Teaching for Mastery Representation and Structure

1 Jayne Webster Secondary Mastery Specialist Enigma Maths Hub

Teaching for Mastery – Representation and Structure

Using manipulatives to expose mathematical structure in Secondary Schools

This booklet is a collection of ideas, resources and examples that I have picked up

on my 5 year mastery journey. Where possible they are research and evidence

based and have all been tried and tested in real classrooms.

Throughout the booklet are links to further research or resources. Please feel free to

use and adapt anything you find in this booklet and the linked resources.

I have tried to group together certain concepts and ideas to show coherence. Some

ideas may not be new to you and some you may disagree with. I hope that you find

something in here that is useful, even if it is just to challenge your own conceptual

understanding.


Contents Teaching for Mastery – Representation and Structure ...............................................................1

Using manipulatives to expose mathematical structure in Secondary Schools ...................................1

The Connective Model ...............................................................................................................................3

Improving mathematics in key stages two and three, EEF .....................................................................5

The 5 big ideas underpinning Teaching for Mastery ...............................................................................6

Mathematical Thinking ..............................................................................................................................6

Fluency ........................................................................................................................................................6

Variation .....................................................................................................................................................7

Conceptual Variation - different representations of the same concept ...............................................7

Conceptual Variation .................................................................................................................................9

Procedural Variation ............................................................................................................................... 10

Coherence – small connected steps ...................................................................................................... 12

Cognitive Load Theory ............................................................................................................................ 13

Representation and Structure................................................................................................................ 15

From Number to Algebra ....................................................................................................................... 17

Four Operations with Integers ............................................................................................................... 18

Adding Integers using zero pairs ............................................................................................................ 19

Minus as Opposite .................................................................................................................................. 21

Collect Like Terms and Simplify Expressions......................................................................................... 22

Subtraction .............................................................................................................................................. 23

Minus Minus ............................................................................................................................................ 24

Multiplying integers using grouping ...................................................................................................... 25

Dividing Integers ..................................................................................................................................... 26

Further Examples of Minus as Opposite ............................................................................................... 27

Place Value .............................................................................................................................................. 28

Dots and Boxes ........................................................................................................................................ 28

Base 2 Blocks ........................................................................................................................................... 30

Dienes Blocks and Algebra Tiles ............................................................................................................. 36

Multiplication .......................................................................................................................................... 38

Division..................................................................................................................................................... 41

Four operations with base 10 and base x counters ............................................................................. 47

Standard Index Form .............................................................................................................................. 50

Multiplicative opposite as fractions in different bases ........................................................................ 54

Cuisenaire Rods ....................................................................................................................................... 56

Sequences ................................................................................................................................................ 68


Algebra Tiles for Equations.............................................................................................101

Simultaneous Linear Equations Examples........................................................................................... 106

Quadratics.............................................................................................................................................. 109

Misconception Busters ......................................................................................................................... 117

The Child and Mathematical Errors ..................................................................................................... 118

The Task and Mathematical Errors ...................................................................................................... 119

The Teacher and Mathematical Errors ................................................................................................ 119

How do we normally ask the question represented by the illustration?

Can you think of a worded question or an algebraic question that could result in this

illustration?

The answer is really clear one blue box i.e. ? = 5

But if this was presented in algebra form 3x+12 =2x+17 how many of our students

would stumble?

I suggest that the visual problem makes more sense intuitively than perhaps the way

this type of problem is often introduced

So we often talk about ‘manipulating the symbols’ but this is an example of

‘symbolising the manipulations’ – i.e. using the bar model to show what the

manipulation ‘subtract x from both sides’ means.

Pete Griffin, Assistant Director (Secondary), NCETM


The Connective Model “when children are engaged in

mathematical activity, they are involved in

manipulating one or more of these four key

components of mathematical experience:

concrete materials, symbols, language and

pictures”

Derek Haylock and Anne Cockburn (2008),

Understanding Mathematics for young children.

The Connective Model Understanding mathematics involves identifying and

understanding connections between mathematical ideas. Haylock and Cockburn

(1989) suggested that effective learning in mathematics takes place when the

learner makes cognitive connections. Teaching and learning of mathematics should

therefore focus on making such connections. The connective model helps to make

explicit the connections between different mathematical representations: symbols,

mathematically structured images, language and contexts.

https://www.babcockldp.co.uk/cms/articles/send-file/7f3bec01-7051-40af-83da-

80cdacf578d5/1

Concrete materials are often referred to as manipulatives. Interactive versions can

be found on https://mathsbot.com/manipulatives/

Dienes blocks algebra tiles double sided counters Cuisenaire rods

I will refer to these manipulatives throughout this booklet.

“symbolise manipulation leads to manipulating

the symbols”


https://www.babcockldp.co.uk/cms/articles/send-file/7f3bec01-7051-40af-83da-80cdacf578d5/1

https://www.babcockldp.co.uk/cms/articles/send-file/7f3bec01-7051-40af-83da-80cdacf578d5/1

https://mathsbot.com/manipulatives/d


Improving mathematics in key stages two and three, EEF


The 5 big ideas underpinning Teaching for Mastery

Representation and structure is one of the 5 big ideas underpinning Teaching for

Mastery. Throughout this booklet you will see examples of the big ideas, however

the main area for development is representation and structure. The next few pages

have a brief description of each of the ideas with examples. For more on each of

the big ideas see https://www.enigmamathshub.co.uk/masterysecondary

Mathematical Thinking Mathematical thinking involves:

• looking for pattern in order to discern structure;

• looking for relationships and connecting ideas;

• reasoning logically, explaining, conjecturing and proving.

Fluency Fluency demands the flexibility to move between different contexts and

representations of mathematics, to recognise relationships and make connections

and to make appropriate choices from a whole toolkit of methods, strategies and

approaches.

Fluency reduces cognitive load and frees up working memory to solve problems and

process new information.

https://www.enigmamathshub.co.uk/masterysecondary


Variation Variation is not the same as variety – careful attention needs to be paid to what aspects are being varied (and what is not being varied) and for what purpose. What’s the same, what’s different and why?

Conceptual Variation - different representations of the same concept Division example

How many groups of 3 are in 12? 12÷3 = 4

4

3 3

4

4

3 12 3 12

4

Compare the array with the area model and the standard short division algorithm

(bus stop method).

part whole model

This example uses Dienes blocks and the part whole model to regroup 10’s and 1’s

and compares them with the standard short division algorithm. This example draws

attention to the exchange of 10s for 1s.


Compare the examples on the next page with the place value counters

and tables and skip counting.

exchange one

10

for ten 1’s.

1 group of 3 tens 4 groups of 3 ones

Skip counting

What’s the same

Division as grouping.

What’s different

Different representations: array, regrouping, place value charts, Dienes blocks, skip

counting and the standard short division algorithm.

Why?

Varying the representation allows students to make connections between the

methods and the standard algorithm.

Starting with the basic idea of grouping leads to a deeper understanding of the

standard division algorithm. See examples of going deeper in the division section.

NCETM Primary PD material - Year 2 Quotitive and Partitive Division:

Division equations can be used to represent ‘grouping’ problems, where the total

quantity (dividend) and the group size (divisor) are known; the number of groups

(quotient) can be calculated by skip counting in the divisor. (quotative division)


For more on division, including dividing polynomials, see page 35 and the

links to the NCETM primary and secondary PD material on this padlet

https://enigmamathshub.padlet.org/websterj9/2ofwz5x083g8

Conceptual Variation - avoid common misconceptions by looking at the

concept and mis-concept simultaneously

Vertically opposite angles example

Show the standard, non-standard and misconception simultaneously.

https://nonexamples.com/compare

https://enigmamathshub.padlet.org/websterj9/2ofwz5x083g8

https://nonexamples.com/compare


Procedural Variation - same model different question

What’s the same

Same number – 30, same ratio and same bar model

What’s different

The questions are all different. 30 could be a part, whole or difference.

Why?

Varying the question allows students the opportunity to see that a variety of

questions could be being asked. This is excellent exam practice. Check the SSDD

problems website

for more

examples.

same diagram

different

questions

https://ssddproblems.com/



Integers example - same digits, different signs. Vary what you want them to pay

attention to and keep the rest the same.

Factorising quadratics example – Keeping the constant term the same draws

attention to the changing coefficient of x. This is often referred to as intelligent or

purposeful practice.

If learners think that mathematical examples are fairly random, or mysteriously

come from the teacher, then they will not have the opportunity to experience

the expectation, confirmation and confidence-building which came from

perceiving variations and then learning that their perceptions are relevant

mathematically – Watson and Mason, 2006


Coherence – small connected steps

• Small connected steps are easier to take.

• Focusing on one key point each lesson allows for deep and sustainable

learning.

• Certain images, techniques and concepts are important pre-cursors to later

ideas. Getting the sequencing of these right is an important skill in planning

and teaching for mastery.

• When introducing new ideas, it is important to make connections with earlier

ones that have already been understood.

When something has been deeply understood and mastered, it can and

should be used in the next steps of learning.


Cognitive Load Theory

https://www.cese.nsw.gov.au/publications-filter/cognitive-load-theory-research-that-

teachers-really-need-to-understand

https://www.cese.nsw.gov.au/publications-filter/cognitive-load-theory-research-that-teachers-really-need-to-understand

https://www.cese.nsw.gov.au/publications-filter/cognitive-load-theory-research-that-teachers-really-need-to-understand


The division example shows the small steps leading to the standard

division algorithm.

Sequencing the small steps can be tricky. Luckily the NCETM have done this for

you.

https://www.ncetm.org.uk/files/108920532/ncetm_secondary_mastery_pd_materials

_structure.pdf

Focused key ideas and making connections

One aspect of the professional development materials is to help you see the learning

journey that students need to go on: what comes first, next and so on. They are

designed to support you in ‘homing in’ on the key idea and avoiding the confusion

that results in mixing up too many ideas and skills in the early stages of learning.

However, as making connections between mathematical ideas is vital to deep

understanding, clearly laying out these key ideas and teaching them sequentially is

not sufficient when teaching for mastery. Each idea must build on and connect with

previous ideas. It is the unifying core concepts within topics that helps you to do this.

https://www.ncetm.org.uk/files/108920532/ncetm_secondary_mastery_pd_materials_structure.pdf

https://www.ncetm.org.uk/files/108920532/ncetm_secondary_mastery_pd_materials_structure.pdf


Representation and Structure 1. The representation needs to clearly show the concept being taught.

It exposes the structure. Dienes blocks expose the geometrical structure of base 10 and are a good visual representation of the relative sizes of digits when linked to place value.

Exploring different bases exposes the structure of place value in any base including base x. Students are more likely to make the connection between the laws of arithmetic and apply them to base x (algebra tiles).

Dienes blocks, place value tables and algebra tiles are all representations. Making

the connections reveals the mathematical structure and enables students to

understand the concept and use the mathematics independently of the manipulatives

and representations.

2. In the end, the students need to be able to do the maths without the

representation.

3. A stem sentence describes the representation and helps the students move to working in the abstract (“ten tenths is equivalent to one whole”) and could be seen as a representation in itself.

...... is divided into groups of ........ There are ....... groups and a remainder of .......


4. There will be some key representations which the students will meet time and again. Guidance material is available on the NCETM website for each of the following key representations. https://www.ncetm.org.uk/resources/53609

• Algebra tiles

• Arrays and area models

• Bar models

• Cuisenaire® rods

• Dienes (and place-value counters)

• Double number lines (and ratio tables)

• The Gattegno chart

• Place-value charts

• Single number lines

5. Pattern and structure are related but different: Students may have seen a pattern without understanding the structure which causes that pattern.

2 x 2 = 4 3 – 1 = 2

2 x 1 = 2 3 – 2 = 1

2 x 0 = 0 3 – 3 = 0

2 x -1 = -2 3 – 4 = -1

2 x -2 = -4 3 – 5 = -2

pattern structure pattern structure

Sequences

2 5 10 17 … Continuing sequences looks at patterns.

3 5 7

…

1+1 4 + 1 9 + 1 16 + 1 n2 + 1

Looking at the geometrical structure exposes the generalisation

https://www.ncetm.org.uk/resources/53609

https://www.ncetm.org.uk/files/110720772/ncetm_ks3_algebra_tiles.pdf

https://www.ncetm.org.uk/files/110720774/ncetm_ks3_arrays_area_models.pdf

https://www.ncetm.org.uk/files/110720775/ncetm_ks3_bar_models.pdf

https://www.ncetm.org.uk/files/110720776/ncetm_ks3_cuisenaire_rods.pdf

https://www.ncetm.org.uk/files/110720777/ncetm_ks3_dienes_pv_counters.pdf

https://www.ncetm.org.uk/files/110720846/ncetm_ks3_double_number_lines_ratio_tables.pdf

https://www.ncetm.org.uk/files/110720847/ncetm_ks3_gattegno_chart.pdf

https://www.ncetm.org.uk/files/110720848/ncetm_ks3_place_value_charts.pdf

https://www.ncetm.org.uk/files/110720850/ncetm_ks3_single_number_lines.pdf


From Number to Algebra

Algebra tiles and discs on mathsbot.com or double-sided counters can be used to

model a number of mathematical concepts. They are particularly useful to

manipulate negative indices and terms and can be used effectively to address

common misconceptions.

The key concept underpinning work with algebra discs/tiles is the zero pair.

Zero pairs sum to zero:

(+1) + (-1) = 0

(+x2) + (-x2) = 0

(+x) + (-x) = 0

The NCETM have produced a booklet on using algebra tiles with further examples

and videos. https://www.ncetm.org.uk/files/110720772/ncetm_ks3_algebra_tiles.pdf

You can create your own tiles using the template at the back of this booklet.

Algebra tiles allow students to visualise and manipulate algebraic expressions

“symbolise manipulation leads to manipulating the symbols”


https://www.ncetm.org.uk/files/110720772/ncetm_ks3_algebra_tiles.pdf


Four Operations with Integers This section shows the small steps leading to working with the four operations with

integers. Depth is added by working with terms and expressions. I have tried to address common

misconceptions as we go through each set of examples.

There are many ways to represent integers and I have looked at many of them. I believe that the

following representations address common misconceptions.

Establishing zero pairs

Why does each set of counters

represent 2?

Stem sentence: There are …….. zero pairs. The zero pairs sum to …….. and the

set of counters simplifies to ……….

What does each pile

represent?

Stem sentence: There are …….. zero pairs. The zero pairs sum to …….. and the

set of counters simplifies to ……….

Use zero pairs to simplify each expression

(+3) + (-3) (+3) + (-1) (-1) + (+3) (-3) + (+1) (+1) + (-3)

Stem sentence: There are …….. zero pairs. The zero pairs sum to zero and the

expression simplifies to ………

Introduce terms and expressions early.

(+2) + ( ) = 0 (-3) + ( ) = 0 ( ) + (-4) = 0

(+x) + ( ) = 0 (-y) + ( ) = 0 ( ) + (+x2) = 0

(-3x) + (+2) + ( ) + ( ) = 0


Adding Integers using zero pairs

Shake 5 double sided counters and throw them. Write down or say what you

see.

Will you ever get 0?

How many ways can you get a positive?

How many ways can you get a negative?

Generalise – Mathematical Thinking

Answer each question with positive, negative or zero.

0 because the positive = negative

0 because the positive = negative

+ because the positive > negative

+ because the positive > negative

- because the positive < negative

- because the positive < negative


Match the expressions with the diagrams.

Common points of confusion

Is the + add or positive?

Is – subtract of negative?

Why do these all mean the same? 3 + -2 (+3) + (-2) 3 + (-2) (-2) + 3

3 + -2 = 1 and 3 – 2 = 1 have the same answer but different meanings.

The minus sign is a potential minefield of misconceptions because it has so many different meanings. The meaning can even change part way through a problem!

3 – x = 1 the minus sign represents subtract x

-x = -2 the same minus sign now represents negative x

Thinking of the minus sign as an opposite takes away the problem of one symbol many meanings.

This is a big statement that I am making! You may disagree with this one but before you dismiss it, give it a try.

The original idea came from James Tanton and his Exploding Dots. https://globalmathproject.org/wp-content/uploads/2019/09/Teaching_Guide_EXPERIENCE_4_GMW2018_Edition.pdf

https://globalmathproject.org/wp-content/uploads/2019/09/Teaching_Guide_EXPERIENCE_4_GMW2018_Edition.pdf



Minus as Opposite

1 x

opp 1 opp x

opp opp 1 opp opp x

opp opp opp 1 opp opp opp x

Conventions

Unnecessary + symbols can be ignored

+3 3

Think of + as “and”

4 + -3 becomes 4 – 3 4 and -3

think of this as collecting the terms 4 and opp 3 which simplifies to 1.

3x -2x+4x simplifies to 5x

think of this as collecting the terms 3x and opp 2x and 4x which simplifies to 5x.

Students make the connections between manipulating number and algebra

using consistent representations and language.


Collect Like Terms and Simplify Expressions

Collect the like terms in a Venn diagram

3a and opp 2a 7b and opp 4b

Compare the Venn diagram with (3-2)a + (7-4)b

Simplify using zero pairs a + 3b

The Venn diagram represents the collecting of like terms. This is quickly replaced by collecting like terms in brackets.

Do not forget to include constant terms

3x -4y + 8 -5x + 2 -7y

(3 -5)x + (-4 -7)y + ( 8 + 2)1

-2x + -11y + 10

-2x - 11y + 10

You could also represent this with algebra discs where the collecting of like terms is

represented by groups of like discs before moving on to the more formal method.


Subtraction

minuend – subtrahend = difference

These examples highlight the problem with subtraction when the subtrahend is

greater than the minuend.

3 – 1 = 2

3 – 2 = 1

3 – 3 = 0

For the next question we need to take away 4 positives from 3 positives.

This is not possible until we add zero pairs.

3 – 4 = -1

3 – 5 = -2

If we think of – as opposite and not subtraction:

3-1 3-4

3 and opp 1 simplifies to 2 3 and opp 4 simplifies to opp 1

3-2 3-5

3 and opp 2 simplifies to 1 3 and opp 5 simplifies to opp 2

Model these as a “take away” with counters.


Minus Minus

Minus as opposite reduces the chance of misconceptions with - - = +

3 – 3 = 0 3 and opp 3 = 0

3 – 2 = 1 3 and opp 2 = 1

3 – 1 = 2 3 and opp 1 = 2

3 – 0 = 3

3 - -1 = 4 3 and opp opp 1 = 4

3 - - 2 = 5 3 and opp opp 2 = 5

Generalise – Mathematical Thinking

Answer each question with positive, negative or zero.

Stem sentences can be used to support mathematical thinking and reasoning.

Larger positive and opp of smaller

positive is the same as …

…larger positive and smaller negative

which simplifies to positive


Multiplying integers using grouping

3 x 2 3 x -2

3 groups 3 groups

of 2 of opp 2

-3 x 2 -3 x -2

opp of 3 groups opp of 3 groups

of 2 of opp 2

Mathematical Thinking

Find three values of a and b so that

a × b = − 24

Find three values of a and b so that

a × b = 24


Dividing Integers

6 ÷ 2 = 3 -6 ÷ -2 = 3

3 groups 3 groups

of 2 of opp 2

make 6 make opp 6

6 ÷ -2 = -3 -6 ÷ 2 = -3

opp of 3 opp of 3

groups groups

of opp 2 of 2

make 6 make opp 6

The use of language is a key representation. We want students to internalise the

language and remove the counters.

4 x -3 “4 groups of opp 4” -12

8 ÷ -2 “how many groups of opp 2 make 8” -4

-5 - 2 “opp 5 and opp 2 simplifies to opp 7” -7

-5 - - 2 “opp 5 and opp opp 2 simplifies to opp 3” -3

3a – 5a “3a and opp 5a simplify to opp 2a” -2a

3x – 4 – (2x -1) “3x and opp 4 and opp 2x and opp opp of 1” x - 3

Try more of your own. I challenge you to find one that does not work!


Further Examples of Minus as Opposite Reflection

f(-x) ≡ f(x) reflected in y axis – reflective opposite

-f(x) ≡ f(x) reflected in x axis – reflective opposite

Inverse function

f-1(x) ≡ opposite of f(x)


Place Value Dots and Boxes adapted from https://www.explodingdots.org/ by

James Tanton

This set of examples lets students explore place value in different bases, leading to a

deeper understanding of base 10. This can be applied to addition, subtraction,

multiplication, standard form and negative indices. To go even deeper you can work

in base x. I use this to introduce algebra tiles.

The 1 ←2 rule: Whenever two counters are in a box together, they stick together

and move one place left. This is a plan view so you cannot see how many counters

are in each pile on these diagrams. Watch this video to get you started.

https://youtu.be/iYTN6iMdGf0

Try this activity with counters to get a real feel for it.

At each stage count up all the

counters in the boxes and below and

you should have 9.

Two counters in the right hand box

and 7 below.

These two counters stick together and

move one place left.

Now this box has a pile of 2

counters and 7 below.

Add two more counters to the right

hand box…

There are now 4 counters in the

boxes and 5 below

The two counters stick together and

move one place left.

https://www.explodingdots.org/

https://youtu.be/iYTN6iMdGf0


There are still 4 counters and 5 below.

The two piles of counters stick

together and move one place left.

This is a pile of 4 counters and 5

below.

Keep adding counters from the

right.

You should end up with a pile of 8

counters and a single counter

The piles of counters can eventually be replaced with a place value chart.


The piles of counters

expose the place

value of base 2 (binary) and the

exponential growth structure.

16 8 4 2 1

Base 2 Blocks https://mathsbot.com/manipulatives/blocks

Using blocks or Cuisenaire rods exposes the geometrical structure of base 2.

Moving 1 place left means x2 so the opposite must be true, moving 1 place the right

means ÷2

It is useful to compare this with moving left and right in base 10, a concept that

students are familiar with.

See this PDF from James Tanton for more ideas for the classroom

https://globalmathproject.org/wp-

content/uploads/2019/09/Teaching_Guide_EXPERIENCE_1_GMW2018_Edition.pdf

https://mathsbot.com/manipulatives/blocks




Base 2 Place Value

Multiplicative Opposite

23 = 1 x 2 x 2 x 2

2-3 = 1 ÷ 2 ÷ 2 ÷ 2

25 = 1 x 2 x 2 x 2 x 2 x 2

2-5 = 1 ÷ 2 ÷ 2 ÷ 2 ÷ 2 ÷ 2

The 1 ←3 rule: Whenever

three counters are in a box

together, they stick together and move one place left.

23 is repeated multiplication

2-3 is repeated division

Minus exponent means opposite of

repeated multiplication


Compare the structures of base 3 to base 2.

27 9 3 1

Using Dienes blocks on mathsbot.com you can create the different bases to

compare both the exponential growth and geometrical structures.

https://mathsbot.com/manipulatives/blocks Copy and paste into Autograph

Base 2 Base 3 y = 2x y = 3x

https://mathsbot.com/manipulatives/blocks

https://completemaths.com/autograph


Base 3 Place Value

35 = 1 x 3 x 3 x 3 x 3 x 3

3-5 = 1 ÷ 3 ÷ 3 ÷ 3 ÷ 3 ÷ 3

Base 5 Place Value

55 = 1 x 5 x 5 x 5 x 5 x 5

5-5 = 1 ÷ 5 ÷ 5 ÷ 5 ÷ 5 ÷ 5

Base 10 Place Value

See later chapters for links to standard form


Base 10 brings us back to Dienes blocks. Most students will

be familiar with these from Primary school.

Using base blocks exposes the geometrical structure common to all bases.

x3 x2 x 1

Base x is more commonly seen as algebra tiles.

https://mathsbot.com/manipulatives/tiles


Algebra tiles can be used to represent the area model for multiplication

and division and much more. I will explain in more detail in later

chapters.

(x + 2)(x + 3) ≡ x2 + 5x + 6

https://youtu.be/qa9ug4qvsXY - Tom Manners. This webinar was on Algebra Tiles -

these have changed my teaching and are ridiculously effective in helping students overcome

misconceptions with algebra, and support conceptual understanding.

https://youtu.be/ZVOhNhCmKRo - Tom Manners. This session delved deeper into how

I like to use algebra tiles, and looked at introducing integers and the four operations (although I

actually used double-sided counters for these!)

https://educationendowmentfoundation.org.uk/tools/guidance-reports/maths-ks-2-3/#closeSignup

https://youtu.be/qa9ug4qvsXY

https://youtu.be/ZVOhNhCmKRo

https://educationendowmentfoundation.org.uk/tools/guidance-reports/maths-ks-2-3/#closeSignup


Dienes Blocks and Algebra Tiles Dienes blocks are an excellent visual representation of base 10 that allow

students to see the connections with base x (algebra tiles). You can use Dienes

blocks to address common misconceptions with standard algorithms.

Addition

123 + 238 → 300 + 50 + 11 → 300 + 60 + 1

Compare the blocks with the standard column method to reinforce the exchange.

The same methods can be used for base x

(x2 + 2x + 3) + (2x2 + 3x + 8)

3x2 + 5x + 11

1 2 3

+ 2 3 8

3 6 1

1

x2 2x 3

+ 2x2 3x 8

3x2 5x 11


Subtraction

238 – 154

Compare the blocks with the standard column method to reinforce the exchange.

The same methods can be used for base x (using – as opposite and zero pairs)

(2x2 + 3x + 8) – (x2 + 5x + 4) “2x2 and 3x and 8 and opp x2 and opp 5x and opp 4”

Adding and simplifying the zero pairs leaves x2 - 2x + 4

12 13 8

- 1 5 5

8 2

2x2 3x 8

-x2 -5x -4

x2 -2x 4


Multiplication 4 X 23 or 4 of 23

20 3

4

20 3

4

https://www.ncetm.org.uk/files/107958600/ncetm_spine2_segment14_y4.pdf

2 3

x 4

9 2

1 80 12



42 X 23 or 42 of 23 or 40 of 23 and 2 of 23

20 3

40

2

NCETM Year 6 PD material


800 120

40 6

2 3

x 4 2

4 6

9 2 0

9 6 6

20 3

40 800 120

920

2 40 6

46

966



Multiplication in base x

(x + 3)(x + 2) ≡ x2 + 5x + 6

Base x: one place left is x times bigger

These 3 representations make connections

between the geometrical area model, the

standard multiplication algorithm and the

commonly used grid method.

Students make the connections between manipulating number and algebra using consistent representations.

x 3

x x 2

2x 6

x2 3x 0

x2 5x 6

x 3

X X2 3x

2 2x 6


Division https://padlet.com/enigmamathshub19/2ofwz5x083g8

Place value counters represent the

exchange…

…compare this to the short

division algorithm.

2 groups of 3 tens and 7 groups of

3 ones

Compare Dienes blocks with

place value counters, the area

model and the standard division

algorithm.

1 group of 3 tens and 3 groups

of 3 ones

https://padlet.com/enigmamathshub19/2ofwz5x083g8


Using Dienes blocks exposes the remainder as a fraction.

Place value tables and charts allow repeated exchanges.


Division with regrouping represents how to partition the dividend

effectively?

1 group of 12 tens and 3 groups of 12 ones


Division in base x

Students make the connections between manipulating number and algebra using consistent representations.


Dienes blocks base x and algebra discs on

https://mathsbot.com/#Manipulatives

https://mathsbot.com/#Manipulatives


Other Applications of Shanghai Style Division


Four operations with base 10 and base x counters

Multiplication

152 x 2

Exchange ten 10’s for 1 hundred

152 x 10

Each counter moves one place to the left (10 times bigger)


2(x2 + 5x + 2) = 2x2 + 10x + 4

x2 x 1 .

x(x2 + 5x + 2) = x3 + 5x2 + 2x

x3 x2 x 1 x3 x2 x 1

each counter moves one place to the left (x times bigger)

(x + 3)(x + 2) ≡ x2 + 5x + 6

x2 x 1

x 3

x 2

2x 6

2(x+3)

x2 3x 0 x(x+3)

x2 5x 6


Division

369 ÷ 3 = 123

1 group of 3 hundreds

2 groups of 3 tens

3 groups of 3 ones

1 2 3

242 ÷ 11 = 22

2 groups of 11 tens

2 groups of 11 ones

(x2 + 5x + 6) ÷ (x + 2) = (x + 3)

x2 x 1

1 group of (x+2) x’s

3 groups of (x+2) ones


Standard Index Form Standard index form is all about base 10.

2000

2 of 103

2 x 103

2100

2 of 103 and 1 of 102

2 of 103 and 1 tenth of 103

2.1 x 103

2140

2 of 103 and 1 of 102 and 4 of 101

2 of 103 and 1 tenth of 103 and 4 hundredths of 103

2.14 x 103


Making sense of standard form misconceptions.

21.4

2 of 101 and 1 of 100 and 4 of 10-1

2 of 101 and 1 tenth of 101 and 4 hundredths of 101

2.14 x 101


0.0214

2 of 10-2 and 1 of 10-3 and 4 of 10-4

2 of 10-2 and 1 tenth of 10-2 and 4 hundredths of 10-2

2.14 x 10-2


Non-standard standard form problems

Write 23 x 102 in standard index form.

Add this to the place value grid as 23 lots of 102

Now exchange 20 hundreds for 2 thousands

2.3 lots of 103

2.3x103


Write 0.23 x 10-2 in standard index form.

Add this to the place value grid as 0.23 lots of 10-2

This is the same as 2.3 lots of 10-3

2.3x10-3


Multiplicative opposite

23 22 21 20 2-1 2-2 2-3

8 4 2 1 1

2

1

4

1

8

33 32 31 30 3-1 3-2 3-3

27 9 3 1 1

3

1

9

1

27

43 42 41 40 4-1 4-2 4-3

64 16 4 1 1

4

1

16

1

64

53 52 51 50 5-1 5-2 5-3

125 25 5 1 1

5

1

25

1

125

103 102 101 100 10-1 10-2 10-3

1000 100 10 1 1

10

1

100

1

1000

a3 a2 a1 a0 a-1 a-2 a-3

1 × 𝑎 × 𝑎 × 𝑎 1 × 𝑎 × 𝑎 1 × 𝑎 1 1

𝑎

1

𝑎 × 𝑎

1

𝑎 × 𝑎 × 𝑎

x3 x2 x1 x0 x-1 x-2 x-3

1 × 𝑥 × 𝑥 × 𝑥 1 × 𝑥 × 𝑥 1 × 𝑥 1 1

𝑥

1

𝑥 × 𝑥

1

𝑥 × 𝑥 × 𝑥


Cuisenaire Rods

https://www.ncetm.org.uk/files/110720776/ncetm_ks3_cuisenaire_rods.pdf This

document from the NCETM gives a good introduction to Cuisenaire rods with lots of

examples. Below are a few more ideas you can try. I recommend that you try these

for yourself with a set of rods or using interactive rods online.

https://mathsbot.com/manipulatives/rods

Fractions

If light green represents 1 what do the other rods represent?

If yellow represents 1 what do the other rods represent?

If green represents 1 what do the other rods represent?

If orange represents 1 what do the other rods represent?

If red represents 1/5 what do the other rods represent?

If red represents 1, show me 2 of ½ and ½ of 2.

If light green represents 1, show me 2 of 1/3 and 1/3 of 2.

This is a nice task to try with years 7 to draw out prior knowledge and

misconceptions. You can link this to equivalent fractions, improper fractions,

decimals and percentages.

Create your own fraction walls.

https://www.ncetm.org.uk/files/110720776/ncetm_ks3_cuisenaire_rods.pdf

https://mathsbot.com/manipulatives/rods


Dividing Fractions

1 ÷ 1

4 How many lots of

1

4 make 1? 4 lots of

1

4 make 1

1 ÷ 1

2 How many lots of

1

2 make 1? 2 lots of

1

2 make 1

1

4 ÷ 1 How many lots of 1 make

1

4 ?

1

4 lots of 1 make

1

4

1

2 ÷ 1 How many lots of 1 make

1

2 ?

1

2 lots of 1 make

1

2

1

2 ÷

1

4 How many lots of

1

4 make

1

2 ? 2 lots of

1

4 make

1

2

1

4 ÷

1

2 How many lots of

1

2 make

1

4 ?

1

2 lots of

1

2 make

1

4

1

3 ÷

1

2 How many lots of

1

2 make

1

3 ? 1

1

2 lots of

1

3 make

1

2

1

3 ÷

1

2 How many lots of

1

2 make

1

3 ?

2

3 lots of

1

2 make

1

3


Dividing by 1 is easy, so use equivalent fractions to demonstrate why you

multiply by the reciprocal. KFC with real depth of understanding!

1

3÷

1

2 ≡

1

31

2

≡ 1

3×

2

11

2×

2

1

≡ 1

3×

2

1

1 ≡

1

3 ×

2

1 ≡

2

3

1

2÷

1

3 ≡

1

21

3

≡ 1

2×

3

11

3×

3

1

≡ 1

2×

3

1

1 ≡

1

2 ×

3

1 ≡

3

2



3

4÷

2

3 ≡

3423

≡

34

×32

23

×32

≡

34

×32

1 ≡

3

4 ×

3

2 ≡

9

8

Have a google search for Cuisenaire rods and you will find lots more on fractions.


Averages

Introduce the three averages and range

What is the average length of the rods?

What is the range?

Put them in order of length and replace the five rods with five rods of the same

length.

This is the mean average.

There are more yellow rods. This is the mode.

The middle rod is yellow. This is the median.

The range is the difference between the largest rod and smallest rod.

Get 5 different rods so that the mean, median and mode are 5. What is the range?

Select any 5 rods and find the mean, median, mode and range.

Select 6 rods and find the mean, median, mode and range.

http://mathmanipulatives1.yolasite.com/resources/Grab%20Bag%20Cuisenaire%20

Rods%20Lesson%20Pre-made%20Lesson%20Plan.pdf

Greater Depth Questions

Select any 5 rods and find the mean, median, mode and range.

Double the length of each rod and find the new mean, median, mode and range.

What do you notice?

Add 2 to the length of each rod and find the new mean, median, mode and range.

What do you notice?

http://mathmanipulatives1.yolasite.com/resources/Grab%20Bag%20Cuisenaire%20Rods%20Lesson%20Pre-made%20Lesson%20Plan.pdf

http://mathmanipulatives1.yolasite.com/resources/Grab%20Bag%20Cuisenaire%20Rods%20Lesson%20Pre-made%20Lesson%20Plan.pdf


Compare Pictogram and Bar Chart

A packet of chocolate chip cookies contains

2 cookies with one chip

4 cookies with 2 chips


1 cookie with 4 chips


15 cookies in total

Make a pictogram to represent the number of chocolate chips per cookie

Compare with the bar chart. What does each bar represent?

Find the mean, median, mode and range of the number of chocolate chips per

cookie.

Which diagram was most useful?

Number of chips

Number of cookies

1 2

2 4

3 5

4 1

5 3

15


Pie Chart

See http://www.croxbyprimary.co.uk/wp-content/uploads/2018/05/Year-6-

Summer-Block-3-Statistics.pdf for more ideas

Make a bar model on squared paper and cut it out.

What proportion of the cookies have 3 chocolate chips?

Make a circle with the bar and mark off each bar on the circumference of the circle.

Now make a pie chart by drawing a line from each mark to the centre of the circle.

Number of chips

Number of cookies

Pie chart

1 2 48

2 4 96

3 5 120

4 1 24

5 3 72

15 360°

Number of chips per cookie

1 2 3 4 5

http://www.croxbyprimary.co.uk/wp-content/uploads/2018/05/Year-6-Summer-Block-3-Statistics.pdf

http://www.croxbyprimary.co.uk/wp-content/uploads/2018/05/Year-6-Summer-Block-3-Statistics.pdf


Create your own bar chart and pie chart for this data.

Number of chips

Number of cookies

Pie chart

1 3

2 5

3 6

4 10

5 6

360°

What is the modal number of chips per cookie?

What is the median number of chips per cookie?

What is the mean number of chocolate chips per cookie?

What is the range?

Interpreting Pie Charts

9 cookies have one chocolate chip.

How many cookies are there?

What fraction of the cookies have 5

chocolate chips?

Number of chips per cookie

1 2 3 4 5

100°


From frequency tables to estimated mean

How do you work out the mean, median

and modal length of this set of rods?

Mode means most – there are more rods of length 3 so mode or modal length is 3cm

Median means middle – first we must put the rods in length order.

There are 5 rods, so the middle is the 3rd rod and it has length 3.

Median length is 3cm.

Mean

These 5 rods of different lengths can be replaced with 5 rods of size 3 cm.

To calculate the mean add to find the total length of all the rods and divide by the

number of rods.

Mean = 1 + 2 + 3 + 3 + 6 = 15 = 3

5 5


Frequency Tables

Length of

rod frequency

1 1

2 1

3 2

6 1

Length of

rod frequency

Total length

of rods

1 1 1x1 = 1

2 1 2x1 = 2

3 2 3x2 = 6

6 1 6x1 = 6

5 15

Which diagram is most helpful to find the mean, median, mode?


Grouped Frequency

Length of

rod (r) frequency

0≤r<4 1

4≤r<8 2

8≤r<12 1

Which set of rods best represents the table and why?

Which set of rods does not represent the table and why?

This activity leads to a nice discussion about midpoint as an estimate.

Length

of rod (r) frequency

Total

length of

rods

0≤r<4 1 2x1=2

4≤r<8 2 6x2=12

8≤r<12 1 10x1=10

4 20

If students can visualise what the table represents, they can relate it back to prior

knowledge of averages.


Sequences All diagrams are created using SMART notebook

Constant – stays the same in every term Variable – varies (changes) in every term

Possible misconceptions

T: How many blocks are in the 100th term?

S: 200n + 1

T: How do you know?

S: Two lots of 100 and 1

T: What does n represent?

S: 100 blocks

T: So how many blocks in total?

S: 201

T: How many blocks in the 17th term?

Spend a lot of time on this problem.

Make sure all understand that

the blue block is the constant

term +1.

n is represented by the

Cuisenaire rods. n represents

the position in the sequence. n is variable.

2 is represented by 2 rods.

The manipulatives help students

to understand the meaning of

the key vocabulary of constant

and variable

Students may come up with an

expression for the nth term but

on later questioning do not

really understand it.

Once students have the nth

term, they want to keep the n

and not substitute it for the term number.


Now repeat with sequences with a similar structure until you are sure that all

understand the structure of the sequence and the meaning of each term.


Change the structure of the sequence.


Greater Depth

This deepens understanding of constant, variable and term. This will stop students

from over generalising and really think about position and nth term.

Equivalent expressions

4(n + 1) + 1 ≡ 4n + 4 + 1 ≡ 4n + 5


Sequences with negatives

*Hot and cold blocks are another way of representing zero pairs

*


Now repeat with sequences with a similar structure until you are sure that

all understand the structure of the sequence and the meaning of each

term.

8n-2 6n + 2(n-1)

Can you see how students might see these two representations?


Greater Depth

Equivalent expressions

2n - 2 + 1 ≡ 2(n – 1) + 1

Which one is represented

here?

Can you represent the

other one using Cuisenaire

rods?

Ask students to show that these expressions are equivalent. Remind them of

expanding brackets if needed.


Different representations (from MEI)

What is the general term?


Different representations

Get the students to see this in their own way first and then discuss the 4 cases

above. Why are all 3 expressions equivalent?


(t + 2)2 – t2 ≡ t2 + 4t + 4 – t2 ≡ 4t + 4


Building sequences

The third term could be 5x3 + 3

5x2 + 3 works for the second term but it is a different structure

Alternative construction without Cuisenaire rods.

Constant term is 8, the initial 8 blocks.

The variable is the number of blocks added. Add 5 blocks to make the next term.

For the fifth term you add 4 lots of 5 because the first term has 8 blocks.

The 20th term is 8 + 5x19. The nth term is 8 + 5(n-1)

Expand to show that this is equivalent to 5n + 3

8 + 5(n-1) ≡ 8 + 5n – 5 ≡ 5n + 3


Bar models

The bar can be a valuable representation to enable students to represent problems in such a way that the mathematical structure is exposed.

This enables students to ‘see’ the problem clearly and to then recognise the strategy they need to solve the problem. NCETM

Avoid the common mistakes! Do not jump straight into the pictorial

representation.

Concrete Pictorial Abstract

It is important to go through all the stages from concrete to pictorial to abstract at the

start so that students can make sense of the problem and build up from something

concrete to an abstract method that they can use fluently.

ALL STEPS ARE IMPORTANT


Primary example from White Rose

Tim has 4 sweets and Ben has 2

sweets.

How many sweets do they have

altogether?

Maths No Problem – One of the recommended Primary Mastery Textbooks

https://mathsnoproblem.com/en/mastery/bar-modelling/

Concrete – real

life objects

Concrete –

handling real

objects

(manipulatives)

blocks,

Cuisenaire rods,

counters etc

Concrete – bar

model

Concrete - real

life objects

Concrete – bar

model

Pictorial – bar

model

https://mathsnoproblem.com/en/mastery/bar-modelling/


Choosing the right model

Part Whole Model Comparison Model

Part Whole Model

https://www.mathplayground.com/tb_fractions/index.html

Kayla bought 7/10 of a gallon of blue paint. She used ½ a gallon to paint the shed.

How much blue paint did she have left?

Comparison Model

https://www.mathplayground.com/ThinkingBlocks/thinking_blocks_decimals_percent_5.html

Thinking Blocks on mathplayground has a wide variety of interactive problems for

students to try along with videos for teachers and a modelling tool.

https://www.mathplayground.com/tb_fractions/index.html

https://www.mathplayground.com/ThinkingBlocks/thinking_blocks_decimals_percent_5.html


Ratio Question(s) – Introducing the bar model

Alfie and Billy go crabbing. Each bucket holds the same number of crabs. Alfie has

one bucket and Billy has 3 buckets. If Alfie has 3 crabs, how many crabs does Billy

have?

Step 1: Get students to use plastic cups and counters to model this.

Step 2: Model using counters

A

B

Step 3: Model using bars

Ans: Billy has 9 crabs

Try these questions using CPA.

Billy has 15 crabs. How many crabs does Alfie have?

There are 28 crabs in total. How many crabs does Billy get?

Billy has 20 more crabs than Alfie. How many crabs are there in total?

When students are familiar with bar models you can start with the bar

model and move on to develop fluency with abstract calculations and

algebra.

1 block = 3 b = 3

3 blocks = 3 x 3 = 9 3b = 9

Concrete - real

Concrete - counters

Pictorial – bar model

Abstract - calculation Abstract -algebra


This is an example of an SSDD question. Keeping the model the same and varying

the questions draws attention to the different types of question that could be asked at

GCSE.

Percentage Question(s) - From bar model to box model

(Van Hiele, proportion matrix, ‘Structure and Insight’ (1986), chapter 28) and

https://drive.google.com/file/d/19OsyPhuoKM_16rEV1cZmOHKLpNUG5WnX/view - Don Steward -

Median)

What is 20% of 160?

100% 160 100% → 160

20% →

20%

Start with single step problems and then move onto two step problems.


https://drive.google.com/file/d/19OsyPhuoKM_16rEV1cZmOHKLpNUG5WnX/view


What is 80% of 160?

100% → 160

80% →

20% of a number is 160. What is the number?

20% → 160

100% →

Start with single step problems and then move onto two step problems.


80% → 160

100% →



12% → 160

100% →

This is another example of an SSDD question. Keeping the numbers the same and

varying the questions draws attention to the different types of question that could be

asked at GCSE.

https://ssddproblems.com/32-2/


Try the following NCETM (adapted) problems using bar models.

Choose the correct model

Annotate the model – Calculation - Algebra

Year 6 Problems

1. Three quarters of a number is 54. What is the number?

2. There are 36 packets of biscuits. One half are chocolate, a ninth are digestive

and a third are wafer biscuits. The rest are ginger nuts. How many biscuits are

ginger nuts?

3. There is 20% off in a sale. How much would a track suit cost, if the normal

price was £44.50?

4. There is 20% off in a sale. The reduced price of the jeans is £36. What was

the original price?

5. At a dance there are 4 girls to every 3 boys. There are 63 children altogether?

How many girls are there?

6. Seven in every nine packets of crisps in a box are salt and vinegar. The rest

are plain. There are 63 packets of salt and vinegar crisps. How many packets

of plain crisps are there?


Key Stage 3 Problems

1. Ralph posts 40 letters, some of which are first class, and some of which are

second class.

He posts four times as many second class letters as first.

How many of each class of letter does he post? (This question appeared on a

GCSE higher tier paper.)

2. A computer game was reduced in a sale by 20% and it now costs £55 What

was the original cost?

3. Sally had a bag of marbles. She gave one-third of them to Rebecca, and then

one quarter of the remaining marbles to John. Sally then had 24 marbles left

in the bag. How many marbles were in the bag to start with?

4. Sam bakes a variety of biscuits.

13 are peanut, 12 are raisin, the remaining 5 were oat. If you choose 1 biscuit

at random, what is the probability that you will get an oat biscuit?

5. Tom spent 30% of his pocket money and put away 45% into his savings. He

was left with £2.50. How much pocket money did he receive?

6. Two numbers are in the ratio 4:5. They both sum to 135. Identify both

numbers.

7. Two numbers are in the ratio 5 : 7. The difference between the numbers is

12. Work out the two numbers.

8. A herbal skin treatment uses yoghurt and honey in the ratio 5 : 3. How much

yoghurt is needed to mix with 130 g of yoghurt?


AQA GCSE Exam Questions

Part whole

60° ° °

°


Comparison

150

?


Comparison and part whole

8

?


Comparison and fraction/percentages

110

30


Comparison and equivalent fractions

1/4

25


Comparisons with changes

Before change

After change

Number of girls remains the same. To compare like for like change 3:4 to 6:8

6



Compare a:b to b:c

To compare like for like b must be the same length (LCM).

5 : 6 and 8 : 11

20 : 24 24 : 33

a + b + c = 20 + 24 + 33 = 77

m : w 5 : 3 35 : 21

w : c 7 : 4 21 : 12

m : w : c = 35 : 21 : 12

35 are men 51.5% are men

68


5 : 6 : 7 18 blocks

7 : 9 : 8 24 blocks

The total number of blocks needs to be the same. LCM of 18 and 24 is 72

20 : 24 : 28 72 blocks

21 : 27 : 24 72 blocks

Rob ends with one more block


1:n and n:1

Should be

1.333333333333…


x:y = a:b to fraction


x:y = a:b to equation

Write as many equations as you can connecting r and g.

r = 2/3g

g = 1.5r g = 3/2r

3r = 2g

If 10b = 7c, b:c = 7:10

a : b 9 : 4 63 : 28

b : c 7 : 10 28 : 40

a : c = 63 : 40


Is a bar model always the most appropriate model?

Try to draw a model for 3a – 4 = 8 – a

Bar modelling & its use in maths teaching alongside

problem solving... and a little cuisenaire too! I hope this will be of interest to any maths teacher who wants to start including more

representation into their lessons, with this session focusing on bar modelling. The content mostly

used topics within the key stages 2 and 3 curricula, but I hope all maths teachers would find the

presentation of interest.

https://youtu.be/LewdrLgehjc, Tom Manners

https://youtu.be/LewdrLgehjc


Algebra Tiles for Equations


The algebra tiles work in a similar way to counters with the added bonus that they

also show the geometrical structure of the concept.

Remember zero pairs sum to zero:

(+1) + (-1) = 0

(+x2) + (-x2) = 0

(+x) + (-x) = 0

The manipulatives on mathsbot disappear when zero pairs are place on top of each

other.

Algebra tiles do not show equality and inequality. Before using algebra tiles to

solve equations represent equality and inequality using a bar model or

dynamic geometry.

x + 5 = 2x + 1

x + 5 > 2x + 1



x + 5 < 2x + 1

Dynamic Geometry https://www.geogebra.org/graphing/e7tx7nsp

Vector equations with sliders

Move the slider and to see how the value of each expression changes.

When is x + 4 = 2x + 3?

When is x – 4 = 2x + 3?

Forming an equation.

https://www.geogebra.org/graphing/e7tx7nsp


The bar (area) model does not work for negative

integers

The vector model will always

work but requires a lot of

thought.

Introducing the balance/elimination method

The bar model represents the equality of both

expressions well but falls down with

negatives.

The balance method leads more intuitively to the standard solving

equations balance method.

I have deliberately chosen to add negatives rather than subtract. See the earlier

section on opposites.


1) 2x – 1 = x +2

2) add -x to both sides

3) x – 1 = 2

4) add +1 to both sides

5) x = 3

Steps 2) and 4) show the zero pairs eliminating x’s and -1’s.

Add multiples of x’s, -x’s, 1’s or -1’s to both expressions to isolate x on one side of

the equation.

This method will always work and leads to the standard balancing method.

3 – x = x – 4

+x +x

3 = 2x - 4

+4 +4

7 = 2x

𝟕

𝟐 = 𝒙



Simultaneous Linear Equations Examples The elimination method is clear to see using algebra tiles and zero pairs

when the coefficients have opposite signs.

Solve 2x + y = 7 and x - y = 2

Add these two equations

y is eliminated and you are left with 3x = 9

x = 3

Substitute into 2x + y = 7

6 + y = 7

y = 1

x = 3 and y = 1

From here you can move on to multiplying one and then both equations before

eliminating by adding the equations.

Difficulties and misconceptions start when the coefficients are the same sign.


Method 1 – subtracting the two equations or finding the difference

Solve 2x + y = 5 and x + y = 1

Find the difference between the two equations or subtract.

x = 4

substitute into x + y = 1

4 + y = 1 y = -3 x = 4 and y = -3

Method 2 – elimination using zero pairs

Add the opposite of the second equation

Now add to eliminate y


Method 2 also avoids the difficulties that arise in questions that involve

subtracting negatives.

Solve 2x + y = 1 and x + y = -2



x = 3 → 2x + y = 1 → 6 + y = 1 → y = -5

Solve 2x – y = 1 and x – y = -2



x = 3 → 2x – y = 1 → 6 – y = 1 → y = 5

This site has more examples of algebra tiles -

http://www.greatmathsteachingideas.com/2015/04/04/algebra-tiles-from-counting-to-completing-

the-square/

http://www.greatmathsteachingideas.com/2015/04/04/algebra-tiles-from-counting-to-completing-the-square/

http://www.greatmathsteachingideas.com/2015/04/04/algebra-tiles-from-counting-to-completing-the-square/


Introducing Quadratics – small steps The representation needs to draw out the concept and expose the

structure.

Start by looking at rectangles as arrays and rectangles made from unit squares.

Make as many rectangles as you can with 12 units.

4

3 x 4 ≡ 12 3

This draws out the concept of factors. Factor x factor = product.

Linear Expressions x + 2

Now introduce units and x’s.

Show me 3x + 6 3

Show me 3x + 7

Which expression will factorise? Why?

3(x + 2) ≡ 3x + 6

Show me 5 expressions that will factorise and write them in expanded and factorised

form.

http://www.google.co.uk/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwjAkavlwdjSAhUESRoKHUUIA94QjRwIBw&url=http://rps2ag2014.primaryblogger.co.uk/2015/03/11/hip-hip-array/&bvm=bv.149397726,d.ZGg&psig=AFQjCNEDLM12Jor5yLAl3jpw_uB6mXACDg&ust=1489667430227643


Quadratic Expressions

Now introduce units, x’s and x². x + 2

Show me x2 + 5x + 6 and x2 + 5x + 3.

Which expression will factorise and why?

x

+

3

(x + 3)(x + 2) ≡ x² + 5x + 6

Show me 5 other expression that will factorise.

Compare the expanded form and factorised form – what do you notice?

Make lots of rectangles and write the factorised and expanded form on the board.

Only use positives at this stage. Students will begin to make generalisations.

Generalise

(x + a)(x + b) x2 + (a + b)x + ab

Compare the grid method to tiles. You could also compare to FOIL.

What’s the same, what’s differerent. Which method do you prefer? Why?


Introducing negatives.

At this stage students should be ready to use the grid method and do not

have to think about negative area.

(x + 5)(x + 2) ≡ x² + 7x + 10

(x + 5)(x - 2) ≡ x² + 3x – 10

(x - 5)(x + 2) ≡ x² - 3x – 10

(x - 5)(x - 2) ≡ x² - 7x + 10

What’s the same and what’s different?

Keeping the numbers the same and changing the sign allows students to focus on

the effect of changing the sign. Working memory is freed up allowing space for

mathematcial thought rather than procedural calculations.

First step to factorise quadratics

What’s the same and what’s different?

Are there any more with constant term 24?

…

How many can you make with constant term 12?


How many can you make with constant term -12?

When is the coefficient of x

positive?

Generalise

x2 + ax – 12 What values of a will give you a quadratic that factorises?

x2 - bx + 12 What values of b will give you a quadratic that factorises?

How many ways can you make quadratics that factorise for each expression?

Factorise each expression. What is special about the circled quadratic

expressions?

Students always forget that 1 is a factor!


Difference of Two Squares

What two values of x sum to 0?

You may need to remind students about zero pairs.

x² - 9 ≡ x² -0x – 9 ≡ (x – 3)(x + 3)

Generalise

(x + a)(x - b) = x2 + px - q when a b

(x + a)(x - b) = x2 - px - q when a b

(x + a)(x - b) = x2 - q when a b

This is another representation of the difference of two squares from

http://www.greatmathsteachingideas.com/2015/07/12/difference-of-two-squares-visual-

representation/

http://www.greatmathsteachingideas.com/2015/07/12/difference-of-two-squares-visual-representation/

http://www.greatmathsteachingideas.com/2015/07/12/difference-of-two-squares-visual-representation/


Compare the graph to the expanded and factorised form of

the quadratic.

y = (x + 3)(x + 2)

y = x2 + 5x + 6

This representation highlights that 6 is constant and x is variable

Zero’s and Solving Quadratic Equations

Zero pairs

When x = -2 and x =-3 the rectangle disappears!

x² + 5x + 6 = 0

(x + 3)(x + 2) = 0

x = -3 or x = -2


Completing the Square

In both sets of questions, start with the perfect square and then add on or take away

1’s. This is an example of variation (change one thing) or intelligent practice. The

practice leads to a generalisation.

Try this activity using algebra tiles on mathsbot. This is a short video to get you on

your way. https://youtu.be/0F83jFoxfXE

https://youtu.be/0F83jFoxfXE


Expand Three Linear Expressions

Expand (x + 3)(x + 4)(x + 1) using Dienes blocks

Plan: (x + 4)(x + 3) = x2 + 7x + 12

Front: (x + 3)(x + 1) = x2 + 4x + 3

Side: (x + 4)(x + 1) = x2 + 5x + 4

(x + 3)(x + 4)(x + 1) ≡ x3 + 8x2 + 19x + 12

x2 7x 12

x x 1

x2 7x 12

x3 7x2 12x 0

x3 8x2 19x 12


Misconception Busters Misconceptions are firmly held beliefs stuck in the long term memory. To

uncover these misconceptions is hard and often left to chance and to undo them has

its own challenges.

“If a schema contains incorrect information – a misconception or an incomplete

model of how a process works – we can’t simply overwrite it. A more primitive

schema can return to dominate unless we unpick and fully re-learn a correct

schema”

Rosenshine’s Principles in Action, 2019, Tom Sherrington

In an ideal world we would show the concept and non-concept at the same time

when the concept if first introduced so that students are able to form a complete

schema. In the real world students enter our classrooms with different experiences

and misconceptions. Some of these have been stored in the long term memory for

years.

Diagnostic questions are an excellent way of exposing common misconceptions.

The uncommon ones are harder to find. Show me questions can often help dig a

little deeper. Giving the “answer” and asking for the method forces that

conversation.

Example

Show me – 2 – 7 = -9

opp 2 and opp 7 simplify to opp 9

negative 2 subtract positive 7 using zero pairs

“The temperature is negative 2 degrees and it gets 7 degrees colder.”


The Child and Mathematical Errors

Experience – Children bring to school different experience. Mathematical errors may

occur when teachers make assumptions about what children already know.

Expertise – When children are asked to complete tasks, there is a certain

understanding of the basic ‘rules’ of the task. Cockburn (1999) takes an example

from Dickson, Brown and Gibson (1984, p331). Percy was shown a picture of 12

children and 24 lollies and asked to give each child the same number of lollies.

Percy’s response was to give each child a lolly and then keep 12 himself.

Misconceptions may occur when a child lacks ability to understand what is required

from the task.

Mathematical knowledge and understanding – When children make errors it may

be due a lack of understanding of which strategies/ procedures to apply and how

those strategies work.

Imagination and Creativity – Mathematical errors may occur when a child’s

imagination or creativity, when deciding upon an approach using past experience,

may contribute to a mathematically incorrect answer.

Mood – The mood with which a task is tackled may affect a child’s performance. If

the child is not in the ‘right mood for working’ or rushed through work, careless errors

may be made.

Attitude and confidence – The child’s self-esteem and attitude towards their ability

in mathematics and their teacher may impact on their performance. For example, a

child may be able in mathematics but afraid of their teacher and therefore not have

the confidence to work to their full potential in that area.


The Task and Mathematical Errors Mathematical complexity – If a task is too difficult, errors may occur

Presentational Complexity – If a task is not presented in an appropriate way, a

child may become confused with what is required from them.

Translational Complexity - This requires the child to read and interpret problems

and understand what mathematics is required as well as understanding the language

used.

“ ‘When it says here, ‘Which angel is the right angel?’ does it mean that the wings should go this way, or that way?’ “ (Dickson et al, 1984 IN Cockburn 1999)

If the task is not interpreted correctly, errors can be made.

The Teacher and Mathematical Errors Attitude and Confidence – As with the child, if a teacher lacks confidence or

dislikes mathematics the amount of errors made within the teaching may increase.

Mood – With the pressures of teaching today, teachers may feel under pressure or

rushed for time and not perform to the best of their ability.

Imagination and Creativity – Where a teacher is creative, they may teach concepts

in a broader manner, looking for applications and alternative approaches thus

reducing the probability of error in learning.

Knowledge – Too much teacher knowledge could result in a teacher not

understanding the difficulties children have whereas too little knowledge could result

in concepts being taught in a limited way.

Expertise – Expertise not only in subject matter but also in communicating with

children and producing effective learning environments. Without this expertise, some

pupils’ mathematics may suffer.

Experience – Knowledge can be gained from making mistakes. Teachers may learn

about children’s misconceptions by coming across them within their teaching.

Misconceptions with the Key Objectives - NCETM

https://www.ncetm.org.uk/public/files/2042723/Misconceptions+with+the+Key+Objectives2.doc


Unpicking Misconceptions

Here are a few unpicking examples:

Standard Index Form

253000 = 253 x 103 or 2.53 x 103 or 2.453

See this section for unpicking tips

Time

3.2 hours = 3 hours 20 minutes

3 hours 15 minutes = 3.15 hours

Represent the clock as a fraction to highlight

this error and show 20 mins as 20/60.

Angles in parallel lines


Numerical answer without reason.

Give students questions that do not need a numerical answer until they are secure

with the reasoning.

https://www.ncetm.org.uk/files/108968564/ncetm_ks3_cc_6_1.pdf

a = b

because………………………………………………………………………………………

c = b

because………………………………………………………………………………………

d = e

because………………………………………………………………………………………

d = f

because………………………………………………………………………………………

a = c

because………………………………………………………………………………………

f = e

because………………………………………………………………………………………

Find the value of each anlge by measuring the least number of angles with a

protractor.

a

e bc d

f

https://www.ncetm.org.uk/files/108968564/ncetm_ks3_cc_6_1.pdf


0.5 x 0.5 = 2.5

Use the area model.

1 0.5

0.5

1

This shows it must be Use 100 square ¼ = 0.25. to show that It also shows it must be <1 each small square is 0.1.

Ratio and fractions

1:3 = 1/3

Use a part whole and comparitive bar model to highlight the ratio and the fraction


Squaring Misconceptions

32 = 6 (3x)2 = 3x2 (x+3)2 = x2 + 9

Make a square of length 3

Make a square of length 3x

Make a square with length (x+3)

The visual representation challenges the misconception.

For more misconceptions see:

https://www.resourceaholic.com/p/misconceptions.html

https://www.risingstars-uk.com/blog/january-2018/overcome-common-

misconceptions-in-maths

https://www.resourceaholic.com/p/misconceptions.html

https://www.risingstars-uk.com/blog/january-2018/overcome-common-misconceptions-in-maths

https://www.risingstars-uk.com/blog/january-2018/overcome-common-misconceptions-in-maths


Algebra Tile Templates to match mathsbot.com


Documents

Teaching for Mastery Representation and Structure