3.1 Introducing Multiplication

Unit Goals

  • Students represent and solve multiplication problems through the context of picture and bar graphs that represent categorical data.

Section A Goals

  • Interpret scaled picture and bar graphs.
  • Represent data using scaled picture and bar graphs.
  • Solve one- and two-step story problems using addition and subtraction.

Section B Goals

  • Represent and solve multiplication problems involving equal groups.
  • Understand multiplication in terms of equal groups.

Section C Goals

  • Represent and solve multiplication problems involving arrays.
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Section A: Interpret and Represent Data on Scaled Graphs

Problem 1

Pre-unit

Practicing Standards:  2.MD.D.10

The table shows how a group of students most enjoy traveling. Use the table to complete the picture graph.

way to travel number of students
airplane 4
hot air balloon 7
sail boat 6
scooter 3
Blank picture graph. Favorite Way to Travel. Horizontal axis: airplane, hot air balloon, sail boat, scooter.

Solution

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Problem 2

Pre-unit

Practicing Standards:  2.MD.D.10

Use the bar graph to answer the questions.

  1. How many students recorded their favorite animal?

  2. How many fewer students chose hamsters than dogs?

Bar graph. Favorite animal. Vertical axis from 0 to 10 by 1's. Horizontal axis labeled hamster, horse, dog, cat. Height of bar: hamster, 2. horse, 5. dog, 9. cat, 6.

Solution

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Problem 3

Pre-unit

Practicing Standards:  2.MD.D.10

The table shows the favorite summer vacation activity for a group of students.

vacation activities number of students
family time 6
playing sports 8
sleeping in 5
reading 3


Use the table to complete the bar graph.

Bar graph. Favorite vacation activities. Horizontal axis labeled family time, playing sports, sleeping in, reading. Vertical axis from 0 to 10 by 1s.

Solution

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Problem 4

Pre-unit

Practicing Standards:  2.NBT.B.5

Find each sum or difference. Show your reasoning.

  1. \(25 + 62\)
  2. \(37 – 9\)
  3. \(24 + 47\)
  4. \(84 – 59\)

Solution

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Problem 5

Pre-unit

Practicing Standards:  2.OA.C.4

How many objects are in each array? Explain or show your reasoning.

  1.  
    Array. 3 rows of 5 dots.
  2.  
    Array. 2 rows of 8 dots.

Solution

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Problem 6

The graph shows some information about flowers in the garden.

Bar graph. Flowers in the Garden. Vertical axis labeled number of flowers from 0 to 10 by ones. Horizontal axis labeled roses, tulips, sunflowers, daffodils. Height of bar: roses, 7. tulips, 5. sunflowers, 9. daffodils, 2.

Flowers.
  1. Write one fact you know based on the data shown in the graph.
  2. Write 2 questions you could answer about the data in the bar graph.

Solution

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Problem 7

The bar graph shows the numbers of different types of flowers in the garden. Use the graph to answer the questions.

  1. How many flowers are represented on the graph?
  2. How many tulips, sunflowers, and daffodils were there altogether?
Bar graph. Flowers in the Garden. Vertical axis labeled Number of Flowers from 0 to 10 by 1s. Horizontal axis labeled roses, tulips, sunflowers, daffodils. Height of bar: Roses, 7. Tulips, 5. Sunflowers, 9. Daffodils, 2.

Solution

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Problem 8

This picture graph records types of balls in the gym. Use the graph to answer the questions.

  1. How many basketballs are in the gym?
  2. How many more basketballs are there than footballs?
  3. Write one other question that you could answer about the balls in the gym using the graph.
Picture Graph. Balls in the Gym. Key: each circle represents 2 balls. baseball, 2 circles. basketball, 6 circles. football, 3 circles. volleyball, 4 circles.  

Solution

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Problem 9

The table shows the favorite sports of some students. Use it to complete the scaled picture graph.

sport number
tennis 6
swimming 6
gymnastics 4
soccer 8
Picture Graph. Key: circle represents 2 people. Tennis, blank. Swimming, blank. Gymnastics, blank. Soccer, blank.

Solution

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Problem 10

The table shows the number of different shapes in a pattern block puzzle. Use it to complete the scaled bar graph.

shape number
triangle 13
trapezoid 10
square 9
hexagon 15
Empty bar graph. Shapes in Puzzle. Vertical axis: number of shapes from 0 to 16 by twos. Horizontal axis: triangle, trapezoid, square, hexagon.

Solution

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Problem 11

The table shows the number of books some students have at home. 

Use the information from the table to create a scaled bar graph.

books number
Elena 25
Andre 9
Tyler 16
Clare 21

Blank bar graph. Books at Home.

Solution

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Problem 12

The graph shows the number of different kinds of nuts in a bowl.

Bar graph. Party Nuts. Vertical axis labeled number of nuts from 0 to 40 by fives. Horizontal axis labeled peanuts, walnuts, pecans, pistachios. Height of bar: peanuts, 27. walnuts, 11. pecans, 18. pistachios, 38.
  1. How many more pistachios are there than walnuts? Explain or show your reasoning.
  2. How many fewer pecans are there than peanuts? Explain or show your reasoning.

Solution

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Problem 13

The scaled bar graph shows how many students are in each class.

Bar graph. Students in Each Class. Vertical axis labeled number of students from 0 to 35 by fives. Horizontal axis labeled class 1, class 2, class 3, class 4. Height of bar: class 1, 24. class 2, 25. class 3, 22. class 4, 31.
  1. How many students are in the 4 classes altogether? Explain or show your reasoning.
  2. How many fewer students are in class 1 than in class 4? Explain or show your reasoning.

Solution

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Problem 14

Exploration

Bar graph. Student Enrollments. Vertical axis labeled number of students. Horizontal axis labeled school A, school B, school C, school D. Height of bars: school A, second lowest. school B, second highest. school C, lowest. school D, highest.

There are 1,000 students at the 4 schools. What is the approximate scale for the bar graph?

Solution

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Problem 15

Exploration

Collect data of interest to you and make a bar graph. You may use the bar graph template if you wish.

Blank bar graph

Solution

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Section B: From Graphs to Multiplication

Problem 1

There are 6 tennis courts. There are 2 players on each tennis court.

Create a drawing or diagram to represent the tennis players. Then, find out how many players are on the tennis courts. Explain or show your reasoning.

Solution

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Problem 2

The picture graph shows the favorite colors of some students.

Picture Graph. Favorite Colors. Key: dot represents 2 people. Purple, 4 dots. Yellow, 8 dots. Blue, 3 dots. Red, 5 dots.

Match each diagram or drawing to the number of people who like each color.

A:
Diagram. 5 equal parts each labeled 2.
B:
Diagram. 8 equal parts each labeled 2.
C:
D:
4 groups of 2 dots.
1: red
2: blue
3: purple
4: yellow

Solution

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Problem 3

Create a drawing or diagram to represent the expression \(4 \times 3\).

Solution

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Problem 4

There are 4 stacks of books on the table. Each stack has 5 books. How many books are on the table? Explain or show your reasoning.

Solution

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Problem 5

There are 6 basketball teams in the gym. There are 5 people on each team. How many people are on the basketball teams in the gym?

  1. Write an equation with a symbol for the unknown to represent the situation.
  2. Find the number that makes the equation true. Show your reasoning.

Solution

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Problem 6

Write an equation for the situation. Use a ? for the unknown. Find the number that makes the equation true.

There are 4 soccer teams. Each soccer team has 10 players. How many players are there altogether?

Solution

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Problem 7

Solve each problem. Explain or show your reasoning.

  1. There are 6 flowers. Each flower has 5 petals. How many petals are there?
  2. There are 50 petals on some flowers. Each flower has 5 petals. How many flowers are there?

Solution

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Problem 8

Exploration

Write an expression for the number of circles in the image. Then, find the number of circles.

6 groups of 6 dots.

Solution

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Problem 9

Exploration

For each image, determine if there is an even or odd number of circles. Explain or show your reasoning.

  1.  

    5 dot cubes of 6.

  2.  

    5 Dot Cubes of 5 dots each.

  3.  

    6 dot cubes of 7.

Solution

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Problem 10

Exploration

Look in your classroom, school, home, or outdoors to find some equal groups of objects.

  1. Describe the objects.
  2. Create a drawing to represent the objects.
  3. Write an equation showing how many objects there are.

Solution

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Section C: Represent Multiplication with Arrays and the Commutative Property

Problem 1

Rearrange the circles to make an array in two different ways.

A group of 12 dots.

Solution

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Problem 2

There are 4 rows of water bottles in the box. There are 5 bottles in each row.

Draw an array representing the situation. Then, write a multiplication expression representing the number of water bottles.

Solution

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Problem 3

There are 5 rows of chairs in the room. There are 4 chairs in each row. How many chairs are in the room?

  1. Write a multiplication equation to represent the situation.

  2. Find the value that makes your equation true.

Solution

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Problem 4

  1. Write a multiplication equation that represents each array.
    Aan array.
    Ban array.
  2. How are the arrays the same? How are they different?

Solution

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Problem 5

  1. Explain or show 2 different ways that you see equal groups in the array.

    Array. 3 rows of 6 dots.

  2. Arrange the dots in an array in a different way.

Solution

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Problem 6

Exploration

Andre says that there are an odd number of circles in this picture.

Do you agree with Andre? Explain or show your reasoning.

5 rows of circles. First row, 1 circle. Second row, 2 circles. Third row, 3 circles. Fourth row, 4 circles. Fifth row, 5 circles.

Solution

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Problem 7

Exploration

Find a collection of objects in the classroom or at home that is arranged in an array.

  1. Describe the objects.
  2. Create a drawing of the objects.
  3. Write an equation showing how many objects there are.

Solution

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