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Understanding Multiplication of Integers

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Examples, videos, and solutions to help Grade 7 students learn learn how to develop rules for multiplying signed numbers.

New York State Common Core Math Grade 7, Module 2, Lesson 10

Download worksheets for Grade 7, Module 2, Lesson 10

Lesson 10 Student Outcomes

• Students practice and justify their understanding of multiplication of integers by using the Integer Game. For example, 3 x 5 corresponds to what happens to your score if you get three 5 cards; 3 x (-5) corresponds to what happens to your score if you get three -5 cards; (-3) x 5 corresponds to what happens to your score if you lose three 5 cards; and (-3) x (-5) corresponds to what happens to your score if you lose three -5 cards.

• Students explain that multiplying by a positive integer is repeated addition and that adding a number multiple times has the same effect as removing the opposite value the same number of times (e.g. 5 x 3 = (-5) x (-3) and 5 x (-3) = (-5) x 3.)

• Students use the properties and facts of operations to extend multiplication of whole numbers to multiplication of integers.

Lesson 10 Summary

Multiplying integers is repeated addition and can be modeled with the Integer Game. If 3 x a corresponds to what happens to your score if you get three cards of value a, then (-3) x a corresponds to what happens to your score if you lose three cards of value a. Adding a number multiple times has the same effect as removing the opposite value the same number of times (e.g. a x b = (-a) x (-b) and a x (-b) = (-a) x b.)

Example 1: Product of a Positive Integer and a Negative Integer
Part A: Instruct students to record the values of their cards on the images in Part A. One of the four card images has a ★ beneath it. The ★ is used to indicate which of the four cards to copy (or multiply) in Part B.
Part B: Instruct students to copy the value of the card with the ★ beneath it from Part A on each card with a ★ beneath it in Part B. The three remaining card values from Part A are entered in the three remaining card images in Part B. Students now have a total of six integer cards.
Use your cards from Part B to answer the questions below:
a. Write a product that describes the three matching cards.
b. Write an expression that represents how each of the ★ cards changes your score.
c. Write an equation that relates these two expressions.
d. Write an integer that represents the total change to your score by the three ★ cards.
e. Write an equation that relates the product and how it affects your score.

Part C: Instruct students to record the values of their cards on the images in Part C. The teacher chooses one of the four images and instructs the class to place a ★ beneath it to indicate which card will be cloned (multiplied) in Part D.



Part D: Instruct students to record the value of the card with the ★ beneath it from Part C on each image with a ★ beneath it in Part D. Also, rewrite the values of the three remaining cards on the other three images. Students now have a total of 8 integer cards.
Use your cards from Part D to answer the questions below:
f. Write a product that describes the five matching cards.
g. Write an expression that represents how each of the ★ cards changes your score.
h. Write an equation that relates these two expressions.
i. Write an integer that represents the total change to your score by the three  cards.
j. Write an equation that relates the product and how it affects your score.
k. Use the expression 5 × 4 to relate the multiplication of a positive valued card to addition.
l. Use the expression 3 × (− 5) to relate the multiplication of a negative valued card to addition.

Example 2: Product of a Negative Integer and a Positive Integer
a. If all of the 4’s from the playing hand on the right are discarded, how will the score be affected? Model this using a product in an equation.
b. What three matching cards could be added to those pictured to get the same change in score? Model this using a product in an equation.
c. Seeing how each play affects the score, relate the products that you used to model them. What do you conclude about multiplying integers with opposite signs?

Example 3: Product of Two Negative Integers
a. If the matching cards from the playing hand on the right are discarded, how will this hand’s score be affected?
b. What four matching cards could be added to those pictured to get the same change in score? Model this using a product in an equation.
c. Seeing how each play affects the score, relate the products that you used to model them. What do you conclude about multiplying integers with the same sign?
Using the conclusions from Examples 2 and 3, what can we conclude about multiplying integers? Write a few examples.

Lesson 10 Problem Set

2. You have the integer cards shown at the right when your teacher tells you to choose a card to multiply four times. If your goal is to get your score as close to zero as possible, which card would you choose?
Explain how your choice changes your score.

Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.


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