Unit 3 Review

Calculate the whole number with remainder answer to 1617 ÷ 13

124 R 5

Write equations to show how the commutative and associative properties of multiplication are involved when you calculate 50 ∙ 800 mentally by relying on basic multiplication facts (such as5 ∙ 8). Write your equations in the form: 50 ∙ 800 = some expression= some expression

50 ∙ 800 = (5 ∙ 10) ∙ (8 ∙ 100)= (5 ∙ 8) ∙ (10 ∙ 100)= 40 ∙ 1000= 40,000

Explain how you could find what 95% of 580 is by using reasoning

100% of 580 is 580. 10% of 580 is 58. 5% is half of 10%, so 5% is 29. Hence,95% = 100% - 5%➔ 580 - 29 = 551

Use the commutative property of multiplication to write an equivalent algebraic expression (𝑥 + 1)(𝑦 + 3)

(𝑦 + 3)(𝑥 + 1)

21/-7

-3

Divide and simplify: − 6/14 ÷ 2/7

-3/2

A small business is open 5 days per week. The business has 52 customers per day. a. Use the Associative Property to write two equivalent expressions for the number of customers that visit the business in w weeks

. (5 ∙ 52) ∙ 𝑤 and 5 ∙ (52 ∙ 𝑤)

Write 1/8 as a decimal using the common method for division

0.125

Solve the multiplication problem 28 ∙ 63 using the partial-products method

1764

Explain how to use the distributive property to make 31 ∙ 25 easy to calculate mentally. Write equations that correspond to your strategy

31 ∙ 25 = (30 + 1) ∙ 25= 30 ∙ 25 + 1 ∙ 25= 750 + 25= 775

Solve the multiplication problem 144 ∙ 22 using the common method / standard algorithm

3168

Solve the multiplication problem 27 ∙ 13 using the lattice method.

351

Write an equivalent expression using the distributive property so that the expression has no parenthesis (5𝑐 − 𝑑)(7𝑎 + 𝑏 − 2)

35𝑎𝑐 + 5𝑏𝑐 − 10𝑐 − 7𝑎𝑑 − 𝑏𝑑 + 2𝑑

Write an equivalent expression using the distributive property so that the expression has no parenthesis (𝑥 + 2)(3𝑦 − 1)

3𝑥𝑦 − 𝑥 + 6𝑦 − 2

-16/-4

4

Draw a large rectangle to represent an array for 43 ∙ 275. Partition the rectangle in a natural way so that the parts of the rectangle correspond to the steps in the partial-products method. Use the areas of the rectangle to write an expression equal to 43 ∙ 275 and solve the multiplication problem

43 ∙ 275 = 40 ∙ 200 + 40 ∙ 70 + 40 ∙ 5 + 3 ∙ 200 + 3 ∙ 70 + 3 ∙ 5 = 11,825

A small business is open 5 days per week. The business has 52 customers per day b. How many customers visit the salon in 4 weeks

5 ∙ 52 ∙ 4 = 1040 customers

Find an expression that has the same value as (7 ∙ 70) + (7 ∙ 5)

7 ∙ (70 + 5) = 7 ∙ 75

Here is Marco's method for calculating 38 ∙ 60: "Four times 6 Is 24, so 40 times 60 is 2400. Then,2 times 6 is 12, so it's 2400-120, which is 2280."Write equations that incorporate Marco's method and that also show why his method is valid. Write your equations in the following form: 38 ∙ 60 = some expression= some expression= 2280

8 ∙ 60 = (40 − 2) ∙ 60= 40 ∙ 60 − 2 ∙ 60= 4 ∙ 10 ∙ 6 ∙ 10 − 2 ∙ 6 ∙ 10= 4 ∙ 6 ∙ 100 − 2 ∙ 6 ∙ 10= 24 ∙ 100 − 12 ∙ 10= 2400 − 120= 2280

Calculate the decimal answer to 5235 ÷ 6

872.5

John currently earns $96 a week doing yard chores after school. After 3 months, his pay will increase by 18 . How much will he then earn per week?

96 ∙ 1/8 = 12. His pay will increase by $12. His new pay will be 96 + 12 = $108

Multiply 2 2/5 ∙ 1 1/7 . Leave your answer as an improper fraction

96/35

Write an Array word problem for 4 ∙ 6 =?. Draw a math picture and solve the problem

A marching band has four rows with 6 people in each row. How many people are in the band? There are 4 ∙ 6 = 24 people in the band

Write an Area word problem for 3 ∙? = 21. Draw a math picture and solve the problem

A painting canvas has a side of 3 feet. It has a total area of 21 𝑓𝑡2. What is the length of the other side?The canvas has side length of 3 ∙? = 21 → ? = 7 f

Anthony is trying to calculate 2 ∙ 57. He makes a math drawing in the picture and concludes from his drawing that 2 ∙ 5/7 = 10/14 , because there are 14 small pieces and 10 of them are shaded. Critique Anthony's reasoning. Is he correct? If he isn't, how could he modify his method?

Anthony is not correct. He doubled the whole picture, but we only double the green pieces andthen look at them in reference to the original whole. He should just double the green pieces andthen fill in the original whole. It takes two pieces to fill in the original, giving one whole and then 3/7 are left over. The correct answer is 2 ∙ 5/7 = 10/7 = 1 3/7

Company A sold 48 ∙ 4 + 84 ∙ 3 − 24 items in 7 days. Company B sold 31 ∙ 2 + 107 ∙ 3 + 29 ∙ 2 − 21 items in the same number of days. Did the companies sell the same number of items

Company A sold 420 items. Company B sold 420 items. So yes, they sold the same number ofitems

Jamie calculates that 7 ÷ 3 = 2 remainder 1. When Jamie is asked to write her answer as a decimal, she simply puts the remainder 1 behind the decimal point: 7 ÷ 3 = 2.1. Is Jamie correct or not? If not, explain why not, and explain to Jamie in a concrete way why the correct answer makes sense

Jamie is not correct. 7 ÷ 3 is the same as 7/3. We can convert this to a mixed number: 7/3 = 2 1/3. The remainder that she found is the part that remains as a fraction. The decimal representation would be 2. 3̅ which is the same as 2 + 1/3

Jenny claims the volume of the right rectangular prism is 30 𝑓𝑡3. Find the correct volume. What mistake might Jenny have made?

Jenny only calculated the area. She forgot to multiply the last side. 𝑉 = (6)(5)(3) = 90𝑓𝑡3

James doubled his profit compared with that of last year. Last year he doubled his profit from the previous year. If his profit this year was $12,000, then what was his profit last year and the year before that?

Last year: $6000, The year before: $3000

Leah is working on the multiplication problem 2.43 ∙ 0.148. Ignoring the decimal points, Leah multiplies 234 ∙ 148 and gets the answer 35,964. But Leah can't remember the rule about whereto put the decimal point in this answer to get the correct answer to 2.43 ∙ 0.148. Explain how Leah can use reasoning about the sizes of the numbers to determine where to put the decimal point

Since 2.43 is between 2 and 3 and 0.148 is between 0 and 1, the decimal answer has to be in therange 2 ∙ 0 = 0 and 3 ∙ 1 = 3. The answer then could be 0.35964, 0.035964, 0.0035964, etc. So,we'll have to make our range smaller. We know 0.148 is between 0.1 and 1. The range isshortened to 2 ∙ 0.1 = 0.2 and 3 ∙ 1 = 3. That tells us that our answer has to be 0.35964

Together, Frank, John, and David earned $14. They want to divide it equally, except that David should only get a half share, since he did half as much work as either Frank or John did (and Frank and John worked equal amounts). Write and solve a division problem to find out how much Frank should get

Since David will only get a half share, we are dividing by 2 1/2 instead of 3. So,14 ÷ 2 1/2 = 14 ÷ 5/2 = 14/1 ∙ 2/5 = 28/5 = 5 3/5 = 5 60/100 = $5.60 Frank and John will both receive $5.60 and David will get 5.602 = $2.80.

Write a Multiplicative Comparison word problem for 3 ∙ 8 =?. Draw a math picture and solve the problem

Robert and Samuel each have a trading card collection. Robert has 3 times as many cards as Samuel. Samuel has 8 cards. How many cards does Robert have? Robert has 3 ∙ 8 = 24 cards.

Your friend says that the quotient 2/3 ÷ 2/9 is 1/3 . What is the correct quotient? What mistake did your friend likely make?

The correct quotient is 3. Your friend flipped the first fraction, not the second one

For the following word problems, write multiplication and division equations that model the problem, using a question mark for the unknown amount. Determine which interpretation of division is involved (how-many-groups or how-many-units-in-1-group) and solve the problem Coach Brown has 72 baseballs. If 8 baseballs fit in a box, how many boxes can Coach Brown fill?

This is a how-many-groups type of problem. ? ∙ 8 = 72 and 72/8 = ?. ? = 9 boxes

For the following word problems, write multiplication and division equations that model the problem, using a question mark for the unknown amount. Determine which interpretation of division is involved (how-many-groups or how-many-units-in-1-group) and solve the problem Carina had 20 cupcakes. She put them into 5 boxes so that there were the same number of cupcakes in each box. How many cupcakes did Carina put into each box?

This is a how-many-units-in-1-group type of problem. 5 ∙? = 20 and 20/5 = ?. ? = 4 cupcake

a. Is 0/5 =? defined? Write an equivalent multiplication expression to help explain why or why not.

This is defined. An equivalent expression would be 5 ∙ ? = 0. When ? = 0 this equation is true

Write an Ordered Pair word problem for 2 ∙ 3 =?. Draw a tree diagram and solve the problem

While packing for vacation, you pack two pairs of pants and three shirts. How many different outfits can you make?You can make 2 ∙ 3 = 6 outfits

Your favorite store is having a 15%-off sale, meaning that the store will take 15% off the price of each item you buy. When the clerk rings up your purchases, she takes 15% off the total (before tax), rather than 15% off each item. Will you get the same discount either way? Is there a property of arithmetic related to this? Explain. Write equations to assist your explanation

Yes, you will get the same discount either way. Separate: 𝑥 − 15% ∙ 𝑥 + 𝑦 − 15% ∙ 𝑦 + 𝑧 − 15% ∙ 𝑧Together: (𝑥 + 𝑦 + 𝑧) − 15%(𝑥 + 𝑦 + 𝑧)Using the distributive property and the commutative property we can see that these two are equal.

Write an equivalent expression using the distributive property so that the expression has no parenthesis - (7𝑎 + 6𝑏 − 4)

−7𝑎 − 6𝑏 + 4

Use the commutative property of multiplication to write an equivalent algebraic expression. a. 17 ∙ 𝑦

𝑦 ∙ 1/7