Welcome to the KS2 Maths workshop

21st February 2017

Aims of the Session

• To build understanding of mathematics and it’s development

throughout KS2 – the four main operations.

• To have a stronger awareness of when and how to progress

from non-formal to formal methods at the appropriate stage

(moving from using concrete to pictoral and finally abstract

representations to support learning)

• To appreciate how fluency, reasoning and problem solving

skills are interlinked in establishing deep understanding of

mathematics in the KS2 curriculum.

Concrete resources

Concrete – students should have the opportunity to use concrete objects and manipulatives to help them understand what they are doing.

Pictorial

Pictorial – students should then build on this concrete approach by using pictorial representations. These representations can then be used to reason and solve problems.

Challenge 1 • 12 + 15 =

• 12 + 11 = • 15 + 14 = • 17 + 12 = • 13 +13 = • 15 + 14 =

Challenge 2 • 16 + 15 =

• 17 + 18 = • 16 + 19 = • 19 + 13 = • 17 + 15 = • 15 + 17 =

Challenge 3 • 31 + 25 =

• 82 + 23 = • 66 + 25 = • 59 + 22 = • 177 + 146 = • 165 + 132 =

Abstract With the foundations firmly laid, students should be able to move to an abstract approach using numbers and key concepts with confidence.

Fluency

• Find the answers:

• 4 x 12 = 5 x 9 = • 7 x 8 = 8 x 11 = • • Fill in the gaps:

• 4 x __ = 12 8 x __ = 64 • 32 = 4 x __ 6 = 24 ÷ __ • • Leila has 6 bags with 5 apples in each. How many apples does

she have altogether? •

Reasoning • □ X □ = 48

• Which pair of numbers could go in the boxes? Could any other numbers go in the boxes ?

• Complete these calculations:

• 7 x 8= 7 x 4 x 2= • 5 x 6 = 5 x 3 x 2= • 12 x 4 = 12 x 2 x 2= • Which calculations have the same answer? Can you explain why?

• • True or False

• 6 x 8 = 6 x 4 x 2 • 6 x 8 = 6 x 4 + 4 • Explain your reasoning.

• Can you write the number 24 as a product of three numbers?

Problem Solving

• Find three possible values for ⃝ and ∆

• ⃝x ∆= 24

• • I am thinking of 2 secret numbers where the sum of the numbers is 16 and the product is 48. What are my secret numbers? Can you make up 2 secret numbers and tell somebody what the sum and product are?

• How many multiplication and division sentences can you write that have the number 72 in them?

•

Whistle stop tour of calculation policy – non formal to formal !

•Addition

•Subtraction

•Multiplication

•Division

Whistle stop tour of calculation policy – non formal to formal !

• A deep understanding of place value and times tables facts are vital for true fluency and understanding of number operation.

Models of Addition

• There are 12 girls and 3 boys. How many children altogether?

• The chocolate bar was 12p last week, but today the price went up by 3p.

What is the price now?

Make up a word problem on a slip of paper that represents

the calculation

12 + 3 = 15

Addition

Do children definitely know what addition is? It could represent an “two items

being totalled” (aggregation) or “add on more to the first” (augmentation)

e.g. 6 + 9

What is the “ideal approach”?

A visual “bar model”.

6 9 ?

Starting Point Before launching in to the expectations of KS2, the following are the new

National Curriculum expectations for year 2:

Solve problems with addition and subtraction:

- Using concrete objects, pictorial representations, including number and measure.

- Apply their increasing knowledge of mental and written methods.

Recall addition & subtraction facts up to 20 fluently, and derive facts up to 100.

Add and subtract numbers using concrete, pictorial & mentally, including:

- TU + U

- TU + T

- TU + TU

- U + U + U

Understand that addition is commutative, but subtraction is not.

Use inverse relationships between addition and subtraction to check calculations & solve missing

number problems

Trading Game – Addition

H T U

30

4

Excellent activity for the end of KS1 which develops conceptually the

“regrouping” required for KS2.

Building the Journey Year 3 Addition and Subtraction up to 3 digits using formal methods

Year 4 Addition and Subtraction up to 4 digits using formal methods

(Solve simple measure and money problems involving decimals to 2 dp)

Year 5 Addition and Subtraction more than 4 digits using formal methods

(Solve problems involving number up to 3 dp)

(They mentally add and subtract tenths, and one-digit whole numbers

and tenths)

(They practise adding and subtracting decimals, including a mix of whole

numbers and decimals, decimals with different numbers of decimal

places, and complements of 1 (for example, 0.83 + 0.17 = 1)).

All years groups also refer to "estimation", "inverse operations" to check,

"problem solving"

Three-Digit Column Addition – Stage 1

Students will still, require concrete resources. This is likely to be Dienes (or

equivalent) to model two digit addition. No re-grouping to take place.

Students record concrete and abstract calculations together.

T U

2 3

4 1 +

T U

Three-Digit Column Addition – Stage 2

Students will still, require concrete resources. This is likely to be Dienes (or

equivalent) to model two digit addition. No re-grouping to take place.

Students record concrete and abstract calculations together.

T U

2 5

4 7 +

T U

Three-Digit Column Addition – Stage 1 remodelled

Students still require concrete resources. Some students will want to

move away from Dienes, and handle resources less cumbersome

(as they now have a feel for “size”) – e.g. place value counters

T U

2 3

4 1 +

T U

10 1 1 1

1

10

10 10 10 10

Three-Digit Column Addition – Stage 2 remodelled

Students still require concrete resources. Students are now ready to

tackle problems requiring “re-grouping”. There are different way

students could effectively communicate their thoughts. In time

students won’t need counters.

T U

2 5 4 7 +

T U

10 1 1 1 10

10 10 10 10

1 1

1 1 1 1 1 1 1

T U

2 5 4 7 +

Three-Digit Column Addition – Stage 3 Students still require concrete resources. Students are now ready to

tackle problems requiring “re-grouping”. There are different way

students could effectively communicate their thoughts. In time

students won’t need counters.

T U

2 5 4 7 +

T U

10 1 1 1 10

10 10 10 10

1 1

1 1 1 1 1 1 1

T U

2 5 4 7 +

10

2

2

1

1

6 0

Three-Digit Column Addition – Stage 3 Students still require concrete resources. Students are now ready to

tackle problems requiring “re-grouping”. There are different way

students could effectively communicate their thoughts. In time

students won’t need counters.

T U

2 5 4 7 +

T U

10 1 10

10 10 10 10 1

T U

2 5 4 7 +

10

2

2

1

1

6 0 7 2

7

10 10

5 2 +

1

1 1 1

1 1

1

1

1 1

1

1

7 4 10

10

10

10

10 10

5 2 +

1

1 1 1

1 1

1

1

1 1

1

1

7 4 10

10

10

10

10 10

5 2 +

1

1 1 1

1 1

1

1

1 1

1

1

7 4 10

10

10

10

10 10

5 2 +

1

1 1 1

1 1

1

1

1 1

1

1

7 4 10

10

10

10

10 10

5 2 +

1

1 1 1

1 1

1

1

1 1

1

1

7 4 10

10

10

10 12

10 10

5 2 +

1

1 1 1

1 1

1

1

1 1

1

1

7 4 10

10

10

10

10

2 1

10 10

+

1

1 1 1

1 1

1

1

1 1

1

1

10

10

10

10

10

2 1

5 2 7 4

10 10

+

1

1 1 1

1 1

1

1

1 1

1

1

10

10

10

10

10

2 1

5 2 7 4

10 10

+

1

1 1 1

1 1

1

1

1 1

1

1

10

10

10

10

10

2 1

5 2 7 4

10 10

+

1

1 1 1

1 1

1

1

1 1

1

1

10

10

10

10

10

2 1

7

5 2 7 4

Year 3 Essentially year 3 becomes a time when more "formal methods are introduced".

:

e.g.

34 + 21 35 + 19

U T

1

1

1

1

1

1

1

1

1 10

10

10

10

10

10

U T

Addition with 3-digits (an end of Year Objective)

U T H

e.g.

345 + 126 283 + 142 364 + 159

Why is this a poor example?

1 10 100

U T H U T H

1 10 100

1 10 100

1 10

1

1 10 100

1 10 100

1 10 100

1 10

10

10

1 10 100

1 10 100

1 10

10

10

10

10

10

Alternatively, set the addition into an application

phase: this is still largely fluency

Find the perimeter of this shape:

Show this pattern goes up by the same amount each time. Then find

the next number in the pattern:

325, 462, 599, …

352

Play the role of the teacher: Reasoning –

why is it incorrect – explain….. Mark the following questions. If they are right give a tick, if they are wrong,

explain why you think the mistake has been made:

Reflection: is column addition the most efficient way to tackle this question?

346 + 300 – 5 Where does planning allow for reflection?

9 5

4 6 +

1 1

1

2

3

9 4

9 5

4 6 +

6 1

2

2

3

9 3

U T H Th 10Th

100

10

1

More than Four-Digit Addition

1000

U T H

10000

Year 5

Year 5 They mentally add and subtract tenths, and one-digit whole numbers and tenths)

They practise adding and subtracting decimals, including a mix of whole numbers

and decimals, decimals with different numbers of decimal places, and

complements of 1 (for example, 0.83 + 0.17 = 1)

Here the most important concept to introduce is 0.9 + 0.1 ≠ 0.10

Two helpful strategies:

http://www.mathsisfun.com/numbers/number-line-zoom.html

Subtraction

Make up a word problem that represents the calculation

9 – 3 = 6

Subtraction problems

• I had 9 apples but my rabbit ate 3 of them. How many did I have left?

• I had 9 apples. My friend Harry had 3 apples. How many more apples did I

have?

3 ? 9

The Bar Model – How does it

support understanding?

Robber maths

43 – 13

The number you

need to subtract

is small enough

to “pick it up and

take it away”

Mind the gap

74 - 69

The gap between the two numbers is

smaller so it is more efficient to find

the difference (probably by counting

on)

Robber maths? – Mind the gap?

101 – 99

63 – 21

84 – 78

1006 – 999

86 – 14

Trading Game

H T U

30

4

10 10

10

10

10

10

10

1 1 7 2 7 4 -

10

1 6

1 1

1

1 1

1 1 1

1

1

10 10

10

10

10

10 7 2 7 4 -

1

1 1 1

1 1

1

1

1 1

1

1

1 6

10 10

10

10

10

10 7 2 7 4 -

1

1 1 1

1 1

1

1

1 1

1

1

1 6

10 10

10

10

10

10 7 2 7 4 -

1

1 1 1

1 1

1

1

1 1

1

1

1 6

10 10

10

10

10

10 7 2 7 4 -

1

1 1 1

1 1

1

1

1 1

1

1

1 6

10 10

10

10

10

10 7 2 7 4 -

1

1 1 1

1 1

1

1

1 1

1

1

1 6

5

10 10

10

10

10

10 7 2

7 4 -

1

1 1 1

1 1

1

1

1 1

1

1

1 6

5

10 10

7 2 7 4 -

1

1 1 1

1 1

1

1

1 1

1

1

1 6

5

10

10

10

10

10 10

7 2 7 4 -

1

1 1 1

1 1

1

1

1 1

1

1

1 6

5

10

10

10

10

10 10

7 2 7 4 -

1

1 1 1

1 1

1

1

1 1

1

1

1 6

5 2 10

10

10

10

10 10

5 2 +

1

1 1 1

1 1

1

1

1 1

1

1

7 4 10

10

10

10

10 10

5 2 +

1

1 1 1

1 1

1

1

1 1

1

1

7 4 10

10

10

10

10 10

5 2 +

1

1 1 1

1 1

1

1

1 1

1

1

7 4 10

10

10

10

10 10

5 2 +

1

1 1 1

1 1

1

1

1 1

1

1

7 4 10

10

10

10

10 10

5 2 +

1

1 1 1

1 1

1

1

1 1

1

1

7 4 10

10

10

10 12

10 10

5 2 +

1

1 1 1

1 1

1

1

1 1

1

1

7 4 10

10

10

10

10

2 1

10 10

5 2 +

1

1 1 1

1 1

1

1

1 1

1

1

7 4 10

10

10

10

10

2 1

10 10

5 2 +

1

1 1 1

1 1

1

1

1 1

1

1

7 4

10

10

10

10

10

2 1

10 10

5 2 +

1

1 1 1

1 1

1

1

1 1

1

1

7 4

10

10

10

10

10

2 1

10 10

5 2 +

1

1 1 1

1 1

1

1

1 1

1

1

7 4

10

10

10

10

10

2 1

7

Year 3 Essentially year 3 becomes a time when more "formal methods are introduced".

:

e.g.

57 - 23 52 - 27

1

1

1

1

1

1

1

1

1

10

10

10

10

10

10

10

10

10

10

U T U T

Expanded method of subtraction 273- 147 =

200 70 3

100 40 7

Subtraction with 3-digits (an end of Year Objective)

e.g.

345 - 236 523 - 136 300 - 159 U T H

1 10 100

U T H U T H

1 10 100

1 10 100

1 10

1

100

100

100

1 10 100

1 10 100

1 100

100

100

Spicing up Addition and Subtraction: Problem

solving through deep reasoning.

Arithmagons Number Walls

Some of these will need real resilience, but the sense of achievement will be

much greater once solved!

Ink Blots/

Missing Digit

Magic Squares Darts?

!

Cryptarithms/

Alphametics

T U 2

2 + 3 5

T U 7

5 + 3 5 1

H

Year 4

The strategies met in year 3, extend into year 4 - with addition and subtraction

now with four digits.

1. Although students might be able to naturally extend the method, revisit the

kinaesthetic examples so they link their new objective to prior learning. If

some students need longer working kinaesthetically than others - fine!

2. Some very visual learners will even remember "counter colours" from the

previous year, so ensure complete consistency between year groups.

Note - all the "livening up" and "problem solving" skills from year 3 should

also be embraced in years 4 & 5. As too should estimation and inverse

operations to check.

Models of multiplication

• I had four bags and they each contained six books. How many books do I

have?

• I had six pens. Tom had four times as many? How many did Tom have?

Year 3

solve problems, including missing number problems, involving

multiplication and division, including positive integer scaling

problems and correspondence problems in which n objects are

connected to m objects

Models for multiplication

74

Scaling

3 times

as tall

This can be generalised

to include any multiplier

including those less than

one – i.e. making

smaller

6 6 6 6 6

6

6

6 6

6 + 6 + 6 + 6

Additive

reasoning

6 x 1 6 x 4

Multiplicative

reasoning

Arrays

2 lots of 3 make 6

3 lots of 2 make 6

There are two 3s in 6

There are three 2s in 6

2 x 3 = 6

3 x 2 = 6

6 divided by 2 = 3

6 divided by 3 = 2

An image for 7 8 = 56

77

Multiplication

At the heart of success of this topic is clearly mastery of times tables. It is the one area

where deviating from your year group and extending has value. Fluency with tables opens

up so many other topics in maths (e.g. fractions and area) and conversely closes off

success in other topics if they haven't been mastered at a young age.

Guide: Yr 2 (2, 5, 10) Yr 3 (3, 4, 8) Year 4-6 (upto 12 x 12)

Is this the wisest plan?

Practice, practice, practice is the key. Use every (daily throughout the year) opportunity

open to you, especially:

- Lining up to assembly, getting changed for PE, etc.

- Parents! They can really support the regularity of practice.

Note: DK Times Table App – many can access at home.

Arrays to Solve Multiplication

10 3

4

10 x 4 = 40 4 x 3 = 12

40 + 12 = 52 13 x 4 = 52

13 x 4 =

Year 4

(set questions involving all their tables targets - though this will need

differentiation throughout the classes)

Always go back to the kinaesthetic example when re-introducing (even if only

for a few seconds)

43 x 6

247 x 3

U T

U T H

Year 5 Multiply upto 4 digits by one-digit or two-digit, using a formal written method,

including long multiplication for two-digit numbers.

re-visit:

Hands-On? Short-Multiplication Long-Multiplication

4253 x 7 4253 x 7 253 x 37

Year 5 Multiply multi-digit numbers up to 4 digits by a 2-digit whole number.

18

13

18 10 8

13

3

10 100 80

24 30

Progressing towards the standard algorithm

1 0 8

1 0

3

1 0 0 8 0

3 0 2 4

10 8

10

3

100 80

30 24

1 8

1 3

1 8 0

5 4

2 3 4

?

When?

How?

Year 5 Multiplying (and Dividing) by 10, 100 and 1000

4 x 10

13 x 10

6 x 100

4.3 x 10

0.12 x 1000

U T H 1

10

1

100

http://www.topmarks.co.uk/Flash.aspx?f=MovingDigitCards

Year 5

Identify multiples and factors, including finding all factor pairs of a number,

and common factors of two numbers.

Know the vocabulary of prime numbers and non-prime numbers.

Differentiation

Year 5

Recognise square numbers and cube numbers - and notation.

Square Numbers Cube Numbers

Year 6 Multiply multi-digit numbers up to 4 digits by a 2-digit whole number.

Identify common factors, common multiples and prime numbers.

Regarding the "mastery learning" approach, if a skill is mastered by the end of

the Autumn or Spring term, consider the following for the Summer Term:

Set questions in a "worded context"

Apply skills to a more "problem solving" question, e.g.

"The school hall measures 12m by 26m. Miss Smith is going to carpet the hall

using square carpet tiles which are 50cm long. How many tiles are needed?“

Extend further??

1

1

Using Inequalities

3 x 42 5 x 21 9 3 2

3

5 4

4 x

Links with other topics…

Area

Pictograms

Measure

Division

At the heart of success of this topic is clearly

mastery of times tables. The more fluent a student

is at their tables, the easier they will find division.

Grouping and Sharing

12 ÷ 3 = 4

Grouping – we know how many are in each group but not how many

groups there will be. The answer is the number of groups.

Sharing - we know how many groups there are but not how many are in

each group. The answer is the number in each group.

Grouping

Building the Journey Year 3 Pupils can derive associated division facts e.g. if 6 ÷ 3 = 2, then 60 ÷ 3 = 20

Pupils develop reliable written methods for division, progressing to the formal written methods

of short division.

Year 4 Pupils can derive associated division facts e.g. if 28 ÷ 7 = 4, then 2800 ÷ 7 = 400

Pupils practise to become fluent in the formal written method of short division

with exact answers

Year 5 Divide numbers up to 4 digits by a one-digit number using the formal written method of short

division and interpret remainders appropriately for the context.

Pupils use multiplication and division as inverses to

support the introduction of ratio in year 6, e.g. in scale drawings or in converting between metric

units.

Year 6 Divide numbers up to 4 digits by a two-digit number using the formal written

method of short division where appropriate, interpreting remainders according to the

context.

Divide numbers up to 4 digits by a two-digit whole number using the formal written method of

long division, and interpret remainders as whole number remainders, fractions, or by rounding,

as appropriate for the context.

Solve problems involving the relative sizes of 2 quantities where missing values can be found

by using integer multiplication and division facts.

Solve problems involving similar shapes where the scale factor is known or can be found.

Ella has 48 plasticine legs to make

animals for a display.

How many cows could she make?

How many beetles could she make?

How many spiders could she make?

An image for 56 7

Either:

• How many 7s can I

see? (grouping)

Or:

• If I put these into 7

groups how many in

each group? (sharing)

An image for 56 7

5 6 7

8 5 6 7

8

The array is an image for division too

363 ÷ 3 =

3 6 3 3

1 2 1

364 ÷ 3 =

3 6 4 3

364 ÷ 3 =

3 6 4 3

1 2 1 rem 1

345 ÷ 3 =

3 4 5 3

1 1 1

5

Year 4 The journey is now about fluency with short division. Although not explicit, three-

digit divided by one-digit seems a sensible goal by the end of the year. There

should be exact answers (no remainders). What if students "master" the process

in the autumn term?

Do you have "tricks" as a teacher to ensure there are no remainders?!

462 ÷ 2

725 ÷ 5

537 ÷ 3

474 ÷ 6

738 ÷ 9

1

10

100 H T U

Year 6 Divide numbers up to 4 digits by a two-digit number using the formal written

method of short division where appropriate, interpreting remainders according

to the context.

Divide numbers up to 4 digits by a two-digit whole number using the formal

written method of long division, and interpret remainders as whole number

remainders, fractions, or by rounding, as appropriate for the context.

(Note: as a mathematician, I never use long division, and do not see its value in

the new National Curriculum... but you have to teach it!)

12 5 4 2 1 12 5 4 2 1

Careful then writing recurring decimals...

Year 6

Solve problems involving the relative sizes of 2 quantities where missing

values can be found by using integer multiplication and division facts.

This is ratio!

"mixing paint"... let your students take ownership of their learning...

(Hidden) Applications of Ratio:

If 3 pencils cost 45p, how much did one pencil cost?

If 2 pencils cost 60p, how much would 5 pencils cost?

If 5 pencils cost 70p, how many pencils could I buy for £2.10?

Ingredients to make 16 gingerbread men

180 g flour

40 g ginger

110 g butter

30 g sugar

How much of each ingredient would you need to make ...... gingerbread men?

Year 6 Solve problems involving similar shapes where the scale factor is known or

can be found.

Language: If two shapes are identical, we say they are ........................

Similar shapes means the two shapes are ..........................

................................................

If you double the sides, does everything double?

Aims of the Session

• To build understanding of mathematics and it’s

development throughout KS2 – the four main

operations.

• To have a stronger awareness of when and how to

progress from non-formal to formal methods at the

appropriate stage (moving from using concrete to

pictorial and finally abstract representations to

support learning)

• To appreciate how fluency, reasoning and problem

solving skills are interlinked in establishing deep

understanding of mathematics in the KS2 curriculum.

Lots more information and detail can be found on the website! Thanks for coming!

Maths activities to support the new curriculum (other subjects too!): http://www.theschoolrun.com/ Various maths games: http://www.transum.org/Software/Game/ http://mathszone.co.uk/number-facts-xd/ http://www.primaryinteractive.co.uk/maths.htm Open-ended maths puzzles: http://nrich.maths.org/ Maths triangles: http://www.helpingwithmath.com/printables/others/fac0201fact_triangle01.htm