Section A Practice Problems

Problem 1

Which number could be labeled with this point on the number line?

<p>Number line. Scale 0 to 100 by 100's. No tick marks. Point between 0 and 100.</p>

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Solution

Problem 2

There are 85 students on the playground. There are 57 fewer students in the classroom than on the playground. How many students are in the classroom? Explain or show your reasoning.

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Solution

28 students. Sample response:

85-50=35

35-7=28

Problem 3

Jada says she can find

87 - 59

by taking away 60 from 87 and adding 1, so it is the same as

27 + 1

or 28. Explain or show why Jada’s method to calculate

87-59

makes sense.

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Solution

Sample response: Jada first takes away 6 tens, leaving 2 tens and 7 ones, $87 - 60 = 27$ . But she took away 1 too many, so she adds that back, $27+1=28$ .

Problem 4

Find the value of $316 + 514$ . Explain or show your reasoning.

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Solution

830. Sample response: I added the hundreds to get 800, then the tens to get 20, and the ones to get 10. I put the 10 together with 20.

Problem 5

Write < or > in the blank to make each statement true.

$197\, \underline{\hspace{1.05cm}}\, 311$
$567\, \underline{\hspace{1.05cm}}\, 555$
$908\, \underline{\hspace{1.05cm}} \,809$

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Solution

Problem 6

Find the value of each expression.

$206 + 543$
$327 + 181$
$674 - 129$

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Solution

Problem 7

Select all representations of the number four hundred twenty-three.

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Solution

B,D

Problem 8

The height of the Empire State Building in New York City is 443 meters. The Burj Khalifa (BURJZH kah-LEE-fah) in Dubai (doo-BAI) is 828 meters. How many meters taller than the Empire State Building is the Burj Khalifa?

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Solution

385. Sample response: $830−400=430$ , $430−40=390$ , $390−3=387$

Problem 9

Find the value of each sum in any way that makes sense to you. Explain or show your reasoning.

$456 + 231$
$372 + 165$

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Solution

687. Sample response: I added the hundreds, then the tens, and then the ones.
537. Sample responses: I added the ones and made a new hundred with the tens. $2 + 5 = 7$ , $70 + 60 = 130$ , $300 + 100 = 400$ , $400 + 130 + 7 = 537$

Problem 10

Here are 3 different ways to find the value of $157 + 436$ .

<p>Two base ten diagrams. Top diagram, 1 hundred, 5 tens, 7 ones. Bottom diagram, 4 hundreds, 3 tens, 6 ones.</p> — A

<p>Addition. One-hundred plus fifty plus seven, plus four-hundred plus thirty plus six, equals five-hundred plus eighty plus thirteen.</p> — B

<p>Addition. One-hundred fifty-seven, plus four-hundred thirty-six, equals thirteen plus eighty plus five-hundred, equals five-hundred ninety-three.</p> — C

How are the methods alike? How are they different? Explain your reasoning.

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Solution

Sample responses: Method A is the only way with a picture. It doesn't show how to combine the numbers or in what order. Methods B and C add up the ones, tens, and hundreds, and then put them all together. Method B shows how the ones, the tens, and the hundreds are put together more explicitly. Only Method C shows the value of the sum.

Problem 11

Here is Elena's algorithm for finding $273 + 481$ .

<p>Addition. Step one. Two-hundred seventy-three plus four-hundred eighty-one, equals four.</p>

<p>Addition. Step 2. Two-hundred seventy-three, plus four-hundred eighty-one, equals 54.</p>

<p>Addition. Step 3. Two-hundred seventy-three plus four-hundred eighty-one equals seven-hundred fifty-four.</p>

Where does the 100 that Elena wrote in Step 2 come from?
Use Elena's method to find $255 + 372$ .

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Solution

Sample response: When she finds $70 + 80$ , that's 150, so Elena records the 5 tens below and the 100 above.
627. Sample response:

Problem 12

What do the 1s above the 2 and the 5 in 253 mean in this calculation?
Use an algorithm or another strategy to find the value of each sum.
1. $572 + 268$
2. $726 + 199$

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Solution

Sample response: The 1 above the 5 in 253 represents 10. When I add 3 and 9 I get 12, so I record the 2 below, and the 10 above as a 1. When I add the 5 tens in 253, the 8 tens in 89, and the 1 ten above, I get 14 tens, so I record 4 of the tens below and the 1 above the 2 represents the other 10 tens or 100.
1. 840. Sample response:
2. 925. Sample response: I added 200 to 726 to get 926, and then I took away 1.

Problem 13

Here is Lin’s strategy to find the value of $596 + 385$ : “I added 600 to 385, and then took away 4.”

Explain why Lin’s strategy works. Then use it to find the value of $596 + 385$ .
For which of these expressions would you use Lin’s strategy? Explain or show your reasoning.
1. $436 + 173$
2. $517 + 255$
3. $787 + 135$
4. $247 + 395$

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Solution

Sample response: Lin is thinking of 596 as 4 less than 600 so adding 600, and then taking away 4 is the same as adding 596. $385+600 = 985$ and $985-4 = 981$ .
Sample response: I might use Lin's strategy for $787 + 135$ , since I can add 800 to get 935 and then take away 13, so $787 + 135 = 922$ . I would use Lin's strategy for $247 + 395$ , which has the same value as $647 - 5$ or 642.

Problem 14

Write an addition problem with 3-digit numbers that you think is well suited for each of the following methods. Then find the value of the sum, using that method.

mental strategies
base-ten blocks
an algorithm

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Solution

Sample responses:

$538 + 299$ . I added 300 and then took away 1 so it's 837.
$213 + 145$ . I chose small numbers so that it was easier to draw a picture or get the blocks out. It is 358 since there are 3 hundreds, 5 tens, and 8 ones.
$467+375$