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New Jersey Center for Teaching and Learning

Progressive Mathematics Initiative

This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others.

Click to go to website:

www.njctl.org

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Place Value

click on the topic to go

to that section

Standard, Numeric and Expanded Forms

Comparing Numbers

Ordering Numbers

Rounding Numbers

Addition

Subtraction

click to return to table of contents

Numbers, like 247, have three digits. Each digit has a different place value.

The right digit is the ones place. It tells you that there are 7 ones.

The middle digit is the tens place. It tells you that there are 4 sets of ten.

The left digit is the hundreds place. It tells you that there are 2 sets of one hundred.

Therefore, there are 2 sets of 100, plus 4 sets of 10, plus 7 ones in the number 247. (200+40+7=247)

2 4 7

What would be the difference if the 2 was in the hundreds place?

Hundreds Tens Ones

What if the 5 was in the tens place?

1 2 5

Click on the image to be directed to a place value identification game.

Roll a die. Copy the numbers below in the order that they are rolled.

_____ _____ _____

Circle the digit in the tens place.

Cross out the digit in the hundreds place.

Underline the digit in the ones place.

Write the number that is 10 more than this number.

_____ _____ _____

Write the number that is 100 less than this number.

_____ _____ _____

erase and try several examples

Standard, Numeric, Expanded Forms

click to return to table of contents

You can write numbers in different ways.

Standard form is writing a number out in digits.

674

Expanded form is writing a number out according to each digits place value.

600+70+4

Word form is writing a number out in words.

six hundred seventy-four

Write the number that the base ten blocks represent in standard form.

___ ___ ___

Write the number 235 in word form.

__________ __________ __________-__________

five

twenty

fifty

two

three

thirty

two hundred

five hundred

three hundred

What is the number 200 + 50 + 4, written in standard form?

10

A two hundred fifty-four

B 254

C 200504

D 200,504

What is the number 108, written in expanded form?

A 100 + 0 + 8

B one hundred eight

C one hundred and eight

D 1 + 0 + 8

What is the number 211, written in word form?

12

A two hundred ten-one

B 200 + 10 + 1

C two hundred eleven

D 2 + 1 + 1

What is the number four hundred two, written in standard form?

13

A 4,002

B 400 + 0 + 2

C 402

D four hundred and two

Which letter has the number 198 written properly in both word form and expanded form?

14

A one hundred ninety-eight, 1 + 9 + 8

B one hundred ninety-eight, 100 + 90 + 8

C one hundred ninety and eight, 100 + 90 + 8

D one hundred ninety and eight, 1 + 9 + 8

click to return to table of contents

There are two symbols we use to compare numbers.

> (greater than)

< (less than)

One number goes on the left of the symbol and another number goes on the right of the symbol.

The number on the left of the ">" shows the larger number.

For example: 2 > 1

The number on the left of the "<" shows the smaller number.

For example: 1 < 2

Remember, one number goes on the left of the symbol and another number goes on the right of the symbol.

The number on the left of the ">" shows the larger number.

For example: 2 > 1

This means that "2 is greater than 1"

The number on the left of the "<" shows the smaller number.

For example: 1 < 2

This means that "2 is less than 1"

3 Hundreds is greater than 2 hundreds. 334 > 245

Compare digits in the same place-value position from left to right.

Hundreds Tens Ones

3 3 4

2 4 5

3 hundreds is greater than 2 hundreds. 334 > 245

The numbers are in order from least to greatest.

245

334

300

340

330

240

250

334 is to the right of 245.

334 > 245

click to return to table of contents

- - write them from least to greatest
- - or from greatest to least.

Step 2 Compare the Tens.

Step 1 Compare the Hundreds.

Hundreds Tens Ones

Hundreds Tens Ones

2 1 0

2 1 0

2 7 3

2 7 3

2 2 6

2 2 6

2 = 2 = 2

1 < 2 < 7

So the order is 210; 226; 273.

210

226

273

250

300

200

Since 273 is to the right of the other numbers, it is the greatest number. 273 > 226 > 210

Roll a die. Copy the numbers below in the order that they are rolled.

_____ _____ _____ _____

Now arrange the numbers to make the largest number.

_____ _____ _____ _____

Now arrange the numbers to make the smallest number.

_____ _____ _____ _____

Which group of numbers are in order from largest to smallest?

22

A 456, 498, 472, 431

B 431, 456, 498, 472

C 431, 456, 472, 498

D 498, 472, 456, 431

Which group of numbers are in order from smallest to largest?

23

A 511, 423, 583, 457

B 511, 423, 457, 583

C 423, 457, 511, 583

D 583, 511, 457, 423

Which group of numbers are in order from smallest to largest?

24

A 365, 321, 359, 328

B 321, 328, 359, 365

C 365, 359, 328, 321

D 321, 359, 328, 365

What number could fill in the blank and have the numbers ordered from least to greatest?

25

34, 45, 125, _____, 132

What number could fill in the blank and have the numbers ordered from greatest to least?

26

329, 313, ____, 301, 295

click to return to table of contents

Rounding makes numbers that are easier to work with in your head.

- Rounded numbers are only approximate.
- An exact answer generally can not be obtained using rounded numbers.
- Use rounding to get an answer that is close but that does not have to be exact.

How to Round Numbers to the Nearest Ten

Make the numbers that end in 1 through 4 into the next lower number that ends in 0. For example 74 rounded to the nearest ten would be 70.

Numbers that end in a digit of 5 or more should be rounded up to the next even ten. The number 88 rounded to the nearest ten would be 90.

Round each to the nearest ten. Use the magnifying glass to check your answers.

50

20

48

19

20

22

33

30

67

81

80

70

80

75

55

60

(click here)

What is 93 rounded to the nearest ten?

What is 12 rounded to the nearest ten?

How to Round Numbers to the Nearest Hundred

To round numbers to the nearest hundred, make the numbers that end in 1 through 49 into the next lower number that ends in 00. For example 424 rounded to the nearest hundred would be 400.

Numbers that have the last two digits of 50 or more should be rounded up to the next even hundred. The number 988 rounded to the nearest hundred would be 1000.

Round each to the nearest hundred. Use the magnifying glass to check your answers.

300

345

550 600

479

500

220 200

773 800

624

600

821 800

945 900

Practice Rounding to the Hundreds

(click here)

1. Put your pencil point under the digit in the tens place.

Look to the right.

2. Is the digit 5 or more?

Yes OR No

3. What happens to the 3?

Increases by 1 OR remains the same

4. What happens to everything to the left of the tens place?

Those digits always remain the same.

5. Write the answer ____________

1. Put your pencil point under the digit in the tens place.

Look to the right.

2. Is the digit 5 or more?

Yes OR No

3. What happens to the 1?

Increases by 1 OR remains the same

4. What happens to everything to the left of the tens place?

Those digits always remain the same.

5. Write the answer ____________

Round 837 to the nearest hundred.

1. Put your pencil point under the digit in the hundreds place.

Look to the right.

2. Is the digit 5 or more?

Yes OR No

3. What happens to the 8?

Increases by 1 OR remains the same

4. What happens to everything to the left of the hundreds place?

Those digits always remain the same.

5. Write the answer ____________

Round 1,439 to the nearest hundred.

1. Put your pencil point under the digit in the hundreds place.

Look to the right.

2. Is the digit 5 or more?

Yes OR No

3. What happens to the 4?

Increases by 1 OR remains the same

4. What happens to everything to the left of the hundreds place?

Those digits always remain the same.

5. Write the answer ____________

What is 875 rounded to the nearest ten?

What is 1,242 rounded to the nearest ten?

What is 1,818 rounded to the nearest hundred?

What is 971 rounded to the nearest ten?

click to return to table of contents

- Addition Properties

click on the topic to go

to that section

- Missing Addends

- Estimate Sums

- Add 2-Digit numbers

- Add 3-Digit numbers

- Addition Story Problems

There are 3 properties of Addition

Communtative Property- You can add numbers in any order and get the same sum (answer).

Identity Property- You can add a zero to any number and the sum will equal the original number.

Associative Property- You can group the addends (numbers being added) in different ways and the sum will be the same.

Label each number sentence with its correct property.

Commutative, Identity, or Associative

4 + 1 = 1 + 4 __________

6 = 8 + 0 __________

0 + 9 = 9 __________

2 + 5 = 5 + 2 __________

100 + ( 5 + 6 ) = ( 100 + 5 ) + 6 __________

50 + 0 = 50 __________

( 1 + 2 ) + 3 = 1 + ( 2 + 3 ) __________

110 + 5 = 5 + 110 __________

( 5 + 8 ) + 50 = 5 + ( 8 + 50 ) __________

Which number sentence equals 15?

A (3 + 5) + 7

B 3 + (5 + 7)

C Both A and B

Associative Property

click here for property

What is the name of the property that allows

23 + 0 to equal 23?

56

A Commutative

B Associative

C Identity

D All of the above

In an addition equation such as 5 + 6 = 11:

5 and 6 are the addends and 11 is the sum.

Sometimes we are given a problem with a missing addend.

4 + ? = 6 12 = 9 +

__+ 5 = 10 7 = a + 3

We have to figure out the number that is missing.

Match the number sentence with its missing addend.

6 + ? = 9 7

x + 4 = 10 2

9 = ? + 4 3

7 = 5 + ? 6

5 + ? = 12 5

There are 16 seats on a roller coaster. If 10 of the seats are already filled, how many more seats can be filled?

62

A 26

B 6

C 10

D

16

When you estimate sums, you round the numbers or use compatible numbers to find an estimate of what the sum may be before you add.

Round each number to the nearest ten.

Then add to find the estimated sum.

138 140

+63+60

19 20

+53+50

Find numbers that are close to the numbers you are working with. These numbers are called compatible numbers.

128 125

+73+75

Press balloon

To make it pop

304 300

+86+85

A family trip involved driving 731 miles to its first stop and 475 miles to its second. About how many miles did the family have to travel?

66

A 200 miles

B 1,000 miles

C 1,200 miles

D 2 ,000 miles

What is the estimated sum for 45 + 68?

How to add two 2-digit numbers without regrouping.

For example, 22 + 56

You will first have to place the numbers on top of one another.

Make sure to line up the place values.

22

+56

Add the ones\' place digits ( 2 + 6 ). This sum is less than 9 so regrouping is not necessary. So place the 8 under the ones place.

Next, add the tens\' place digits ( 2 + 5 ). Place the 7 under the tens place.

How to add two 2-digit numbers with regrouping.

For example, 38 + 55

You will first have to place the numbers on top of one another.

Make sure to line up the place values.

38

+55

Add the ones\' place digits ( 8 + 5 ). This sum is more than 9 so you need to regroup. Place the 3 in the ones\' place and the 1 in the tens place.

Next, add the tens\' place digits (1+ 3 + 5 ). Place the 9 under the tens place.

What is the sum? 42 + 29 = _____

What is the sum? ___ = 75 + 25

What is the sum? 23 + 42 = ___

To add two 3-digit numbers, place one 3-digit number on top of the other 3-digit number.

For example, 462 +557 becomes:

462

+557

Remember to:

Make sure to line up the digits based on their place values.

Add the ones\' digit place first. If the sum is less than 9, no regrouping is necessary.

Next, add the tens\' place. If the sum is greater than 9, you will need to regroup and place a 1 on the hundreds place.

Solve each problem. Pass the magnifying glass over to see it you are correct.

1414

992

1577

754

1066

967

956

195 + 461 = 562 + 852 =

159 + 534 = 724 + 268 =

982 + 644 = 756 + 821 =

654 + 987 = 597 + 157 =

365 + 482 = 495 + 571 =

625 + 274 = 856 + 111 =

421 + 685 = 165 + 791 =

656

693

1626

1641

847

899

1106

Find the sum of 556 + 654.

A sporting goods store sold 159 baseballs and 354 golf balls last month. How many balls did the store sell last month in all?

80

When solving story problems, make sure you read the entire problem.

You may have to reread the problems to make sure you understand what it is asking you.

Be careful of "invisible" numbers. Words like week, dozen, double, etc. can stand for numbers.

Two turtles laid eggs on the beach. One laid 151 eggs and the other laid 206 eggs. How many eggs were there in all?

What is the number sentence?

A school placed an order for 500 pencils and 700 colored pencils. How many pencils were ordered in all?

Number sentence:

There are 365 days in a year. How many days are in two years?

Number sentence:

A farmer planted 120 tomato seeds and 56 pepper seeds. How many plants were planted in all?

Number sentence:

On the way to her birthday party, Mara dropped 1 box of a dozen cupcakes. She still had two more boxes. How many cupcakes does she still have?

Number Sentence:

Jason read 45 pages on Monday night and 26 pages on Tuesday night. He read a total of 71 pages.

81

A True

B False

The second grade collected 243 bottle caps and the third grade collected 534. Did they collect more than 800 caps?

82

A Yes

B No

Gavin had $300 in his bank account. His grandmother deposited $150. How much does he have now?

83

A $150

B $550

C $450

D $250

We have to drive 48 miles to reach Philadelphia, PA. How many miles will we have traveled there and back?

84

How many feet of wood would be needed to finish a 276 foot long bridge, if 155 feet has already been completed?

85

click to return to table of contents

Estimate Differences

click on the topic to go

to that section

2-Digit Subtraction

3-Digit Subtraction

Subtraction Across Zeros

Checking Subtraction with Addition

Click to Return to Subtraction Table of Contents

When you estimate differences, you round the numbers or use compatible numbers to find an estimate of what the difference may be before you subtract.

Round each number to the nearest ten.

Then add to find the estimated difference.

138 140

- 63- 60

53 50

- 19- 20

Find numbers that are close to the numbers you are working with. These numbers are called compatible numbers.

174 175

-28- 25

304 300

- 86- 85

When you need to estimate 256 while estimating differences, could you estimate it to 250?

A Yes

B No

A family trip involved driving 731 miles. They have already traveled 475 miles. About how many more miles did the family have to travel?

A 200 miles

B 1,000 miles

C 1,200 miles

D 2,000 miles

What is the estimated difference for 68-45?

Click to Return to Subtraction Table of Contents

2-digit Subtraction without regrouping

When you subtract two 2-digit numbers, it is important to line up the digits based on their place values.

For example, 45 - 14 = should be written as:

45

- 14

First, subtract the ones\' column. 5 is greater than 4, so regrouping is not needed.

Then, subtract the tens\' column.

2 digit Subtraction with Regrouping

When you subtract two 2-digit numbers, it is important to line up the digits based on their place values.

For example, 54 - 26 = should be written as:

54

- 26

First, subtract the ones\' column. 4 is less than 6, so regrouping is needed.

Then, subtract the tens\' column.

Circle the problems that need regrouped. Put a box around the

problems that do not need regrouping.

14 - 7 =

84 - 35

54 - 21 =

98 - 21 =

32 - 15 =

87 - 51 =

87 - 15 =

98 - 54 =

27 - 14 =

74 - 14 =

32 - 13

71 - 65

27 - 19 =

32 - 24

What is 77- 19?

What is the estimated and actual

difference of 46 - 23?

A 20 and 23

B 30 and 69

C 30 and 32

D 30 and 23

Samantha sold 89 shirts. She has delivered 24 of them. How many does she still have to deliver?

Click to Return to Subtraction Table of Contents

When finding the difference of two 3-digit numbers, you must make sure the digits are lined up based on their place value.

For example, 574-159 should be written as:

574

-159

Find the difference in the ones\' column. 4 is not larger than 9, therefor we need to regroup.

Find the difference in the tens\' column. 6 is larger than 5, therefor we do not need to regroup.

Find the difference in the hundreds\' column.

Yesterday, 915 people went to the zoo. Today, only 710 came. How many more people visited yesterday?

What is the difference of 150 and 25?

Click to Return to Subtraction Table of Contents

An example of "Subtracting Across Zeroes" is the problem, 500-159.

When this problem is written with its digits lined up based on their place values, it looks like:

500

-159

First, look at the numbers in the ones\' column. 0 is not larger than 9, so regrouping is needed.

However, we can not regroup the tens\' column because it, too, has a zero on top.

So we then must regroup the hundreds\' column.

500

-159

Regroup the hundreds\' column by crossing out the 5 and making it a 4. Then, make the tens\' column 0 a 10.

410

500

-159

Regroup the tens\' column by crossing out the 10 and making it a 9. Then make the ones\' column 0 a 10.

9

4 10 10

500

-159

What is the difference of 800 and 799?

Sarah needs 500 tickets to get a prize at the arcade. She has already received 161 tickets from playing her favorite games. How many more tickets does she need?

105

Sometimes, mistakes are made when we subtract. Estimating is one way to check the reasonableness of our work. But, we can also use addition to check our subtraction.

You already know that you can use fact families. For example, the subtraction problem 9-5=4 can be checked with the addition problem 4+5=9. The same can be done with larger numbers.

When checking a subtraction problem with addition, add the number that is subtracted and the answer. You should end up with the original number you started with.

45

-12 12

33 +33

45

451

-122 122

329 +329

451

How many ones are in the number 359?

A 354 + 194 = 160

B 194 - 354 = 160

C 160 + 354 = 194

D 194 + 160 = 354

When checking the subtraction problem, 365 - 65 = 300, what number could you use write first in your addition problem?

110

New Jersey Center for Teaching and Learning

Progressive Mathematics Initiative (PMI)

For additional NJCTL Math content, visit http://njctl.org/courses/math/.

Progressive Science Initiative (PSI)

For NJCTL Science content, visit http://njctl.org/courses/science/.

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