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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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Vocabulary

Click on a topic

to go to that section

Multi-digit Addition

Multi-digit Subtraction

Subtraction across zeros

Properties & Inverse Operations of Addition and Subtraction

click to reveal

click to reveal

click to reveal

click to reveal

click to reveal

click to reveal

click to reveal

Addends: Numbers you add

Sum: The total after you add

Plus sign: The sign + that tells you to add

Equal sign: The sign = that means is equal to

Subtract: To take away one quantity from another

Difference: The result after a subtraction

Minus sign: The sign - that tells you to subtract

Review of addition and subtraction facts

Addition is the "putting together" of two groups of objects

and finding how many in all.

Subtractiontells "how many are left" or "how many more or less."

We say addition and subtraction are inverse operations because one

operation can "undo" the other operation.

Adding 3 and 5 to get 8 is the opposite of 8minus 5 leaving 3.

3 + 5 = 8 and 8 - 5 = 3

addends sum difference

There are 25 students in one fourth grade classroom and 22 in another. How many total students are in both classes?

Solve the addition problem: 25 + 22 =

Step 1: Line up place values of the numbers

and write it vertically.

25

+ 22

Step 2: Add working right to left

(ones then tens)

Rewrite the problems vertically and then add. in another. How many total students are in both classes?

41 + 22 = 16 + 11 = 30 + 50 =

41 16 30

+ 22+ 11+ 50

click

click

click

There are 203 students in one school and 84 students in another. What is the total number of students in the two schools?

Solve the addition problem: 203 + 84 =

Step 1: Line up place values of the numbers

and write it vertically.

203

+ 84

Step 2: Add working right to left

(ones, then tens, then hundreds)

Be sure to line this 4 under the 3

(they are both in the ones column).

Rewrite the problems vertically and then add. another. What is the total number of students in the two schools?

426 + 71 = 4,105 + 62 = 54 + 704 =

426 4105 54

+ 71+ 62+ 704

click

click

click

Match the sum to the correct problem. another. What is the total number of students in the two schools?

501 + 317 =

541 + 301 =

5,401 + 37 =

541 + 37 =

51 + 307 =

5,041 + 317 =

1 another. What is the total number of students in the two schools?

When you add 64 + 25 you get 89

Yes

No

2 another. What is the total number of students in the two schools?

When adding 340 + 9 the nine should be lined up under the zero, because they are both in the ones position.

True

False

3 another. What is the total number of students in the two schools?

The sum of 30 + 33 is?

A

62

63

B

C

64

4 another. What is the total number of students in the two schools?

The sum of 122 + 21 is?

5 another. What is the total number of students in the two schools?

The sum of 230 + 35 is?

If Noah buys 15 science fiction books and 11 puzzle books, how many books did Noah purchase in all?

6

Miguel read 274 pages in his novel and Jeff read only 25 pages. How many pages did they read altogether?

7

8 pages. How many pages did they read altogether?

Violet ran 834 yards on Monday and 1,120 yards on Wednesday. How many total yards did she run?

9 pages. How many pages did they read altogether?

Larry made $46 raking leaves and $120 trimming trees. How much did he earn?

10 pages. How many pages did they read altogether?

A school collects labels to raise money. The total collected for two months is in the table.

What is the total collected for the two months?

Addition with Regrouping pages. How many pages did they read altogether?

Add 135 + 278

Hundreds

Ones

Tens

When you add a column that has more than 10 blocks, such as the ones column, you change the ones into a ten block. Let's look to see what happens.

135 + 278 pages. How many pages did they read altogether?

Hundreds

Ones

Tens

10 of the

13 ones

move left.

100 of the

tens move

left.

Now we will look at the new model.

Hundreds pages. How many pages did they read altogether?

Tens

Ones

135 + 278 = 400 + 10 + 3 = 413

Click

National Library of Virtual Manipulatives Web Site shows exchange of base ten blocks.

Click for link.

Steps for addition with regrouping 367 + 74 = exchange of base ten blocks.

367

+74

Step 1: Line up the numbers vertically.

step 2: Add the ones column. If the sum is more than 9 regroup to the tens column.

Step 3: Add the tens column. If the sum is more that 9 regroup to the hundreds column.

Step 4: Add the hundreds column.

Steps for addition with regrouping 367 + 74 = exchange of base ten blocks.

1

367

+ 74

1

Step 1: Line up the numbers vertically.

Step 2: Add the ones column. If the sum is more than 9 regroup to the tens column.

Step 3: Add the tens column. If the sum is more that 9 regroup to the hundreds column.

Step 4: Add the hundreds column.

Steps for addition with regrouping 367 + 74 = exchange of base ten blocks.

11

367

+ 84

51

Step 1: Line up the numbers vertically.

Step 2: Add the ones column. If the sum is more than 9 regroup to the tens column.

Step 3: Add the tens column. If the sum is more that 9 regroup to the hundreds column.

Step 4: Add the hundreds column.

Steps for addition with regrouping 367 + 74 = exchange of base ten blocks.

11

367

+ 84

451

Step 1: Line up the numbers vertically.

Step 2: Add the ones column. If the sum is more than 9 regroup to the tens column.

Step 3: Add the tens column. If the sum is more that 9 regroup to the hundreds column.

Step 4: Add the hundreds column.

Solve each problem. exchange of base ten blocks.

1 372 + 218 =

2 159 + 222 =

3 867 + 323 =

4 573 + 618 =

5 448 + 338 =

3, exchange of base ten blocks.405 + 839 =

13

73 exchange of base ten blocks. + 946 =

14

The answer to an addition problem is called? exchange of base ten blocks.

15

+

sum

A

difference

B

product

C

Two groups of boy scouts are camping. There are 143 campers at one site and 219 at a second site. How many boy scouts are camping altogether?

16

17 at one site and 219 at a second site. How many boy scouts are camping altogether?

Jordan has 258 coins in his collection. His uncle gives him 71 more coins. How many coins does he have now?

Tonia created a survey and asked her classmates which season is their favorite. How many students did she survey?

19

Do you find it easier to add numbers vertically or horizontal?

20

vertically (up and down)

A

horizontal (across)

B

either

C

Words to Remember horizontal?

Subtract:To take away one quantity

from another 5 - 2

Difference:The result after subtraction

5 - 2 = 3

Minus sign:The sign - that tells you to

subtract

Example 1 horizontal?Finding How Many Are Left

Hannah picks 14 flowers in her garden to put in vases. She puts 8 flowers in one vase. How many flowers are left for a second vase?

Step 1Draw how many you start with. Cross out the ones you subtract.

Start with 14 flowers subtract 8

Step 2 Count how many are left. 14 - 8 = 6

Answer There are 6 flowers left for the second vase.

Example 2 horizontal? Comparing with subtraction

Diana has 19 friends listed in her instant messaging icon. Sixteen friends are currently online. How many more friends are online than offline?

Try this horizontal?

Fill in the missing information to solve the problem.

Susan's family had eaten 11 of the 19 bananas she bought at the grocery store. How many are left?

remove

move

remove

remove

19 - 11 = 8 bananas

Fill in the missing operation signs. horizontal?

14 15 = 29 95 4 = 91

29 8 = 21 29 14 = 15

55 11 = 44 30 8 = 22

22 8 = 30 38 22 = 60

73 9 = 82 59 44 = 103

23 33 = 56 87 22 = 65

6 horizontal?2 - 41 =

21

4 horizontal?56 - 34 =

22

3 horizontal?,524 - 1,220 =

23

2 horizontal?4

Gary has $368 in his savings account. He purchased a video game for $43. How much money does he have left in his savings account.

There are 437 students in a school. One Monday 32 of them are absent. How many students are in school that Monday?

25

Subtraction with Regrouping are absent. How many students are in school that Monday?

Subtract 243 - 128

Ones

Hundreds

Tens

When you can not subtract from a column, you regroup from the column to the left.

Subtract 243 - 128 are absent. How many students are in school that Monday?

Ones

Hundreds

Tens

Take a ten and split it into ten ones.

Subtract 243 - 128 are absent. How many students are in school that Monday?

Ones

Hundreds

Tens

X

3 13

243

- 128

115

X

/ /

X

X

X

X

X

X

X

X

X

X

X

Now we can subtract 8 from the ones column.

National Library of Virtual Manipulatives Web Site shows splitting and moving of base ten blocks. Click for link.

Word Problems splitting and moving of base ten blocks. Click for link.

What number do you need to add to 13 to get 28?

Hints: Solve 28 - 13 = _____

Answer15

What number do you need to add to 24 to get 43?

Answer19

What number do you need to add to 45 to get 57?

Answer12

Remove

Remove

Remove

3 splitting and moving of base ten blocks. Click for link.72 - 199 =

27

1 splitting and moving of base ten blocks. Click for link.,257 - 182 =

28

Xavier has 348 songs on his MP3 player and his brother has 284 songs on his. How many more songs does Xavier have than his brother?

29

The park had 1,736 visitors on Saturday and 1,588 visitors on Sunday. How many more visitors were there on Saturday?

30

Subtraction Across Zeros on Sunday. How many more visitors were there on Saturday?

Click to Return to

Table of Contents

Subtraction Across Zeros on Sunday. How many more visitors were there on Saturday?

Example:

What is 600- 243

Math Concept:

While subtracting across zeros, you may need to regroup the

hundreds first.

Step-1

5 10

Are there enough ones in 600 to subtract 3 ones? No (3 > 0)

600

- 243

Since there are 0 tens, we can't borrow from tens.

So, borrow 1 hundred and regroup as 10 tens.

6 hundreds 0 tens = 5 hundreds 10 tens.

Step-2 on Sunday. How many more visitors were there on Saturday?

9

51010

Borrow 1 ten and regroup as 10 ones

600

- 243

10 tens 0 ones = 9 tens 10 ones

Step-3

9

51010

Subtract the ones (10 - 3 = 7)

600

- 243

Subtract the tens (9 - 4 = 5)

Subtract the hundreds (5 - 2 = 3)

Solve the problems on the next page using the web site. Click for web site.

300 800 400 Click for web site.

- 200- 425- 174

5 Click for web site.00 - 406 =

34

3 Click for web site.00 - 113 =

36

There are 500 students at an elementary school, 229 of those students are involved in sports. How many are not involved in sports?

37

W students are involved in sports. How many are not involved in sports?hat is 3,500 – 1,875

38

1,625

A

2,375

B

C

2,425

There are 201 members of the soccer club. 155 of them sold items for the fundraiser. How many soccer players did not fund raise?

39

One thousand kids were surveyed. 287 of them do not enjoy cooking. How many kids do enjoy cooking?

40

Another way to think of it.... cooking. How many kids do enjoy cooking?

600

- 461

Step 1: Underline the 0 in the tens place, since there is nothing to regroup with, keep going to the 6.

Step 2: Now you have 60 underlined, regroup 60 into 59 + 1, put the 1 with the 0 in the ones place.

Step 3: Subtract.

600 cooking. How many kids do enjoy cooking?

- 461

59

1

Step 1: Underline the 0 in the tens place, since there is nothing to regroup with, keep going to the 6.

Step 2: Now you have 60 underlined, regroup 60 into 59 + 1, put the 1 with the 0 in the ones place.

Step 3: Subtract.

600 cooking. How many kids do enjoy cooking?

- 461

139

59

1

Step 1: Underline the 0 in the tens place, since there is nothing to regroup with, keep going to the 6.

Step 2: Now you have 60 underlined, regroup 60 into 59 + 1, put the 1 with the 0 in the ones place.

Step 3: Subtract.

9 cooking. How many kids do enjoy cooking?,000 - 2,481 =

43

4 cooking. How many kids do enjoy cooking?4

There are 600 students in the school district. Each morning, 482 students come early to have breakfast. How many students do not come early?

4 cooking. How many kids do enjoy cooking?5

The distance across the United States is about 3,000 miles. If you drive 372 miles in one day, how many more miles do you have to drive?

Properties & cooking. How many kids do enjoy cooking?

Inverse Operations of

Addition and Subtraction

Click to Return to

Table of Contents

Number Properties cooking. How many kids do enjoy cooking?

Identity Property of Addition

When you add zero to a number,

the sum is that same number.

7 + 0 = 7

Number Properties cooking. How many kids do enjoy cooking?

Commutative Property

The order of the numbers does not

change the sum of an addition problem.

Addition

a + b = b + a

6 + 3 = 3 + 6

9 = 9

Number Properties cooking. How many kids do enjoy cooking?

Associative Property

The grouping of the numbers

does not change the sum of an

addition problem.

Addition

(a + b) + c = a + (b + c)

(2 + 3) + 4 = 2 + (3 + 4)

5 + 4 = 2 + 7

9 = 9

"Stick People" cooking. How many kids do enjoy cooking?

in

Flight

Addition

Property

Shown

Number

in

Flight

2 + 0 = 2

Identity (Zero)

0 + 3 = 3

3 + 2 = 5

Commutative

(Order)

2 + 3 = 5

(3 + 2) + 4 = 9

Associative

(Grouping)

3 + (2 + 4) = 9

Use the indicated property to cooking. How many kids do enjoy cooking?

complete the following.

1. Identity 7 + 0 =__7__

2. Commutative 6 + 2 = __2__+ 6

3. Associative (1 + 6) + 3 = __10__

1 + (6 + _3_) = __10__

4. Identity 0 + 11 = 11

5. Commutative_15_+ 2 = _2_+ 15

6. Associative_8_+ (5 + 4) = _17_

(8 + 5) + 4 = _17_

4 cooking. How many kids do enjoy cooking?6

The property of 7 + 4 = 4 + 7 is _________

identity (zero)

A

commutative (order)

B

associative (grouping)

C

4 cooking. How many kids do enjoy cooking?7

The property of (8+ 2) + 4 = 8 + (2 + 4) is ___________

identity

A

commutative

B

associative

C

The property of 9 + (5 + 3) = (9 + 5) + 3 is ________ cooking. How many kids do enjoy cooking?

48

identity

A

commutative

B

associative

C

49 cooking. How many kids do enjoy cooking?

The property to 7 + 4 = 4 + 7 is _______

identity

A

commutative

B

associative

C

Addition and subtraction are related operations. cooking. How many kids do enjoy cooking?

7 + 4 = 11 Addition sentences

4 + 7 = 11

11 - 7 = 4 Related subtraction

11 - 4 = 4 sentences

Use a related subtraction sentence to find a missing addend.

Missing addend

___ + 9 = 12

12 - 9 = 3

3 + 9 = 12

Related subtraction sentence

Check to be sure

Undoing an Addition Problem cooking. How many kids do enjoy cooking?

If 19 + 7 = 25, what is 25 - 7?

Solution

Ask yourself, what number added to 7 has a sum of 25?

X

18 + 7 = 25, so the opposite must be true: 25 - 7 = 18.

Fill in the blanks. cooking. How many kids do enjoy cooking?

Addition

50 cooking. How many kids do enjoy cooking?

Which is related to this addition sentence?

19 + 6 = 25

19 - 6 = 25

A

B

25 - 6 = 19

C

19 - 6 = 25

5 cooking. How many kids do enjoy cooking?1

Which is a related subtraction sentence?

120 + 30 = 150

120 - 30 = 150

A

B

150 - 20 = 120

150 - 30 = 120

C

Since addition and subtraction are "opposite" operation, cooking. How many kids do enjoy cooking?

they are call fact families. If you add any two numbers,

then subtract one of the addends from the sum, the difference

will be the other addend. These 3 numbers, along with the

operations of addition and subtraction, form a factfamily.

Example: Write the addition and subtraction fact family

for 7, 2, and 9

7 + 2 = 9 9 - 7 = 2

2 + 7 = 9 9 - 2 = 7

Directions: For numbers 1 through 4, fill in the blanks to complete the fact familty.

1. 4 + 6 = 10 2. ____________________

___________________ _____________________

___________________ _____________________

___________________ 8 - 3 = 5

3. ___________________ 4. _____________________

3 + 2 = 5 ______________________

____________________ 16 - 7 = 9

____________________ _____________________

5 complete the fact familty.2

Which is the missing fact for this family?

539 + 76 = 615

615 - 539 = 76

615 - 76 = 539

539 + 76 = 615

A

B

615 - 15 = 600

C

539 - 76 = 615

D

76 + 539 = 615

5 complete the fact familty.3

Which is the missing fact for this family?

88 + 88 = 176

A

123 + 88 = 211

211 - 123 = 88

211 - 88 = 123

B

123 - 211 = 88

C

88 + 123 = 211

D

123 + 0 = 123

5 complete the fact familty.4

Which is the missing fact for this family?

25 + 38 = 63

38 + 25 = 63

63 - 25 = 38

0 + 63 = 63

A

63 - 38 = 25

B

25 + 25 = 50

C

D

63 - 33 = 30

Philip solved a subtraction problem this way: complete the fact familty.

4,003

–2,564

1,439

Which number sentence shows the inverse of his subtraction problem?

55

1439 + 2564 = 4003

A

4003 - 1439 = 2564

B

1439 + 4003 = 2564

C

Jerry solved the following problem on his math test: complete the fact familty.

12,462

+ 5,980

18,442

Which equation shows the inverse of addition?

56

12,462 - 5980 = 6,482

A

18,442 - 5,980 = 12,462

B

18,442 + 5.980 = 24,422

C

5 complete the fact familty.7

Fill in the missing addend by relating the addition to subtraction.

___ + 146 = 156

5 subtraction. 9

Fill in the missing addend by relating the addition to subtraction.

1507 + ____ = 1653

6 subtraction. 0

Fill in the missing addend by relating the addition to subtraction.

843 + ____ = 8,000

6 subtraction. 1

Fill in the missing number by relating the subtraction to addition.

7,032 - ____ = 4,392

6 subtraction. 2

Matt has saved $12 to buy a video game. The video game costs $34. How much more will Matt have to save to buy the video game? Use the following open number sentence to find the answer.

34 = m + 12

$16

A

$18

B

$22

C

Linda completed 102 math problems last week. Linda and Sal completed a total of 238 math problems altogether. How many math problems did Sal complete last week? Which equation will help solve this problem?

(Let n = the number of math problems completed

by Sal last week.)

63

102 + n = 238

A

102 - n = 238

B

102 × n = 238

C

6 completed a total of 238 math problems altogether. How many math problems did Sal complete last week? Which equation will help solve this problem?4

Lorraine received $30 for her birthday. She bought earrings for $12.63 pocketbook and a T-shirt for $12.80. How much of her birthday money does Lorraine have left?

30 = 12.63 + 12.80 + m

A

$17.37

B

$14.57

C

$ 4.57

Abe has saved $13 to buy a new soccer ball. The soccer ball costs $37. How much more will Abe have to save to buy the ball? Use the following open number sentence to find the answer.

37 = m + 13

65

$22

A

$23

B

$24

C

T costs $37. How much more will Abe have to save to buy the ball? Use the following open number sentence to find the answer. he hot dog stand at the carnival had 2,000 hot dogs. They sold 1,259 of them. How many hot dogs do they have left?

1,259 + h = 2,000

66

$2.23 costs $37. How much more will Abe have to save to buy the ball? Use the following open number sentence to find the answer.

$4.67

$5.03

You purchase a toy train and a drum. You hand the cashier a ten dollar bill. What will be your change be?

67

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

Sometimes, mistakes are made when we subtract. Estimating number that is subtracted and the answer. You should end up with the original number you started with.

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 a student adds 543 + 328 the answer gotten is 861. Is the student correct?

Check the addition with subtraction.

861

- 543

318 This should be 328, so the answer is wrong.

11

5

When a student subtracts 3,471 - 1,850 the answer gotten is 1,621. Is the student correct?

Check the subtraction with addition.

1621

+ 1850

3471 This is what was subtracted from, so the answer is correct

6 1,621. Is the student correct?8

Use inverse operations to figure out if the following problem is correct?

453 +3,271 = 3,624

Yes

It is correct

check

No

It is wrong

3624

-3271

______

Use inverse operations to figure out if the following problem is correct?

69

4,300 -1,283 = 3183

It is correct

Yes

check

3183

+1283

No

It is wrong

______

7 problem is correct?0

Use inverse operations to figure out if the following problem is correct?

23,521 +6,356 = 29,877

It is correct

Yes

check

No

It is wrong

29877

-6356

________

7 problem is correct?1

You have $729 in your bank account. You rake leaves for several of your neighbors and then deposit $134 into your account. The bank teller says you now have a balance of $853. Is this correct?

Yes

No

You combine trading cards with your friend. You have 341 cards and your friend has 492 cards. Together you will have 833 cards. Is this correct?

72

Yes

No

Soccer camp is 524 miles away. If you drive 385 miles the first day, you will need to drive 139 miles on the second day. Is this correct?

73

Yes

No

7 first day, you will need to drive 139 miles on the second day. Is this correct?4

The football team ran for 253 yards during the game one Saturday. The next Saturday they ran 198 yards. They ran for a total of 341 yards during these two games. Is this correct?

Yes

No

For the next set of responder questions, first day, you will need to drive 139 miles on the second day. Is this correct?

check your answer before responding.

7 first day, you will need to drive 139 miles on the second day. Is this correct?5

Many of the students at Lake Elementary buy school lunch. 367 of the students ordered chicken nuggets on Monday, while 189 of the students ordered peanut butter and jelly. How many students ordered lunch on Monday?

178

A

556

B

546

C

7 first day, you will need to drive 139 miles on the second day. Is this correct?6

The Jones Middle School was having a school dance on Friday night. Only 350 students were allowed to sign up for the dance. 147 signed up to go. How many more students were able to sign up before Friday night?

7 first day, you will need to drive 139 miles on the second day. Is this correct?7

Massachsetts occupies 10,655 square miles of land and water. If 2,673 square miles is water, how many square miles of Massachsetts is land?

Rudy uses 156 ml of salad dressing for his salad. Mike uses 208 ml on his salad. How much dressing do they use altogether?

78

Ivan is putting books in his bookcase. He has put 134 on the shelves. He has 408 books in all. How many more books does he need to put on the shelves?

79

You have paid the cashier at the store with four $20 bills. The item you want to buy is $67. How much change will you get back?

80

8 The item you want to buy is $67. How much change will you get back?1

Kenny went to the Boardwalk for a week in July. His mom bought 400 tickets for him for the amusement rides. He used 315 of them. How many tickets does Kenny have left?

8 The item you want to buy is $67. How much change will you get back?2

The football stadium holds 80,000 people. 65,876 people bought tickets to go to the game. How many more tickets can be sold to fill the stadium?

Benny had a collection of marbles. He has exactly 450 marbles. For Christmas, his uncle Fred gave him 37 more marbles. How many marbles does Benny have in his collection now?

83

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