Subtraction and computation

Dr. Calvin J Irons

When/where do we need to be able to subtract?

What types of situations lend themselves to subtraction?

I have $125. If I buy the $89 ticket, how much money will I have left?

What pictures do students have of numbers/operations?

What pictures do students need to calculate mentally with subtraction?

The mathematical structure of numbers include:

Counting – a discrete representation:ungroupedgroupedplace value

Measurement – a continuous representation:lineararea

Counting is not an efficient strategy.

The students do (and we stress) too much counting.

Counting

$54$38

How do we calculate if our picture of numbers is one that relates more to counting than some of the other representations?

Find the difference

Place value is an abstract system that enables us to read and write numbers.

Place value

$54

How do you calculate if the main picture of number is place value?

$38Find the difference

Remember, place value is just one of the structures. In some situations other representations may be more useful – particularly for mental strategies involving addition and subtraction (and of course for fractions and even decimals).

Place value

The only model that can be used with ALL numbers.

The best model for many strategies Not enough time (probably no time) is spent to develop this model.

And then, the true linear aspects of the model are not stressed.

Linear

0 10 20 30 40 50 60 70 80 90 100 110 120

How does a number line show a number?

What is 38 on this number line?

When asked, most people say a number such as three is at a particular ‘point.’

A number is a length (starting at 0).

When students are asked to work with a number such as thirty-eight, what ‘picture’ do they use?

How would you work out the difference between the prices of these two items?

$38

$53

Some important and less structured features of numbers include being able to :

Partition a number in multiple ways:

Double/halve with confidence:

Are their other pictures that we use to

$190$160

Are their other pictures that we use which are not based on a strict mathematical structure?

Find the difference? How could you think?

How could we work out the difference in price of these two items?

$3.75 $1.95$1.75 $1.95

Addition StrategiesBegin by extending a fact strategy

Strategies

Count-on6 + 19 + 2

Use doubles7 + 76 + 5

Bridge-ten9 + 4

First extension

Count-on16 + 119 + 2

Use doubles25 + 2526 + 25

Bridge-ten39 + 4

Further extensions

Count-on26 + 2129 + 12

Use doubles27 + 27

126 + 125

Bridge-ten198 + 25

decimal extensions

Count-on3.6 + 2.12.9 + 1.2

Use doubles2.5 + 2.5

1.26 + 1.25

Bridge-ten1.98 + 0.6

Subtraction StrategiesBegin by extending a fact strategy

Strategies

Take small6 – 19 – 2

Use addition6 – 59 – 7

12 – 615 – 714 – 9

First extension

Take small16 – 1

59 – 21

Use addition26 – 2519 – 17

120 – 6030 – 1523 – 9

Further extensions

Use addition

no bridging

67 – 53bridging

85 – 59 126 – 98

What influences the teaching sequence for subtraction?

The number combinations involved?

The subtraction situations?

Joel has $75 for the day at Maze World. How much does he have left after paying the $24 admission charge?

The type of problem and choice of numbers influence how you think. Change either, and you may want to use a different strategy.

$29

take-away subtraction

What is the difference between the prices of these computer games?

$75 $29

difference subtraction

How much more do you need?

$35 missing addend subtraction

How much more do you need?

$8.35 missing addend subtraction

Cube A: 8, 8, 8, 9, 9, 9Cube B:11,12,13,14,15,16

2 2 3 3

4 5 5 5

6 7 7 7

3

6

8

4

6

8

Bridging: Subtraction factsUse one board between 2 players. Roll both cubes. Write the numbers with the answer. Get three in a line.

Cube A: 8, 18, 28, 9, 19, 29Cube B:31,32,33,34,35,36

Bridging: beyond subtraction factsUse one board between 2 players. Roll both cubes. Write the numbers with the answer. Get three in a line.

2 3 4 5

12 13 14 15

22 23 24 25

6

16

26

7

17

27

8

18

28

0.2 0.3 0.4 0.5 0.6 0.7 0.8

1.2 1.3 1.4 1.5 1.6 1.7 1.82.2 2.3 2.4 2.5 2.6 2.7 2.80.2 0.3 0.4 0.5 0.6 0.7 0.81.2 1.3 1.4 1.5 1.6 1.7 1.82.2 2.3 2.4 2.5 2.6 2.7 2.8

Cube A: 1.8, 2.8, 3.8, 1.9, 2.9, 3.9Cube B: 4.1, 4.2, 4.3, 4.4, 4.5, 4.6

Bridging: subtracting decimals

Mental strategies for subtraction

1. Use addition

$190$160

Mental strategies for subtraction

2. Use place value (in some way)

$190$160

Mental strategies for subtraction

3. Use partitioning (other than place value) – and possibly use another strategy

$190$160

Where is subtraction used outside the realm of whole numbers and decimals?

0 21

0 21