## Engage NY Eureka Math 7th Grade Module 2 Lesson 11 Answer Key

### Eureka Math Grade 7 Module 2 Lesson 11 Example Answer Key

Example 1.
Extending Whole Number Multiplication to the Integers
Part A: Complete quadrants I and IV of the table below to show how sets of matching integer cards will affect a player’s score in the Integer Game. For example, three 2’s would increase a player’s score by 0+2+2+2=6 points.

a. What patterns do you see in the right half of the table?
The products in quadrant I are positive and the products in quadrant IV are negative. When looking at the absolute values of the products, quadrants I and IV are a reflection of each other with respect to the middle row.

b. Enter the missing integers in the left side of the middle row, and describe what they represent.
The numbers represent how many matching cards are being discarded or removed

Part B: Students complete quadrant II of the table.
Students describe, using an Integer Game scenario, what quadrant II of the table represents and record this in the student materials.

Part B: Complete quadrant II of the table.

c. What relationships or patterns do you notice between the products (values) in quadrant II and the products (values) in quadrant I?
The products in quadrant II are all negative values. Looking at the absolute values of the products, quadrant I and II are a reflection of each other with respect to the center column.

d. What relationships or patterns do you notice between the products (values) in quadrant II and the products (values) in quadrant IV?
The products in quadrants II and IV are all negative values. Each product of integers in quadrant II is equal to the product of their opposites in quadrant IV.

e. Use what you know about the products (values) in quadrants I, II, and IV to describe what quadrant III will look like when its products (values) are entered.
The reflection symmetry of quadrant I to quadrants II and IV suggests that there should be similar relationships between quadrant II, III, and IV. The number patterns in quadrants II and IV also suggest that the products in quadrant III are positive values.

Part C: Discuss the following question. Then instruct students to complete the final quadrant of the table.
→ In the Integer Game, what happens to a player’s score when he removes a matching set of cards with negative values from his hand?
→ His score increases because the negative cards that cause the score to decrease are removed.
Students describe, using an Integer Game scenario, what quadrant III of the table represents and complete the quadrant in the student materials.

Part C: Complete quadrant III of the table.

Students refer to the completed table to answer parts (f) and (g).

f. Is it possible to know the sign of a product of two integers just by knowing in which quadrant each integer is located? Explain.
Yes, it is possible to know the sign of a product of two integers just by knowing each integer’s quadrant because the signs of the values in each of the quadrants are consistent. Two quadrants contain positive values, and the other two quadrants contain negative values.

g. Which quadrants contain which values? Describe an Integer Game scenario represented in each quadrant.
Quadrants I and III contain all positive values. Picking up three 4’s is represented in quadrant I and increases your score. Removing three (-4)’s is represented in quadrant III and also increases your score. Quadrants II and IV contain all negative values. Picking up three (-4)’s is represented in quadrant IV and decreases your score. Removing three 4’s is represented in quadrant II and also decreases your score.

### Eureka Math Grade 7 Module 2 Lesson 11 Exercise Answer Key

Exercise 1.
Multiplication of Integers in the Real World
Generate real-world situations that can be modeled by each of the following multiplication problems. Use the Integer Game as a resource.

a. -3 × 5
I lost three $5 bills, and now I have -$15.

b. -6 × (-3)
I removed six (-3)’s from my hand in the Integer Game, and my score increased 18 points.

c. 4 × (-7)
If I lose 7 lb. per month for 4 months, my weight will change -28 lb. total.

### Eureka Math Grade 7 Module 2 Lesson 11 Problem Set Answer Key

Question 1.
Complete the problems below. Then, answer the question that follows.

Which row shows the same pattern as the outlined column? Are the problems similar or different? Explain.

The row outlined shows the same pattern as the outlined column. The problems have the same answers, but the signs of the integer factors are switched. For example, 3×(-1)=-3×1. This shows that the product of two integers with opposite signs is equal to the product of their opposites.

Question 2.
Explain why (-4) × (-5) = 20. Use patterns, an example from the Integer Game, or the properties of operations to support your reasoning.
Answers may vary. Losing four -5 cards will increase a score in the Integer Game by 20 because a negative value decreases a score, but the score increases when it is removed.

Question 3.
Each time that Samantha rides the commuter train, she spends $4 for her fare. Write an integer that represents the change in Samantha’s money from riding the commuter train to and from work for 13 days. Explain your reasoning. Answer: If Samantha rides to and from work for 13 days, then she rides the train a total of 26 times. The cost of each ride can be represented by -4. So, the change to Samantha’s money is represented by -4 × 26 = -104. The change to Samantha’s money after 13 days of riding the train to and from work is -$104.

Question 4.
Write a real-world problem that can be modeled by 4 × (-7).
Answers will vary. Every day, Alec loses 7 pounds of air pressure in a tire on his car. At that rate, what is the change in air pressure in his tire after 4 days?

Challenge:

Question 5.
Use properties to explain why for each integer a, -a = -1 × a. (Hint: What does (1 + (-1))×a equal? What is the additive inverse of a?)
0 × a = 0 Zero product property
(1 + (-1)) × a = 0 Substitution
a + (-1 × a) = 0 Distributive property
Since a and (-1 × a) have a sum of zero, they must be additive inverses. By definition, the additive inverse of a is -a, so (-1 × a) = -a.

### Eureka Math Grade 7 Module 2 Lesson 11 Exit Ticket Answer Key

Question 1.
Create a real-life example that can be modeled by the expression -2 × 4, and then state the product.