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Texas Go Math Grade 6 Lesson 6.2 Answer Key Dividing Integers
Reflect
Question 1.
Make a Conjecture Make a conjecture about the quotient of two integers with different signs. Make a conjecture about the quotient of two integers with the same sign.
Answer:
The quotient of two integers with different signs is negative.
Rewrite the expression
a ÷ b = c
as multiplication
b × c = a
If b is negative and a is positive, factor c must be negative because the product (a) of two integers with the same sign (b × c) is positive.
If b is positive and a is negative, factor e must be negative because the product (a) of two integers with opposite signs (b × c) is negative
The quotient of two integers with the same sign is positive.
Rewrite the expression
a ÷ b = c
as multiplication
b × c = a
If b is negative and a is negative, factor c must be positive because the product (a) of two integers with opposite signs (b × C) is negative.
If b is positive and a is positive, factor c must be positive because the product (a) of two integers with the same sign (b × c) is positive.
Your Turn
Find each quotient.
Question 2.
0 ÷ (- 6) ____________
Answer:
First, determine if the quotient will be positive or negative.
Since dividend is 0 and divisor – 6 is not 0. the quotient is 0.
The result of division 15:
0 ÷ (- 6) = 0
Result is 0
Find the sign of the quotient (opposite signs of integers then -, the same sign of integers then +) and divide given integers.
Question 3.
38 ÷ (- 19) ____________
Answer:
First, determine if the quotient will be positive or negative
Since 38 is positive and – 19 is negative (they have opposite signs), the quotient will be negative.
Divide given integers:
38 ÷ (- 19) = – 2
Result is – 2
Find the sign of the quotient (opposite signs of integers then -, the same sign of integers then +) and divide given integers.
Question 4.
– 13 ÷ (- 1) ____________
Answer:
First determine if the quotient will be positive or negative
Since – 13 is negative and – 1 is negative (they have the same sign), the quotient will be positive.
Divide given integers:
– 13 ÷ (- 1) = 13
Result is 13
Find the sign of the quotient (opposite signs of integers then -, the same sign of integers then +) and divide given integers.
Your Turn
Question 5.
A penalty in Meteor-Mania is -5 seconds. A penalty in Cosmic Calamity is -7 seconds. Yolanda had penalties totaling -25 seconds in a game of Meteor-Mania and -35 seconds in a game of Cosmic Calamity. In which game did Yolanda receive more penalties? Justify your answer.
Answer:
First, calculate how many penalties Yolanda received in first game. Find the quotient:
– 25 ÷ (- 5)
Determine if the quotient will be positive or negative.
Since – 25 is negative and – 5 is negative (they have the same sign), the quotient will be positive.
Divide given integers:
– 25 ÷ (- 5) = 5
Then, calculate how many penalties she received in second game Find the quotient:
– 35 ÷ (- 7)
Determine if the quotient will be positive or negative.
Since – 35 is negative and – 7 is negative (they have the same sign), the quotient will be positive.
Divide given integers:
– 35 ÷ (- 7) = 5
Compare number of penalties in both games:
5 = 5
Yolanda received the same number of penalties in each game, 5 penalties in first and 5 in the second game.
Calculate the number of penalties in each game and compare the results to find in which game she received more.
Texas Go Math Grade 6 Lesson 6.2 Guided Practice Answer Key
Find each quotient.
Question 1.
\(\frac{-14}{2}\) ____________
Answer:
First, determine if the quotient will be positive or negative
Since – 14 is negative and 2 is positive (they have opposite signs), the quotient will be negative.
Divide given integers:
\(\frac{-14}{2}\) = – 7
Result is – 7
Find the sign of the quotient (opposite signs of integers then -, the same sign of integers then +) and divide given integers.
Question 2.
21 ÷ (- 3) ____________
Answer:
First, determine if the quotient will be positive or negative
Since 21 is positive and – 3 is negative (they have opposite signs), the quotient will be negative.
Divide given integers:
21 ÷ (- 3) = – 7
Result is – 7
Find the sign of the quotient (opposite signs of integers then -, the same sign of integers then +) and divide given integers.
Question 3.
\(\frac{26}{- 13}\) ____________
Answer:
First, determine if the quotient will be positive or negative
Since 26 is positive and – 13 is negative (they have opposite signs), the quotient will be negative.
Divide given integers:
\(\frac{26}{-13}\) = – 2
Result is – 2
Find the sign of the quotient (opposite signs of integers then -, the same sign of integers then +) and divide given integers.
Question 4.
0 ÷ (- 4) ____________
Answer:
First, determine if the quotient will be positive or negative
Since 21 is dividend is 0 and divisor – 4 is not 0, the quotient is 0.
Divide given integers:
0 ÷ (- 4) = 0
Result is 0
Find the sign of the quotient (opposite signs of integers then -, the same sign of integers then +) and divide given integers.
Question 5.
\(\frac{-45}{-5}\) ____________
Answer:
First, determine if the quotient will be positive or negative
Since – 45 is negative and – 5 is negative (they have same sign), the quotient will be positive.
Divide given integers:
\(\frac{- 45}{-5}\) = 9
Result is 9
Find the sign of the quotient (opposite signs of integers then -, the same sign of integers then +) and divide given integers.
Question 6.
– 30 ÷ (10) ____________
Answer:
First, determine if the quotient will be positive or negative
Since -30 is negative and 10 is positive (they have opposite signs), the quotient will be negative.
Divide given integers:
– 30 ÷ (10) = – 3
Result is – 3
Find the sign of the quotient (opposite signs of integers then -, the same sign of integers then +) and divide given integers.
Question 7.
\(\frac{-11}{-1}\) ____________
Answer:
First, determine if the quotient will be positive or negative
Since – 11 is negative and – 1 is negative (they have same signs), the quotient will be positive.
Divide given integers:
\(\frac{-11}{-1}\) = 11
Result is 11
Find the sign of the quotient (opposite signs of integers then -, the same sign of integers then +) and divide given integers.
Question 8.
– 31 ÷ (- 31) ____________
Answer:
First, determine if the quotient will be positive or negative
Since – 31 is negative and – 31 is negative (they have same signs), the quotient will be positive.
Divide given integers:
– 31 ÷ (- 31) = 1
Result is 1
Find the sign of the quotient (opposite signs of integers then -, the same sign of integers then +) and divide given integers.
Question 9.
\(\frac{0}{-7}\) ____________
Answer:
First, determine if the quotient will be positive or negative
Since dividend is 0 and divisor – 7 is not 0, the quotient is 0.
The result of division is:
\(\frac{0}{-7}\) = 0
Result is 0
Find the sign of the quotient (opposite signs of integers then -, the same sign of integers then +) and divide given integers.
Question 10.
\(\frac{-121}{-11}\) ____________
Answer:
First, determine if the quotient will be positive or negative
Since – 121 is negative and – 11 is negative (they have the same sign). the quotient will be positive.
Divide given integers:
\(\frac{-121}{-11}\) = 11
Result is 11
Find the sign of the quotient (opposite signs of integers then -, the same sign of integers then +) and divide given integers.
Question 11.
84 ÷ (- 7) ____________
Answer:
First, determine if the quotient will be positive or negative
Since 84 is positive and – 7 is negative (they have opposite signs), the quotient will be negative.
Divide given integers:
84 ÷ (- 7) = – 12
Result is – 12
Find the sign of the quotient (opposite signs of integers then -, the same sign of integers then +) and divide given integers.
Question 12.
\(\frac{500}{-25}\) ____________
Answer:
First, determine if the quotient will be positive or negative
Since 500 is positive and – 25 is negative (they have opposite signs. the quotient will be negative
Divide given integers:
\(\frac{500}{-25}\) = – 20
Result is – 20
Find the sign of the quotient (opposite signs of integers then -, the same sign of integers then +) and divide given integers.
Question 13.
– 6 ÷ (0) ____________
Answer:
Rewrite given expression as a multiplication problem:
0 × ? = – 6
Since the product of 0 and any integer is 0, that multiplication problem does not nave a solution.
Therefore, the division by 0 is undefined, it is not possible
The division by 0 is undefined because there is no such a number you can multiply with 0 and get a product different from 0.
Question 14.
\(\frac{-63}{-21}\) ____________
Answer:
First, determine if the quotient will be positive or negative
Since – 63 is negative and – 21 is negative (they have the same sign), the quotient wilt be positive
Divide given integers:
\(\frac{-63}{-21}\) = 3
Result is 3
Find the sign of the quotient (opposite signs of integers then -, the same sign of integers then +) and divide given integers.
Write a division expression for each problem. Then find the value of the expression.
Question 15.
Clark made four of his truck payments late and was fined four late fees. The total change to his savings from late fees was – $40. How much was one late fee?
Answer:
Total change to Clark’s savings was – $40 so that would be the dividend.
He made four payments late so the divisor would he 4.
The value of the expression
– 40 ÷ 4
represents the amount of one late fee.
First, determine if the quotient will be positive or negative.
Since – 40 is negative and 4 is positive (they have opposite signs), the quotient will be negative.
Divide given integers:
– 40 ÷ 4 = – 10
One late fee was – $10.
The value of the expression – 40 ÷ 4 represents the amount of one late fee.
Find the sign of the quotient (opposite signs of integers then -, the same sign of integers then +) and divide given integers.
Question 16.
Jan received – 22 points on her exam. She got 11 questions wrong out of 50 questions. How much was Jan penalized for each wrong answer?
Answer:
Jane received 22 negative points so the dividend is – 22
She got 11 questions wrong so that would be the divisor
The value of the expression
– 22 ÷ 11
represents penalty for each wrong answer
First determine if the quotient will be positive or negative
Since – 22 is negative and 11 is positive (they have opposite signs), the quotient will be negative.
Divide given integers:
– 22 ÷ 11 = – 2
She was penalized – 2 points for each wrong answer
The value of the expression – 22 ÷ 11 represents penalty for each wrong answer.
Find the sign of the quotient (opposite signs of integers then -, the same sign of integers then +) and divide given integers.
Question 17.
Allen’s score in a video game was changed by – 75 points because he missed some targets. He got – 15 points for each missed target. How many targets did he miss?
Answer:
His score was changed by – 75 so the dividend is – 75.
The divisor is – 15 because he got 15 negative points for each missed target.
The value of the expression
– 75 ÷ (- 15)
represents the number of targets he missed.
First, determine if the quotient will be positive or negative
Since – 75 is negative and – 15 is negative (they have the same sign), the quotient will be positive
Divide given integers:
– 75 ÷ (- 15) = 5
Allen missed 5 targets.
The value of the expression – 75 ÷(-15) represents the number of targets he missed.
Find the sign of the quotient (opposite signs of integers then -, the same sign of integers then +) and divide given integers.
Question 18.
Louisa’s savings change by – $9 each time she goes bowling. In all, it changed by – $99 during the summer. How many times did she go bowling in the summer?
Answer:
Total change to lier savings is – $ 99 so that would be the dividend.
She spends $9 for each time she goes bowling so the divisor would be – 9.
The value of the expression
– 99 ÷ (- 9)
represents the number of times she went bowling.
First, determine if the quotient will be positive or negative.
Since – 99 is negative and – 9 is negative (they have the same sign), the quotient will be positive.
Divide given integers:
– 99 ÷ (- 9) = 11
Louisa went bowling 11 times in the summer.
The value of the expression – 99 ÷ (- 9) represents the number of times she went bowling.
Find the sign of the quotient (opposite signs of integers then -, the same sign of integers then +) and divide given integers.
Essential Question Check-In
Question 19.
How is the process of dividing integers similar to the process of multiplying integers?
Answer:
You can simplify the division problem by rewriting it as a multiplication problem.
You are supposed to find the quotient c value of the expression
a ÷ b
Find a number that when multiplied with b gives a product a:
b × c = a
and that would be the quotient c for the previous division problem.
Further, rule for the sign of the quotient is the same as rule for the sign of the product:
If both integers have the same sign, the quotient would be positive (same as product of two integers with same sign).
If integers have opposite signs, the quotient would be negative (same as product of two integers with opposite signs).
You can simplify the division probLem by rewriting it as a multiplication problem
Rule for the sign of the quotient is the same as rule for the sign of the product
Question 20.
Walter buys a bus pass for $30. Every time he rides the bus, money is deducted from the value of the pass. He rode 12 times and $24 was deducted from the value of the pass. How much does each bus ride cost?
Answer:
The amount that was deducted front value of the pass is $ 24 so the dividend is 24. Waiter rode 12 times so the divisor is 12.
The value of the expression
24 ÷ 12
represents the cost of each bus ride.
First, determine if the quotient will he positive or negative.
Since 24 is positive and 12 is positive (they have the same sign), the quotient will be positive.
Divide given integers:
24 ÷ 12 = 2
Each bus ride cost $2.
Find the sign of the quotient (opposite signs of integers then -, the same sign of integers then +) and divide given integers.
Question 21.
Analyze Relationships Elisa withdrew $20 at a time from her bank account and withdrew a total of $140. Francis withdrew $45 at a time from his bank account and withdrew a total of $270. Who made the greater number of withdrawals? Justify your answer.
Answer:
Use negative integers to represent the amounts because Elisa withdrew money.
The value of the expression
– 140 ÷ (- 20)
represents the number of withdrawals Elisa made.
First, determine if the quotient will be positive or negative
Since -140 is negative and -20 is negative (they have the same sign), the quotient will be positive
Divide given integers:
– 140 ÷ (- 20) = 7
Elisa made 7 withdrawals
Use negative integers to represent the amounts because Francis withdrew money.
The value of the expression
– 270 ÷ (- 45)
represents the number of withdrawals Francis made
First, determine if the quotient will be positive or negative
Since – 270 is negative and – 45 is negative (they have the same sign), the quotient will be positive
Divide given integers:
– 270 ÷ (- 45) = 6
Francis made 6 withdrawals
Compare results to see who made more withdrawals
7 > 6
Elisa made 1 withdrawal more than Francis.
Find the sign of the quotient (opposite signs of integers then -, the same sign of integers then +) and divide given integers.
Question 22.
MuItistep At 7 p.m. last night, the temperature was 10°F, At 7a.m. the next morning, the temperature was – 2 °F.
a. By how much did the temperature change from 7 p.m. to 7 a.m.?
Answer:
The difference
– 2 – 10
represents how much the temperature changed.
Subtracting 10 is equal to adding -lo so you can find the sum:
– 2 + (- 10)
First, find absolute values of given integers:
|- 2| = 2 and |- 10| = 10
Then, sum that absolute values
2 + 10 = 12
Final result would be 12 or -12, depending on sign of given integers (both positive then +, both negative then -).
Given integers are negative so final result is – 12.
The temperature changed by – 12 °F.
b. The temperature changed by a steady amount overnight. By how much did it change each hour?
Answer:
Find the quotient
– 12 ÷ 12
to find how much the temperature changed each hour
Determine if the quotient wilt be positive or negative.
Since – 12 is negative and 12 is positive (they have opposite signs), the quotient will be negative.
Divide given integers:
– 12 ÷ 12 = – 1
The temperature changed by – 1 °F each hour.
Find the sign of the quotient (opposite signs of integers then -, the same sign of integers then +) and divide given integers.
Question 23.
Analyze Relationships Nola hiked down a trail at a steady rate for 10 minutes. Her change in elevation was – 200 feet. Then she continued to hike down for another 20 minutes at a different rate. Her change in elevation for this part of the hike was – 300 feet. During which portion of the hike did she walk down at a faster rate? Explain your reasoning.
Answer:
First calculate a rate of Nora’s hiking during first 10 minutes.
The quotient
– 200 ÷ 10
shows her change in elevation each minute.
Determine if the quotient will be positive or negative.
Since -200 is negative and 10 is positive (they have opposite signs)., the quotient will be negative
Divide given integers:
– 200 ÷ 10 = – 20
Note that the quotient is negative because she walked down a trail.
Nora walked down at a rate of 20 feet per minute during first 10 minutes.
Then, calculate a rate of Nora’s hiking during another 20 minute&
The quotient
– 300 ÷ 20
shows her change in elevation each minute
Determine if the quotient will be positive or negative.
Since -300 is negative and 20 is positive (they have opposite signs), the quotient will be negative
Divide given integers:
– 300 ÷ 20 = – 15
Again, the quotient is negative because she walked down a trail.
Nora walked down at a rate of 15 feet per minute during those 20 minutes.
Compare the results:
20 > 15
Nora walked down at a faster rate during first part of the hike.
She walked down at a rate of 20 feet per minute during the first part and she walked down at a rate of 15 feet per minute during the second part of the hike.
Question 24.
Write a real world description to fit the expression – 50 ÷ 5.
Answer:
John has a debt of $50 in bank. During 5 months, equal amounts of money are taken from his account as a payment of the debt. Calculate the change in his account each month.
Use negative integer to model a debt he has.
The expression you get is:
– 50 ÷ 5
First, determine if the quotient will be positive or negative
Since – 50 is negative and 5 is positive (they have opposite signs), the quotient wilt be negative.
Divide given integers:
– 50 ÷ 5 = – 10
The change in his account is – $10 each month.
John has a debt of $50 in bank. During 5 months, equal amounts of money are taken from his account as a payment of the debt. Calculate the change in his account each month.
The change in his account is – $10 each month.
Find the sign of the quotient (opposite signs of integers then -, the same sign of integers +) and divide given integers.
Question 25.
Communicate Mathematical Ideas Two integers, a and b. have different signs. The absolute value of integer a is divisible by the absolute value of integer b. Find two integers that fit this description. Then decide if the product of the integers is greater than or less than the quotient of the integers. Show your work.
Answer:
You need to find two integers with different signs, for example, the positive one, a, and the negative one, b.
You know that you can divide absolute values of those two integers:
\(\frac{|a|}{|b|}\) = c
Take integers 27 and – 9:
a = 27, b = – 9
To find a value of c, find absolute values of given integers and divide them:
|27| = 27
|- 9| = 9
\(\frac{27}{9}\) = 3
Therefore, |27| is divisible by |- 9|.
The quotient of 27 and – 9 would be negative because they have opposite signs so you get;
\(\frac{27}{-9}\) = – 3
Now, find the product:
27 × (- 9)
and compare it with the quotient
Use rules for multiplication integers to find the result:
27 × (- 9) = – 243
Compare the quotient with the product:
– 3 > – 243
The product is less than the quotient of these two integers.
Numbers that fit given description are 27 and – 9.
Find the sign of the quotient (opposite signs of integers then -, the same sign of integers +) and divide given integers.
Determine if each statement is true or false. Justify your answer.
Question 26.
For any two nonzero integers, the product and quotient have the same sign.
Answer:
You know that the product will be positive if integers have the same sign and that it will be negative if integers have opposite signs.
You use the same rule to determine if the quotient will be positive or negative.
If integers have the same sign, the quotient will be positive and if integers have opposite signs, it will be negative.
The statement is true.
You use the same rule to determine if the product or the quotient wilt be positive or negative.
Question 27.
Any nonzero integer divided by 0 equals 0.
Answer:
The statement is false.
Rewrite the expression as a multiplication problem
0 × ? = a
Since the product of 0 and any other integer is 0, this multiplication problem does not have a solution.
Therefore, the division by 0 is undefined, it is not possible.
The statement is false
The division by 0 is undefined because there is no such a number you can multiply with 0 and get a product different from 0.
H.O.T. Focus on Higher Order Thinking
Question 28.
Multi-step A perfect score on a test with 25 questions is 100. Each question is worth the same number of points.
a. How many points is each question on the test worth? _____________
Answer:
a. A test has 25 questions and it is worth 100 points. The value of the expression:
100 ÷ 25
shows how many points each question is worth.
First determine if the quotient wiLt be positive or negative
Since 100 is positive and 25 is positive (they have the same sign), the quotient will be positive.
Divide given integers:
100 ÷ 25 = 4
Each question is worth 4 points.
b. Fred got a score of 84 on the test. Write a division sentence using negative numbers where the quotient represents the number of questions Fred answered incorrectly. _____________
Answer:
The difference
84 – 100
represents number of points that Fred lost on the test.
Use negative integer – 4 to represent number of points he lost in each incorrectly answered question.
The quotient
(84 – 100) ÷ (- 4)
represents number of question lie answered incorrectly.
Find the difference in brackets using rules for subtracting integers:
– 16 ÷ (- 4)
Use rules for dividing integers to find the final result:
– 16 ÷ (- 4) = 4
Fred answered 4 questions incorrectly.
Use rules for dividing integers and rules for subtracting integers to find the results.
Question 29.
Persevere in Problem Solving Colleen divided integer a by – 3 and got 8. Then she divided 8 by integer b and got – 4. Find the quotient of integer a and integer b.
Answer:
She got an expression:
a ÷ (-3) = 8
Rewrite that expression as a multiplication to get the value of a.
You get the expression:
– 3 × 8
Use rules for multiplying integers to find the result:
– 3 × 8 = – 24
a = – 24.
Then, she got an expression:
8 ÷ b = – 4
Divide 8 by – 4 to get the value of b.
You get the expression:
8 ÷ (- 4)
Use rules for dividing integers to find the result:
8 ÷ (- 4) = – 2
b = – 2
Find the quotient:
– 24 ÷ (- 2)
Determine if the quotient will be positive or negative.
Since – 24 is negative and – 2 is negative (they have the same sign), the product will, be positive.
Divide given integers:
– 24 ÷ (- 2) = 12
The quotient is 12.
The quotient of a and b is 12.
Find the values of a and b using rules for multiplying integers and rules for dividing integers and then find the quotient a ÷ b.
Question 30.
Justify Reasoning The quotient of two negative integers results in an integer. How does the value of the quotient compare to the value of the original two integers? Explain.
Answer:
You have the quotient of two negative integers, a and b:
a ÷ b
Since both integers are negative (they nave the same sign). the quotient will be positive.
Both integers are negative so they are less than 0
and the quotient is positive so it is greater than 0
Therefore, the quotient is greater than the original integers.
The quotient is greater than the original integers because integers are negative so they are less than 0 and the quotient is positive so it is greater than 0.