mental computation

Common Purpose and Goals for Teaching Mathematics at SMOT:

We aim to ensure that all students are given an equal opportunity to achieve and to be challenged to reach their full potential.

We aim to teach the fundamental mathematical skills needed to function in everyday life. Learning mathematical skills will provide our students with the opportunity to become proficient in their understanding of a variety of mathematical concepts, to help them problem solve and apply strategies to life situations.

Mental ComputationAdapted from Bern Long and Angela

Rogers presentation, 2013

K. Chiodi

Discuss with the person next to you what you think mental computation involves?

When should mental computation be taught?

What is mental computation?

Mental computation is a calculation performed entirely in the head, with only the answer being written (McIntosh, 2005)

Reading: Mental Computation and Estimation Read and then discuss at your tableVictoria Department of Education and Early Childhood Development, 2009

Resource:Mental Computation: A Strategies ApproachAlistair McIntosh, 2004

What is mental computation?

How are you teaching mental computation in your classroom?

Mental computation is based on understanding.

Mental arithmetic is based on speed and accuracy related to memory.

Research by Biggs (1967) revealed that:“Allocation of time to mental arithmetic bore no relation to attainment”

“In other words, these daily speed and accuracy tests did not make the children noticeably more competent, but it did make them slightly more neurotic about numbers” (McIntosh, 2004, 1)

Mental Computation Vs. Mental Arithmetic

Warm Up activity using the Westwood Addition and Subtraction test.

Mental Computation Vs. Mental Arithmetic

Introduce and make explicit the strategies we use to help us complete mental computations.

Memorisation of some basic facts required. It is certainly desirable for children to know the

addition facts to 20. Mental computation strategies must be efficient

and always allow us to arrive at the correct answer. Strategies are necessary because they allow

students not only to calculate simple 1-digit facts but also to calculate much bigger equations

e.g. 6 + 4 …….. 66 + 24

How do we teach mental computation?

Level Mental Computation Link

F Subitise small collections of objects (ACMNA003)1 Represent and solve simple addition and subtraction problems

using a range of strategies including counting on, partitioning and rearranging parts (ACMNA015)

2 Explore the connection between addition and subtraction (ACMNA029)Solve simple addition and subtraction problems using a range of efficient mental and written strategies (ACMNA030)

3 Recognise and explain the connection between addition and subtraction (ACMNA054)

Recall addition facts for single-digit numbers and related subtractionfacts to develop increasingly efficient mental strategies for computation (ACMNA055)

Mental Computation in the National Curriculum

Level Mental Computation Link

4 Recall multiplication facts up to 10 × 10 and related division facts (ACMNA075)

Develop efficient mental and written strategies and use appropriatedigital technologies for multiplication and for division where there is no remainder (ACMNA076)

5 Use efficient mental and written strategies and apply appropriatedigital technologies to solve problems (ACMNA291)

Use equivalent number sentences involving multiplication and divisionto find unknown quantities (ACMNA121)

6 Select and apply efficient mental and written strategies andappropriate digital technologies to solve problems involving all fouroperations with whole numbers (ACMNA123)

The Addition and Subtraction strategies to be taught F – 6

Count on in onesTens FactsDoubles (for addition)Doubles (for subtraction)Near Doubles (for addition)Near Doubles (for subtraction)Bridging to 10Adding 10Communitivity (counting on from the larger number)

Count backCount down toCount up fromTens Facts (for subtraction)Subtract 10Bridging 10 (for subtraction)Inverse – Think ‘+’

You can use your fingers to count on 0, 1, 2, 3

3+0=3

3+1=4

3+2=5

3+3=6 (also a double)

2. Count on 0, 1, 2, 3

Add 1,2,3 to a multi digit no. e.g. 3+35=

Add 10, 20, 30 to a multiple of 10 up to 90. e.g. 80+30=

Add 10, 20, 30 to a multi digit number e.g. 34+30=

Count on 1, 2, 3Using the single digit strategies year 3-6 move to:

Add 100, 200, 300 to a hundreds number

E.g. 500+200= Add 100, 200, 300 to a

multi digit number e.g. 34+300

Add 1000, 2000, 3000 to a single digit number

E.g. 3 000+9= Add 1000, 2000, 3000 to

a multi digit number e.g. 41+2 000=

If you spin around addition equations you get the same answer.

This shows children the commutativity of addition equations

3. Spin Arounds (commutativity)

2 + 4 =6

4 + 2 = 6

Single digit with multi-digit E.g. 4+64=64+4

Multi-digit with multi-digit E.g. 97+123=123+97

3. Spin AroundsYear 3-6 move to:

Start with real world We need to learn these doubles.

1+1=2 6+6=12 2+2=4 7+7=14 3+3=6 8+8=16 4+4=8 9+9=18 5+5=10 10+10=20

4. Doubles

Double multiples of 10 up to 90

e.g. 50+50Think…5 tens+5 tens= 10 tens=100 Double multiples of 100

up to 900e.g. 600+600=Think…6 hundred+6 hundred=12 hundred= 1 200

Doubles Using single digit strategiesyear 3-6 move to:

Double multiples of 1 000 up to 9 000

e.g. 6 000+6 000Think…6 thousand+6 thousand=12 thousand= 12 00

Use doubles strategy with multi-digit and single digit numbers

e.g. 64+4 Use doubles strategy when adding multi-

digit with multi-digit64+24356+36

Doubles Using single digit strategiesyear 3-6 move to:

If we remember the doubles, we can work out these sums.

6+7=13 5+6=11 4+3=7 9+8=17

5. Near Doubles

Near doubles with multiples of 10 up to 90 e.g. 50+60

Think…5 tens +5tens is tens and 1 more ten is 11 tens=110 Near doubles with multiples

of 100 to 900 e.g. 400+500 Near doubles with multiples

of 1000 up to 9000e.g. 7000+6000

Near Doubles Using single digit strategiesyear 3-6 move to:

Use near doubles strategy with multi-digit and single digit numbers e.g. 64+5

Use near doubles strategy when adding multi-digit with multi-digit

64+25356+37

These equations add to 10.

1+9=10 9+1=10 2+8=10 8+2=10 3+7=10 7+3=10 4+6=10 6+4=10 5+5=10 5+5=10

6. Tens Facts

Tens facts with single digits that add to 20

E.g. 6+14 Tens facts with

multiples of 10 that add to 100

E.g. 60+40 Tens facts with

multiples of 100 that add to 1000

E.g. 200+800

Tens factsUsing single digit strategiesyear 3-6 move to:

Tens facts with multiples of 1000 that add to 10 000

E.g. 4000+6000 Tens facts with single

digit that add with multi digit numbers

E.g. 6+34 Tens facts with multi

digit with multi digitE.g. 64+36 129+211

Seven, Eight, Nine are close to Ten.

9+2=11 8+6=14 7+6=13 9+3=12 8+7=15 7+5=12 9+4=13 8+8=16 7+4=11

7. Bridging Ten

Bridging with multiples of 10 e.g. 40+90 think…4 tens +9 tens is 13 tens = 130400+9004000+9000 Bridging single digit numbers with multi digit

numbers E.g. 43+9 Bridging multi digit numbers with multi-digit

numberse.g. 43+59256+349

Bridging 10Using single digit strategies year 3-6 move to:

When we add ten, the ones number stays the same.

2+10=12 6+10=16 3+10=13 7+10=17 4+10=14 8+10=18 5+10=15 9+10=19

8. Adding Ten

Add 10 to multi-digit numberse.g. 10+25 10+257 Add 100 to single digit e.g. 100+5 Add 100 to multi-digit e.g. 100+27 Add 1000 to single digite.g. 1 000+6= Add 1000 to multi-digite.g. 56+1 000

Adding 10Using single digit strategies year 3-6 move to:

To teach with understanding all strategies must be taught using visual aids and at any time when students are experiencing difficulties teachers must return to visual aids e.g. tens frames, bead strings, place value cards etc.

Important to note!

2 dice are rolled

Write the number sentence on your playing board that corresponds with the strategy you used to work out the answer

The player to fill up their playing board first calls ‘Bingo’ and reads out their answers

Strategy Game Bingo!

Subtraction: Basic Facts

You can count back 0, 1, 2, 3 using your fingers or in your head

5-2=3 6-1=5

1. Count Back 0, 1, 2, 3

Count down to: Beginning at the total, count down to the number

being taken away. The answer is the number of steps this takes.

For example, 18-13 start at 18 and count down to 13, 17, 16, 15, 14, 13= 5 numbers counted back

Count up from: Beginning at the number being taken away, count

up to the total. The answer is the number of steps this takes.

For example, 21-17 start at 17 and count up to 21. 18, 19, 20, 21 = 4 numbers counted up

Tens facts can help us work out the answer when we subtract from ten.

E.g.. 10=5=5 10-6=4

Tens Facts

If you know 6+6=12, then you also know 12-6=6

E.g.: 4+4=8 so 8-4=4

10+10=20 so 20-10=10

3. Doubles

You can take away the ones number and it leaves you with just the tens.

E.g.: 13-3=10

25-5=20

4. Subtract 10

Count back to the nearest tens number, then its easy to take away what is left.

E.g.. 11-3= First do 11-1=10 Then 10-2=8 (tens fact)

5. Bridging Ten

If you know the doubles the near doubles can also help.

E.g.. 12-7= I know 12-6 (double)=6 Take one more = 5

6. Near Doubles

While you play… Reflect on these Mental Computation Questions (Brian Tickle)

Explain How did you figure it out?Justify How did you do it like that?Compare Is there another way?

Which one do you like?I really like this strategy. What other problems will this strategy work for? Will it always work?

Reasoning and reflecting

Is your answer reasonable/ Could that be the answer?How do you know your right?

Application What would you use this for in the real world?