3.4 Relating Multiplication to Division

Unit Goals

  • Students learn about and use the relationship between multiplication and division, place value understanding, and the properties of operations to multiply and divide whole numbers within 100. They also represent and solve two-step word problems using the four operations.

Section A Goals

  • Represent and solve “how many groups?” and “how many in each group?” problems.

Section B Goals

  • Understand division as a missing-factor problem.
  • Use properties of operations to develop fluency with single-digit multiplication facts, and their related division facts.

Section C Goals

  • Use properties of operations and place value understanding to develop strategies to multiply within 100 and to multiply one-digit numbers by a multiple of 10.

Section D Goals

  • Use properties of operations, place value understanding, and the relationship between multiplication and division to divide within 100.
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Section A: What is Division?

Problem 1

Pre-unit

Practicing Standards:  3.OA.A.1

Array. 4 rows of 5 dots.
  1. Write a multiplication expression that represents the array.
  2. Write a multiplication equation that represents the array.

Solution

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Problem 2

Pre-unit

Practicing Standards:  3.MD.C.7

Find the area of each rectangle.

AArea diagram.
BA rectangle. Left side labeled 4 centimeters. Top labeled 10 centimeters. Other sides unlabeled.

Solution

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Problem 3

Pre-unit

Practicing Standards:  3.MD.C.7

The area of the rectangle is 40 square centimeters.

Find the missing side length of the rectangle. Explain your reasoning.

A rectangle. Left side labeled 5 centimeters. Top labeled question mark. Other sides unlabeled.

Solution

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Problem 4

Pre-unit

Practicing Standards:  3.OA.A.4

Find the number that makes each equation true.

  1. \(8 \times 5 = \underline{\hspace{1cm}}\)
  2. \(5 \times \underline{\hspace{1cm}} = 35\)
  3. \( \underline{\hspace{1cm}} \times 2 = 18\)

Solution

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Problem 5

Pre-unit

Practicing Standards:  3.OA.A.3

There are 6 volleyball teams in the gym. Each team has 10 players. How many volleyball players are there altogether?

  1. Make a drawing of the situation.
  2. Write an equation with a “?” for the unknown that represents the situation.
  3. Solve the problem.

Solution

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Problem 6

For each problem, show your thinking using a drawing or a diagram.

  1. There are 40 apples packed into boxes. If there are 8 apples in each box, how many boxes are there?
  2. There are 40 apples packed into boxes. If there are 10 apples in each box, how many boxes are there?

Solution

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Problem 7

For each problem, show your thinking using a drawing or a diagram.

  1. There are 30 oranges. If they are packed into 5 bags with the same amount of oranges in each bag, how many oranges are in each bag?
  2. There are 30 oranges. If they are packed into 3 bags with the same amount of oranges in each bag, how many oranges are in each bag?

Solution

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Problem 8

  1. 10 people go to the movies in cars. Two people go in each car. How many cars are there? Show your thinking using a drawing or a diagram.
  2. 10 other people go to the movies in cars. They ride in 2 cars with the same number in each car. How many people are in each car? Show your thinking using a drawing or diagram.
  3. How are the two situations the same? How are they different? How are the diagrams the same? How are they different?

Solution

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Problem 9

There are 20 desks in the class. They are divided equally into 5 groups. How many desks are in each group?

  1. Which expression represents this situation: \(20\div4\) or \(20\div5\)? Explain your reasoning.

  2. Choose the diagram that represents this situation. Explain your reasoning.
    A4 groups of 5 dots.

    B5 groups of 4 dots.

Solution

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Problem 10

Mai’s family picked 40 pounds of peaches. They put 5 pounds in each bag.

  1. Write a division expression that represents the situation.
  2. How many bags of peaches did Mai’s family pick? Explain or show your reasoning.

Solution

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Problem 11

Complete each story by putting a number in the blank that makes sense. Then, answer the questions. Draw a diagram to solve each problem.

  1. Mai has __________ stickers. She is going to put the same number of stickers on each of her 5 notebooks. How many stickers will be on each notebook?
  2. Andre has __________ cards. He is going to arrange them in rows of __________ cards. How many rows will Andre's cards make?

Solution

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Problem 12

Exploration

Write a division situation to match each diagram.

A2 groups of 7 dots.
B9 groups of 3 dots.
C6 groups of 4 dots.

Solution

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Section B: Relate Multiplication and Division

Problem 1

There are 35 books on the bookcase. There are 7 books on each shelf. How many shelves are there? Explain how the equations \(35 \div 7 = {?}\) and \({?} \times 7 = 35\) both represent the situation.

Solution

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Problem 2

There are 24 eggs in the container. There are 6 in each row. How many rows of eggs are there?

Write an equation that represents the situation. Use a symbol for the unknown. Then, answer the question.

Solution

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Problem 3

For each multiplication equation, write a related division fact you know from the multiplication equation.

  1. \(8 \times 5 = 40\)

  2. \(2 \times 9 = 18\)

Solution

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Problem 4

Lin knows \(8 \times 5 = 40\). Explain how she can use this fact to find \(8 \times 4\).

Solution

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Problem 5

  1. Highlight parts of the diagram to show the expression \((5 \times 7) + (2 \times 7)\).
    Area diagram.
  2. Explain how you could use the diagram to calculate \(7 \times 7\).

Solution

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Problem 6

Mark or shade the rectangle to show a strategy for finding its area. Then, explain how to use the diagram to find the area.

Solution

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Problem 7

Exploration

Noah finds \(9 \times 8\) by calculating \((10 \times 8) - (1 \times 8)\).

  1. Make a drawing showing why Noah's calculation works.
  2. Use Noah's method to calculate \(9 \times 8\).

Solution

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Section C: Multiplying Larger Numbers

Problem 1

  1. How many tens are there in 50?

  2. How many tens are there in \(7 \times 50\)? Explain your reasoning.

  3. What is the value of \(7 \times 50\)? Explain your reasoning.

Solution

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Problem 2

There are 4 lunch tables. There are 12 students at each table. How many students are there at the tables? Show your thinking using objects, a drawing, or a diagram.

Solution

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Problem 3

  1. What do the 60 and 24 in the diagram represent?

    Area diagram. Rectangle divided into 2 parts. One part labeled 60 with a top measurement of 10, the other labeled 24, with a top measurement of 4. Left side measurement 6.
  2. Explain how to use the diagram to calculate \(14 \times 6\).

Solution

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Problem 4

There were 14 days of school in the month. There were 7 hours of school each day. How many hours of school were there during the month?

Solution

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Problem 5

Find the value of each expression. Explain or show your reasoning.

  1. \(2 \times 47\)

  2. \(3 \times 25\)

Solution

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Problem 6

A rope is 640 inches long. Andre cuts off 5 pieces of rope that are 16 inches each. How much rope is left?

Solution

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Problem 7

Exploration

Here is Mai’s strategy for calculating \(4 \times 21\): “First I double 21 and that’s 42. Then I double 42 and that’s 84.”

  1. Explain why Mai’s strategy works.

  2. Use Mai’s strategy to find \(4 \times 23\).

Solution

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Problem 8

Exploration

Multiplication table.
  1. Make a list of the numbers less than 20 that do not appear in the multiplication table.

  2. What do these numbers have in common?

  3. Choose one of these numbers and count out that number of objects. Can you make an array out of the objects?

Solution

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Problem 9

Exploration

Look at the two different diagrams of the same multiplication expression:

Area diagram. Rectangle divided into 2 parts. One part labeled 60 with a top measurement of 10, the other labeled 18, with a top measurement of 3. Left side measurement 6.
Base ten blocks. 6 tens. 18 ones.
  1. What multiplication expression do the two diagrams represent?
  2. Can you show a third way to represent the same multiplication expression?
  3. What is the value of the expression?
  4. Write a story problem to match the expression.

Solution

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Section D: Dividing Larger Numbers

Problem 1

There are 85 chairs in the gym. They are arranged in 5 rows with the same number of chairs in each row. How many chairs are in each row? Show your thinking using diagrams, symbols, or other representations.

Solution

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Problem 2

  1. Find the value of \(96 \div 6\). Use base-ten blocks if they are helpful.
  2. Find the value of \(52 \div 4\). Use base-ten blocks if they are helpful.

Solution

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Problem 3

  1. Find the value of \(78 \div 6\). Draw a diagram if it is helpful.
  2. Find the value of \(42 \div 3\). Draw a diagram if it is helpful.

Solution

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Problem 4

Find the value of each quotient.

  1. \(96 \div 6\)
  2. \(87 \div 3\)

Solution

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Problem 5

There are 240 people at the park for the soccer games. There are 150 fans. The rest of the people are on 6 soccer teams with an equal number of players. How many players are on each soccer team?

  1. Write an equation to represent this situation. Use a letter for the unknown quantity.
  2. Solve the problem. Explain or show your reasoning.

Solution

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Problem 6

Exploration

To find the value of \(96 \div 3\), Diego divides \(9\) by 3 and \(6\) by 3 and says the answer is 32.

  1. Explain why Diego's method is correct. Use equations or drawings to support your reasoning.
  2. Does Diego's method work to find the value of \(78 \div 3\)? Explain your reasoning.

Solution

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Problem 7

Exploration

What are the different ways you can divide 48 objects into equal groups?

  1. Make a list.
  2. Write a multiplication or division equation for each different way.

Solution

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