Making SubtractionConcepts Meaningful

Rosemary Reuille IronsSenior LecturerQueensland University of Technologyr.irons@qut.edu.au or mathmates@ozemail.com.au

What is a concept?

A concept is the picture in your mind of an idea.

Images built through language experiences help develop concepts?

What steps do we follow to develop operation concepts?

• Child’s Language• Materials Language• Mathematical Language• Symbols

CONCRETE/VISUAL

VERBAL

SYMBOLIC

Student Language oral written

Representations

Eight mice are playing by the cheese. Two mice run away. How many mice are playing now?

CONCRETE/VISUAL

VERBAL

SYMBOLIC

Student Language oral written

Materials Language oral and written

Representations

Concrete/pictorial materials – take away

8

take out 2

8 cover up 2

CONCRETE/VISUAL

VERBAL

SYMBOLIC

Student Language oral written

Materials Language oral and written

Mathematical Language oral and written

Representations

The new mathematical words that are used with the concept

8 5spend 3 leaves 8 5take 3is

8 5subtract 3equals

CONCRETE/VISUAL

VERBAL

SYMBOLIC

Student Language oral written

Materials Language oral and written

Mathematical Language oral and written

Symbolic Language written

Representations

The mathematical abbreviations and formulae.

8 5 3=

Teaching the Subtraction Concept

Subtraction Concept

Finding the missing part.

The missing part could be what is left after a take away.

The missing part could be how many to add on.

The missing part could be the difference in number.

Rosie needs 12 apples. She has picked 7 apples. How many more apples does she need?

Rosie had 12 apples in a bag. She took out 7 apples. How many apples are in the bag now?

Rosie has 12 red apples and 7 green apples. How many fewer green apples does she have?

Take AwayChild’s language

Materials language

Mathematical language

Symbolic language

Missing addendChild’s language

Materials language

Mathematical language

Symbolic language

DifferenceChild’s language

Materials language

Mathematical language

Symbolic language

Child's language

Everyday language – take away

Eight mice are playing? Two mice run away? How many mice are playing now?

Child's language

Everyday language – missing addendThere are 8 mice altogether. How many mice are hiding in the cheese?

Four cars in the carpark. How many more will drive in to make ten cars in the carpark?

Child's language

Everyday language – differenceEight mice are playing in front of the cheese. Two mice are playing in the back. How many more mice are playing in front?

Materials language

Concrete/pictorial materials – take away

spend 2

Materials language

Concrete/pictorial materials – take away

8

take out 2

8 cover up 2

Materials language

Concrete/pictorial materials – missing addend

There are 8 altogether. How many are covered?

Materials language

Concrete/pictorial materials – difference

8 cover up 2How much more is 8 than 2?

Make the number of objects to represent the two groups.

Cover the parts of the groups that are the same to show the difference.

Mathematical languageThe new mathematical words that are used with the concept

subtract

[Try to avoid using the word minus. In mathematics this is best associated with negative numbers.]

Symbol language

The mathematical abbreviations and formulae.

8 5 3=

Rosie needs 12 apples. She has picked 7 apples. How many more apples does she need?

Rosie had 12 apples in a bag. She took out 7 apples. How many apples are in the bag now?

Rosie has 12 red apples and 7 green apples. How many fewer green apples does she have?

What are the features of the stories that make them all subtraction?

For each subtraction situation, the total and number in one part of the total are known. The unknown value is the other part of the total.

For addition, 2 or more parts are known. The unknown value is the total.

Stories provide the opportunity to relate the operations.Make sure that both are introduced when the addition concept is developed.

Relate subtraction to addition

How can you work out the number of covered dots?

6

13 altogether

Build links to addition during the work with missing addend subtraction.

5 +

8=

8 5 =

Teaching the number fact

strategies

The approach to number factsNumber facts are best

learned in clusters. Each cluster is organised

around one strategy – a strategy that can be used to learn facts and then with numbers beyond the facts.

The stages for each cluster

• introduce the strategy

• reinforce the strategy

• practice the facts

• extend to examples beyond the fact range.

Cluster 1: Count on

Count on 1

Count on 2 and for some students,

Count on 3

6

Cluster 2: Use Doubles

Double

Double-add-1

Double-add-2

Cluster 3: Make Ten

Number facts in this cluster have one addend close to 10.

9 + 4 = ____

is the same as

10 + 3 = ____

Teaching the subtraction number

facts

Use the sequence for addition facts to plan the sequence for subtraction facts

Count on factsUse doubles factsMake to 10 or bridge to 10 facts

For each subtraction cluster, encourage students to use the strategy ‘think addition.’

The connection between addition and subtraction is essential.

Begin the links to subtraction when the addition concept is taught.

The stages for each cluster

• introduce the strategy• reinforce the strategy• practice the facts• extend to examples beyond the fact range.

8

Introduce the strategy There were 8 cubes in the cup.I have taken out 2 cubes. How many cubes are still in the cup? What are all of the ways you know?

How can you work out the number of covered dots?

6

8 altogether

Count on/Count back subtraction facts

The addition facts6 + 2 = ___ 2 + 6 = ___

are in the count-on cluster.

The related subtraction facts are8 – 2 = ___ 8 – 6 = ___.

Initially, students might work out 8 – 2 =__ using a count back strategy.

Ask questions such as: How would you work out the answer?

11 - 9 = ___

How could you work out the number that is covered?

2 + = 10

Reinforce the strategy

Use addition to plan the sequenceCount on facts

6 + 2 = __ 8 – 2 = __8 – 6 = __

Doubles facts6 + 7 = __ 13 – 6 = __

13 – 7 = __

Use doubles subtraction facts How can you work out the number of

covered dots?

6

13 altogether

Use addition to plan the sequenceCount on facts

9 + 2 = __ 11 – 2 = __11 – 9 = __

Doubles facts6 + 8 = __ 14 – 6 = __

14 – 8 = __Make to ten facts

6 + 9 = __ 15 – 9 = __15 – 6 = __

Make to 10 subtraction facts How can you work out the number of

covered dots?

6

15 altogether

Consideration of interests does not mean indulging children or abdicating responsibility. It means that children are more likely to find curriculum meaningful and engaging when it relates to and respects their interests.

NAEYC- Developmentally

Appropriate Practice 1997

Learning never ends and as teachers we should approach each day –

the same way as a child does

everything is a new discovery.

Discover something new each day about each child in your learning environment.