## Engage NY Eureka Math 6th Grade Module 4 Lesson 8 Answer Key

### Eureka Math Grade 6 Module 4 Lesson 8 Example Answer Key

Example 1.
g + 0 = g

Remember a letter in a mathematical expression represents a number. Can we replace g with any number?
Yes

Choose a value for g, and replace g with that number in the equation. What do you observe?
The value of g does not change when 0 is added to g.

Repeat this process several times, each time choosing a different number for g.

Will all values of result in a true number sentence?
Yes

Write the mathematical language for this property below:
g + 0 = g, additive identity property of zero. Any number added to zero equals itself.

Example 2.
Multiplicative Identity Property of One
g × 1 = g
Remember a letter in a mathematical expression represents a number. Can we replace g with any number?
Yes

Choose a value for g, and replace g with that number in the equation. What do you observe?
The value of g does not change when g is multiplied by 1.

Will all values of result in a true number sentence? Experiment with different values before making your claim.
Yes

Write the mathematical language for this property below:
g × 1 = g, multiplicative identity property of one. Any number multiplied by one equals itself.

Example 3.
Commutative Property of Addition and Multiplication
3 + 4 = 4 + 3
3 × 4 = 4 × 3

Replace the 3’s in these number sentences with the letter
a + 4 = 4 + a
a × 4 = 4 × a

Choose a value for a, and replace with that number in each of the equations. What do you observe?
The result is a true number sentence.

Will all values of result in a true number sentence? Experiment with different values before making your claim.
Yes, any number, even zero, can be used in place of the variable

Now, write the equations again, this time replacing the number 4 with a variable, b.
a + b = b + a
a × b = b × a

Will all values of and result in true number sentences for the first two equations? Experiment with different values before making your claim.
Yes

Write the mathematical language for this property below:
a + b = b + a commutative property of addition. Order does not matter when adding.
a × b = b × a commutative property of multiplication. Order does not matter when multiplying.

Example 4.
3 + 3 + 3 + 3 = 4 × 3
3 ÷ 4 = $$\frac{3}{4}$$

Replace the 3’s in these number sentences with the letter
a + a + a + a = 4 × a
a ÷ 4 = $$\frac{a}{4}$$

Choose a value for a, and replace a with that number in each of the equations. What do you observe?
The result is a true number sentence.

Will all values of a result in a true number sentence? Experiment with different values before making your claim.
Yes, any number, even zero, can be used in place of the variable

Now, write the equations again, this time replacing the number 4 with a variable, b.
a + a + a + a = b × a
a ÷ b = $$\frac{a}{b}$$, b ≠ 0

Will all values of and result in true number sentences for the equations? Experiment with different values before making your claim.
In the equation a + a + a + a = b × a, any value can be substituted for the variable a, but only 4 can be used for b since there are exactly 4 copies of a in the equation.
It is true for all values of and all values of b ≠ 0.

### Eureka Math Grade 6 Module 4 Lesson 8 Problem Set Answer Key

Question 1.
State the commutative property of addition using the variables a and b.
a + b = b + a

Question 2.
State the commutative property of multiplication using the variables a and b.
a × b = b × a

Question 3.
State the additive property of zero using the variable b.
b + 0 = b

Question 4.
State the multiplicative identity property of one using the variable b.
b × 1 = b

Question 5.
Demonstrate the property listed in the first column by filling in the third column of the table.

Question 6.
Why is there no commutative property for subtraction or division? Show examples.
Answers will vary. Examples should show reasoning and proof that the commutative property does not work for subtraction and division. An example would be 8 ÷ 2 and 2 ÷ 8. 8 ÷ 2 = 4, but 2 ÷ 8 = $$\frac{1}{4}$$.

### Eureka Math Grade 6 Module 4 Lesson 8 Exit Ticket Answer Key

Question 1.
State the commutative property of addition, and provide an example using two different numbers.
Any two different addends can be chosen, such as 5 + 6 = 6 + 5.

Question 2.
State the commutative property of multiplication, and provide an example using two different numbers.
Any two different factors can be chosen, such as 4 × 9 = 9 × 4.

Question 3.
State the additive property of zero, and provide an example using any other number.
Any nonzero addend can be chosen, such as 3 + 0 = 3.

Question 4.
State the multiplicative identity property of one, and provide an example using any other number.
Any nonzero factor can be chosen, such as 12 × 1 = 12.

### Eureka Math Grade 6 Module 4 Lesson 8 Opening Exercise Answer Key

4 + 0 = 4
4 × 1 = 4
4 ÷ 1 = 4
4 × 0 = 0
1 ÷ 4 = $$\frac{1}{4}$$

How many of these statements are true?
All of them

How many of those statements would be true if the number was replaced with the number in each of the number sentences?
All of them

Would the number sentences be true if we were to replace the number with any other number?

What if we replaced the number 4 with the number 0? Would each of the number sentences be true?
No. The first four are true, but the last one, dividing by zero, is not true.

What if we replace the number 4 with a letter g? Please write all 4 expressions below, replacing each 4 with a g.
g + 0 = g
g × 1 = g
g ÷ 1 = g
g × 0 = 0
1 ÷ g = $$\frac{1}{g}$$

Are these all true (except for g= 0) when dividing?
Yes

### Eureka Math Grade 6 Module 4 Lesson 8 Division of Fractions II Answer Key

Division of Fractions II – Round 1

Directions: Determine the quotient of the fractions and simplify.

Question 1.
$$\frac{4}{10} \div \frac{2}{10}$$
$$\frac{4}{2}$$ = 2

Question 2.
$$\frac{9}{12} \div \frac{3}{12}$$
$$\frac{9}{3}$$ = 3

Question 3.
$$\frac{6}{10} \div \frac{4}{10}$$
$$\frac{6}{4}=\frac{3}{2}=1 \frac{1}{2}$$

Question 4.
$$\frac{2}{8} \div \frac{3}{8}$$
$$\frac{2}{3}$$

Question 5.
$$\frac{2}{7} \div \frac{6}{7}$$
$$\frac{2}{6}=\frac{1}{3}$$

Question 6.
$$\frac{11}{9} \div \frac{8}{9}$$
$$\frac{11}{8}=1 \frac{3}{8}$$

Question 7.
$$\frac{5}{13} \div \frac{10}{13}$$
$$\frac{5}{10}=\frac{1}{2}$$

Question 8.
$$\frac{7}{8} \div \frac{13}{16}$$
$$\frac{14}{13}=1 \frac{1}{13}$$

Question 9.
$$\frac{3}{5} \div \frac{7}{10}$$
$$\frac{6}{7}$$

Question 10.
$$\frac{9}{30} \div \frac{3}{5}$$
$$\frac{9}{18}=\frac{1}{2}$$

Question 11.
$$\frac{1}{3} \div \frac{4}{5}$$
$$\frac{5}{12}$$

Question 12.
$$\frac{2}{5} \div \frac{3}{4}$$
$$\frac{8}{15}$$

Question 13.
$$\frac{3}{4} \div \frac{5}{9}$$
$$\frac{27}{20}=1 \frac{7}{20}$$

Question 14.
$$\frac{4}{5} \div \frac{7}{12}$$
$$\frac{48}{35}=1 \frac{13}{35}$$

Question 15.
$$\frac{3}{8} \div \frac{5}{2}$$
$$\frac{6}{40}=\frac{3}{20}$$

Question 16.
$$3 \frac{1}{8} \div \frac{2}{3}$$
$$\frac{75}{16}=4 \frac{11}{16}$$

Question 17.
$$1 \frac{5}{6} \div \frac{1}{2}$$
$$\frac{22}{6}=\frac{11}{3}=3 \frac{2}{3}$$

Question 18.
$$\frac{5}{8} \div 2 \frac{3}{4}$$
$$\frac{20}{88}=\frac{5}{22}$$

Question 19.
$$\frac{1}{3} \div 1 \frac{4}{5}$$
$$\frac{5}{27}$$

Question 20.
$$\frac{3}{4} \div 2 \frac{3}{10}$$
$$\frac{30}{92}=\frac{15}{46}$$

Question 21.
$$2 \frac{1}{5} \div 1 \frac{1}{6}$$
$$\frac{66}{35}=1 \frac{31}{35}$$

Question 22.
$$2 \frac{4}{9} \div 1 \frac{3}{5}$$
$$\frac{110}{72}=\frac{55}{36}=1 \frac{19}{36}$$

Question 23.
$$1 \frac{2}{9} \div 3 \frac{2}{5}$$
$$\frac{55}{153}$$

Question 24.
$$2 \frac{2}{3} \div 3$$
$$\frac{8}{9}$$

Question 25.
$$1 \frac{3}{4} \div 2 \frac{2}{5}$$
$$\frac{35}{48}$$

Question 26.
$$4 \div 1 \frac{2}{9}$$
$$\frac{36}{11}=3 \frac{3}{11}$$

Question 27.
$$3 \frac{1}{5} \div 6$$
$$\frac{16}{30}=\frac{8}{15}$$

Question 28.
$$2 \frac{5}{6} \div 1 \frac{1}{3}$$
$$\frac{51}{24}=2 \frac{3}{24}=2 \frac{1}{8}$$

Question 29.
$$10 \frac{2}{3} \div 8$$
$$\frac{32}{24}=\frac{4}{3}=1 \frac{1}{3}$$

Question 30.
$$15 \div 2 \frac{3}{5}$$
$$\frac{75}{13}=5 \frac{10}{13}$$

Division of Fractions II – Round 2

Directions: Determine the quotient of the fractions and simplify.

Question 1.
$$\frac{10}{2} \div \frac{5}{2}$$
$$\frac{10}{5}$$ = 2

Question 2.
$$\frac{6}{5} \div \frac{3}{5}$$
$$\frac{6}{3}$$ = 2

Question 3.
$$\frac{10}{7} \div \frac{2}{7}$$
$$\frac{10}{2}$$ = 5

Question 4.
$$\frac{3}{8} \div \frac{5}{8}$$
$$\frac{3}{5}$$

Question 5.
$$\frac{1}{4} \div \frac{3}{12}$$
$$\frac{3}{3}$$ = 1

Question 6.
$$\frac{1}{4} \div \frac{3}{12}$$
$$\frac{14}{3}=4 \frac{2}{3}$$

Question 7.
$$\frac{8}{15} \div \frac{4}{5}$$
$$\frac{8}{12}=\frac{2}{3}$$

Question 8.
$$\frac{5}{6} \div \frac{5}{12}$$
$$\frac{10}{5}$$ = 2

Question 9.
$$\frac{3}{5} \div \frac{7}{9}$$
$$\frac{27}{35}$$

Question 10.
$$\frac{3}{10} \div \frac{3}{9}$$
$$\frac{27}{30}=\frac{9}{10}$$

Question 11.
$$\frac{3}{4} \div \frac{7}{9}$$
$$\frac{27}{28}$$

Question 12.
$$\frac{7}{10} \div \frac{3}{8}$$
$$\frac{56}{30}=\frac{28}{15}=1 \frac{13}{15}$$

Question 13.
$$4 \div \frac{4}{9}$$
$$\frac{36}{4}$$ = 9

Question 14.
$$\frac{5}{8} \div 7$$
$$\frac{5}{56}$$

Question 15.
$$9 \div \frac{2}{3}$$
$$\frac{27}{2}=13 \frac{1}{2}$$

Question 16.
$$\frac{5}{8} \div 1 \frac{3}{4}$$
$$\frac{20}{56}=\frac{5}{14}$$

Question 17.
$$\frac{1}{4} \div 2 \frac{2}{5}$$
$$\frac{5}{48}$$

Question 18.
$$2 \frac{3}{5} \div \frac{3}{8}$$
$$\frac{104}{15}=6 \frac{14}{15}$$

Question 19.
$$1 \frac{3}{5} \div \frac{2}{9}$$
$$\frac{72}{10}=7 \frac{2}{10}=7 \frac{1}{5}$$

Question 20.
$$4 \div 2 \frac{3}{8}$$
$$\frac{32}{19}=1 \frac{13}{19}$$

Question 21.
$$1 \frac{1}{2} \div 5$$
$$\frac{3}{10}$$

Question 22.
$$3 \frac{1}{3} \div 1 \frac{3}{4}$$
$$\frac{40}{21}=1 \frac{19}{21}$$

Question 23.
$$2 \frac{2}{5} \div 1 \frac{1}{4}$$
$$\frac{48}{25}=1 \frac{23}{25}$$

Question 24.
$$3 \frac{1}{2} \div 2 \frac{2}{3}$$
$$\frac{21}{16}=1 \frac{5}{16}$$

Question 25.
$$1 \frac{4}{5} \div 2 \frac{3}{4}$$
$$\frac{36}{55}$$

Question 26.
$$3 \frac{1}{6} \div 1 \frac{3}{5}$$
$$\frac{95}{48}=1 \frac{47}{48}$$

Question 27.
$$3 \frac{3}{5} \div 2 \frac{1}{8}$$
$$\frac{144}{85}=1 \frac{59}{85}$$

Question 28.
$$5 \div 1 \frac{1}{6}$$
$$\frac{30}{7}=4 \frac{2}{7}$$
$$3 \frac{3}{4} \div 5 \frac{1}{2}$$
$$\frac{30}{44}=\frac{15}{22}$$
$$4 \frac{2}{3} \div 5 \frac{1}{4}$$
$$\frac{56}{63}=\frac{8}{9}$$