# 4.4 From Hundredths to Hundred-thousands

## Unit Goals

• Students read, write and compare numbers in decimal notation. They also extend place value understanding for multi-digit whole numbers and add and subtract within 1,000,000.

### Section A Goals

• Represent, compare, and order decimals to the hundredths by reasoning about their size.
• Write tenths and hundredths in decimal notation.

### Section B Goals

• Read, represent, and describe the relative magnitude of multi-digit whole numbers up to 1 million.
• Recognize that in a multi-digit whole number, the value of a digit in one place represents ten times what it represents in the place to its right.

### Section C Goals

• Compare, order, and round multi-digit whole numbers within 1,000,000.

### Section D Goals

• Add and subtract multi-digit whole numbers using the standard algorithm.

### Problem 1

#### Pre-unit

Practicing Standards:  3.NBT.A.1

Round each number to the nearest 10 and to the nearest 100.

1. 63

2. 350

3. 485

### Solution

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### Problem 2

#### Pre-unit

Practicing Standards:  3.NBT.A.1

A number $$P$$ is located on the number line.
1. Round $$P$$ to the nearest multiple of 100. Explain your reasoning.
2. Can you tell what $$P$$ is if rounded to the nearest multiple of 10? Explain your reasoning.

### Solution

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### Problem 3

#### Pre-unit

Practicing Standards:  3.NBT.A.2

Find the value of each expression. Show your reasoning.

1. $$523 + 278$$
2. $$418 - 235$$

### Solution

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### Problem 4

#### Pre-unit

Practicing Standards:  2.NBT.A.1

Here are three numbers: 265, 652, and 526. For each question, explain your reasoning.

1. Does the digit 6 have a greater value in 265 or 652?
2. Does the digit 5 have a greater value in 265 or 652?
3. In which number does the digit 2 have the greatest value? In which one does it have the least value?

### Solution

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### Problem 5

Each large square represents 1.

1. Write a fraction and a decimal that represent the shaded part of the large square.

Fraction: __________

Decimal: __________

2. Shade a part of each square to represent each given number.

Fraction: $$\frac{13}{100}$$

Decimal: __________

Fraction: __________

Decimal: 0.44

### Solution

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### Problem 6

Select all the numbers equivalent to $$\frac{2}{10}$$.

A:  0.5
B:  0.2
C:  $$\frac{20}{100}$$
D:  $$\frac{25}{100}$$
E:  0.20

### Solution

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### Problem 7

1. Locate and label 0.6 and 0.35 on the number line.

2. Compare 0.6 and 0.35 using < or >.

### Solution

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### Problem 8

Order the numbers from least to greatest:

5.90

9.05

5.95

0.59

5.59

### Solution

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### Problem 9

Order the numbers from least to greatest:

$$\frac{13}{10}$$

1.25

1.46

$$\frac{7}{5}$$

$$\frac{155}{100}$$

### Solution

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### Problem 10

#### Exploration

The table shows the distances, in miles, some students walked during the school week.

Order the numbers from least to greatest.

student distance (miles)
Han $$5\frac{3}{4}$$
Tyler $$5\frac{7}{8}$$
Mai 5.95
Elena $$5\frac{8}{10}$$
Andre 5.79

### Solution

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### Problem 11

#### Exploration

In a recent lesson, you learned about the lengths of the jumps made by Carl Lewis and other athletes.

Create and label a number line to show the distances of all ten jumps made by the athletes.

### Solution

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### Problem 1

1. Write the name of the number 8,500 in words.
2. How many hundreds are there in 8,500? Explain how you know.

### Solution

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### Problem 2

1. Count by 10,000 starting at 6,500 and stopping at 66,500. Record each number:
2. Pick two numbers from your list and write their names in words.

### Solution

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### Problem 3

1. If each small square represents 1, what number does the picture represent?
2. If each small square represents 10, what number does the picture represent?

### Solution

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### Problem 4

1. Write the names of the numbers 702,150, and 73,026 in words.
2. How is the value of the 7 in 702,150 related to the value of the 7 in 73,026?

### Solution

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### Problem 5

1. What is the value of the 6 in 65,247?
2. What is the value of the 6 in 16,803?
3. Write multiplication and division equations to represent the relationship between the value of the 6 in 65,247 and the value of the 6 in 16,803.

### Solution

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### Problem 6

1. Locate and label each number on the number line:

• 100,000
• 10,000
• 1,000

2. Which numbers were easiest to locate? Which were most difficult? Why?

### Solution

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### Problem 7

#### Exploration

For each question, use only the digits 1, 0, 5, 9, and 3. You may not use a digit more than once and you do not need to use all the digits.

1. Can you make three numbers greater than 3,000 but less than 3,500?
2. Can you make three numbers greater than 9,000 but less than 10,000?
3. Which numbers can you make that are greater than 39,500 but less than 40,000?

### Solution

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### Problem 8

#### Exploration

Estimate the value of the number labeled A on the number line. Explain your reasoning.

### Solution

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### Problem 1

Jada writes the same digit in the two blanks to make the statement true. Which digits could she write?

$$\large \boxed{6} \ \boxed{\phantom{8}} \ , \ \boxed{4} \ \boxed{3} \ \boxed{2} < \boxed{6} \ \boxed{5} \ ,\!\boxed{\phantom{8}} \ \boxed{9} \ \boxed{8}$$

### Solution

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### Problem 2

1. Order these numbers from least to greatest:

• 98,107

• 102,356

• 752,031

• 88,207

• 99,653

2. How did you pick the smallest number? Explain your reasoning.

### Solution

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### Problem 3

1. Which multiple of 10,000 is closest to 132,256?

2. Which multiple of 100,000 is closest to 132,256?

3. Which multiple of 100,000 is closest to the number labeled A?

### Solution

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### Problem 4

For the number 583,642:

1. What is the nearest multiple of 100,000?

2. What is the nearest multiple of 10,000?

3. What is the nearest multiple of 1,000?

4. What is the nearest multiple of 100?

5. What is the nearest multiple of 10?

### Solution

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### Problem 5

1. Describe the numbers that are 460,000 when rounded to the nearest 10,000.

2. Where are these numbers located on the number line?

### Solution

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### Problem 6

When rounded to the nearest 1,000, Airplane X is flying at 30,000 feet, Airplane Y at 31,000 feet, and Airplane Z at 32,000 feet.

1. Could Airplanes X and Y be within 1,000 feet of each other? If you think so, give some examples. If you don't think so, explain why not.
2. Explain why Airplanes X and Z could not be within 1,000 feet of each other. Use a number line if you find it helpful.

### Solution

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### Problem 7

#### Exploration

Rounded to the nearest 10 pounds, one bag of sand weighs 50 pounds.

Jada wants at least 1,000 pounds of sand for a sandbox. How many bags of sand does Jada need to buy to be sure that she has enough sand?

### Solution

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### Problem 8

#### Exploration

You will need a set of digit cards 0–9 for this exploration.

Shuffle your cards and stack them face down. Turn over 6 digit cards.

Can you put the 6 digits in the blanks so that all three statements are true?

1. $$\large \boxed{4} \ , \boxed{\phantom{8}} \ \boxed{2} \ \boxed{3} > \boxed{\phantom{8}} \ , \boxed{9} \ \boxed{7} \ \boxed{8}$$
2. $$\large \boxed{\phantom{8}} \ \boxed{2} \ , \boxed{4} \ \boxed{0} \ \boxed{3} > \boxed{4} \ \boxed{2} \ , \boxed{\phantom{8}} \ \boxed{0} \ \boxed{1}$$
3. $$\large \boxed{4} \ \boxed 3 \ \boxed{\phantom{8}} \ , \boxed{2} \ \boxed{5} \ \boxed{7} > \boxed{4} \ \boxed{\phantom{8}} \ \boxed{5} \ , \boxed{{9}} \ \boxed{3} \ \boxed{7}$$

### Solution

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### Problem 9

#### Exploration

To answer these riddles, think about rounding to the nearest 10, 100, 1,000, or 10,000. Use a number line if it is helpful.

1. I can be rounded to 100 or to 140. What number could I be?
2. I can be rounded to 7,500 or to 8,000. What number could I be?
3. I can be rounded to 60,000 or to 57,000. What number could I be?

### Solution

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### Problem 1

Clare took 11,243 steps on Saturday and 12,485 steps on Sunday.

1. How many steps did Clare take altogether on Saturday and Sunday?
2. How many more steps did Clare take on Sunday than on Saturday?

### Solution

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### Problem 2

1. Find the value of the sum. Explain your calculations.

​​​​​​

2. Find the value of the difference. Explain your calculations.

### Solution

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### Problem 3

Find the value of each sum and difference using the standard algorithm.

### Solution

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### Problem 4

Here is how Han found $$300,\!526 - 4,\!472$$

1. How can you tell by estimating that Han has made an error?
2. What error did Han make?
3. Find the value of $$300,\!526 - 4,\!472$$.

### Solution

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### Problem 5

In 2018 the population of Boston is estimated as 694,583 and the population of Seattle is estimated as 744,995.

1. Is the population difference between Boston and Seattle more or less than 100,000? Explain how you know.
2. Is the population difference more or less than 50,000? Explain how you know.
3. Find the difference in the populations of the two cities.

### Solution

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### Problem 6

#### Exploration

Han says he has a method to find the value of $$1,\!000,\!000 - 267,\!923$$ without any carrying: “I just write 1,000,000 as $$999,\!999 + 1$$.”

1. How might rewriting 1,000,000, as Han suggested, help with finding the difference of $$1,\!000,\!000 - 267,\!923$$?
2. Try Han's method to find $$1,\!000,\!000 - 267,\!923$$.

### Solution

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### Problem 7

#### Exploration

Use the information to determine when the airplane, telephone, printing press, and automobile were first invented.

• The airplane was invented in 1903.
• The printing press was invented 453 years before the most recent invention.
• The automobile was invented 15 years before 1900.
• It was 426 years after the invention of the printing press that the telephone was invented.
• The automobile and telephone were invented the closest together in time with only 9 years between them.

### Solution

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