## Envision Math 6th Grade Textbook Answer Key Topic 2.5 Mental Math

Mental Math

How can you break apart numbers to compute mentally?
Jo has to read 45 history pages and 46 science pages by the end of next week.

Question.
How many total pages must Jo read?

Another Example
What other strategies can you use to compute mentally?
Look for compatible numbers and use properties of operations to compute mentally.

Use compensation to create compatible numbers that are easy to compute mentally.

Explain It
Question 1.
When you use compensation to subtract, why must you add to the difference you find?
You must add back to the difference the extra amount you subtracted.
Explanation:

Question 2.
When you use compensation to multiply, which property are you using?
Distributive Property
Explanation:

Use mental math.

Combine the sums. 80 + 11 = 91
So, 45 + 46 = 91.

Explain why breaking apart the numbers works.
40 + 5) + (40 + 6) ← Break apart numbers.
(40 + 40) + (5 + 6) ← Use Commutative and Associative Properties.
Jo has 91 pages to read.

Guided Practice

Do you know HOW?
Compute mentally.
Question 1.
89 + 32 + 8
(32 + 8) + 89 = 129
Explanation:

Question 2.
76 + 59 + 6
See margin.
Explanation:

Question 3.
2 × 9 × 20
(2 × 20)9 = 360
Explanation:

Question 4.
5 × 31 × 2
(5 × 2)31 = 310
Explanation:

Question 5.
8 × 39
See margin.
Explanation:

Question 6.
48 + 52
See margin.
Explanation:

Question 7.
453 – 397
See margin.
Explanation:

Question 8.
6(42)
6(40) + 6(2) = 252
Explanation:

Do you UNDERSTAND?
Question 9.
When you use the Compensation Strategy in a problem such as 5 x 698, you turn one multiplication problem into two and then combine them.
Why does this idea make sense?
See margin.
Explanation:

Question 10.
Jo has to read 2 science chapters for each of the next 5 weeks. Each chapter is 45 pages long. How many pages does she have to read?
2 × 5 × 45 = (2 × 5)45 = 450
Explanation:

Independent Practice

Compute mentally.
Question 11.
10 + 23 +130
10 + 130 + 23 = 163
Explanation:

Question 12.
721 – 395
(721 + 5) – (395 + 5) = 326
Explanation:

Question 13.
2 × 38 × 5
(2 × 5)38 = 380
Explanation:

Question 14.
28 + 26 + 32 + 14
See margin.
Explanation:

Question 15.
5 × 3 × 40
(5 × 40)3 = 600
Explanation:

Question 16.
856 – 403
See margin.
Explanation:

Question 17.
6(69)
6(70) – 6(1) = 414
Explanation:

Question 18.
80 × 10 × 5
(80 × 10)5 = 4,000
Explanation:

Question 19.
44 + 56
See margin
Explanation:

Question 20.
840 + 260 + 72
See margin.
Explanation:

Question 21.
495 + 75 + 14
See margin
Explanation:

Question 22.
397 + 255
See margin.
Explanation:

Question 23.
8(82)
8(80) + 8(2) = 656
Explanation:

Question 24.
4 × 5 × 25
(4 × 25) × 5 = 500
Explanation:

Problem Solving

Use the data table for 25 and 26. Find the answers mentally.

Question 25.
How much did Jacob make during the first three weeks?
$45 +$32 + $55 = ($45 + $55) +$32 = $100 +$32 = $132 Explanation: Question 26. Jamal earned$5 for every dollar Jacob earned during week 4 on his paper route. How much money did Jamal earn?
$5($64) = $5($60 + $4) =$300 + $20 =$320
Explanation:

Question 27.
Writing to Explain Avis swam her first lap in 32 seconds and her second lap in 45 seconds. Explain the steps you can use to mentally calculate her total time.
See margin.
Explanation:

Question 28.
Use the break-apart strategy to mentally solve 81 + 43 + 2.
(80 + 1) + (40 + 3) + 2 = 120 + 6 = 126
Explanation:

Question 29.
Draw It Copy and complete the bar diagram to show how to use the Distributive Property to mentally compute the problem below.

Four friends are having a snack. They each have 8 strawberries and 22 blueberries. How many berries do they have altogether?
The friends have 4 × 30 = 120 berries in all.
Explanation:

Question 30.
Number Sense What kind of numbers would make 3-factor multiplication problems easy to do mentally?
Numbers in which two of them combine to a multiple of 10
Explanation:

Question 31.
Writing to Explain Explain how you can use the Distributive Property to multiply 6 and 82.
See margin.
Explanation:

One of the world’s tallest fountains shoots water 171 m into the air once per hour. Use this information for 32 and 33.

Question 32.
Think About the Process Which expression shows a break-apart strategy to mentally calculate the total of the heights of the water this fountain shoots in the air in 5 hours?
A. 171 × 5
B. 5(100 × 71)
C. 170 × 5 + 1
D. 5(100 + 70 + 1)
D. 5(100 + 70 + 1)
Explanation:

Question 33.
The fountain sprays water for 15 minutes during every hour between 10:00 A.M. and 9:00 P.M. If the fountain sprayed water 10 times higher than it does, how high would the water shoot up? Solve mentally, and then check your answer.
1,710m
Explanation:

Mixed Problem Solving

Counting Calories
Calories come from the food you eat. Your body converts those calories into the energy you need every day.
After class, the sixth grade had the snack foods shown in the chart. For 1 through 5, write an expression to show how many calories the person ate. Then, evaluate each expression.

Tips: Make sure to watch the units on the chart compared to the units in the problems.

Question 1.
Wen-Wei ate a peach and an apple for lunch. After class, he ate 2 cups of celery and 8 tablespoons of peanuts. How many calories did he eat after class?
2(20) + 8(50) = 440 calories
Explanation:

Question 2.
Andrea ate 3 tablespoons of Brazil nuts. Then she ate 2 more tablespoons. Later, she ate another 4 tablespoons.
3(60) + 2(60) + 4(60) = 540 calories
Explanation:

Question 3.
Gloria ate 2 ounces of low fat cottage cheese, a cup of carrots, and 6 tablespoons of cashews. Then, she went back and got 4 more tablespoons of cashews and an orange. She ate everything but half of the orange.
See margin
Explanation:

Question 4.
Ayesha ate 4 cups of celery, 3 oranges, and 2 ounces of Swiss cheese.
4(20) + 3(60) + 2(105) = 470 calories
Explanation:

Question 5.
Daryl took 2 oz dried apricots, 4 tbsp almonds, and 2 cups of pretzels. He ate all of it, except for 1 oz of apricots.
2(40) + 4(55) + 2(60) – 40 = 380 calories
Explanation:

For 6 and 7, write an expression that describes the situation, and then evaluate the expression.
Question 6.
Paulo took 4 bananas, 6 ounces of dates, 4 ounces of cheddar cheese, and 2 ounces of Swiss cheese. He split it all evenly with Alex. How many calories did Paulo eat?