 3 How many pears are there? 4 Which box has 0 toys in it?

Question 5.

Answer:

Explanation:
I marked the option C as it shows 4 leaves.One more than the counters shown.

Question 6.

Answer:

Explanation:
I counted the butterflies and colored a box as i counted each buttetrfly.There are 2 butterflies.

Question 7.

Answer:

Explanation:
I drew counters to show the numbe rof dots.

Directions Have students: 5 listen to the story: Tina is counting leaves. She uses these counters to show how many leaves she has counted so far. Which of these shows the next number of leaves she will count? 6 count the butterflies, and then color the boxes to show how many; 7 count the number of dots and then draw counters in the box to show the same number of dots.

Question 8.

Answer:

Explanation:
I wrote the number 0, 1, 2, 3 and 4 as i counted the number of snowflakes given.

Question 9.

Answer:

Explanation:
I countec and wrote the number of red  and yellow counters.

Directions Have students: 8 count the snowflakes, and then write the number to tell how many; 9 listen to the story: Jack has some red and some yellow counters. He uses the counters to show one way to make 4. Write numbers to tell how many of each color he used.

Question 10.

Answer:

Explanation:
I counted the plates and wrote the numbe 3 as there are 3 plates.

Question 11.

Answer:

Explanation:
I didnot color any apple as there are no apples in the plate.

Question 12.

Answer:

Explanation:
I drew 5 marbles to show the number 5.

Directions Have students: 10 count the plates, and then write the number to tell how many; 11 count the number of apples on the plate, and then color the apples to show how many; 12 draw 5 marbles, and then write the number to tell how many.

Topic 1 Performance Task

Question 1.

Answer:

Explanation:
I counted and wrote the number of flowers in the vases.

Directions Flower Cart Say: Michael’s family sells flowers from a flower cart, 1 Have students count how many of each kind of flower, and then write the number to tell how many.

Question 2.

Answer:

Explanation:
As there are no flowers in the vase so, i wrote the number 0.

Question 3.

Answer:

Explanation:
I counted the flowers and colored a box for each flower i count.There are 4 flowers and i wrote the number 4 to show it.

Question 4.

Answer:

Explanation:
Michael can give his mother sunflowers.I drew 5 flowers andwrote the number 5 to show the number of flowers.

Directions 2 Say: How can Michael show the number of flowers in the green vase? Have students show two different ways. 3 Have students look at the 4 flowers, and then draw another way to show 4. 4 Say: Michael wants to give his mother 5 flowers from the cart. What kinds of flowers can he give her? Have students show one way to make an arrangement of 5 flowers, and write the number to tell how many.

Go through the enVision Math Common Core Grade 4 Answer Key Topic 6 Use Operations with Whole Numbers to Solve Problems regularly and improve your accuracy in solving questions.

enVision Math Common Core 4th Grade Answers Key Topic 6 Use Operations with Whole Numbers to Solve Problems

Essential Questions:
How is comparing with multiplication different from comparing with addition? How can you use equations to solve multi-step problems?
Answer:
In an additive comparison, we find the relation between two amounts by asking or telling how much more is one compared to the other. In multiplicative comparison, there are two different sets being compared. The first set contains a certain number of items. The second set contains multiple copies of the first set
Example:
Additive Comparison:
2 + 2 + 5 + 3 + … n
Multiplicative Comparison:
3 × 4 × 5 × … n

enVision STEM Project: Energy and Multiplication
Do Research
Use the Internet or other sources to find and describe 3 examples of renewable energy?
Answer:
The examples of renewable energy are:
a. Solar energy
b. Wind energy
c. Tidal energy
d. Geothermal energy

Journal: Write a Report that Includes what you found. Also in your report:
A)
Solar panels are made up of smaller modules or sections called cells. Find a picture of a solar panel. How many cells are in 6 solar panels? How many cells are in 9 solar panels? How many more cells are in the 9 solar panels than the 6 solar panels?
Answer:
It is given that
Solar panels are made up of smaller modules or sections called cells
Now,
We know that,
A solar power panel is made of photovoltaic cells arranged in a configuration that can contain 32, 36, 48, 60, 72, and 96 cells
Now,
Let a solar panel contains 32 cells
So,
The number of cells that present in 6 solar panels = (The number of solar panels) × (The number of cells present in a solar panel)
= 6 × 32
= 6 × (30 + 2)
= (6 × 30) + (6 × 2)
= 180 + 12
= 192 cells
So,
The number of cells that present in 9 solar panels = (The number of solar panels) × (The number of cells present in a solar panel)
= 9 × 32
= 9 × (30 + 2)
= (9 × 30) + (9 × 2)
= 270 + 18
= 288 cells
So,
The difference between the two solar panels = 288 – 192
= 96 cells
Now,
The picture of a solar panel is:

Hence, from the above,
We can conclude that
The difference of the cells between 9 solar panels and 6 solar panels is: 96 cells

B)
Find examples of other items that use solar power?
Answer:
Some of the items that use solar power are:
A) Solar charger.
B) Solar cell phone charger. Strawberry Tree.
C) Solar chimney.
D) Solar-powered waste compacting bin.
E) Solar cooker.
F) Solar dryer.
G) Solar-powered fan.
H) Solar furnace.

Review What You Know

Vocabulary
Choose the best term from the box. Write it on the blank.

• Associative Property of Multiplication
• Commutative Property of Multiplication
• Compatible numbers
• Distributive Property

Question 1.
The __________ says that factors can be multiplied in any order and the product remains the same.
Answer:
We know that,
The “Commutative Property of Multiplication” says that factors can be multiplied in any order and the product remains the same.
Hence, from the above,
We can conclude that the best term that is suitable for the given definition is: Cumulative Property of Multiplication

Question 2.
Breaking apart a multiplication problem into the sum or difference of two simpler multiplication problems is an example of using the __________
Answer:
We know that,
Breaking apart a multiplication problem into the sum or difference of two simpler multiplication problems is an example of using the “Distributive Property”
Hence, from the above,
We can conclude that the best term that is suitable for the given definition is: Distributive Property

Question 3.
According to the ________, factors can be regrouped and the product remains the same.
Answer:
We know that,
According to the “Associative property of Multiplication”, factors can be regrouped and the product remains the same.
Hence, from the above,
We can conclude that the best term that is suitable for the given definition is: Associative Property of Multiplication

Dividing by 1-Digit Numbers

Estimate each quotient.
Question 4.
16 ÷ 3
Answer:
The given expression is: 16 ÷ 3
Now,
By using the Long Division method,

Hence, from the above,
We can conclude that the quotient value for the given expression is: 5

Question 5.
25 ÷ 4
Answer:
The given expression is: 25 ÷ 4
Now,
By using the Long Division method,

Hence, from the above,
We can conclude that the quotient value for the given expression is: 6

Question 6.
155 ÷ 4
Answer:
The given expression is: 155 ÷ 4
Now,
By using the Long Division method,

Hence, from the above,
We can conclude that the quotient value for the given expression is: 38

Question 7.
304 ÷ 3
Answer:
The given expression is: 304 ÷ 3
Now,
By using the Long Division method,

Hence, from the above,
We can conclude that the quotient value for the given expression is: 101

Question 8.
1,283 ÷ 6
Answer:
The given expression is: 1,283 ÷ 6
Now,
By using the Long Division method,

Hence, from the above,
We can conclude that the quotient value for the given expression is: 213

Question 9.
1,999 ÷ 4
Answer:
The given expression is: 1,999 ÷ 4
Now,
By using the Long Division method,

Hence, from the above,
We can conclude that the quotient value for the given expression is: 499

Multiplying by 1-Digit Numbers

Find each product.
Question 10.
53 × 9
Answer:
The given expression is: 53 × 9
Now,
By using the partial Products method,
53 × 9
= (50 + 3) × 9
= (50 × 9) + (3 × 9)
= 450 + 27
= 477
Hence, from the above,
We can conclude that the product value for the given expression is: 477

Question 11.
1,127 × 7
Answer:
The given expression is: 1,127 × 7
Now,
By using the partial Products method,
1,127 × 7
= (1,000 + 100 + 27) × 7
= (1,000 × 7) + (100 × 7) + (27 × 7)
= 7,000 + 700 + 189
= 7,889
Hence, from the above,
We can conclude that the product value for the given expression is: 7,889

Question 12.
2,769 × 5
Answer:
The given expression is: 2,769 × 5
Now,
By using the partial Products method,
2,769 × 5
= (2,000 + 700 + 60 + 9) × 5
= (2,000 × 5) + (700 × 5) + (60 × 5) + (9 × 5)
= 10,000 + 3,500 + 300 + 45
= 13,845
Hence, from the above,
We can conclude that the product value for the given expression is: 13,845

Question 13.
3 × 215
Answer:
The given expression is: 3 × 215
Now,
By using the partial Products method,
3 × 215
= 3 × (200 + 15)
= (3 × 200) + (3 × 15)
= 600 + 45
= 645
Hence, from the above,
We can conclude that the product value for the given expression is: 645

Question 14.
914 × 5
Answer:
The given expression is: 914 × 5
Now,
By using the partial Products method,
914 × 5
= (900 + 14) × 5
= (900 × 5) + (14 × 5)
= 4,500 + 70
= 4,570
Hence, from the above,
We can conclude that the product value for the given expression is: 4,570

Question 15.
1,238 × 5
Answer:
The given expression is: 1,238 × 5
Now,
By using the partial Products method,
1,238 × 5
= (1,000 + 200 + 30 + 8) × 5
= (1,000 × 5) + (200 × 5) + (30 × 5) + (8 × 5)
= 5,000 + 1,000 + 150 + 40
= 6,190
Hence, from the above,
We can conclude that the product value for the given expression is: 6,190

Problem Solving

Question 16.
Construct Arguments Explain why the array represents 3 × 21

Answer:
The given array is:

Now,
We know that,
An “Array” is a collection of rows and columns
Now,
From the given figure,
We can observe that
The number of rows is: 21
The number of columns is: 3
So,
To find the number of cells in a given array,
We have to multiply the rows and columns
Hence, from the above,
We can conclude that the given array represents 21 × 3 since it represents the number of rows and number of columns

Question 17.
James multiplies 38 by 55. He finds three of the four partial products: 40, 150, and 400. Which partial product is James missing? What is the solution?
Answer:
It is given that
James multiplies 38 by 55. He finds three of the four partial products: 40, 150, and 400.
Now,
The given expression is: 38 × 55
Now,
By using the partial Products method,
38 × 55
= (30 + 8) × (50 + 5)
= (30 × 50) + (30 × 5) + (8 × 50) + (8 × 5)
= 1,500 + 150 +400 + 40
= 2,090
Hence, from the above,
We can conclude that
The Partial product James is missing is: 1,500
The solution for the given expression is: 2,090

Pick a Project

PROJECT 6A
How tall is tall?
Project: Model the Height of a Redwood Tree

PROJECT 6B
What are some interesting number facts about a manatee?
Project: Make a Presentation with Manatee Number Facts

PROJECT 6C
How does temperature affect an alligator egg?
Project: Create a Bar Diagram

PROJECT 6D
How many eggs does the loggerhead turtle lay?
Project: Model a Turtle’s Egg

Lesson 6.1 Solve Comparison Problems

Solve & Share
Sarah is making a square pillow with edges that each measure 18 inches long. She needs a strip of fabric 4 times as long as one edge of the pillow to make a border around the pillow. How long does the strip of fabric need to be? Solve this problem any way you choose.
I can … use multiplication or addition to compare one quantity to another.

Answer:
It is given that
Sarah is making a square pillow with edges that each measure 18 inches long. She needs a strip of fabric 4 times as long as one edge of the pillow to make a border around the pillow
Now,
Let the length of the strip of fabric be x
So,
According to the given data,
The length of the strip (x) = 4 × (The length of the pillow at the edges)
= 4 × 18
= 72 inches
Hence, from the above,
We can conclude that
The length of the fabric needed is: 72 inches

Look Back!
Model with Math
How could a bar diagram help you write and solve an equation for the problem?
Answer:
It is given that
Sarah is making a square pillow with edges that each measure 18 inches long. She needs a strip of fabric 4 times as long as one edge of the pillow to make a border around the pillow
Now,
The representation of the given situation in the form of a bar diagram is:

So,
From the above bar diagram,
We can observe that
The length of the string = 4 × (The length of the edge of the pillow)
= 4 × 18
= 72 inches
Hence, from the above,
We can conclude that
By using the bar diagram, the length of the string is: 72 inches

Essential Question
How Is Comparing with Multiplication Different from Comparing with Addition?
Answer:
In an additive comparison, we find the relation between two amounts by asking or telling how much more is one compared to the other. In multiplicative comparison, there are two different sets being compared. The first set contains a certain number of items. The second set contains multiple copies of the first set
Example:
Additive Comparison:
2 + 2 + 5 + 3 + … n
Multiplicative Comparison:
3 × 4 × 5 × … n

Visual Learning Bridge
Max said the Rangers scored 3 times as many runs as the Stars. Jody said the Rangers scored 8 more runs than the Stars. Could both Max and Jody be correct?

Compare with Multiplication
Find 3 times as many as 4 runs.

m = 3 × 4
m = 12 runs
According to Max, the Rangers scored 12 runs.

Compare with Addition
Find 8 more than 4 runs.

j = 4 + 8
j = 12 runs
According to Jody, the Rangers scored 12 runs.
Max and Jody are both correct.

Convince Me!
Construct Arguments Describe when you would use multiplication and when you would use addition to make a comparison?
Answer:
When distinguishing multiplicative comparison from additive comparison,
We should note that “Additive comparisons” focus on the difference between two quantities whereas “Multiplicative comparisons” focus on the difference between two qualities

Guided Practice

Do You Understand?
Question 1.
Use the information from the previous page. If the Rangers scored 5 times as many runs as the Stars, how many runs would the Rangers have scored?
a. Compare using multiplication. Write and solve an equation.
Answer:
It is given that
Rangers scored 5 times as many runs as the Stars
Now,
The given figure is:

Now,
The representation of the given situation in the form of a bar diagram is:

So,
The total number of runs rangers scored = 5 × (The number of runs Stars scored)
= 5 × 4
= 20 runs
Hence, from the above,
We can conclude that the number of runs scored by Rangers is: 20

b. How could you use addition to compare the runs for the two teams?
Answer:
It is given that
Rangers scored 5 times as many runs as the Stars
Now,
The given figure is:

Now,
The representation of the given situation in the form of a bar diagram is:

So,
The total number of runs rangers scored = (The number of runs Stars scored) + (The number of runs Stars scored) + (The number of runs Stars scored) + (The number of runs Stars scored) + (The number of runs Stars scored)
= 5 + 5 + 5 + 5 +5
= 20 runs
Hence, from the above,
We can conclude that the number of runs scored by Rangers is: 20

Do You Know How?
For 2-3, complete the comparison sentence. Find the value of the variable that makes the sentence true.
Question 2.
Sam’s baseball team has 4 times as many helmets as Ed’s team. Ed’s team has 21 helmets. Let h = the number of helmets Sam’s team has.
h is ___________ times as many as ________.
h = _________
Answer:
It is given that
Sam’s baseball team has 4 times as many helmets as Ed’s team. Ed’s team has 21 helmets.
So,
The number of helmets Sam’s team has (h) = 4 × (The number of helmets Ed’s team has)
= 4 × 21
= 4 × (20 + 1)
= (4 × 20) + (4 × 1)
= 80 + 4
= 84 helmets
Hence, from the above,
We can conclude that
h is 4 times as many as the number of helmets Ed’s team has
The number of helmets Sam’s team has is: 84 helmets

Question 3.
There are 128 more trees in the park than at Ty’s house. There are 3 trees at Ty’s house. Let t = the number of trees in the park.
_______ more than _____ is t.
t = __________
Answer:
It is given that
There are 128 more trees in the park than at Ty’s house. There are 3 trees at Ty’s house.
So,
The number of trees in the park (t) = (The number of trees at Ty’s house) + (The number of more trees in the park)
= 3 + 128
= 131 trees
Hence, from the above,
We can conclude that
128 trees more than 3 trees is t
The total number of trees in the park is: 131 trees

Independent Practice

Use times as many as or more than to compare the amounts.
For 4-9, write a comparison sentence. For 4-7, find the value of the variable that makes the sentence true.

Question 4.
Katy has 6 times as many nickels as Shaun. Shaun has 18 nickels. Let n = the number of nickels Katy has.
n is _____________
n = ___________.
Answer:
It is given that
Katy has 6 times as many nickels as Shaun. Shaun has 18 nickels.
So,
The number of nickels Katy has (n) = 6 × (The number of nickels Shaun has)
= 6 × 18
= 6 × (10 + 8)
= (6 × 10) + (6 × 8)
= 60 + 48
= 108 nickels
Hence, from the above,
We can conclude that
n is 6 times as many as the number of nickels Shaun has
The number of nickels Katy has: 108 nickels

Question 5.
Kyle has watched 238 movies. Jason has watched 49 more movies than Kyle. Let m = the number of movies Jason has watched.
m is _____________
m = _____________
Answer:
It is given that
Kyle has watched 238 movies. Jason has watched 49 more movies than Kyle
So,
The number of movies Jason has watched (m) = The number of movies Kyle watched + 49
= 238 + 49
= 287 movies
Hence, from the above,
We can conclude that
m is 49 mere than the number of movies Kyle has watched
The number of movies Jason has watched is: 287 movies

Question 6.
Amber tied 89 knots to make a macrame wall hanging. Hunter tied 3 times as many knots as Amber. Let k= the number of knots Hunter tied. k is _____________
k = _____________
Answer:
It is given that
Amber tied 89 knots to make a macrame wall hanging. Hunter tied 3 times as many knots as Amber
So,
The number of knots Hunter tied (k) = 3 × (The number of knots Amber tied)
= 3 × 89
= 3 × (80 + 9)
= (3 × 80) + (3 × 9)
= 240 + 27
= 267
Hence, from the above,
We can conclude that
k is 3 times as many knots as Amber
The number of knots Hunter tied is: 267 knots

Question 7.
Tia sells 292 newspapers. Tess sells 117 more newspapers than Tia. Let n = the number of newspapers Tess sells.
n is _____________
n = _____________
Answer:
It is given that
Tia sells 292 newspapers. Tess sells 117 more newspapers than Tia
So,
The number of newspapers Tess sells (n) = 117 + (The number of newspapers tia sells)
= 117 + 292
= 409 newspapers
Hence, from the above,
We can conclude that
n is 117 more than Tia’s newspapers
The number of newspapers Tess sells is: 409 newspapers

Question 8.
Trent has 48 markers. Sharon has 8 markers.
Answer:

Question 9.
Lucy has 317 bottles. Craig has 82 bottles.
Answer:

Problem Solving

Question 10.
Model with Math
Roger swam 19 laps in the pool. Anna Maria swam 4 times as many laps as Roger. How many laps did Anna Maria swim? Draw a bar diagram and write an equation to solve the problem.
Answer:
It is given that
Roger swam 19 laps in the pool. Anna Maria swam 4 times as many laps as Roger
So,
The representation of the given situation in the form of a bar diagram is:

So,
The number of laps did Anna maria swam = 4 × (The number of laps did Roger swam)
= 4 × 19
= 4 × (10 + 9)
= (4 × 10) + (4 × 9)
= 40 + 36
= 76 laps
Hence, from the above,
We can conclude that the number of laps did Anna swam is: 76 laps

Question 11.
Critique Reasoning Nina says the equation 600 = 12 × 50 means 600 is 12 times as many as 50. Julio says the equation means 600 is 50 times as many as 12. Who is correct? Explain.
Answer:
It is given that
Nina says the equation 600 = 12 × 50 means 600 is 12 times as many as 50. Julio says the equation means 600 is 50 times as many as 12
Now,
We know that,
According to Commutative Property of Multiplication,
a × b = b × a
So,
12 × 50 is the same as 50 × 12
“12 times as many as 50” is the same as “50 times as many as 12”
Hence, from the above,
We can conclude that
Both Nina and Julio are correct

Question 12.
Vocabulary The amount that is left after dividing a number into equal parts is called the _________.
13 ÷ 4 = ________

Answer:
We know that,
The amount that is left after dividing a number into equal parts is called the “Remainder”
Now,
The given expression is: 13 ÷ 4
Now,
By using the Long division method,

Hence, from the above,
We can conclude that the remainder for the given expression is: 1

Question 13.
Higher-Order Thinking A shirt is on sale for d dollars. The regular price is 4 times as much. Todd has enough money to buy 2 shirts at the regular price. How many shirts can Todd buy at the sale price? Explain.
Answer:
It is given that
A shirt is on sale for d dollars. The regular price is 4 times as much. Todd has enough money to buy 2 shirts at the regular price
So,
The regular price of a shirt = 4 × (The sale price of a shirt)
= 4 × d
= 4d
Now,
According to the given information,
The sale price of the shirts that Todd can buy = (Regular price of a shirt) × (The number of shirts Todd can buy)
= 4d × 2
= 8d
So,
The number of shirts Todd can buy = \(\frac{The sale price of the shirts Todd can buy}{The sale price of a shirt}\)
= \(\frac{8d}{d}\)
= 8
Hence, from the above,
We can conclude that Todd can buy 8 shirts at the sale price

Assessment Practice

Question 14.
Select all the sentences that involve a comparison using multiplication.
☐ k is 26 times as many as 7.
☐ u is 18 more than 314.
☐ Tom ran 4 miles. Cindy ran 2 more miles than Tom. How many miles did Cindy run?
☐ Yuan has 2 dogs and Jon has 3 times as many dogs. How many dogs does Jon have?
☐ Kris has 4 times as many pairs of shoes as her brother. Her brother has 8 pairs of shoes.
Answer:
The sentences that involve a comparison using multiplication are:

Question 15.
Select all the statements that can be represented by the equation 5 × 9 = w.
☐ w is 9 more than 5.
☐ w is 5 times as much as 9.
☐ Henry sang 9 songs during practice. He sang 5 times as many as Joe, who sang w songs.
☐ Greg carried 9 buckets of water to his sister’s baby pool. His mother carried w buckets, which were 5 times as many.
☐ Tom has 9 pens. Joan has w pens, which is 5 fewer.
Answer:
The sentences that can be represented by the equation 5 × 9 = w are:

Lesson 6.2 Continue to solve Comparison Problems

Solve & Share
The students in Ms. Chang’s fourth-grade class plant a tree every year. One tree they planted is now 288 inches tall. This height is 6 times as great as when the tree was first planted. How tall was the tree when it was first planted? Solve this problem any way you choose. Show your work.
I can … use multiplication or division to compare one quantity to another.

Answer:
It is given that
The students in Ms. Chang’s fourth-grade class plant a tree every year. One tree they planted is now 288 inches tall. This height is 6 times as great as when the tree was first planted
Now,
Let the height of the tree when first planted be x
Now,
According to the given information,
6 × x = 288
x = \(\frac{288}{6}\)
x = 48 inches
Hence, from the above,
We can conclude that
The height of the tree when it was first planted is: 48 inches

Look Back! Compare the heights of the 2 trees using addition.
Answer:
From part (a),
We can observe that
The height of the tree when it was first planted is: 48 inches
The current height of the tree is: 288 inches
So,
The difference between the heights of the two trees = (The current height of the tree) – (The height of the tree when it was first planted)
= 288 – 48
= 240 inches
Hence, from the above,
We can conclude that the current height of the tree is 240 inches more than the height of the tree when it was first planted

Essential Question
How Can You Solve a Problem Involving Multiplication as Comparison?
Answer:
Interpret a multiplication equation as a comparison
Example:
Interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5.

Visual Learning Bridge
Carla and Calvin are twins that attend different colleges. Carla’s college is four times as far from home as Calvin’s college. How far does Calvin travel to college?

The number of miles Carla travels, or 192 miles, is 4 times as far as Calvin travels.

Write a multiplication equation to find the number of miles Calvin travels to college.
192 = 4 × m
m = 48 miles
What number times 4 equals 192?

If 192 = 4 × m, them m = 192 ÷ 4.

Calvin travels 48 miles to college.

Convince Me!

Use Structure When do you use division to make a comparison?
Answer:
A ratio is a comparison of two numbers by division. The value of a ratio is the quotient that results from dividing the two numbers.
Example:
The value of the ratio 35: 7 is 5, which you find by computing 35 ÷ 7 = 5.

Guided Practice

Do You Understand?
Question 1.
Carla travels 3 times as far to college as her friend Mandy travels to college. Write and solve a related multiplication and division equation to find how far Mandy travels to college.
Answer:
It is given that
Carla travels 3 times as far to college as her friend Mandy travels to college
Now,
From the example on the previous page,
We can observe that
Carla travels 192 miles to college
Now,
Let x be  how far Mandy travels to college
So,
From the given information,
192 = 3 × x
x = \(\frac{192}{3}\)
x = 64 miles
Hence, from the above,
We can conclude that Mandy travels 64 miles to the college

Do You Know How?
Question 2.
Complete the comparison sentence. Find the value of the variable that makes the sentence true.
If 3 × m = 48, then m = 48 ÷ 3.
3 times as many as m is 48.
m = 16
Answer:
The given expression is:
3 × m = 48
m = \(\frac{48}{3}\)
m = 16
Hence, from the above,
We can conclude that
3 times as many as m is 48
The value of m is: 16

Independent Practice

For 3-8, write a comparison sentence and an equation. Find the value of the variable that makes the sentence true.
Question 3.
Connor has 77 magazines. That is 7 times as many magazines Kristen has. How many magazines, n, does Kristen have?
Answer:
It is given that
Connor has 77 magazines. That is 7 times as many magazines Kristen has
Now,
Let n be the number of magazines Kristen has
So,
According to the given information,
7 × n = 77
n = \(\frac{77}{7}\)
n = 11
Hence, from the above,
We can conclude that
The number of magazines that Kristen has: 11

Question 4.
Eric completed 75 math problems. That is 5 times as many math problems as Katie completed. How many math problems, m, did Katie complete?
Answer:
It is given that
Eric completed 75 math problems. That is 5 times as many math problems as Katie completed
Now,
Let m be the math problems Katie completed
So,
According to the given information,
5 × m = 75
m = \(\frac{75}{5}\)
m = 15
Hence, from the above,
We can conclude that
The number of math problems Katie completed is: 15

Question 5.
Clare counted 117 different colors at the paint store. That is 9 times as many as the number James counted. How many different colors, c, did James count?
Answer:
It is given that
Clare counted 117 different colors at the paint store. That is 9 times as many as the number James counted.
Now,
Let c be the number of colors James counted
So,
According to the given information,
9 × c = 117
c = \(\frac{117}{9}\)
c = 13
Hence, from the above,
We can conclude that
The number of different colors James counted is: 13

Question 6.
Alisa has 153 dominoes. That is 3 times as many dominoes as Stan has. How many dominoes, d, does Stan have?
Answer:
It is given that
Alisa has 153 dominoes. That is 3 times as many dominoes as Stan has
Now,
Let d be the number of dominoes does Stan have
So,
According to the given information,
3 × d = 153
d = \(\frac{153}{3}\)
d = 51
Hence, from the above,
We can conclude that
The number of dominoes does Stan have is: 51

Question 7.
Justin practiced piano for h hours. His sister practiced for 12 hours, which was 3 times as many hours as Justin practiced. Write and solve an equation to find how many hours Justin practiced piano.
Answer:
It is given that
Justin practiced piano for h hours. His sister practiced for 12 hours, which was 3 times as many hours as Justin practiced
Now,
Let h be the number of hours Justin practiced piano
So,
According to the given information,
h × 3 = 12
h = \(\frac{12}{3}\)
h = 4
Hence, from the above,
We can conclude that
The number of hours that Justin practiced piano is: 4 hours

Question 8.
Mary practiced violin for 2 hours and her brother practiced trombone t times as long or 8 hours. Write and solve an equation to find how many times as long Mary’s brother practiced trombone.
Answer:
It is given that
Mary practiced violin for 2 hours and her brother practiced trombone t times as long or 8 hours
Now,
Let t be the number of hours Mary’s brother practiced trombone
So,
According to the given information,
2 × t = 8
t = \(\frac{8}{2}\)
t = 4
Hence, from the above,
We can conclude that
The number of hours mary’s brother practiced trombone is: 4 hours

Problem Solving

Question 9.
Model with Math
Dave is making soup that includes 12 cups of water and 3 cups of broth. How many times as much water as the broth will be in the soup? Draw a bar diagram and write and solve an equation.
Answer:
It is given that
Dave is making soup that includes 12 cups of water and 3 cups of broth.
Now,
Let x be the number of cups of water present in the broth
So,
According to the given information,
3 × x = 12
x = \(\frac{12}{3}\)
x = 4

Hence, from the above,
We can conclude that
The number of cups of water present in the broth is: 4

Question 10.
Trevor wants to buy three light fixtures that cost $168 each. He has $500. Does he have enough money to buy the three light fixtures? Use a comparison sentence to explain your reasoning.
Answer:
It is given that
Trevor wants to buy three light fixtures that cost $168 each. He has $500.
So,
The cost of three light fixtures that Trevor wants to buy = 3 × (The cost of each light fixture)
= 3 × 168
= 3 × (100 + 68)
= (3 × 100) + (3 × 68)
= 300 + 204
= $504
So,
From the given information,
$500 < $504
Hence, from the above,
We can conclude that
The cost of the three light fixtures that Trevor wants to buy is $4 more than the amount of money Trevor has
He does not have enough money to buy the three light fixtures

Question 11.
Miranda has 4 times as many leaves in her collection as Joy Joy has 13 more leaves than Armani. Armani has 10 leaves in his collection. How many leaves does Miranda have in her collection? Explain.
Answer:
It is given that
Miranda has 4 times as many leaves in her collection as Joy Joy has 13 more leaves than Armani. Armani has 10 leaves in his collection
Now,
Let the number of leaves Miranda has in her collection be x
Let the number of leaves Joy has be j
Let the number of leaves Armani has be a
So,
According to the given information,
a = 10
j = a + 13
4 × j = x
Now,
j = 10 + 13
j = 23
So,
4 × 23 = x
x = 4 × (20 + 3)
x = (4 × 20) + (4 × 3)
x = 80 + 12
x = 92 leaves
Hence, from the above,
We can conclude that the number of leaves Miranda has in her collection is: 92

Question 12.
Higher-Order Thinking Jordan needs $9,240 for her first year of college tuition. Each of her two grandfathers said they would match what she saves. She has 8 years before she goes to college. How much does Jordan need to save on her own each year to have enough for her first year with the help from her two grandfathers?

Answer:
It is given that
Jordan needs $9,240 for her first year of college tuition. Each of her two grandfathers said they would match what she saves. She has 8 years before she goes to college
So,
The amount of money that will be saved by Jordan and her 2 grandfathers = \(\frac{$9,240}{3}\)
= $3,080
So,
The amount of money Jordan has to save each year so that she would have enough money for her first year = \(\frac{$3,080}{8}\)
= $385
Hence, from the above,
We can conclude that
Jordan needs to save on her own each year to have enough for her first year with the help from her two grandfathers will be: $385

Assessment Practice

Question 13.
Tina walked 20 miles for a fundraiser. Lia walked m miles. Tina walked 4 times as far as Lia. Which equation can be used to find m, the number of miles Lia walked?
A. m = 4 × 20
B. 20 = 4 × m
C. 20 = m ÷ 4
D. m = 20 + 4
Answer:
It is given that
Tina walked 20 miles for a fundraiser. Lia walked m miles. Tina walked 4 times as far as Lia
So,
According to the given information,
4 × m = 20
Hence, from the above,
We can conclude that the number of miles Lia walked is:

Question 14.
Jason and Raul kept a reading log for the year. Jason read 7 books and Raul read 105. How many times as many books as Jason did Raul read?
A. 13 times
B. 14 times
C. 15 times
D. 16 times
Answer:
It is given that
Jason and Raul kept a reading log for the year. Jason read 7 books and Raul read 105
Now,
Let the number of times Jason read as many as Raul did be x
So,
(The number of books Jason read) × (The number of times Jason read as many as Raul did) = The number of books Raul did
7 × x = 105
x = \(\frac{105}{7}\)
x = 15
Hence,from the above,
We can conclude that the number of times Raul read as many books as Jason is:

Lesson 6.3 Model Multi-Step Problems

Solve & Share
Last year, 18 people went on a family camping trip. This year, three times as many people went. How many more people went this year than last year? Complete the bar diagram and show how you solve it?
I can … find hidden questions and use bar diagrams and equations to model and solve multi-step problems.

Answer:
It is given that
Last year, 18 people went on a family camping trip. This year, three times as many people went
So,
The number of people that went on a family camping trip this year = 3 × (The number of people that went on a family camping trip last year)
= 3 × 18
= 3 × (10 + 8)
= (3 × 10) + (3 × 8)
= 30 + 24
= 54 people
So,
The representation of the number of people that went on a camping trip in the form of a bar diagram is:

Hence, from the above,
We can conclude that the number of people that went on a camping trip this year is: 54

Look Back!
How can you use estimation to decide if your answer is reasonable?
Answer:
From the above problem,
We can observe that
The expression we used to find the number of people that went on a camping trip this year is:
3 × 18
Now,
Estimate:
3 × 18
= 3 × 20
= 3 × 2 ×10
= (3 × 2) × 10
= 6 × 10
= 60
So,
The difference between the number of people we obtained in the previous problem and in this problem is not large
Hence, from the above,
We can conclude that your estimation for the answer is reasonable

Essential Question
How Can You Use Diagrams and Equations to Solve Multi-Step Problems?
Answer:
The process we can use to solve Multi-step problems is:
Step 1:
Circle and underline. Circle only the necessary information and underline what ultimately needs to be figured out.
Step 2:
Figure out the first step/problem in the paragraph and solve it.
Step 3:
Find the answer by using the information from Steps 1 and 2

Visual Learning Bridge
A school uses 8 buses to transport the fourth-grade students and 6 vans to transport the third-grade students on a field trip. How many fewer third-grade students than fourth-grade students are on the field trip?

Step 1
Find and answer the hidden questions.
Hidden Question: How many fourth-grade students are on the field trip?

F = 8 × 24
= 192 fourth-grade students
Hidden Question: How many third-grade students are on the field trip?

T= 6 × 15
= 90 third-grade students

Step 2
Use the answers to the hidden questions to answer the original question.
Original Question: How many fewer third-grade students than fourth-grade students are on the field trip?
S = the number fewer third-grade students than fourth-grade students

S = 192 – 90
= 102
102 fewer third-grade students than fourth-grade students are on the field trip.

Convince Me!
Construct Arguments Amanda says she can solve the problem using the equation below. Do you agree? Explain.
S = (8 × 24) – (6 × 15)
Answer:
It is given that
Amanda says she can solve the problem using the equation below.
S = (8 × 24) – (6 × 15)
Now,
From the example on the previous page,
We have to find out the number of fewer third-grade students than the fourth-grade students
So,
We can find the difference between the third-grade and the fourth-grade students by finding the total number of third-grade and the fourth-grade students
So,
The total number of fourth-grade students = (The number of buses) × (The number of students in each bus)
= 8 × 24
The total number of third-grade students = (The number of vans) × (The number of students in each van)
= 6 × 15
So,
The difference between the fourth-grade and the third-grade students = (8 × 24) – (6 × 15)
Hence, from the above,
We can conclude that we can agree with Amanda

Another Example!
The Lopez family gets 2 pizzas and 4 hot dogs at Eat-It-Up. What is the total cost?

Guided Practice

Do You Understand?
Question 1.
Doug says he can use the equation T = 18 + 18 + 6 + 6 + 6 + 6 to solve the problem above. Is Doug correct? Explain.
Answer:
It is given that
Doug says he can use the equation T = 18 + 18 + 6 + 6 + 6 + 6 to solve the problem above.
Now,
From the example on the previous page,
We can observe that
The cost of each pizza is $18 and there are 2 pizzas
The cost of each hot-dog is $6 and there are 4 hot-dogs
So,
The total cost = (The total cost of pizzas) + (The total cost of hot-dogs)
= [(The cost of each pizza) × (The total number of pizzas)] + [(The cost of each hot-dog) × (The total number of hot-dogs)]
= [18 × 2] + [6 × 4]
= [18 + 18] + [6 + 6 + 6 + 6]
Hence, from the above,
We can conclude that Doug is correct

Question 2.
Write one or more equations you could use to find the total cost C, of 2 turkey wraps, 2 hamburgers, and 4 drinks.

Answer:
It is given that
There are 2 turkey wraps, 2 hamburgers, and 4 drinks
Now,
The given table is:

So,
The total cost of the three items (C) = [(The cost of each turkey wrap) × (The number of turkey wraps)] + [(The cost of each hamburger) × (The number of hamburgers)] + [(The cost of each drink) × (The number of drinks)]
= (2 × $4) + (2 × $8) + (4 × $2) ———-> 1 way of writing the equation to find the total cost
= ($4 + $4) + ($8 + $8) + ($2 + $2 + $2 + $2) —– > Other way of writing the equation to find the total cost
= $8 + $16 + $8
= $32
Hence, from the above,
We can conclude that
The total cost of all the given items is: $32

Do You Know How?
Question 3.
In the problem on the previous page, suppose the school fills 11 buses and lets the remaining students ride in vans. How many students ride in vans? Use one or more bar diagrams and equations to show how you solve.
Answer:
It is given that
In the problem on the previous page, the school fills 11 buses and the remaining students ride in vans.
Now,
From the example on the previous page,
The total number of students = (The number of fourth-grade students that travel in buses) + (The number of third-grade students that travel in vans)
= 192 + 90
= 282 students
Now,
We know that,
The number of students each bus carries is: 24
So,
The total number of students present in 11 buses = (The number of students present in each bus) × (The number of buses)
= 24 × 11
= 24 × (10 + 1)
= (24 × 10) + (24 × 1)
= 240 + 24
= 264 students
So,
The number of remaining students who travel in vans = (The total number of students) – (The number of students who travel in 11 buses)
= 282 – 264
= 18 students
Now,
The representation of the given problem in the form of bar diagrams is:


Hence, from the above,
We can conclude that the number of students that ride in vans is: 18

Independent Practice

Draw a bar diagram, and write an equation to solve the problem. Use a variable to represent an unknown quantity and tell what the variable represents.
Question 4.
Four boys and five girls went to the movies together. Between them, they had $120 to spend. Tickets cost $8 each. How much money did they have to buy refreshments?
Answer:
It is given that
Four boys and five girls went to the movies together. Between them, they had $120 to spend. Tickets cost $8 each
So,
The total number of people that went to the movies = 4 + 5
= 9
Now,
We know that,
The total money spent by 9 people = (The amount of money spent on Tickets) + (The amount of money spent to buy refreshments)
$120 = [(The total number of people) × (The cost of each ticket)] + (The amount of money spent to buy refreshments)
$120 = (9 × $8) + (The amount of money spent to buy refreshments)
The amount of money spent to buy refreshments = $120 – $72
= $48
Now,
The representation of the given information in the form of a bar diagram is:

Hence, from the above,
We can conclude that the amount of money spent on refreshments is: $48

Problem Solving

Question 5.
Model with Math
A power plant has 4 tons of coal. A ton of coal produces 2,460 kilowatt-hours of electricity. The plant reserves enough electricity to power 9 light bulbs for a year. How many additional kilowatt-hours of electricity is produced? Draw the bar diagrams and write one or more equations to show how you solve them. Tell what your variables represent.

Answer:
It is given that
A power plant has 4 tons of coal. A ton of coal produces 2,460 kilowatt-hours of electricity. The plant reserves enough electricity to power 9 light bulbs for a year
Now,
The energy produced by 4 tons of coal = (The energy produced by a ton of coal) × (The number of tons of coal)
= 2,460 × 4
(2,000 + 400 + 60) × 4
= (2,000 × 4) + (400 × 4) + (60 × 4)
= 8,000 + 1,600 + 240
= 9,280 Kilowatt-hours
Now,
The energy produced by 9 light bulbs for a year = (The energy produced by a light bulb for a year) × (The number of light bulbs)
= 876 × 9
= (800 + 70 + 6) × 9
= (800 × 9) + (70 × 9) + (6 × 9)
= 7,200 + 630 + 54
= 7,884 Kilowatt-hours
So,
The number of additional Kilowatt-hours produced = (The energy produced by 4 tons of coal) – (The energy produced by 9 light bulbs for a year)
= 9,280 – 7,884
= 1,396 Kilowatt-hours
Now,
The representation of the given information in the form of a bar diagram is:

Hence, from the above,
We can conclude that the amount of additional Kilowatt-hours produced is: 1,39 Kilowatt-hours

Question 6.
Higher-Order Thinking You have $360 to buy 26 tickets for a baseball game. You need to buy some of each kind of seat. You want to spend most of the money. How many of each type of ticket can you buy? Find two different solutions to the problem. Use an equation to show each solution.

Answer:
It is given that
You have $360 to buy 26 tickets for a baseball game. You need to buy some of each kind of seat. You want to spend most of the money.
Now,
The given cost for each type of seat is:

Now,
Let the number of tickets for Box seats be b
Let the number of tickets for Upper deck be u
So,
In terms of total number of tickets,
b + u = 26 —- (1)
So,
In terms of cost of the tickets,
Total cost =[ (The cost of each ticket of box seat) × (The number of tickets of box seats)] + [(The cost of each ticket of Upper deck) × (The number of tickets of Upper deck)]
18b + 12u = 360 —- (2)
So,
b = 26 – u —- (3)
Now,
By using the method of substitution,
18 (26 – u) + 12u = 350
(18 × 26) – 18u + 12u = 350
[18 × (20 + 6)] – 6u = 350
[(18 × 20) + (18 × 6)] – 6u = 350
[360 + 108] – 6u = 350
468 – 6u = 350
6u = 468 – 360
6u = 108
u = \(\frac{108}{6}\)
u = 18 tickets
So,
b = 26  -u
= 26  -18
= 8 tickets
Hence, from the above,
We can conclude that
The number of tickets of box seat is: 8
The number of tickets of Upper desk is: 18

Assessment Practice

Question 7.
The gym teacher has $250 to spend on volleyball equipment. She buys 4 volleyball nets for $28 each volleyball costs $7 each. How many volleyballs can she buy? Explain how you solve. Use one or more equations and bar diagrams in your explanation. Tell what your variables represent.
Answer:
It is given that
The gym teacher has $250 to spend on volleyball equipment. She buys 4 volleyball nets for $28 each volleyball costs $7 each.
Now,
Let the number of volleyballs be x
So,
The total cost of volleyball equipment = (The total cost of volleyball nets) + (The total cost of volleyballs)
$250 = [(The cost of each volleyball net) × (The number of volleyballs)] + [(The number of volleyballs) × (The cost of each volleyball)]
$250 = (4 × $28) + (x × $7)
$250 = 4 × (20 + 8) + x × 7
$250 = (4 × 20) + (4 × 8) + x× 7
$250 = 80 + 32 + 7x
$250 = 112 + 7x
7x = 250 – 112
7x = 138
x = \(\frac{138}{7}\)
x = 19.7
x ≈ 20
Now,
The representation of the given information in the form of a bar diagram is:

Hence, from the above,
We can conclude that the number of volleyballs is: 20

Lesson 6.4 More Model MultiStep Problems

Solve & Share
Three eight-year-old children, one adult, and one senior had $125 saved to go to the Happy Days amusement park. After buying tickets, how much money did they have left to buy refreshments? Complete the bar diagram, and use it to write one or more equations to solve the problem. Tell what your variables represent.
I can … Solve multi-step problems by drawing bar diagrams and writing expressions and equations.

Answer:
It is given that
Three eight-year-old children, one adult, and one senior had $125 saved to go to the Happy Days amusement park
Now,
The given table is:

Now,
From the given table,
The ticket cost of 1 8-year old child is: $12
The ticket cost of 1 adult is: $25
The ticket cost of 1 senior is: $18
So,
The total amount of money saved by the given people = (The total ticket cost of the people) + (The total cost of buying refreshments for the people)
= $125 = [(3 × The ticket cost of each 8-year old child) + (The ticket cost of 1 adult) + (The ticket cost of 1 senior)] + (The total cost of buying refreshments for the people)
$125 = [(3 × $12) + $25 + $18] + (The total cost of buying refreshments for the people)
$125 = ($36 + $25 + $18) + (The total cost of buying refreshments for the people)
The total cost of buying refreshments for the people = $125 – $79
The total cost of buying refreshments for the people = $46
Now,
The representation of the given information in the form of a bar diagram is:

Hence, from the above,
We can conclude that
The total cost spent on refreshments by the given people is: $46

Look Back! What expressions and equations did you use to solve this problem? What hidden questions did you need to answer?
Answer:
The equation used to solve the above problem is:
The total amount of money saved by the given people = (The total ticket cost of the people) + (The total cost of buying refreshments for the people)
Now,
The hidden question we have to answer for the above problem is:
What is the total cost of the tickets for 3 8-year old children, 1 adult, and 1 senior?

Essential Question
How Can You Model and Solve Multi-Step Problems?
Answer:
The steps to solve Multi-steps Problems are:
Step 1:
Circle and underline. Circle only the necessary information and underline what ultimately needs to be figured out.
Step 2:
Figure out the first step/problem in the paragraph and solve it.
Step 3:
Find the answer by using the information from Steps 1 and 2

Visual Learning Bridge
Chef Angela needs 8 cartons of eggs to make the cakes that are ordered. She has 2 cartons of eggs and 4 single eggs in the refrigerator. How many more eggs does she need to make all of the cakes?

One Way
How many eggs does the chef have?
(2 × 12) + 4 = 24 + 4
= 28 eggs
How many more eggs does the chef need?
N = number of additional eggs needed
N = (8 × 12) – 28
= 96 – 28 = 68
The expression 8 × 12 is how many eggs the chef needs in all.
The chef needs 68 more eggs.
Since 68 is close to 72, the answer is reasonable.

Another Way
How many full cartons should the chef buy?
8 – 2 = 6 full cartons
How many more eggs does the chef need?
N = number of additional eggs needed
N =(6 × 12) – 4
= 72 – 4 = 68
The expression 6 × 12 is the number of eggs in 6 cartons.
The chef needs 68 more eggs.
Since 68 is close to 72, the answer is reasonable.

Convince Me!
A model with Math
Draw the bar diagrams to represent a solution to the problem above.
Answer:
The representations of the bar diagrams for the above problem are:

Guided Practice

Do You Understand?
Question 1.
Explain the meaning of the expression (2 × 12) + 4 in the problem on the previous page.
Answer:
In the problem on the previous page,
The meaning of the expression (2 × 12) + 4 is:
2 × 12:
The number of single eggs present in the 2 cartons
4:
The number of single eggs that are present separately
Hence, from the above,
We can conclude that
(2 × 12) + 4:
The total number of single eggs present for chef Angela to make a cake

Question 2.
In the problem on the previous page, how many eggs will the chef have leftover? Explain.
Answer:
In the problem on the previous page,
It is given that
The total number of cartons the chef needed is: 8 cartons
The number of cartons the chef have: 2 cartons
So,
The remaining number of cartons the chef needed = (The total number of cartons the chef needed) – (The number of cartons the chef have)
= 8 – 2
= 6 cartons
Now,
We know that,
A single carton has 12 eggs
So,
The number of eggs present in 6 cartons = 12 × 6
= 72 eggs
But,
It is given that the 4 single eggs are present
So,
From 72 eggs, only 8 single eggs are needed along with 2 cartons
Hence, from the above,
We can conclude that
The number of eggs will the chef have leftover is: 4 eggs

Do You Know How?
Question 3.
Carrie’s stamp book has 20 pages and each page can hold 15 stamps. She has 45 international stamps. She has 4 times as many U.S. stamps. Can all her stamps fit in her book? Use the bar diagram below to help solve this problem. Draw other bar diagrams as needed. Show the equations used to solve this problem.

Answer:
It is given that
Carrie’s stamp book has 20 pages and each page can hold 15 stamps. She has 45 international stamps. She has 4 times as many U.S. stamps
So,
The total number of stamps present in Carrie’s stamp book = (The number of pages present in Carrie’s stamp book) × (The number of stamps that each stamp can hold)
= 20 × 15
= 20 × (10 + 5)
= (20 × 10) + (20 × 5)
= 200 + 100
= 300 stamps
Now,
According to the given information,
The number of U.S stamps Carrie have = 4 × (The number of International stamps)
= 4 × 45
= 4 × (40 + 5)
= (4 × 40) + (4 × 5)
= 160 +20
= 180 U.S stamps
So,
300 > 180
Now,
The representation of the given information in the form of a bar diagram is:

Hence, from the above,
We can conclude that all of Carrie’s stamps fit in her book since the number of stamps Carrie’s book can accommodate is greater than the number of stamps Carrie have

Independent Practice

For 4-5, draw the bar diagrams, and write equations to solve each problem. Use variables to represent unknown quantities and tell what each variable represents.
Question 4.
Five toymakers each carved 28 blocks and 17 airplanes. Three other toymakers each carved the same number of airplanes and twice as many blocks. How many toys did the eight carve in all?
Answer:
It is given that
Five toymakers each carved 28 blocks and 17 airplanes. Three other toymakers each carved the same number of airplanes and twice as many blocks.
So,
The total number of toys the eight people carved = (The number of toys 5 toymakers carved) + (The number of toys the remaining 3 toymakers carved)
= ([The number of blocks] + [The number of airplanes]) + ([The number of airplanes as the same as 5 toymakers made] + [The number of blocks as the same as 5 toymakers made] × 2)
= (28 + 17) + (17 + [28 × 2])
= 45 + (17 + 56)
= 45 + 73
= 118 toys
Now,
The representation of the given information in the form of a bar diagram is:

Hence, from the above,
We can conclude that
The total number of toys that the eight toymakers carved are: 118 toys

Question 5.
Kendra is using 27 blue patches and some white patches to make a quilt. The quilt has a total area of 540 square inches. Each patch has an area of 9 square inches. How much of the area of the quilt is white?
Answer:
It is given that
Kendra is using 27 blue patches and some white patches to make a quilt. The quilt has a total area of 540 square inches. Each patch has an area of 9 square inches.
Now,
Let the area of the white patch be x
Now,
The total number of patches = (The number of blue patches) + (The number of white patches)
= (27 + x)
Now,
The total area of the patches (or) The total area of the quilt = (The total number of patches) × (The area of each patch)
= (27 + x) × 9 square inches
Now,
According to the given information,
(27 + x) × 9 = 540
27 + x = \(\frac{540}{9}\)
27 + x = 60
x = 60 – 27
x = 33 square inches
Now,
The representation of the given information in the form of a bar diagram is:

Hence, from the above,
We can conclude that
The area of the white patch is: 33 square inches

Problem Solving

Question 6.
Make Sense and Persevere A ticket to a movie for a student is $7. The cost for an adult is $2 more than for a student. How much would it cost 5 adults and 29 students for tickets to the movie?
Answer:
It is given that
A ticket to a movie for a student is $7. The cost for an adult is $2 more than for a student
Now,
The cost of a ticket for an adult = $2 + (The cost of a ticket for a student)
= $2 + $7
= $9
So,
The cost of the tickets for 5 adults = 5 × (The cost of a ticket for an adult)
= 5 × $9
= $45
So,
The cost of the tickets for 29 students = 29 × (The cost of a ticket for a student)
= 29 × $7
= (20 + 9) × $7
= (20 × $7) + (9 × $7)
= 140 + 63
= $203
So,
The total cost of the tickets for 5 adults and 29 students = $45 + $203
= $248
Hence, from the above,
We can conclude that the total cost of the tickets for 5 adults and 29 students to a movie is: $248

Question 7.
Vocabulary Give an example of an expression. Then give an example of an equation.
Answer:
An example of an expression is:
25 × 3
An example of an equation is:
20 × 3 + a × 3 = 250
Where,
‘a’ is an unknown variable

Question 8.
Higher-Order Thinking Cody and Max both solve the problem below correctly. Explain how each solves.
Emma has $79 to spend at the toy store. She wants to buy a building set, a board game, and 2 action figures from her favorite movie. What else can she buy?

Answer:
It is given that
Emma has $79 to spend at the toy store. She wants to buy a building set, a board game, and 2 action figures from her favorite movie
Now,
The given table is:

Now,
According to Max:
Let the money Emma has after she bought the mentioned items be x
So,
The amount of money Emma has left = (The total amount of money Emma has) – (The total amount of money spent on buying the items)
= $79 – ($32 + $19 + 2 ($8))
= $79 – ($32 + $19 + $16)
= $79 – 67
= $12
So,
From the given table,
We can observe that
For $12, we can buy a doll or an action figure
Now,
According to Cody:
The amount of money Emma have after she bought a building set = (The total amount of money Emma have) – (The amount of money spent by Emma on buying a building set)
= $79 – $32
= $47
The amount of money Emma have after she bought a board game = (The remaining amount of money Emma have after she bought a building set) – (The amount of money spent by Emma on buying a board game)
= $47 – $19
= $28
The amount of money Emma have after she bought 2 action figures = (The remaining amount of money Emma have after she bought a board game) – (The amount of money spent by Emma on buying 2 action figures)
= $28 – 2 ($8)
= $28 – $16
= $12
Hence, from the above,
We can conclude that Emma can buy a doll or an action figure

Assessment Practice

Question 9.
A company has 2 geothermal plants which can power a total of 2,034 homes. After they build 3 additional, more powerful, geothermal plants, they can power a total of 5,799 homes. How many homes does each of the new plants’ power? Explain how you solve. Use one or more equations and bar diagrams in your explanation. Tell what your variables represent.
Answer:
It is given that
A company has 2 geothermal plants which can power a total of 2,034 homes. After they build 3 additional, more powerful, geothermal plants, they can power a total of 5,799 homes
So,
The total number of geothermal plants = 2 + 3
= 5 geothermal plants
The number of homes that can be powered by all the geothermal plants =(The number of homes powered by 2 geothermal plants) + (The number of homes powered by 3 geothermal plants)
= 2,034 + 5,799
= 7,833 homes
So,
The number of homes powered by each geothermal plant = \(\frac{The number of homes powered by all the geothermal plants}{5}\)
= \(\frac{7,833}{5}\)
= 1,766.5
≅ 1,777 homes
Now,
The representation of the given information in the form of a bar diagram is:

Hence, from the above,
We can conclude that the number of homes provided by each geothermal plant is: 1,777 homes

Lesson 6.5 Solve Multi-Step Problems

Solve & Share
A farmer needs $3,500 to buy a previously owned truck. If she sells 45 maple trees and 27 pine trees, will she earn enough to buy the truck? If not, how much more money does she need? Solve this problem any way you choose.
I can … solve multi-step problems by finding and solving hidden questions and by writing expressions and equations.

Answer:
It is given that
A farmer needs $3,500 to buy a previously owned truck and she sells 45 maple trees and 27 pine trees
Now,
The given table is:

Now,
The amount earned by selling 45 maple trees = (The number of maple trees) × (The cost of each maple tree)
= 45 × $56
= (40 + 5) × (50 + 6)
= (40 × 50) + (40 × 6) + (5 × 50) + (5 × 6)
= 2,000 + 240 + 250 + 30
= $2,520
Now,
The amount earned by selling 27 pine trees = (The number of pine trees) × (The cost of each pine tree)
= 27 × $33
= (20 + 7) × (30 + 3)
= (20 × 30) + (20 × 3) + (7 × 30) + (7 × 3)
= 600 + 60 + 210 + 21
= $891
So,
The total amount of money earned by selling 45 maple trees and 27 pine trees = $2,520 + $891
= $3,411
But,
A farmer needs $3,500
Hence, from the above,
We can conclude that she will not earn enough money to buy the truck

Look Back! In the question above, how many more of each tree could the farmer sell to get enough money to buy the truck? Explain.
Answer:
From the above problem,
We can observe that
The total amount of money earned by selling 45 maple trees and 27 pine trees is: $3,411
She will need $3,500
Now,
The extra money she have = $3,500 – $3,411
= $189
Hence, from the above,
We can conclude that
With the remaining money,
She can buy
The number of maple trees she can buy are: 2
The number of pine trees she can buy are: 2

Essential Question
How Can You Use Equations to Solve Multi-Step Problems?
Answer:
The steps to solve Multi-Step problems using equations are:
a. Multiply to clear any fractions or decimals.
b. Simplify each side by clearing parentheses and combining like terms.
c. Add or subtract to isolate the variable term—you may have to move a term with the variable.
d. Multiply or divide to isolate the variable.
e. Check the solution.

Visual Learning Bridge
The students in the fourth and fifth grades are going to a concert. There are 178 students. How many rows are needed for the fourth graders?

Step 1
Find and solve the first hidden question.
Hidden Question: How many fifth graders are there?
12 × 8 = s
S = 96
There are 96 fifth graders.

Step 2
Find and solve the second hidden question.
Hidden Question: How many fourth graders are there?
178 – 96 = f
f = 82
There are 82 fourth graders.

Step 3
Answer the original question.
Original Question: How many rows are needed for the fourth graders?
82 ÷ 8 = r
r = 10 R2
Ten rows will be filled with 2 fourth graders left over. So, 11 rows will be needed.

Convince Me!
Construct Arguments Does the answer of 11 rows makes sense for the problem above? Explain.
Answer:
From the above problem,
We can observe that
The number of rows needed for the fourth graders is: 10 R 2
Now,
The meaning of 10 R 2 in the given situation is:
10 rows with the remainder 2 seats and we have to arrange that remainder in the next row
So,
The total number of rows is: 11
Hence, from the above,
We can conclude that the answer of 11 rows makes sense for the above problem

Guided Practice

Do You Understand?
Question 1.
On the previous page, suppose there were only 11 rows of fifth-graders, but the same total number of students. Do you need to solve the whole problem again to find how many rows are needed for the fourth graders? Explain.
Answer:
It is given that
On the previous page, there were only 11 rows of fifth-graders, but the same total number of students
Now,
In the example on the previous page,
It is given that
There are 8 students in each row and the total number of students are 178
So,
The number of rows needed for the fourth-grade students = [(The total number of students) – (The number of rows needed for the fifth-graders) × (The number of students in each row)] ÷ 8
Hence, from the above,
We can conclude that
We need to solve the whole problem again to find the number of rows needed for the fourth-graders

Do You Know How?
Question 2.
Show another way to solve the problem on the previous page.
Answer:
The given figure is:

Now,
In the previous example on the previous page,
It is given that
There are a total of 178 students
Now,
The number of students in the fifth-grade = (The number of rows filled by the fifth-graders) × (The number of students in each row)
= 12 × 8
= (10 + 2) × 8
= (10 × 8) + (2 × 8)
= 80 + 16
= 96 students
So,
The number of students in the fourth-grade = (The total number of students) – (The number of students in the fifth grade)
= 178 – 96
= 82 students
So,
The number of rows filled by the fourth-graders = \(\frac{The number of students in the fourth-grade}{8}\)
= \(\frac{82}{8}\)
= 10 R 2
Hence,from the above,
We can conclude that the number of rows filled by the fourth-grade students is: 11

Independent Practice

For 3-4, solve each multi-step problem. Write equations to show how you solve. Draw the bar diagrams to help if needed. Use estimation to decide if your answer is reasonable.

Question 3.
Vanya bought 5 medium packages of buttons and 3 small packages of buttons. What was the total number of buttons Vanya bought?
Answer:
The given data is:

So,
The total number of buttons Vanya bought = (The number of buttons present in 5 medium packages of buttons) + (The number of buttons present in 3 small packages of buttons)
= (5 × The number of buttons present in a medium package of buttons) + (3 × The number of buttons present in a  small package of buttons)
= (5 × 38) + (3 × 18)
= 5 × (30 + 8) + 3 × (10 + 8)
= (5 × 30) + (5 × 8) + (3 × 10) + (3 × 8)
= 150 + 40 + 30 + 24
= 244 buttons
Hence, from the above,
We can conclude that
The total number of buttons Vanya bought is: 244 buttons

Question 4.
Vance bought 2 packages of large beads, 1 package of medium beads, 2 packages of large buttons, and 2 packages of medium buttons. How many more beads than buttons did Vance buy?
Answer:
The given data is:

So,
The total number of beads Vance bought = (The total number of beads present in 2 packages of large beads) + (The total number of beads present in 1 package of medium beads)
= (2 × 96) + (1 × 64)
= 2 × (90 + 6) + 64
= (2 × 90) + (2 × 6) + 64
= 180 + 12 + 64
= 256 beads
So,
The total number of buttons Vance bought = (The total number of buttons present in 2 packages of large buttons) + (The total numbr of buttons present in 2 packages of medium buttons)
= (2 × 56) + (2 × 38)
= 2 × (50 + 6) + 2 × (30 + 8)
= (2 × 50) + (2 × 6) + (2 × 30) + (2 × 8)
= 100 + 12 + 600 + 16
= 728 buttons
So,
The difference between the number of buttons and beads = 728 – 256
= 472 beads
Hence, from the above,
We can conclude that there are 472 more buttons than beads that Vance bought

Problem Solving

Question 5.
en Vision® STEM How much more does it cost to generate 9 megawatt-hours of electricity with conventional coal than with wind energy? Write one or more equations to show how you solve. Tell what your variables represent.

Answer:
The given figure is:

Now,
The cost to generate 9 megawatt hours of electricity with wind energy = 9 × (The cost to generate 1 megawatt hours of electricity with wind energy)
= 9 × $87
= 9 × (80 + 7)
= (9 × 80) + (9 × 7)
= 720 + 63
= $783
Now,
The cost to generate 9 megawatt hours of electricity with conventional coal = 9 × (The cost to generate 1 megawatt hours of electricity with conventional coal)
= 9 × $105
= 9 × (100 + 5)
= (9 × 100) + (9 × 5)
= 900 + 45
= $945
So,
The difference between the energy produced with conventional coal and the wind energy = $945 – $783
= $162
Hence, from the above,
We can conclude that
It takes $162 more to generate 9 megawatt-hours of electricity with conventional coal than the wind energy

Question 6.
A model with Math Anna earns $8 an hour babysitting and $6 an hour working in the garden. Last month, she worked 15 hours babysitting and 8 hours in the garden. How much more money does she need to buy a robot which costs $199? Explain how you solve. Use one or more equations in your explanation. Tell what your variables represent.
Answer:
It is given that
Anna earns $8 an hour babysitting and $6 an hour working in the garden. Last month, she worked 15 hours babysitting and 8 hours in the garden.
So,
The amount of money earned by Anna for babysitting = (The number of hours Anna did babysitting) × (The amount of money earned by Anna for 1 hour of babysitting)
= 15 × $8
= (10 + 5) × $8
= (10 × 8) + (5 × 8)
= 80 + 40
= $120
So,
The amount of money earned by Anna for working in the garden = (The number of hours Anna did work in the garden) × (The amount of money earned by Anna for 1 hour working in the garden)
= 8 × $6
= $48
So,
The total amount of money earned by Anna = $120 + $48
= $168
But,
She needed to buy a robot which costs $199
So,
The amount of more money she needed to buy a robot = $199 – $168
= $31
Hence, from the above,
We can conclude that
She needs $31 more to buy a robot which costs $199

Question 7.
Higher-Order Thinking Show two different ways to find the answer to the problem below.
Dog and cat food are sold in 20-pound bags. There are 14 bags of dog food and 12 bags of cat food on the store shelves. How many pounds of dog and cat food is on the shelves?
Answer:
It is given that
Dog and cat food are sold in 20-pound bags. There are 14 bags of dog food and 12 bags of cat food on the store shelves
So,
The number of pounds of dog food = (The number of bags of dog food) × (The number of pounds each bag contains)
= 14 × 20
= (10 + 4) × 20
= (10 × 20) + (4 × 20)
= 200 + 80
= 280 pounds
The number of pounds of cat food = (The number of bags of cat food) × (The number of pounds each bag consists)
= 12 × 20
= (10 + 2) × 20
= (10 × 20) + (2 × 20)
= 200 + 40
= 240 pounds
Hence, from the above,
We can conclude that
The number of pounds of dog food present on the shelves is: 280 pounds
The number of pounds of cat food present on the shelves is: 240 pounds

Assessment Practice

Question 8.
Chris needs $858 to buy a computer. She has already saved $575. She gets $15 an hour for babysitting and will babysit 12 hours in the next month. She can save $8 a week from her allowance. How many weeks of allowance will it take Chris to save enough to buy the computer? Explain how you solve. Use one or more equations in your explanation. Tell what your variables represent.
Answer:
It is given that
Chris needs $858 to buy a computer. She has already saved $575. She gets $15 an hour for babysitting and will babysit 12 hours in the next month. She can save $8 a week from her allowance
So,
The amount of money she required to buy a computer = (The amount of money Chris needs to buy a computer) – (The amount of money she already saved)
= $858 – $575
= $283
Now,
The amount of money she will earn for babysitting = (The number of hours she did babysitting) × (The amount of money she gets for babysitting for an hour)
= 12 × $15
= (10 + 2) × $15
= (10 × 15) + (2 × 15)
= 150 + 30
= $180
So,
The remaining money she will need to buy a computer = $283 – $180
= $103
Now,
The amount of money she earned from allowance = (The number of weeks) × (The amount of money she will earn for a week as an allowance)
$103 = x × $8
x = \(\frac{103}{8}\)
x = 12.8
x ≈ 13 weeks
Hence, from the above,
We can conclude that
About 13 weeks of allowance will it take Chris to save enough to buy the computer

Lesson 6.6 Problem Solving

Make Sense and Persevere
Solve & Share
Ms. Valenzuela had her students design a snake house for the zoo. In the design shown, the anaconda has 538 more square feet than the python. The python has twice as many square feet as the rattlesnake. How much of the house’s area does each snake have? Solve this problem any way you choose.
I can … make sense of problems and keep working if I get stuck.

Thinking Habits
Be a good thinker! These questions can help you.

• What do I need to find?
• What do I know?
• What’s my plan for solving the problem?
• What else can I try if I get stuck?
• How can I check that my solution makes sense?

Answer:
It is given that
Ms. Valenzuela had her students design a snake house for the zoo. In the design shown, the anaconda has 538 more square feet than the python. The python has twice as many square feet as the rattlesnake
Now,
The given figure is:

Now,
Let the area of the rattlesnake be r square feet
Let the area of the python be p square feet
So,
The area of the python = 2 × (The area of the rattlesnake)
= 2r square feet
Now,
According to the given information,
(The area of the anaconda) – (The area of the python) = 538
So,
1,928 – p = 538
p = 1,928 – 538
p = 1,390 square feet
So,
The area of the rattlesnake = \(\frac{The area of the python (p)}{2}\)
= \(\frac{1,390}{2}\)
= 695 square feet
Hence, from the above,
We can conclude that
The area of an anaconda is: 1,928 square feet
The area of python is: 1,390 square feet
The area of the rattlesnake is: 695 square feet

Look Back!
Make Sense and Persevere How can you check that your solution makes sense?
Answer:
To check whether the given solution makes sense or not,
Estimate the solution
Now,
By the process of Estimation,
The area of the anaconda is: 1,900 square feet
So,
(The area of the anaconda) – (The area of python) = 540
The area of python = 1,900 – 540
= 1,360 square feet
So,
The area of rattlesnake = \(\frac{The area of python}{2}\)
= \(\frac{1,360}{2}\)
= 685 square feet
So,
The area of the anaconda is: 1,900 square feet
The area of python is: 1,360 square feet
The area of the rattlesnake is: 685 square feet
Now,
By observing the answers in the above problem and the answers for this problem,
We can say that your answer is reasonable
Hence, from the above,
We can conclude that your solution makes sense

Essential Question
How Do You Make Sense of a Multi-Step Problem and Persevere in Solving It?
Answer:
The steps to make sense of a Multi-Step Problem and Persevere in Solving it are:
a. Identify the Problem. Begin by determining the scenario the problem wants you to solve.
b. Gather Information.
c. Create an Equation.
d. Solve the Problem.
e. Verify the Answer.

Visual Learning Bridge
Bryan and Alex have to buy their own instruments for the band. Alex made $1,025 from a fundraiser. He has a part-time job that pays $8 an hour. How many hours does Alex need to work to have enough money to buy his instrument?

What do you need to do to solve this problem?
I need to determine how much Alex’s tuba costs and how much Alex needs to earn to buy the tuba.

How can I make sense of and solve this problem?
I can

• identify the quantities given.
• understand how the quantities are related.
• choose and implement an appropriate strategy.
• check to be sure my work and answer make sense.

Here’s my thinking
Use the cost of the trumpet to find the cost of the tuba.
$159 x 9 = $1,431 Alex’s tuba costs $1,431.
Find how much more money Alex needs to earn.
$1,431 – $1,025 = $406 Alex needs to earn $406.
Find how many hours Alex needs to work.
$406 ÷ 8 = 50 R6
Alex will not have enough money by working 50 hours, so he needs to work 51 hours.

Convince Me!
Make Sense and Persevere How can you check to make sure the work and answer given above make sense?
Answer:
To check whether the given answer is reasonable or not,
Estimate the answer by using Estimation methods
Now,
The given figure is:

Now,
From the given figure,
We can observe that
The cost of Bryan’s trumpet is about $160
So,
The cost of Alex’s tuba = 9 × (The cost of Bryan’s trumpet)
= 9 × $160
= 9 × (100 + 60)
= (9 × 100) + (9 × 60)
= 900 + 540
= $1,440
Now,
The more money Alex must have to buy his instrument = (The cost of Alex’s tuba) – (The money raised by fundraiser)
= $1,440 – $1,025
= $415
Now,
The number of hours needed for Alex to earn more money = \(\frac{The more money Alex must have to buy his instrument}{8}\)
= \(\frac{$415}{8}\)
= 51 R 7
Hence, from the above,
We can conclude that Alex needs about 7 hours for Alex to earn more money
Your solution is reasonable since the difference between the number of hours in the above example and this problem is not large

Guided Practice

Make Sense and Persevere In the problem on the previous page, suppose Alex wanted to know how many weeks it would take him to work
Think about 51 hours. Alex works 3 hours a day and 4 days a week. this question to help you

Question 1.
What are you asked to find and what is the hidden question What’s a good plan for solving question?
Answer:
The hidden question to answer is:
How many weeks it would take him to work?
Now,
A good plan for solving the given question is:
a. The number of hours in a day that Alex works
b. The number of days in a week that Alex works
c. The number of weeks Alex will take to complete the work

Question 2.
Write and solve equations to solve the problem. Be sure to tell what each variable represents.
Answer:
It is given that
Alex works 3 hours a day and 4 days a week
So,
The number of hours in a week that Alex will work = (The number of hours a day) × (The number of days in a week that Alex will work)
= 3 × 4
= 12 hours a week
So,
The number of weeks Alex will need to complete the work = \(\frac{The total number of hore Alex completed the week}{The number of hours in a week that Alex will work}\)
= \(\frac{51}{12}\)
= 4 R 3
Hence, from the above,
We can conclude that Alex will need about 4 weeks to complete the work

Question 3.
Does your answer make sense? Explain.
Answer:
To make whether your answer makes sense or not,
Estimate the solution using the Estimation methods
Now,
The number of hours in a week that Alex will work = (The number of hours a day) × (The number of days in a week that Alex will work)
= 3 × 4
= 12 hours a week
So,
The number of hours in a week that Alex will work is about 10 hours
So,
The number of weeks Alex will need to complete the work = \(\frac{The total number of hore Alex completed the week}{The number of hours in a week that Alex will work}\)
= \(\frac{51}{10}\)
= 5 R 1
So,
We can conclude that Alex will need about 5 weeks to complete the work
Hence, from the above,
We can conclude that your answer is reasonable since the difference between the actual answer and the estimated answer is not large

Independent Practice

Make Sense and Persevere The high school tennis team is selling tennis balls to raise $500 for new equipment. They sell the balls for $2 each. Will they make enough money if they sell 4 cases?

Question 4.
What are the hidden questions? Write equations to solve each.
Answer:
It is given that
The high school tennis team is selling tennis balls to raise $500 for new equipment. They sell the balls for $2 each
So,
The hidden question for the given problem is:
What is the cost of each case?
Now,
The cost of each case = (The number of packages in each case) × (The number of tennis balls in each package) × (The cost of each tennis ball)
= 24 × 3 × $2
= 24 × $6
= (20 + 4) × 6
= (20 × 6) + (6 × 4)
= 120 + 24
= $144
So,
The cost of 4 cases = 4 × (The cost of each case)
= 4 × $144
= 4 × (100 + 40 + 4)
= (4 × 100) + (4 × 40) + (4 × 4)
= 400 + 160 + 16
= $576
But,
It is given that the high school tennis team is selling tennis balls to raise $500 for new equipment
Hence, from the above,
We can conclude that they will not make enough money if they sell 4 cases

Question 5.
Will the team makes enough money? Explain.
Answer:
From the above problem,
We can observe that
The high school tennis team is selling tennis balls to raise $500 for new equipment
The cost of tennis balls present in 4 cases is: $576
So,
$500 < $576
Hence, from the above,
We can conclude that the team will not make enough money

Problem Solving

Performance Task
Designing a Flag Rainey’s group designed the flag shown for a class project. They used 234 square inches of green fabric. After making one flag, Rainey’s group has 35 square inches of yellow fabric left. How can Rainey’s group determine the total area of the flag?

Question 6.
Make Sense and Persevere What hidden question(s) do you need to answer first?
Answer:
It is given that
Rainey’s group designed the flag shown for a class project. They used 234 square inches of green fabric. After making one flag, Rainey’s group has 35 square inches of yellow fabric left
Now,
The hidden questions you need to answer first are:
a. Find the remaining colors present in the flag?
b. What is the area of the remaining colors in the flag?
c. What is the total area of the flag?

Question 7.
Appropriate Tools Draw diagrams and write equations to represent the hidden question(s). Be sure to tell what each variable represents.

Answer:
The given figure is:

Now,
It is given that
They used 234 square inches of green fabric
So,
The area of the orange fabric = 2 × (The area of the green fabric)
= 2 × 234
= 2 × (200 + 30 + 4)
= (2 × 200) + (2 × 30) + (2 × 4)
= 400 + 60 + 8
= 468 square inches
So,
The area of the yellow fabric = 3 × (The arae of the green fabric)
= 3 × 234
= (2 × 234) + (1 × 234)
= 468 + 234
= 702 square inches
But,
It is given that Rainey’s group has 35 square inches of yellow fabric left
So,
The area of the yellow fabric = 702 – 35
= 667 square inches
So,
The total area of the flag = (The area of the green flag) + (The area of the orange flag) + (The area of the yellow flag)
= 234 + 702 + 667
= 1,603 square inches
Hence, from the above,
We can conclude that
The total area of the flag is: 1,603 square inches

Question 8.
Be Precise Use your drawings and equations to find the total area of the flag. Explain carefully, using the correct units.
Answer:
From the above problem,
We can observe that
The total area of the flag can be found out by adding the areas of the three colors
So,
The total area of the flag = (The area of the green flag) + (The area of the orange flag) + (The area of the yellow flag)
= 234 + 702 + 667
= 1,603 square inches
Hence, from the above,
We can conclude that
The total area of the flag is: 1,603 square inches

Question 9.
Make Sense and Persevere What information was not needed to solve the problem?
Answer:
There is not any unnecessary information present that is not used to solve the given problem

Topic 6 Fluency Practice Activity

Find a Match
Work 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.
I can … subtract multi-digit whole numbers.

Clues
A. The difference is exactly 528.
B. The difference is between 550 and 560.
C. The difference is between 800 and 900.
D. The difference is exactly 614.
E. The difference is between 100 and 105.
F. The difference is between 470 and 480.
G. The difference is exactly 392.
H. The difference is between 70 and 80.

Answer:

Topic 6 Vocabulary Review

Understand Vocabulary
Word List

• addition comparison
• Associative Property of Multiplication
• Commutative Property of Multiplication
• Distributive Property of Multiplication
• equation
• multiplication comparison
• product
• variable

Write T for true and F for false.
Question 1.
__________ Addition comparison is used when you can multiply to find how one quantity is related to another.
Answer:
We know that,
Additive comparisons focus on the difference between two quantities
Hence, from the above,
We can conclude that the given statement is: False

Question 2.
___________ Multiplication comparison is used when one quantity is x times more than another quantity.
Answer:
We know that,
In multiplicative comparison, there are two different sets being compared. The first set contains a certain number of items. The second set contains multiple copies of the first set.
Hence, from the above,
We can conclude that the given statement is: False

Question 3.
__________ A number sentence that uses an equal sign to show that two expressions have the same value is called an equation.
Answer:
We know that,
A number sentence that uses an equal sign to show that two expressions have the same value is called an “Equation”
Hence, from the above,
We can conclude that the given statement is: True

Question 4.
___________ The answer to a subtraction problem is called the product.
Answer:
We know that,
The answer to a multiplication problem is called the “Product”
Hence, from the above,
We can conclude that the given statement is: True

Question 5.
_________ A symbol or letter that stands for a number is called a variable.
Answer:
We know that,
A “Variable” is a symbol that works as a placeholder for expression or quantities that may vary or change
Hence, from the above,
We can conclude that the given statement is: True

Label each example with a term from the Word List.
Question 6.
(3 × 4) × 5 = 3 (4 × 5) _______________
Answer:
We know that,
According to Associative Property of Multiplication,
a × (b × c) = (a × b) × c
So,
we can observe that
The given expression is also in the same form
Hence, from the above,
We can conclude that the term that is suitable for the given problem is: Associative Property of Multiplication

Question 7.
3 × (4 + 5) = (3 × 4) + (3 × 5) _______________
Answer:
We know that,
According to the Distributive Property of Multiplication,
a × (b + c) = (a × b) + (a × c)
So,
we can observe that
The given expression is also in the same form
Hence, from the above,
We can conclude that the term that is suitable for the given problem is: Distributive Property of Multiplication

Question 8.
3 × 4 × 5 = 4 × 3 × 5 _______________
Answer:
We know that,
According to the Commutative Property of Multiplication,
a × b = b × a
So,
We can observe that
The given expression is also in the same form
Hence, from the above,
We can conclude that the term that is suitable for the given problem is: Commutative Property of Multiplication

Use Vocabulary in Writing
Question 9.
Seth wrote and solved the following comparison:
Find 6 times as many as 5.
6 × 5 = n
n = 30
Use at least 3 terms from the Word List to describe Seth’s comparison.
Answer:
It is given that
Find 6 times as many as 5 ———-> Multiplication comparison
So,
The given sentence can be written as:
6 × 5 = n ——-> Equation
n = 30 ——-> Product
Where,
n ——> variable

Topic 6 Reteaching

Set A pages 225-232

Write an equation for each comparison. Find the value of the variable that makes the equation true.
k is 9 times as many as 3.
k = 9 × 3
k = 27
m is 6 more than 21.
m = 6 + 21
m = 27
There are 30 apples and 6 bananas in a basket. How many times as many apples as bananas are in the basket?
Let t = the number of times as many apples as bananas.
t times as many as 6 is 30.
t × 6 = 30
Since 5 × 6 = 30, there are 5 times as many apples as bananas in the basket.

Remember to use addition or subtraction when you know how much more, and multiplication or division when you know how many times as many.

Write and solve an equation to match each comparison.
Question 1.
x is 21 more than 21.
Answer:
It is given that
x is 21 more than 21
So,
According to the Property of Addition comparison,
x = 21 + 21
x = 42
Hence, from the above,
We can conclude that the value of x is: 42

Question 2.
Macon has 32 rocks in his collection. He has 4 times as many rocks as his brother. How many rocks, r, does Macon’s brother have?
Answer:
It is given that
Macon has 32 rocks in his collection. He has 4 times as many rocks as his brother
Now,
Let the number of rocks Macon’s brother has to be: r
So,
According to the given information,
r × 4 = 32
r = \(\frac{32}{4}\)
r = 8 rocks
Hence from the above,
We can conclude that Macon’s brother has 8 rocks

Question 3.
Pam has 24 pencils and 6 erasers. How many times, t, as many pencils as erasers does Pam have?
Answer:
It is given that
Pam has 24 pencils and 6 erasers
Now,
Let the number of erasers be: x
So,
According to the given information,
x × 6 = 24
x = \(\frac{24}{6}\)
x = 4 erasers
Hence, from the above,
We can conclude that
Pam has the number of pencils 4 times as many as erasers

Set B pages 233-240

At a restaurant, a children’s meal costs $5, and an adult meal costs $9. Four children and 2 adults order meals. The family has a $25 gift certificate to use. How much will their total be before tax?

Remember to first find and answer any hidden questions.

Question 1.
There are 64 singers in the choir. The tenors and sopranos are in separate rows. There are 8 singers in each row. There are 4 rows of tenors. How many rows of sopranos are there?
Answer:
It is given that
There are 64 singers in the choir. The tenors and sopranos are in separate rows. There are 8 singers in each row. There are 4 rows of tenors.
Now,
Let the number of rows of sopranos be x
So,
The total number of rows = (The number of rows of sopranos) + (The number of rows of tenors)
= (x + 4)
Now,
According to the given information,
The total number of rows = \(\frac{The total number of singers}{The number of singers in each row}\)
= \(\frac{64}{8}\)
= 8
So,
x + 4 = 8
x = 8 – 4
x = 4 rows
Now,
The representation of the given information in the form of a bar diagram is:

Hence, from the above,
We can conclude that
The total number of rows of sopranos is: 4 rows

Question 2.
Samantha has $600 saved for a trip. She buys an airline ticket for $120 and reserves a hotel room for $55 each night for 4 nights. If Samantha’s trip lasts for 5 days and she spends the same amount each day, how much can Samantha spend each day?
Answer:
It is given that
Samantha has $600 saved for a trip. She buys an airline ticket for $120 and reserves a hotel room for $55 each night for 4 nights and Samantha’s trip lasts for 5 days and she spends the same amount each day
Now,
The amount of money Spent by Samantha = (The amount of money spent to buy an airline ticket) + (The amount of money spent to reserve a hotel room for 4 nights)
= $120 + ($55 × 4)
= $120 + (50 + 5) × 4
= 120 + (50 × 4) + (5 × 4)
= 120 + 200 + 20
= $120 + $220
= $340
So,
The amount of money spent by Samantha for 5 days = $340 × 5
= (300 + 40) × 5
= (300 × 5) + (40 × 5)
= 1,500 + 200
= $1,700
But,
Samantha can buy the airline ticket only once
So,
The amount of money Samantha spent for 5 days = $1,700 – $120
= $580
So,
The amount of money spent by Samantha each day = \(\frac{$580}{5}[latex]
= $116
Hence, from the above,
We can conclude that the amount of money spent by Samantha each day is: $116

Set C pages 241-244

There are 13 girls and 14 boys signed up to join volleyball. Each team needs 6 players. How many teams can be formed, and how many more players are needed to form one more team?
Find the total number of players signed up.
13 + 14 = p
27 = p
Divide to find the number of teams that can be formed.
27 ÷ 6 = t
4 R 3 = t
There are 3 students left not on a team. Subtract to find the number of players still needed to make a team.
6 – 3 = p
3 = p
4 teams can be formed. 3 more players are needed to make one more team.

Remember to draw the bar diagrams to help if needed.

Solve each multi-step problem. Write equations to show how you solve.
Question 1.
Kevin made $96 in necklace sales and half that amount in bracelet sales. How much money did Kevin make in necklace and bracelet sales?
Answer:
It is given that
Kevin made $96 in necklace sales and half that amount in bracelet sales
So,
The amount made by Kevin in bracelet sales = (The amount made by Kevin in necklace sales) ÷ 2
= [latex]\frac{96}{2}\)
= $48
So,
The amount of money made by Kevin in necklace and bracelet sales = (The amount of money made by Kevin in necklace sales) + (The amount of money made by Kevin in bracelet sales)
= $96 + $48
= $144
Hence, from the above,
We can conclude that
The amount made by Kevin in necklace and bracelet sales is: $144

Question 2.
A dog and cat rescue use 40 pounds of dog food and 15 pounds of cat food to feed its animals each day. How many pounds of dog and cat food do they use in seven days?
Answer:
It is given that
A dog and cat rescue use 40 pounds of dog food and 15 pounds of cat food to feed its animals each day
So,
The number of pounds of dog and cat food they use each day = (The number of pounds of dog food) + (The number of pounds of cat food)
= 40 + 15
= 55 pounds
So,
The number of pounds of dog and cat food they use in 7 days = 7 × (The number of pounds of dog and cat food they use each day)
= 7 × 55
= 7 × (50 + 5)
= (7 × 50) + (7 × 5)
= 350 + 35
= 385 pounds
Hence, from the above,
We can conclude that
The number of pounds of dog and cat food they use in 7 days is: 385 pounds

Set D pages 245-248

Think about these questions to help you make sense of the problem.
Thinking Habits

• What do I need to find?
• What do I know?
• What’s my plan for solving the problem?
• What else can I try if I get stuck?
• How can I check that my solution makes sense?
  

Remember to make sense of the problem before starting to solve it.

At a local shelter, each large dog can be adopted for $10 and each small dog for $5. There are 17 large dogs up for adoption. If all the dogs are adopted, the shelter will make $215. How many small dogs were at the shelter?
Question 1.
Find the hidden questions. Write equations to solve each.
Answer:
It is given that
At a local shelter, each large dog can be adopted for $10 and each small dog for $5. There are 17 large dogs up for adoption. If all the dogs are adopted, the shelter will make $215.
So,
The hidden questions for the given problem are:
a. What is the cost for the adoption of 17 large dogs?
b. What is the cost for the adoption of the small dogs?

Question 2.
How many small dogs are at the shelter?
Answer:
It is given that
At a local shelter, each large dog can be adopted for $10 and each small dog for $5. There are 17 large dogs up for adoption. If all the dogs are adopted, the shelter will make $215.
Now,
Let the number of small dogs be x
So,
The total cost for the adoption of all dogs = (The total cost for the adoption of large dogs) + (The total cost for the adoption of small dogs)
$215 = [(The number of large dogs) × (The cost for the adoption of each large dog)] + [(The number of small dogs) × (The cost for the adoption of each small dog)]
$215 = (17 × $10) + (x × $5)
$215 = (17 × 1 × 10) + (x × $5)
$215 = $170 + (x × $5)
x × $5 = $215 – $170
x × $5 = $45
x = \(\frac{45}{5}\)
x = 9
Hence, from the above,
We can conclude that the total number of small dogs is: 9 small dogs

Topic 6 Assessment Practice

Question 1.
Jason and his 3 brothers want to buy a gift for their mother. They have $314 saved. Each of them will save $17 a week until they have at least $515 for her gift. How much money will they save after 3 weeks? Will they have enough money to buy the gift?
A. What are the hidden questions?
Answer:
It is given that
Jason and his 3 brothers want to buy a gift for their mother. They have $314 saved. Each of them will save $17 a week until they have at least $515 for her gift
Hence,
The hidden questions for the given problem are:
a. How much money Jason and his 3 brothers will save each week?
b. How much amount of money do Jason and his brothers need to save?

B. Write an equation that can be used to answer each hidden question. Then solve.
Answer:
It is given that
Jason and his 3 brothers want to buy a gift for their mother. They have $314 saved. Each of them will save $17 a week until they have at least $515 for her gift
Now,
The amount of money jason and his 3 brothers will save each week = 4 × $17
= 4 × (10 + 7)
= (4 × 10) + (4 × 7)
= 40 + 28
= $68
Now,
The amount of enough money Jason and his 3 brothers will have to save = $515 – $314
= $201
Hence, from the above,
We can conclude that
The amount of money Jason and his 3 brothers will save each week is: $68
The amount of enough money Jason and his 3 brothers will have to save is: $201

C. Write and solve an equation to find how much money they will save after 3 weeks. Will they have enough to buy the gift? Explain.
Answer:
From part (a),
We can observe that
The amount of money Jason and his 3 brothers will save each week is: $68
So,
The amount of money Jason and his 3 brothers will save in 3 weeks = 3 × $68
= 3 × (60 + 8)
= (3 × 60) + (3 × 8)
= 180 + 24
= $204
So,
The amount of money Jason and his 3 brothers will save after 3 weeks = $204 – $201
= $3
So,
The amount of money Jason and his 3 brothers will have = $515 + $3
= $518
Hence, from the above,
We can conclude that Jason and his 3 brothers will have enough money to buy the gift

Question 2.
Marco ordered 2 reclining chairs for $230 each and a coffee table for $350. The shipping cost is $100. Marco is going to pay off his bill in 5 equal payments. How much is each of Marco’s payments?
Answer:
It is given that
Marco ordered 2 reclining chairs for $230 each and a coffee table for $350. The shipping cost is $100. Marco is going to pay off his bill in 5 equal payments
So,
The total payment made by Marco = (The payment made by Marco for 2 reclining chairs) + (The payment made by Marco for a coffee table) + (The shipping cost)
= (2 × 230) + 350 + 100
= [(2 × 23) × 10] + 450
= 460 + 450
= $910
Now,
The payment made by Marco in each equal installment = \(\frac{$910}{5}\)
= $182
Hence, from the above,
We can conclude that each of Marco’s payments is: $182

Question 3.
Mitchell wants to beat the record for the most points scored in a season. This season, he has scored 51 points. If he scores 27 points at each of the next 7 games, he will break the record by 1 point. How many total points will break the record by 1 point?

Mitchell will break the record by 1 point if he scores a total of ________ points.
Answer:
It is given that
Mitchell wants to beat the record for the most points scored in a season. This season, he has scored 51 points. and if he scores 27 points at each of the next 7 games, he will break the record by 1 point.
Now,
The number of points scored by Mitchell in the next 7 games = 27 × 7
= (20 + 7) × 7
= (20 × 7) + (7 × 7)
= 140 + 49
= 189 points
So,
The total number of points scored by Mitchell in this season = 189 + 51
= 240 points
So,
The representation of the given situation in the form of a bar diagram is:

Hence, from the above,
We can conclude that
Mitchell will break the record by 1 point if he scores a total of 240 points.

Question 4.
Select all the sentences that describe a comparison using multiplication.
☐ 9 is 3 times as many as p.
☐ 27 more than r is 41.
☐ A bus can travel 3 times as fast as a boat.
☐ It costs d dollars for 9 packages.
☐ There are 4 times as many girls as boys.
Answer:
The sentences that describe a comparison using multiplication are:

Question 5.
Choose the correct word from the box to complete each statement.

Answer:
The completed statements with the suitable word from the given box are:

Question 6.
Dee has $120 to spend. She went to the grocery store and spent $55. She then bought 3 potted flowers for $18 each at the nursery. How much money did Dee have left? Select each correct equation or set of equations that could be used to solve the question.
☐ $120 – $55 = M; M – (3 × $18)
☐ $120 – $55 – (3 × $18) = M
☐ (3 × $18) – $55 – $120 = M
☐ ($120 – $55) – (3 × $18) = M
☐ $120 + $55 – (3 × $18) = M.
Answer:
It is given that
Dee has $120 to spend. She went to the grocery store and spent $55. She then bought 3 potted flowers for $18 each at the nursery.
Now,
The cost of the 3 potted flowers = 3 × $18
= 3 × (10 + 8)
= (3 × 10) + (3 × 8)
= 30 + 24
= $54
So,
The amount of money spent by Dee = $55 + $54
= $109
So,
The amount of money Dee left = (The amount of money Dee has to spend) – (The amount of money Dee has spent)
= $120 – $109
= $11
Hence, from the above,
We can conclude that
Each correct equation or set of equations that could be used to solve the given question is:

Question 7.
Darcy ordered 18 boxes of red balloons and 12 boxes of blue balloons for a party. She ordered a total of 240 balloons. How many balloons are in each box?
A. 6
B. 8
C. 7
D. 9
Answer:
It is given that
Darcy ordered 18 boxes of red balloons and 12 boxes of blue balloons for a party. She ordered a total of 240 balloons
So,
The total number of balloons present in 2 boxes = (The number of red balloons) + (The number of blue balloons)
= 18 + 12
= 30 balloons
So,
The number of balloons present in each box = \(\frac{The total number of balloons}{The total number of balloons present i 2 boxes}\)
= \(\frac{240}{30}\)
= 8 balloons
Hence, from the above,
We can conclude that
The number of balloons present in each box is:

Question 8.
Select all of the sentences that are true for the number 9.
☐ 18 × _____ = 162
☐ 20 more than _____ = 180
☐ _____ times as many as 16 is 145.
☐ 315 is _____ times as many as 35.
☐ 16 × _______ = 128
Answer:
The sentences that are true for the number 9 are:

Question 9.
Select all the expressions that are equal to the product of 14 and 9.
☐ (2 × 7) + 9
☐ 9 times as many as 14
☐ 14 × 9
☐ 14 more than 9
☐ 9 less than 14
Answer:
We know that,
According to the Commutative Property of Multiplication,
a × b = b × a
So,
14 × 9 = 14 × 9
Hence, from the above,
We can conclude that
All the expressions that are equal to the product of 14 and 9 are:

Question 10.
Maggie collected 63 pounds of paper for recycling. Carl collected 9 pounds. How many times as many pounds did Maggie collect as Carl?
A. 3 times
B. 5 times
C. 7 times
D. 8 times
Answer:
It is given that
Maggie collected 63 pounds of paper for recycling. Carl collected 9 pounds
Now,
Let the number of pounds as many times collected be x
So,
According to the given information,
9 × x = 63
x = \(\frac{63}{9}\)
x = 7
Hence, from the above,
We can conclude that
The number of times as many pounds did Maggie collect as Carl are:

Topic 6 Performance Task

Ski Jumping
Jackie and her older brother Robert went skiing. The Ski Jumping table shows their best jumps, including the distance they jumped and the total length of their run.

Question 1.
Jackie wants to find how much farther Robert’s run was than her run.
Part A
What is the total length of Robert’s run? Draw a bar diagram and write and solve an equation to represent the problem. Does this situation use addition or multiplication to compare?
Answer:
The given table is:

Now,
From the given table,
We can observe that
The total length of Robert’s run = 3 × (The jump distance of Robert)
= 3 × 297
= 3 × (200 + 90  + 7)
= (3 × 200) + (3 × 90) + (3 × 7)
= 600 + 270 + 21
= 891 feet
Now,
The representation of the given information about Robert in the form of a bar diagram is:

So,
The given situation uses multiplication to compare
Hence, from the above,
We can conclude that the total length of Robert’s run is: 891 feet

Part B
How much farther was Robert’s run than Jackie’s run? Write equations to represent each step of the problem. Tell what your variables represent.
Answer:
The given table is:

Now,
From the given table,
The total length of Jackie’s run = 3 × (The jump distance of Jackie)
= 3 × 189
= 3 × (100 + 80 + 9)
= (3 × 100) + (3 × 80) + (3 × 9)
= 300 + 240 + 27
= 567 feet
Now,
From part (A),
The total length of Robert’s run is: 891 feet
So,
The difference between Robert and Jackie’s length = 891 – 567
= 324 feet
Hence, from the above,
We can conclude that Robert’s run was 324 feet farther than Jackie’s run

Question 2.
Jackie’s run was the same on two different jumps. In the first, she jumped 145 feet and went an additional 167 feet after the jump. In the second, she jumped 135 feet. How many feet, f, did she go after the jump?
Answer:
It is given that
Jackie’s run was the same on two different jumps. In the first, she jumped 145 feet and went an additional 167 feet after the jump. In the second, she jumped 135 feet
So,
The total run of Jackie in the first jump = (The total run of Jackie before she jumped) + (The additional run of Jackie after she jumped)
= 145 + 167
= 312 feet
Now,
Let the additional run of Jackie after the second jump be x
So,
According to the given information,
312 = 135 + x
x = 312 – 135
x = 177 feet
Hence, from the above,
We can conclude that Jackie had gone 177 feet after her second jump

Question 3.
Use the Beginner Jump information to find how much longer the total length of the ski jump hill for an advanced jump is than a beginner jump. The advanced jump distance is 408 feet.

Part A
What are the hidden questions you need to answer to solve the problem? Name a variable for each question.
Answer:
It is given that
The advanced jump distance is 408 feet.
Now,
The given information is:

Hence, from the above,
We can conclude that
The hidden questions you need to answer to solve the given problem are:
a. What is the jump distance of the beginner jump distance?
b What is the total length of a beginner jump?

Part B
How much longer is d, the total length of an advanced jump hill, then a beginner jump hill? Write equations and explain how to solve the hidden questions and the original question.
Answer:
It is given that
The advanced jump distance is 408 feet.
Now,
The given table is:

Now,
Let the advanced jump distance is x times as the beginner jump distance
So,
According to the given information,
x × 8 = 408
x = \(\frac{408}{8}\)
x = 51 feet
So,
The beginner jump distance is: 51 feet
Now,
According to the given information,
x × 3 = 51
x = \(\frac{51}{3}\)
x = 17 feet
So,
The toatl length of a beginner jump is: 17 feet
So,
The length of an advanced jump hill, greater than a beginner jump hill = 51 – 17
= 34 feet
Hence, from the above,
We can conclude that
The length of an advanced jump hill is 34 feet longer than a beginner jump hill

Go through the enVision Math Common Core Kindergarten Answer Key Topic 5 Classify and Count Data regularly and improve your accuracy in solving questions.

Envision Math Common Core Kindergarten Answers Key Topic 5 Classify and Count Data

Essential Question:
How can classifying data help answer questions?

enVision STEM Project: Sorting Animals
Directions Read the character speech bubbles to students. Find Out! Have students find out about animals that can be organized by color. Say: Talk to friends and relatives about animals. Talk about how an animal can be one color, but another of the same animal can be a different color. Journal: Make a Poster Have students make a poster. Have them choose one animal they learned about, and then draw a group of 6-10 animals. Ask them to color the animals using two different colors, and then write the numbers to tell how many of each color.

Review What You Know

Directions Have students: 1 draw a circle around the group with 10 bugs; 2 draw a circle around the group that has a number of birds that is less than 5; 3 draw a circle around the group that has a number of birds that is greater than 5; 4 – 6 count the frogs or bugs in each group, and then write the number to tell how many.

Question 1.

Answer:

Question 2.

Answer:

Question 3.

Answer:

Question 4.

Answer:

Question 5.

Answer:

Question 6.

Answer:

Pick A Project

Directions Say: You will choose one of these projects. Look at picture A. Think about this question: What would our class flag look like? If you choose Project A, you will design a flag. Look at picture B. Think about this question: How do you go? If you choose Project B, you will make a model. Look at picture C. Think about this question: How does an instrument make music? If you choose Project C, you will act out playing instruments and making music.

3-ACT MATH PREVIEW

Directions Read the robot’s speech bubble to students. Generate Interest Ask students what they use to color pictures at home. Say: What colors would you use to color a dog? What colors would you never use? Have them draw their own dogs and color them in creative or unusual ways.

I can … model with moth to make equal groups lo solve a problem.

Lesson 5.1 Classify Objects into Categories

Solve & Share

Directions Say: Carlos’ kindergarten class ¡s having a pet fair. The pets need to be put into Iwo tents. One tent is for pets with q legs. The other tent is for pets that do NOT have q legs. Draw pictures of 5 pets. How many animals are in the q legs tent? How many animals are in the NOT q legs tent? Count the legs and talk about your pictures.

I can ………. classify objects into categories and tell why they are in each category.

Answers:

Explanation:
In the pet fair he want to classify the pets in 2 tents
one with 4 legs and the other with not 4 legs
classified accordingly

Visual Learning Bridge

Guided Practice

Directions 1 Have students draw a circle around the animals that have feathers, and then mark an X on the animals that do NOT have feathers.

Question 1.

Answer:

Explanation:
peacock, bird, hen, and cock are having feathers
snakes and frog doesn’t have frogs

Directions 2 enVision® STEM Say: What can most animals with wings do? Have students draw a circle around the animals that have wings, and then mark an X on the animals that do NOT have wings. Have students: 3 draw a circle around the rabbits that are white, and then mark an X on the rabbits that are NOT white; 4 draw a circle around the cows that are brown, and then mark an X on the cows that are NOT brown; 5 draw a circle around the dogs that have spots, and then mark an X on the dogs that do NOT have spots.

Question 2.

Answer:

Explanation:
The animals can fly with the help of wings or feathers
wings gives the support to birds .
circled the animals which are having wings and crossed the animals which are not having wings.

Question 3.

Answer:

Explanation:
total rabbits are 8
2 rabbits white in color which are circled
6 are crossed which are brown in color.

Question 4.

Answer:

Explanation:
There  are 6 cows in total
3 are in brown which are circled
3 are crossed which are not in brown.

Question 5.

Answer:

Explanation:
number of dogs 9
4 are with dots and
5 are without dots

Independent Practice

Directions Have students: 6 draw a circle around the birds that are green, and then mark an X on the birds that are NOT green; 7 draw a circle around the animals that have tails, and then mark an X on the animals that do NOT have tails, 8 Higher Order Thinking Say: The animals have been classified into two categories. How were the animals classified? Have students draw a picture of an animal that belongs in each category.

Question 6.

Answer:

Explanation:
there are 9 birds in which 4 are green and circled
5 are crossed which are not in green.

Question 7.

Answer:

Explanation:
circled the animals which are having tails
and crossed the animals which are not having the tails

Question 8.

Answer: They were classified as wild animals and pet animals and images are drawn
wolf is a wild animal where as cow is a pet

Lesson 5.2 Count the Number of Objects in Each Category

Solve & Share
Directions Say: Carlos goes outside and sees some creatures. How many creatures does he see on the ground? How many does he see that are NOT on the ground? Tell how you know you counted oil of the creatures.

I can …….. count how many objects are in different categories.

Visual Learning Bridge

Guided Practice

Directions 1 Hove students draw lines in the chart as they count the animals that ore in the pond and the animals that are NOT in the pond, and then write the numbers to tell how many in another chart.

Question 1.

Answer:

Explanation:
6 were in the pond and 5 are out of the pond

Directions 2 Vocabulary Have students draw lines in the chart as they count the animals that have 8 legs and the animals that do NOT have 8 legs, and then write the numbers to tell how many are in each category in another chart. 3 Have students draw lines in the chart as they count the birds that are in the trees and the birds that are NOT in the trees, and then write the numbers to tell how many are in each category in another chart.

Question 2.

Answer:

Explanation:
7 insects has 8 legs and 4 has different number of legs

Question 3.

Answer:

Explanation:
4 birds on the tree
3 birds are not there on the tree.

Independent Practice

Directions Have students: 4 draw lines in the chart as they count the ants that are red and the ants that are NOT red, and then write the numbers to tell how many in another chart; 5 draw lines in the chart as they count the bees that are on flowers and the bees that are NOT on flowers, and then write the numbers to tell how many in another chart. 6 Higher Order Thinking Say: These mice are sorted into two categories. How are the mice on the left different than the mice on the right? Have students draw a picture in the chart to show the categories, and then write the numbers to tell how many mice are in each category.

Question 4.

Answer:

7 ants are red in color
7 ants are not in red color

Question 5.

Answer:

Explanation:
6 bees are not on flower
2 bees are on the flower

Question 6

Answer:

5 mice are in white color
4 mice are in brown and ash color

Lesson 5.3 Sort the Categories by Counting

Solve & Share
Directions Say: Carlos’s kindergarten class has a new playground area. Sort the new playground into toys that have wheels and toys that do NOT have wheels. Count the toys in each category. Write numbers to tell how many. Draw a circle around the category that is greater than the other category. Tell how you know.

I can … use counting to compare how many objects are in categories.

The objects with wheels are 6
and the objects without wheels are 7
so, 7 is circled

Visual Learning Bridge

Guided Practice

Directions 1 Have students sort the crayons into crayons that are blue and crayons that are NOT blue, count them, and then write numbers in the chart to tell how many. Have students draw a circle around the category that is less in number than the other category and tell how they know.

Question 1.

Directions Have students: 2 sort the blocks into blocks that have letters and blocks that do NOT have letters, count them, and then write numbers in the chart to tell how many. Then have students draw a circle around the category that is greater in number than the other category and tell how they know; 3 sort the books into books that are open and books that are NOT open, count them, and then write numbers in the chart to tell how many. Then have students draw a circle around the category that is less in number than the other category and tell how they know.

Question 2.

Answer:

Explanation:
sorted the blocks into blocks that have letters and blocks that do NOT have letters, counted them, and then write numbers in the chart. Then students have drawn a circle around the category that is greater in number than the other category
number of blocks with A are 4 and number of blocks without A are 6

Question 3.

Answer:

Explanation:
sorted the books into books that are open and books that are NOT open, counted them, and then written numbers in the chart
and circled the highest number.

Independent Practice

Directions Have students: 4 sort the balls into balls that are yellow and balls that are NOT yellow, count them, and then write numbers in the chart to tell how many. Then have students draw a circle around the category that is greater in number than the other category and tell how they know; 5 sort the pencils into pencils that are short and pencils that are NOT short, count them, and then write numbers in the chart to tell how many. Then have students draw a circle around the category that is greater in number than the other category and tell how they know. 6 Higher Order Thinking Say: The fish are sorted into fish that have spots and fish that do NOT have spots. Have students draw fish so the categories have an equal number of fish, and then write the number of fish in each category. Ask: How do you know the categories have an equal number of fish?

Question 4.

Answer:

Explanation:
sorted the balls into balls that are yellow and balls that are NOT yellow, counted them, and then written numbers in the chart
and circled the highest number

Question 5.

Answer:

Explanation:

Sorted the pencils into pencils that are short and pencils that are NOT short, counted them, and then written numbers in the chart
and circled the highest number.

Question 6.

Answer:

Explanation:
The fish are sorted into fish that have spots and fish that do NOT have spots. students have drawn fish so the categories have an equal number of fish, and then written the number of fish in each category. black fishes are having spots and 3 same as green categories have an equal number of fish.

Lesson 5.4 Problem Solving

Critique Reasoning
Solve & Share

Directions Say: Carlos says that the number of blue cubes is equal to the number of cubes that are NOT blue. Does his answer make sense? Use numbers, pictures, or words to explain your answer.

I can ……….. tell whether the way objects have been sorted, counted, and compared makes sense. I can explain how I know.

Explanation:
by using the numbers we have sorted
number of blues are 10 and number of not blues are 9
Visual Learning Bridge

Guided Practice

Directions 1 Say: Gabbi says that the category of airplanes is greater in number than the category that is NOT airplanes. Does her answer make sense? Have students draw a circle around yes or no, and then use the sorting and counting of each category to explain their reasoning.

Question 1.

Answer:
The students have drawn circles around no

Independent Practice

Directions Have students listen to each problem, draw a circle around yes or no, and then use the sorting and counting of each category to explain their reasoning. 2 Damon says that he counted 8 yellow train cars and 6 train cars that are NOT yellow. Does his answer make sense? 3 Malinda says that the category of yellow train cars is less than the category of train cars that are NOT yellow. Does her answer make sense? 4 Aaron says that the category of red train cars is greater than the category of train cars that are NOT red. Does his answer make sense?

Question 2.

Answer:

no Damon says that there are 8 yellow but there are only 6 yellow cars.

Question 3.

Answer:

yes, Malinda is correct.

Question 4.

Answer:

Explanation:
Aaron says correct number of reds are higher than the not reds.

Problem Solving

Performance Task
Directions Read the problem aloud. Then have students use multiple problem-solving methods to solve the problem. Say: Alex says that if there was I fewer orange ball, then the category of orange balls would be equal in number to the category of balls that are NOT orange. Does his answer make sense? 5 Reasoning Think about it. How many orange balls would there be if there were I fewer orange ball? Use numbers, tools, or draw a picture to show how many orange balls there would be. 6 Be Precise Is the number of orange balls equal to the number of balls that are NOT orange? 7 Critique Reasoning Use the sorting and counting of each category to explain your reasoning.

Answer:
Explanation:
1:
what alex says is wrong because oranges more than the not oranges
2:

alex says 1 fewer of oranges is equal to not oranges
he is not true
as 6 is greater than 5.

Topic 5 Vocabulary Review

Directions Understand Vocabulary Have students: 1 draw an animal that fits each category, and then tell how the groups are organized; 2 sort books into books that are open and books that are NOT open. Have them draw tally marks in the chart as they count, and then write the number in another chart.

Question 1.

Answer:

Explanation:
They are sorted according to 4 legs and not 4 legs.

Question 2.

Answer:

Explanation:
Books with open are 3 and books without open are 4.

Directions Understand Vocabulary Have students: 3 classify the dogs by drawing circles and marking Xs, and then explain how they organized them; 4 draw lines in the chart to show how many in each group, and then draw a circle around the group that is greater in number than the other group.

Question 3.

Answer:

Explanation:
Dogs are sorted according to dots and not having dots
having dots are circled and no t having dots are crossed.

Question 4.

Answer:

Explanation:
circled the highest one.

Topic 5 Reteaching

Directions Have students: 1 draw a circle around the animals that walk on 2 legs, and then mark an X on the animals that do NOT walk on 2 legs; 2 draw lines in the chart as they count the toys that are on the rug and the toys that are NOT on the rug. Then have them write the numbers to tell how many are in each group in another chart.

Set A

Question 1.

Answer:

Explanation:
Circled which walk on 2 legs
and crossed which will not walk on 2 legs.

Set B

Question 2.

Answer:

Explanation:
4 on the rug and 4 are not on the rug.

Directions 3 Have students sort the balls into balls that are white and balls that are NOT white, and then write numbers in the chart to tell how many. Then have students draw a circle around the group that is greater in number than the other group and tell how they know. 4 Say: Malik says that the group of airplanes is greater than the group that is NOT airplanes. Does his answer make sense? Have students draw a circle around yes or no, and then use the sorting and counting of each category to explain their reasoning.

Set C

Question 3.

Answer:

Explanation:
11 is greater so circled.

Set D

Explanation:

Question 4.

Answer: No
Aero planes are 4 and boats are 5
so, what he is saying is not true.

Topic 5 Assessment Practice

Directions Have students: 1 A. draw lines ¡n the chart as they count the fish that are yellow and the fish that are NOT yellow; B. compare the number of fish that are yellow to the number of fish that are NOT yellow. Say: Draw a circle around the category that is less in numbers, 2 draw a circle around the animals that fly, and mark an X on the animals that do NOT fly. 3 A. Say: The animals have been classified into two categories. Mark all the animals that belong in the category of animals inside the circle. B. Then hove students compare the number of animals inside the circle with the number of animals outside the circle. Say: Which statement correctly describes the picture?

Question 1.

Answer:

Answer:
drawn lines in the chart and circled the not yellow fishes because they are less in numbers.

Question 2.

Answer:

Explanation:
drawn circles among the birds which can fly
and crossed the animals which cannot fly

Question 3.

A. The number outside is greater than the number inside.
B. The number inside is less than the number outside.
C. The number inside is equal to the number outside.
D. The number outside is less than the number inside.
Answer: D

Directions Have students: 4 draw lines in the chart as they count the buttons that are green and the buttons that are NOT green, and then write the numbers to tell how many in another chart; 5 sort the balls into balls that are tennis balls and balls that are NOT tennis balls, count them, and then write numbers in the chart to tell how many. Then have students draw a circle around the category that is less than the other category. 6 listen to the problem, draw a circle around yes or no, and then use numbers, pictures, or words to explain how they know whether the answer makes sense. Say: Dana says that the category of blue beads is greater than the category of beads that are NOT blue. Does her answer make sense?

Question 4.

Answer:

Explanation:
Drawn lines in the chart and 6 are green in color and 3 are not green in color.

Question 5.

Answer:

Explanation:
4 are tennis balls and 5 are not tennis balls and circled which are less

Question 6.

Answer:

Explanation:
blue beads are 5 and pink beads are 7
blue beads are not greater than pink beads

Topic 5 Performance Task

Directions Works of Art Say: A kindergarten class uses paintbrushes and paint to draw pictures. Have students: 1 draw a circle around the little paintbrushes, and then mark an X over the paintbrushes that are NOT little; 2 draw lines in the first chart as they count the paintbrushes that are little and the paintbrushes that are NOT little. Then have them write the number to tell how many are in each group in the second chart, and draw a circle around the number of the group that is less than the number of the other group.

Question 1.

Answer:

Explanation:
drawn circles around little paint brushes and crossed among the bigger painting brushes

Question 2.

Answer:

EXplanation:
Completed the chart and counted and written the numbers.

Directions 3 Have students show one way to organize the jars of paint they see on the page before, and then explain how they sorted them. 4 Say: Tina says that the number of small jars of paint is equal to the number of large jars of paint. Does her answer make sense? Have students look at the paint on the page before, draw a circle around yes or no, and then use the sorting and counting of each category to explain their reasoning.

Question 3.
Answer:
sorted by the color
No , they didn’t sorted

Question 4.

Answer: NO

Explanation:
Tina says that the number of small jars of paint is equal to the number of large jars of paint.No, her answer is wrong. No, the students have not  look at the paint on the page before.
Drawn circle among the no.

Go through the enVision Math Common Core Kindergarten Answer Key Topic 4 Compare Numbers 0 to 10 regularly and improve your accuracy in solving questions.

Envision Math Common Core Kindergarten Answers Key Topic 4 Compare Numbers 0 to 10

Essential Question: How can numbers from 0 to 10 be compared and ordered?

enVision STEM Project: Weather Changes

Directions Read the character speech bubbles to students. Find Out! Have students find out about weather changes. Say: The weather changes from day to day. Talk to friends and relatives about the weather. Ask them to help you record the number of sunny days and rainy days during the week. Journal: Make a Poster Have students make a poster. Have them draw up to 10 lightning bolts above one house and up to 10 lightning bolts above another house. Ask them to write the number of lightning bolts above each house, and then draw a circle around the number that is greater than the other, or draw a circle around both numbers if they are the same.

Review What You Know

Question 1.

Answer:

Explanation:
I circled the group of birds that are lessthan the other group.3 is lessthan 4.

Question 2.

Answer:

Explanation:
I circled the group of dogs that is greater than the ther group.5 is greater than 1.

Question 3.

Answer:

Explanation:
I circled 2 groups of marbles that are equal in number.3 is equal to 3.

Question 4.

Answer:

Explanation:
There are 6 objects in the above picture.So, i counted and wrote 6.

Question 5.

Answer:

Explanation:
There are 8 objects in the above picture.So, i counted and wrote 8.

Question 6.

Answer:

Explanation:
There are 10 objects in the above picture.So, i counted and wrote 10.

Directions Have students: 1 draw a circle around the group of birds that is less than the other group; 2 draw a circle around the group of dogs that is greater than the other group; 3 draw a circle around the two groups that have an equal number of marbles; 4-6 count the number of objects, and then write the number to tell how many.

Pick a Project

Directions Say: You will choose one of these projects. Look at picture A. Think about this question: How can you train to go into space? If you choose Project A, you will act out an exercise skit. Look at picture B, Think about this question: What kinds of fruit would you put into a fruit salad? If you choose Project B, you will create a fruit salad recipe.

Directions Say: You will choose one of these projects. Look at picture C. Think about this question: What is the most exciting ride at a theme park? If you choose Project C, you will design a ride. Look at picture D. Think about this question: What do you like to do on a vacation? If you choose Project D, you will make a list.

Lesson 4.1 Compare Groups to 10 by Matching

Solve & Share

Directions Say: Work with a partner. Take turns drawing one cube from the bag and placing it on your page in the rectangle of the same color. When the bag is empty, do you have more red or blue cubes? How do you know? Draw a picture of your cubes in The rectangles showing which color is more.

Visual Learning Bridge

Guided practice

Question 1.

Answer:

Explanation:
I drew a line from each chick in the top group to a chick in the bottom group, and then drew a circle around the group that is greater in number than the other group.7 is greater than 6.

Directions 1 Have students compare the groups, draw a line from each chick in the top group to a chick in the bottom group, and then draw a circle around the group that is greater in number than the other group.

Question 2.

Answer:

Explanation:
I compared the groups, drew a line from each chick in the top group to a chick in the bottom group, and then drew a circle around the group that is greater in number than the other group.8 is greater than 3.

Question 3.

Answer:

Explanation:
I compared the groups, drew a line from each chick in the top group to a chick in the bottom group, and then drew a circle around the group that is less in number than the other group.4 is less than 5.

Question 4.

Answer:

Explanation:
I compared the groups, drew a line from each chick in the top group to a chick in the bottom group, and then drew a circle around the group that is less in number than the other group.7 is less than 9.

Directions 2 envision^® STEM Say: Chicks live in coops. Coops protect chicks in different types of weather. Have students compare the groups, draw a line from each chick in the top group to a chick in the bottom group, and then draw a circle around the group that is greater in number than the other group. 3 and 4 Have students compare the groups, draw a line from each chick in the top group to a chick in the bottom group, and then draw a circle around the group that is less in number than the other group.

Independent Practice

Question 5.

Answer:

Explanation:
I compared the groups, drew a line from each bucket in the top group to a bucket in the bottom group, and then drew a circle around the group that is greater in number than the other group.8 is greater than 6.

Question 6.

Answer:

Explanation:
I compared the groups, drew a line from each bucket in the top group to a bucket in the bottom group, and then drew a circle around the group that is less in number than the other group.2 is less than 3.

Question 7.

Answer:

Explanation:
I compared the groups, drew a line from each bucket in the top group to a bucket in the bottom group, and then drew a circle around the group that is less in number than the other group.4 is less than 7.

Question 8.

Answer:

Explanation:
I counted the numbe rof buckets, there are 5 buckets in the top group.
I drew a group of 9 buckets that is greater than the buckets in the top group.
9 is greater than 5.

Directions Have students: 5 compare the groups, draw a line from each bucket in the top group to a bucket in the bottom group, and then draw a circle around the group that is greater in number than the other group; 6 and 7 compare the groups, draw a line from each bucket in the top group to a bucket in the bottom group, and then draw a circle around the group that is less in number than the other group, 8 Higher Order Thinking Have students draw a group of buckets that is greater in number than the group shown.

Lesson 4.2 Compare Numbers Using Numerals to 10

Answer:

Explanation:
Emily is planting seedlings, or little plants. She plants 5 red pepper seedlings and 7 yellow pepper seedlings.
I drew counters to show groups of seedlings and i wrote the numbers 5 and 7.I circled 7 as it is greater than 5.

Directions Say: Emily is planting seedlings, or little plants. She plants 5 red pepper seedlings and 7 yellow pepper seedlings. Use counters to show the groups of seedlings. Write the numbers, and then circle the number that tells which group has more.

Visual Learning Bridge

Guided Practice

Question 1.

Answer:

Explanation:
I drew a line from each watering can in the top group to a watering can in the bottom group, and then draw a circle around the number 4 as it is greater than 3.

Directions 1 Have students count the watering cans in each group, write the number to tell how many, draw a line from each watering can in the top group to a watering can in the bottom group, and then draw a circle around the number that is greater than the other number.

Question 2.

Answer:

Explanation:
I counted the vegetables in each group, wrote the numbers 4 and 5 to tell how many, drew a line from each vegetable in the top group to a vegetable in the bottom group, and then marked an X on the number 4 as it is less than 4.

Question 3.

Answer:

Explanation:
I counted the vegetables in each group, i drew 3 more pea pods to make the groups equal, wrote the numbers 6 to tell how many in each group, and then drew a line from each vegetable in the top group to a vegetable in the bottom group to compare.

Directions 2 Have students count the vegetables in each group, write the number to tell how many, draw a line from each vegetable in the top group to a vegetable in the bottom group, and then mark an X on the number that is less than the other number. 3 Number Sense Have students count the vegetables in each group, draw more pea pods to make the groups equal, write the numbers to tell how many in each group, and then draw a line from each vegetable in the top group to a vegetable in the bottom group to compare.

Independent Practice

Question 4.

Answer:

Explanation:
I counted the seed packets in each group, wrote the numbers 10 and 7 to tell how many, drew a line from each seed packet in the top group to a seed packet in the bottom group, and then mark an X on the number 7 as it  is less than 10.

Question 5.

Answer:

Explanation:
I counted the flowers in the group, drew a group of 3 flowers that is less than the group shown, and then write the numbers 6 and 3 to tell how many.

Directions 4 Have students count the seed packets in each group, write the number to tell how many, draw a line from each seed packet in the top group to a seed packet in the bottom group, and then mark an X on the number that is less than the other number. 5 Higher Order Thinking Have students count the flowers in the group, draw a group of flowers that is less than the group shown, and then write the numbers to tell how many.

Lesson 4.3 Compare Groups to 10 by Counting

Solve & Share

Answer:

Explanation:
I counted and placed counters on goldfish and tetras, there are 6 goldfish and 9 tetras.I drew a circle around the number 9 as it is greater than 6.

Directions Say: The class aquarium has two kinds of fish, goldfish and tetras. Place counters on the fish as you count how many of each kind. Write numbers to tell how many of each kind. Draw a circle around the fish that has a number greater than the other. Tell how you know you are right.

Visual Learning Bridge

Guided Practice

Question 1.

Answer:

Explanation:
I counted and wrote the number of fishes each color, there are 10 pink fishes and 8 purple fishes.I circled 10 as it is greater than 8.

Directions 1 Have students count the number of each color fish, write the numbers to tell how many, and then draw a circle around the number that is greater than the other number. Use the number sequence to help find the answer.

Question 2.

Answer:

Explanation:
I counted and wrote the number of fishes each color, there are 6 green fishes and 7 yellow fishes.I circled 7 as it is greater than 6.

Question 3.

Answer:

Explanation:
I counted wrote the number of fishes each color, there are 8 blue fishes and 9 gold fishes.I marked X on both  the numbers as they are NOT equal.

Question 4.

Answer:

Explanation:
I counted wrote the number of fishes each color, there are 8 brown fishes and 7 green fishes.I marked X on the number 7 as it is lessthan 8.

Question 5.

Answer:

Explanation:
I counted wrote the number of fishes each color, there are 9 purple fishes and 10 gold fishes.I marked X on the number 7 as it is lessthan 10.

Directions Have students count the number of each color fish, write the numbers to tell how many, and then: 2 draw a circle around the number that is greater than the other number; 3 draw a circle around both numbers if they are equal, or mark an X on both numbers if they are NOT equal; 4 and 5 mark an X on the number that is less than the other number. Use the number sequence to help find the answer for each problem.

Independent Practice

Question 6.

Answer:

Explanation:
I counted and wrote the number of each critter.I wrote the numbers 6 to tell that there are 6 yellow and 6 green critters.I circled both the numbers as they are equal.

Question 7.

Answer:

Explanation:
I counted and wrote the number of each critter.I wrote the numbers 9 and 8 to tell that there are 9 blue and 8 peach critters.I marked X on the numbe 8 as it is lessthan 9.

Question 8.

Answer:

Explanation:
There are 7 trantulas in the above picture so, i drew 9 spiders that is two more than the number of trantulas.
Then i wrote the numbers 7 and 9.

Question 9.

Answer:

Explanation:
I counted the number of butterflies.There are 6 butterflies.I wrote all the numbers up to 10 that are greater than the number of butterflies shown.The numbers that are greater than 6 are 7, 8, 9 and 10.

Directions Have students count the number of each critter, write the numbers to tell how many, and then: 6 draw a circle around both numbers if they are equal, or mark an X on both numbers if they are NOT equal; 7 mark an X on the number that is less than the other number; 8 draw a group of spiders that is two greater in number than the number of tarantulas shown, and then write the number to tell how many. 9 Higher Order Thinking Have students count the butterflies, and then write all the numbers up to 10 that are greater than the number of butterflies shown. Use the number sequence to help find the answer for each problem.

Lesson 4.4 Compare Numbers to 10

Solve & Share

Directions Say: Emily’s mother asked her to bring the towels in off the line. Her basket can hold less than 7 towels. How many towels might Emily bring in? You can give more than one answer. Show how you know your answers are right.
Answer:
Emily’s mother asked her to bring the towels in off the line. Her basket can hold less than 7 towels.
Emily can bring 1, 2, , 4, 5 or 6 towels in her basket.

Visual Learning Bridge

Guided Practice

Question 1.

Answer:

Explanation:
I counted the numbers 1 to 10. I used the number sequence and drew lines from the numbers to the sequence and i circled 8 a sit is greater than 7.

Question 2.

Answer:

Explanation:
I drew counters to show the numbers 6 and 4.I drew a circle around the number 6 as it is greater than the number 4.

Directions Have students: 1 count the numbers 1 to 10 and use the number sequence to show how they know which number is greater than the other, and then draw a circle around the number that is greater; 2 draw counters in the ten-frames to show how they know which number is greater than the other, and then draw a circle around the number that is greater.

Question 3.

Answer:

Explanation:
I drew pictures to show the numbers 6 and 9.9 is greater than 6.

Question 4.

Answer:

Explanation:
I drew counters to show the numbers 8.
I circled both the numbers as they are equal.

Question 5.

Answer:

Explanation:
I used the number sequence to find the greater number.I marked X on the number 9 as it is lessthan 10.

Question 6.

Answer:

Explanation:
I drew the pictures to show the numbers 9 and 8.I marked X on the number 8 as it is lessthan 9.

Directions Have students: 3 draw pictures to show how they know which number is greater than the other, and then draw a circle around the number that is greater; 4 draw counters in the ten-frames to show how they know if the numbers are equal, and then draw a circle around both numbers if they are equal, or mark an X on both numbers if they are NOT equal; 5 use the number sequence to show how they know which number is less than the other number, and then mark an X on the number that is less; 6 draw pictures to show how they know which number is less than the other number, and then mark an X on the number that is less.

Independent Practice

Question 7.

Answer:

Explanation:
I drew the pictures to show the numbers 6 and 8.I marked X on the number 6 as it is lessthan 8.

Question 8.

Answer:

Explanation:
I used the number sequence to find the greater number.I marked X on the number 7 as it is lessthan 9.

Question 9.

Answer:

Explanation:
I wrote the next two numbers that are greater than 8. the numbers that are greater than 8 are 9 and 10.

Question 10.

Answer:

Explanation:
I wrote the number 7 as it is greater than 5 and lessthan 9.

Directions Have students: 7 draw pictures to show how they know which number is less than the other number, and then mark an X on the number that is less; 8 use the number sequence to show how they know which number is less than the other number, and then mark an X on the number that is less. 9 Higher Order Thinking Have students write the next two numbers that are greater than the number shown, and then tell how they know. 10 Higher Order Thinking Have students write a number that is greater than the number on the left, but less than the number on the right.

Lesson 4.5 Repeated Reasoning

Problem Solving

Solve & Share

Answer:

Directions Say: There are 7 fish in a bowl. Emily puts I more fish in the bowl. How many fish are in the bowl now? How can you solve this problem?

Visual Learning Bridge

Guided Practice

Question 1.

Answer:

Explanation:
I drew counters for 4 frogs and i drew one more.The number that is one greater than 4 is 5.

Directions 1 Say: Carlos sees 4 frogs at the pond. Then he sees 1 more. How many frogs are there now? Have students use reasoning to find the number that is 1 greater than the number of frogs shown. Draw counters to show the answer, and then write the number. Have students explain their reasoning.

Independent Practice

Question 2.

Answer:

Explanation:
I drew counters for 7 frogs and i drew one more.The number that is one greater than 7 is 8.

Question 3.

Answer:

Explanation:
I drew counters for 2 frogs and i drew one more.The number that is one greater than 2 is 3.

Question 4.

Answer:

Explanation:
I drew counters for 8 frogs and i drew one more.The number that is one greater than 8 is 9.

Question 5.

Answer:

Explanation:
I drew counters for 6 frogs and i drew one more.The number that is one greater than 6 is 7.

Directions Say: Alex sees frogs at the pond. Then he sees 1 more. How many frogs are there now? 2-5 Have students use reasoning to find the number that is 1 greater than the number of frogs shown. Draw counters to show the answer, and then write the number. Have students explain their reasoning.

Problem Solving

Performance Task

Question 6, 7, 8.

Answer:

Explanation:
I drew counters for 5 pets of Marta, i drew 1 more counter.The number that is one greater than 5 is 6.Marta will now have 6 pets.

Directions Read the problem aloud. Then have students use multiple problem-solving methods to solve the problem. Say: Marta’s family has 5 pets. Then her family gets 1 more. How many pets do they have now? 6 Generalize Does something repeat in the problem? How does that help? 7 Use Tools What tool can you use to help solve the problem? Use the tool to find the number of pets in Marta’s family now. 8 Make Sense Should the answer be greater than or less than 5?

Topic 4 Vocabulary Review

Question 1.

Answer:

Explanation:
I circled the number 9 as it is greater than 7.

Question 2.

Answer:

Explanation:
I counted the counters given, there are 7 counters in all.so, i wrote the number 7 in the blank.

Question 3.

Answer:

Explanation:
The number that means NONE is 0.

Question 4.

Answer:

Explanation:
There are 2 red cubes and 3 yellow cubes.I circled the red cubes as they are less than yellow cubes.I wote the number 2 as it is lessthan 3.

Directions Understand Vocabulary Have students: 1 draw a circle around the number that is greater than 7; 2 count the counters, and then write the number to tell how many; 3 write the number that means none; 4 count how many of each color cube there is, draw a circle around the group that has a number of cubes that is less than the other group, and then write the number to tell how many there are in that group.

Question 5.

Answer:

Explanation:
I circled the number 8 as it is greater than 3 and marked X on the number 3 as it is lessthan 8.

Question 6.

Answer:

Explanation:
I wrote 4 as it is greater than 3 and lessthan 5.

Question 7.

Answer:

Explanation:
I drew counters for the given number 5.

Question 8.

Answer:

Explanation:
I wrote the missing numbers 3, 4, 6, 7 and 8 in the gien blanks.

Directions Understand Vocabulary Have students: 5 compare the numbers, draw a circle around the number that is greater, and then mark an X on the number that is less; 6 write a number that is greater than 3, but less than 5; 7 draw 5 counters in a row and then write the number to tell how many; 8 write the missing numbers in order.

Topic 4 Reteaching

Set A

Question 1.

Answer:

Explanation:
I counted the number of counters, there are 6 red counters and 10 yellow counters.I circled red counters frame as they are less in number than the yellow counter frame.

Set B

Question 2.

Answer:

Explanation:
I drew lines from the top group to a piece of fruit in the bottom group, and then i drew a circle around the number 7 as it is greater than 5.

Directions Have students: 1 compare the groups, and draw a circle around the group that is less in number than the other group; 2 count the fruit in each group, write the numbers that tell how many, draw a line from each piece of fruit in the top group to a piece of fruit in the bottom group, and then draw a circle around the number that is greater than the other number.

Set C

Question 3.

Answer:

Explanation:
I counted and wrote the number of blue critter and peach critters.They are 6 peach critters and 4 blue critters.
I marked X on the number 4 as it is lessthan 6.

Set D

Question 4.

Answer:

Explanation:
I drew counters for the 6 frogs and drew one more counter.1 greater than 6 is 7.

Directions Have students: 3 count the number of each critter, write the numbers, and then mark an X on the number that is less than the other number; 4 Say: April sees frogs at The pond. Then she sees 1 more. How many frogs does she see now? Have students use reasoning to find the number that is 1 greater than the number of frogs shown. Draw counters to show the answer, and then write the number.

Topic 4 Assessment Practice

Question 1.

Answer:

Explanation:
There are 8 yellow birds.I marked group ‘A’ as it has more number of blue birds than the yellow birds.There are 10 blue bieds in group A.

Question 2.

Answer:

Explanation:
I marked the numbers 6, 5 and 3.The numbers 3, 5 and 6 as they are lessthan the number 7.

Question 3.

Answer:

Explanation:
I counted and wrote the number of lemons and limes, there are 7 lemons and 5 limes.
I circled the number 7 as it is greater than 5.

Directions Have students mark the best answer. 1 Which group of blue birds is greater in number than the group of yellow birds? 2 Look at the number line. Then mark all the numbers that are less than the number on the card. 3 Have students count the number of lemons and limes, write the number that tells how many of each, and then draw a circle around the number that is greater.

Question 4.

Answer:

Explanation:
The number of the first card is 7, the number before 7 is 6.I counted forward from 6 and wrote the numbers 7, 8, 9 and 10 in the blanks.

Question 5.

Answer:

Explanation:
There are 6 sandwiches and I drew the 4 juice boxes as 4 is lessthan 6.

Question 6.

Answer:

Explanation:
I drew counters for 7 beads and drew 1 more counter.one greater than the number 7 is 8.
Kayla will now have 8 beads to make a bracelet.

Directions Have students: 4 write the number that is counted first among the 4 number cards, and then count forward and write the number that is 1 greater than the number before; 5 count the sandwiches in the group, draw a group of juice boxes that is less than the group of sandwiches shown, and then write the numbers to tell how many. 6 Say: Kayla has 7 beads to make a bracelet. Then she buys 1 more. How many beads does she have now? Have students use reasoning to find the number that is 1 greater than the number of beads shown. Draw counters to show the answer, and then write the number to tell how many.

Topic 4 Performance Task

Question 1.

Answer:

Explanation:
There are 9 shunks and 8 raccoons in the forest.
Using number sequence i found that number 8 is lessthan number 9 so, i marked X on the number 8.

Directions Forest Animals Say: The forest is home to woodland animals. One part of the forest has many different animal homes in it. 1 Have students study the picture. Say: How many skunks live in this part of the forest? How many raccoons live in this part of the forest? Count the number of each type of animal and write the numbers. Then have students draw a circle around the number that is greater than the other number and mark an X on the number that is less than the other number. Have them use the number sequence to help find the answers.

Question 2.

Answer:

Explanation:
There are 6 foxes in the forest.I counted and wrote the numbers between 6 and 10.They are 7, 8, 9 and 10.

Question 3.

Answer:

Explanation:
I drew counters for the number 5, I drew circle around both the numbers as they are equal.

Question 4.

Answer:

Explanation:
There are 7 birds in the forest.I drew counters for 7 birds and one more counter as 1 bird flies into the forest.
One greater than the number 7 is 8.

Directions Have students look at the picture on the page before. 2 Say: How many foxes live in this part of the forest? Count how many and write the number. Then have students write all the numbers through 10 that are greater than the number of foxes. 3 Say: 5 chipmunks and 5 frogs move out of this part of the forest. Draw a circle around both numbers if they are equal, or mark an X on both numbers if they are NOT equal. Show how you know you are correct. 4 Say: How many birds live in this part of the forest? Count how many and write the number. I more bird flies into the forest. How many birds are in this part of the forest now? Have students use tools to solve the problem and write the number. Then have them show how they found the answer.

Go through the enVision Math Common Core Grade 6 Answer Key Topic 2 Integers and Rational Numbers regularly and improve your accuracy in solving questions.

Envision Math Common Core 6th Grade Answers Key Topic 2 Integers and Rational Numbers

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

3-ACT MATH

The ULTIMATE THROW

The Ultimate Throw
Have you ever played ultimate? It’s a team sport played with a flying disc. The goal is to score the most points by passing the disc to your opponent’s end zone. Ultimate is played by millions of people across the globe, from casual games to professional leagues.

There are many ways to throw a flying disc. It takes a lot of practice to learn each type of throw. If you want the disc to travel a specific path and distance, you need to try different throws with different amounts of spin and power. Think about this during the 3-Act Mathematical Modeling lesson.

enVision STEM Project

Did You Know?

Your Task: Improve Your School

Now that you have defined the problem, or improvement needed, you and your classmates will apply the engineering design process to propose solutions.

Topic 2 GET READY!

Review What You Know!

Vocabulary

Choose the best term from the box to complete each definition.

decimal
denominator
fraction
numerator

Question 1.
A ____ names part of a whole, part of a set, or a location on a number line.
Answer:
We know that,
A “Fraction” names part of a whole, part of a set, or a location on a number line.
Hence, from the above,
We can conclude that
The best term that is suitable for the given definition is: Fraction

Question 2.
The number above the fraction bar that represents the part of the whole is the ____
Answer:
We know that,
The number above the fraction bar that represents the part of the whole is the “Numerator”
Hence, from the above,
We can conclude that
The best term that is suitable for the given definition is: Numerator

Question 3.
The number below the fraction bar that represents the total number of equal parts in one whole is the ____
Answer:
We know that,
The number below the fraction bar that represents the total number of equal parts in one whole is the “Denominator”
Hence, from the above,
We can conclude that
The best term that is suitable for the given definition is: Denominator

Fractions and Decimals
Write each fraction as a decimal.

Question 4.
\(\frac{2}{5}\)
Answer:
The given fraction is: \(\frac{2}{5}\)
Hence, from the above,
We can conclude that
The representation of the given fraction in the form of a decimal is:

Question 5.
\(\frac{3}{4}\)
Answer:
The given fraction is: \(\frac{3}{4}\)
Hence, from the above,
We can conclude that
The representation of the given fraction in the form of a decimal is:

Question 6.
\(\frac{10}{4}\)
Answer:
The given fraction is: \(\frac{10}{4}\)
Hence, from the above,
We can conclude that
The representation of the given fraction in the form of a decimal is:

Question 7.
\(\frac{12}{5}\)
Answer:
The given fraction is: \(\frac{12}{5}\)
Hence, from the above,
We can conclude that
The representation of the given fraction in the form of a decimal is:

Question 8.
\(\frac{3}{5}\)
Answer:
The given fraction is: \(\frac{3}{5}\)
Hence, from the above,
We can conclude that
The representation of the given fraction in the form of a decimal is:

Question 9.
\(\frac{15}{3}\)
Answer:
The given fraction is: \(\frac{15}{3}\)
Hence, from the above,
We can conclude that
The representation of the given fraction in the form of a decimal is:

Division with Decimals

Divide.

Question 10.
1.25 ÷ 0.5
Answer:
The given Division Expression is: 1.25 ÷ 0.5
Now,
By using the Long Division,

Hence, from the above,
We can conclude that
1.25 ÷ 0.5 = 2.5

Question 11.
13 ÷ 0.65
Answer:
The given Division Expression is: 13 ÷ 0.65
Now,
By using the Long Division,

Hence, from the above,
We can conclude that
13 ÷ 0.65 = 20

Question 12.
12.2 ÷ 0.4
Answer:
The given Division Expression is: 12.2 ÷ 0.4
Now,
By using the Long Division,

Hence, from the above,
We can conclude that
12.2 ÷ 0.4 = 30

Ordered Pairs
Write the ordered pair for each point shown on the graph.

Question 13.
J
Answer:
The given coordinate plane is:

Now,
We know that,
“Ordered pairs” are often used to represent two variables. The number which corresponds to the value of x is called the x-coordinate and the number which corresponds to the value of y is called the y-coordinate.
Hence, from the above,
We can conclude that
The ordered pair J is: (4, 3)

Question 14.
K
Answer:
The given coordinate plane is:

Now,
We know that,
“Ordered pairs” are often used to represent two variables. The number which corresponds to the value of x is called the x-coordinate and the number which corresponds to the value of y is called the y-coordinate.
Hence, from the above,
We can conclude that
The ordered pair K is: (0, 6)