Addition and Subtraction. Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that you may.

Addition and Subtraction

Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that you may be better able to discover with accuracy the peculiar bent of the genius of each.

Plato

Arithmetic Today Arithmetic has generally been learned

through basic algorithms, but it has great potential through problem solving techniques.

Current Traditional Algorithm

Addition1

47+28 75

“7 + 8 = 15. Put down the 5 and carry the 1. 4 + 2 + 1 = 7”

Subtraction 7 13

83- 37 46

“I can’t do 3 – 7. So I borrow from the 8 and make it a 7. The 3 becomes 13. 13 – 7 = 6. 7 – 3 = 4.”

Expanded Column Method

Number Line Method

Add on Tens, Then Add Ones

46 + 38

46 + 30 = 76 76 + 8 = 76 + 4 + 4

76 + 4 = 8080 + 4 = 84

Partitioning Using Tens Method

Nice Numbers Method

Lattice Method

First arrange the numbers in a column-like fashion.

Next, create squares directly under each column of numbers.

Then split each box diagonally from the bottom-left corner to the top-right corner. This is called the lattice.

Now add down the columns and place the sum in the respective box, making the tens place in the upper box and the ones place in the lower box.

Lastly, add the diagonals, carrying when necessary.

Counting Down Using Tens Method

Partitioning Using Tens Method

Nice Numbers Method

The Counting-Up Method

The Counting-Up Method

Nines Complement

827 → 827

- 259 → 740 (nines complement)

+ 1 (to get the ten's complement)

1568

568 (Drop the leading digit)

Benefits of Alternative Algorithms

Place value concepts are enhanced They are built on student understanding Students make fewer errors

Suggestions for Using/Teaching

Traditional Algorithms We are not saying that the traditional algorithms

are bad. The problems occur when they are introduced

too early, before students have developed adequate number concepts and place value concepts to fully understand the algorithm.

Then they become isolated processes that stop students from thinking.

Integers

Integers can be easily approached by thinking in regards of basic addition/subtraction and determining its position on the number lineIs the final result positive or negative?

Integer Addition Rules

Rule #1 – If the signs are the same, pretend the signs aren’t there. Add the numbers and then put the sign of the addends in front of your answer.

9 + 5 = 14-9 + -5 = -14

Integer Addition Rules

Rule #2 – If the signs are different pretend the signs aren’t there. Subtract the smaller from the larger one and put the sign of the one with the larger absolute value in front of your answer.

-9 + 5 =

9 - 5 = 4

Larger abs. value

Answer = - 4

One Way to Add Integers Is With a Number Line

0 1 2 3 4 5 6-1-2-3-4-5-6

When the number is positive, count to the right.

When the number is negative, count to the left.

+-

One Way to Add Integers Is With a Number Line

0 1 2 3 4 5 6-1-2-3-4-5-6+

-

+3 + -5 = -2

One Way to Add Integers Is With a Number Line

0 1 2 3 4 5 6-1-2-3-4-5-6

-

+

-3 + +7 = +4

Adding Integers with Tiles

We can model integer addition with tiles. Represent -2 with the fewest number of

tiles

Represent +5 with the fewest number of tiles.

ADDING INTEGERS What number is represented by combining

the 2 groups of tiles?

Write the number sentence that is illustrated.

-2 + +5 = +3

+3

ADDING INTEGERS

Use your red and yellow tiles to find each sum.

-2 + -3 = ?

+ = -5

ADDING INTEGERS -6 + +2 = ?

+ = - 4

+ = +1

-3 + +4 = ?

SUBTRACTING INTEGERS

We often think of subtraction as a “take away” operation.

Which diagram could be used to compute

+3 - +5 = ?

+3 +3

SUBTRACTING INTEGERS

We can’t take away 5

yellow tiles from this

diagram. There is not

enough tiles to take

away!!

This diagram also represents

+3, and we can take away +5.

• When we take 5 yellow tiles

away, we have 2 red tiles left.

SUBTRACTING INTEGERS

Use your red and yellow tiles to model each subtraction problem.

--2 - 2 - --4 = ?4 = ?

SUBTRACTING INTEGERS

This representation

of -2 doesn’t have

enough tiles to

take away -4.

Now if you add 2 more reds

tiles and 2 more yellow tiles

(adding zero) you would

have a total of 4 red tiles and

the tiles still represent -2.

Now you can take away 4 red tiles.

-2 - -4 = +2

2 yellow tiles are left, so the answer is…

SUBTRACTING INTEGERS

Work this problem.

+3 - -5 = ?

SUBTRACTING INTEGERS

• Add enough red and yellow pairs so you can take away 5 red tiles.

• Take away 5 red tiles, you have 8 yellow tiles left.

+3 - -5 = +8

Although children learn addition of whole numbers with ease, addition of fractions — though conceptually the same as addition of whole numbers — is much harder.

It requires knowledge of fraction equivalencies.

To add two fractions, you have to know that they must be thought of in terms of like units.

We take this for granted when we add whole numbers: 3 + 5 is really 3 ones + 5 ones

— but not when we add fractions: 3 halves + 5 fourths is, for purposes of addition, 6 fourths + 5 fourths.

Why is adding fractions a difficult concept for students to grasp?

Let’s Eat Pizza

The pizza is currently 8 pieces

What if I wanted to eat one eighth of the pizza?

One fourth of the pizza?

One sixteenth of the pizza?

One twelfth of the pizza?

Addition of Fractions

The objects must be of the same type We combine bundles with bundles and sticks

with sticks.

Addition means combining objects in two or more sets

In fractions, we can only combine pieces of the same size

In other words, the denominators must be the same

Click to see animation

+ = ?

Example:

Addition of Fractions

8

3

8

1

Example:

Addition of Fractions

+ =

8

3

8

1

+ =

Example:

The answer is which can be simplified to

Addition of Fractions

8

3

8

1

8

4

8

)31(

2

1

Addition of Fractions with equal denominators

More examples

5

1

5

2

5

3

10

7

10

6

10

13

15

8

15

6

15

14

10

31

With different denominators

In this case, we need to first convert them into equivalent fraction with the same denominator.

Example:

15

5

53

51

3

1 15

6

35

32

5

2

5

2

3

1

An easy choice for a common denominator is 3×5 = 15

Therefore,

15

11

15

6

15

5

5

2

3

1

Addition of Fractions

• When the denominators are bigger, we need to find the least common denominator by factoring.

• If you do not know prime factorization yet, you have to multiply the two denominators together.

With different denominators

Addition of Fractions

More Exercises:

8

1

24

23

8

1

4

3

7

2

5

3

9

4

6

5

=

=

=

57

52

75

73

69

64

96

95

=

=

=

8

1

8

6=

8

7

8

16

35

10

35

21

35

31

35

1021

=

54

24

54

45

54

151

54

69

54

2445

=

Subtraction of Fractions Subtraction means taking objects away

Objects must be of the same type we can only take away apples from a

group of apples

In fractions, we can only take away pieces of the same size. In other words, the denominators must be the same.

Subtraction of Fractions

Example:

12

3

12

11

This means to take away

12

11

take away

equal denominators

12

3

12

311

12

3

12

11

3

2

12

8

More examples:

16

7

16

15

16

715

2

1

16

8

9

4

7

6

79

74

97

96

63

28

63

54

63

2854 26

63

23

11

10

7

1023

1011

2310

237

2310

1011237

230

110161

230

51

Did you get all the answers right?

Subtraction of Fractions

Adding/Subtracting

38 8

138 4

1= =-- 18

882=

c

ba

c

b

c

a Fraction Addition/Subtraction

Fraction Simplification

2811 2

7+ ++

=

==

Common Denominator = ??????

28

2811

7

244

2811

288

28+11 8 =

2819

c

ba

c

b

c

a Fraction Addition/Subtraction

???728

211

Adding/Subtracting

Fractions: Steps for Success

1. Know the fraction rules and how to apply them

2. Show your work and write out each step3. Check your work for errors or careless

mistakes