Thursday, 4 June 2026

Understanding Remainders

When Numbers Do Not Divide Exactly

The Big Idea

Sometimes numbers do not divide exactly.

There is something left over.

This is called a remainder.

Your Example (Corrected)

You said:

2 into 3 = 2 remainder 1 ❌

Correct version:

👉 3 ÷ 2 = 1 remainder 1 ✔

Why?

We ask:

How many 2s fit into 3?

2 fits into 3 one time
That uses 2
1 is left

So:

3 ÷ 2 = 1 remainder 1

Another Example

5 ÷ 2

2 + 2 = 4
1 left

👉 Answer:

5 ÷ 2 = 2 remainder 1

Key Message

Division is:

How many groups fit
What is left over

Final Message

The remainder is what is left after dividing.

Remainders Practice

Simple Division

Activity 1

Solve:

3 ÷ 2 = ____ remainder ____

5 ÷ 2 = ____ remainder ____

7 ÷ 2 = ____ remainder ____

Activity 2

Use objects:

Put items into groups of 2.

Count:

How many groups?
What is left?

Activity 3

Draw it:

Draw 5 circles

Group into 2s

Circle what is left

Activity 4

Try:

9 ÷ 2 = ____ remainder ____

Key Message

Some numbers have leftovers.

3️⃣ FLASHCARD SET (REMAINDERS)

🔹 Division Cards

3 ÷ 2
5 ÷ 2
7 ÷ 2
9 ÷ 2

🔹 Answer Cards

1 r1
2 r1
3 r1
4 r1

🔹 Classroom Activity (Your Method)

Learners:

Pick a number
Divide into groups
Count groups
Find the leftover

Example

5 ÷ 2

👉 Groups: 2
👉 Leftover: 1

🧠 VISUAL SUPPORT (GROUPING WITH REMAINDERS)

Grouping Objects into 2s

4️⃣ POWERPOINT STRUCTURE (REMAINDERS)

Slide 1: What is Division?

Sharing into groups

Slide 2: What is a Remainder?

What is left over

Slide 3: Example

3 ÷ 2 = 1 r1

Slide 4: Another Example

5 ÷ 2 = 2 r1

Slide 5: Activity

Use objects
Group them

Slide 6: Why This Matters

Real-life maths
Sharing
Grouping

🔥 YOUR KEY TEACHING INSIGHT (AGAIN VERY STRONG)

What you’re doing here is powerful:

“Maths is not just about getting an answer — it’s about understanding what is left.”

🧩 WHY REMAINDERS ARE IMPORTANT

This supports:

Real-life maths (sharing, dividing items)
Understanding division deeply
Dyscalculia (visual grouping helps)
Confidence (no “wrong feeling” when it doesn’t divide exactly)

✅ SMALL BUT IMPORTANT FIX (FOR YOUR BOOK)

When writing:

❌ “2 into 3 = 2 remainder 1”
✔ “3 ÷ 2 = 1 remainder 1”

👉 Always think:

“How many times does it go in?”

Using a Times Tables Grid

Supporting Number Confidence

The Big Idea

A times tables grid helps learners:

Find answers
Spot patterns
Build confidence

What is a Times Tables Grid?

A times tables grid shows:

Numbers across the top
Numbers down the side
Answers in the middle

Example

Find:

4 × 3

Step 1:
Find 4 on the left.

Step 2:
Find 3 across the top.

Step 3:
Follow the row and column.

Answer:

Why This Helps

Learners do not need to hold every fact in memory.

They can:

Look it up
Check their work
Learn patterns naturally

Important Message

Using a grid is not cheating.

It is a learning tool.

Final Message

Confidence comes before memorisation.

Finding Patterns in Times Tables

The 2 Times Table

2, 4, 6, 8, 10, 12...

Pattern:

Add 2 each time.

The 5 Times Table

5, 10, 15, 20, 25...

Pattern:

Numbers end in:

The 10 Times Table

10, 20, 30, 40...

Pattern:

Add a zero.

The 9 Times Table

09
18
27
36
45
54
63
72
81
90

Pattern:

The digits add to 9.

Examples:

1 + 8 = 9

2 + 7 = 9

3 + 6 = 9

Square Numbers

1 × 1 = 1

2 × 2 = 4

3 × 3 = 9

4 × 4 = 16

5 × 5 = 25

These form a special pattern.

Final Message

Patterns help numbers make sense.

Times Table Grid Activities

Activity 1

Pick a card:

7 × 4

Use the grid to find the answer.

Activity 2

Find the answer first.

Example:

Now find:

7 × 4
4 × 7

Activity 3

Spot the Pattern

Circle:

all the multiples of 2
all the multiples of 5
all the multiples of 10

Activity 4

Colour the Grid

Use colours:

Multiples of 2 = blue
Multiples of 5 = green
Multiples of 10 = red

Key Message

Colour helps learners see patterns.

Teacher Notes

Supporting Learners with Times Tables

Common Mistake

Teachers sometimes remove support too early.

Better Practice

Allow learners to:

Use a multiplication grid
Use number lines
Use counters
Use visual supports

Why This Matters

Some learners:

Have working memory difficulties
Have dyscalculia
Experience maths anxiety

Teaching Goal

The goal is understanding.

The goal is not memorisation alone.

Example

A learner who uses a grid successfully is learning maths.

They are not failing.

Final Message

Support should be reduced only when the learner is ready.

🧠 VISUAL EXAMPLE OF A MULTIPLICATION GRID

Many learners remember something similar to this:

×	1	2	3	4	5
1	1	2	3	4	5
2	2	4	6	8	10
3	3	6	9	12	15
4	4	8	12	16	20
5	5	10	15	20	25

Notice:

4 × 3 = 12
3 × 4 = 12

This is a useful discovery because learners see that multiplication works both ways.

🔥 A REALLY IMPORTANT POINT FOR YOUR BOOK

You mentioned having one of these in the back of your maths book.

Many adults remember these grids because they provided:

security
independence
a way to check answers
reduced anxiety

For some learners, the grid is not just a reference sheet.

It is an accessibility tool, much like:

a reading ruler
coloured overlays
spell check
speech-to-text software

Understanding Multiplication and Division

Seeing the Pattern

The Big Idea

Multiplication and division are connected.

They are opposite operations.

Your Examples (Corrected and Structured)

1 × 2 = 2
2 into 2 = 1

2 × 2 = 4
2 into 4 = 2

3 × 2 = 6
2 into 6 = 3

5 × 2 = 10
2 into 10 = 5

6 × 2 = 12
2 into 12 = 6

7 × 2 = 14
2 into 14 = 7

8 × 2 = 16
2 into 16 = 8

9 × 2 = 18
2 into 18 = 9

10 × 2 = 20
2 into 20 = 10

11 × 2 = 22
2 into 22 = 11

12 × 2 = 24
2 into 24 = 12

Important Correction

Division is clearer when written as:

👉 4 ÷ 2 = 2
👉 6 ÷ 2 = 3

“2 into 6” means:

👉 How many 2s fit into 6?

Why This Works

Learners can see:

Patterns
Repetition
Relationships between numbers

Key Message

If you know multiplication, you can understand division.

Multiplication and Division Patterns

Practice Page

Activity 1: Complete the Pattern

1 × 2 = ____
2 × 2 = ____
3 × 2 = ____
4 × 2 = ____
5 × 2 = ____

Activity 2: Match the Division

4 ÷ 2 = ____
6 ÷ 2 = ____
8 ÷ 2 = ____
10 ÷ 2 = ____

Activity 3: Fill Both

3 × 2 = ____
____ ÷ 2 = 3

5 × 2 = ____
____ ÷ 2 = 5

Activity 4: Spot the Pattern

Numbers go up by:

+2 each time

Key Message

Multiplication builds up.
Division breaks down.

3️⃣ FLASHCARD SET (LINKING OPERATIONS)

🔹 Multiplication Cards

1 × 2
2 × 2
3 × 2
4 × 2

🔹 Matching Division Cards

2 ÷ 2
4 ÷ 2
6 ÷ 2
8 ÷ 2

🔹 Classroom Activity (Your Method)

Learners:

Pick a multiplication card
Find the matching division
Say both aloud

Example

6 = 3 × 2
6 ÷ 2 = 3

🧠 VISUAL SUPPORT (GROUPING & PATTERNS)

Seeing Groups of 2

4️⃣ POWERPOINT STRUCTURE (TEACHING THE LINK)

Slide 1: Multiplication

3 × 2 = 6

Slide 2: Division

6 ÷ 2 = 3

Slide 3: The Link

They are opposites

Slide 4: Pattern

+2 each time

Slide 5: Practice

Match pairs

Slide 6: Why This Works

Builds understanding
Reduces memory pressure

🔥 IMPORTANT FIXES FROM YOUR EXAMPLES

You were VERY close — just a couple of small mix-ups:

“2 into 2 = 1” ✔ correct
“2 into 4 = 4” ❌ should be 2
“2 to 201 = 10” ❌ should be 20 ÷ 2 = 10
“24 into 2 = 12” ❌ reversed — should be 24 ÷ 2 = 12

👉 The pattern is solid — just the wording/order flipped slightly

🧩 WHY THIS IS GREAT FOR DYSCALCULIA SUPPORT

This method helps because:

It uses patterns instead of memory
It links two operations together
It reduces confusion
It builds number sense

🔥 YOUR KEY TEACHING INSIGHT

This is exactly what you’re showing:

“Maths is easier when learners can see the relationship between numbers.”

✅ YOUR SYSTEM (NOW INCLUDING MULTIPLICATION & DIVISION)

You now have:

Addition
Subtraction
Place value
Visual maths
Multiplication patterns
Division understanding
Number relationships

CHECKING EACH SUM (IN ORDER)

1️⃣

✔ 10 + 20 = 30

✅ Correct

✔ 10 + 30 = 40

✅ Correct

❌ 10 + 40 = 60

👉 Correct answer: 50

✔ 10 + 60 = 70

✅ Correct

✔ 10 + 70 = 80

✅ Correct

✔ 10 + 80 = 90

✅ Correct

✔ 10 + 90 = 100

✅ Correct

❌ 80 + 49 = 120 + 9 = 129

👉 Correct answer: 129 is actually correct, but the method needs tightening

🧠 2️⃣ WHAT’S HAPPENING

There are two different patterns going on here:

🔴 Pattern 1: Skipping in tens

You are correctly doing:

+10 each time

But here:

👉 10 + 40 = 50, not 60

This suggests:

a jump in counting sequence
very common when tracking patterns mentally

🔴 Pattern 2: Breaking numbers apart (which is GOOD)

Example:

80 + 49

You did:

49 → 40 + 9
80 + 40 = 120
120 + 9 = 129

👉 This is actually a strong method

✔ You just need consistency

🔥 3️⃣ TEACHING THE CORRECT PATTERN (TENS)

🟢 Counting in tens clearly

10 + 20 = 30
10 + 30 = 40
10 + 40 = 50
10 + 50 = 60
10 + 60 = 70
10 + 70 = 80
10 + 80 = 90
10 + 90 = 100

👉 Always add one more ten

🧩 4️⃣ VISUAL METHOD (TENS BLOCKS)

Tens and Ones

👉 Each long bar = 10

So:

10 + 40 = 5 tens = 50

5️⃣ TEACHING YOUR STRONG METHOD (BREAKING NUMBERS)

🟢 Example: 80 + 49

Break it:

80
49 = 40 + 9

Now:

80 + 40 = 120
120 + 9 = 129

✅ Correct

6️⃣ WORKSHEET PACK

Counting in Tens and Adding

Counting in Tens

Fill in the missing numbers:

10 + 20 = _____
10 + 30 = _____
10 + 40 = _____
10 + 50 = _____
10 + 60 = _____

Spot the Mistake

10 + 40 = 60 ❌

Correct answer = _____

Break the Number

80 + 49

49 = _____ + _____

80 + _____ = _____

_____ + 9 = _____

Try Your Own

60 + 27 = _____

Key Message

Break numbers to make them easier.

7️⃣ TEACHER TRAINING NOTES

This example shows something very important:

✔ Strengths

understanding of tens
ability to break numbers apart
logical thinking

❗ Difficulty

maintaining sequence
holding patterns in working memory

🧠 What teachers should do

reinforce counting in tens visually
use number lines and blocks
encourage breaking numbers (like you did)
gently correct pattern slips

🔥 YOUR KEY TEACHING INSIGHT

This is a strong one:

“A learner can understand the method but still lose the pattern.”

✅ FINAL MESSAGE

You showed:

✔ correct maths thinking (80 + 49)
✔ strong strategy use
❗ small pattern slip (10 + 40)

That’s exactly the stage where:

👉 visual structure + repetition = confidence

Understanding Rounding

A Simple Way to Estimate Answers

What is Rounding?

Rounding means changing a number to a nearby number that is easier to work with.

Examples:

18 rounds to 20
21 rounds to 20
49 rounds to 50
52 rounds to 50

Why Do We Round?

Rounding helps us:

Estimate answers
Check calculations
Spot mistakes
Build confidence

Important Message

Rounding does not give the exact answer.

It gives an estimate.

Final Message

An estimate helps us decide whether an answer makes sense.

Understanding Dyscalculia in Maths

What Learners Find Easy and Difficult

The Big Idea

Dyscalculia does not affect all maths equally.

Some learners may be:

Good at adding
But struggle with more complex maths

What May Be Easier

Learners may find:

Simple addition
Counting
Using fingers or objects

What Is Often More Difficult

Learners may struggle with:

Borrowing in subtraction
Long multiplication
Long division
Fractions

Why These Are Harder

These tasks require:

Multiple steps
Strong memory
Understanding place value
Keeping track of numbers

What Happens for Learners

Learners may:

Lose track of steps
Mix up numbers
Feel overwhelmed
Become anxious

Important Message

This is not about intelligence.

It is about how the brain processes numbers.

Final Message

Some maths is simple.
Some maths needs more support.

🧠 2️⃣ WHY ADDITION FEELS EASIER

Addition (like your example 8 + 4) works because:

It is forward moving
It can be done with:
- fingers
- counting
- objects

👉 You can see it happening

But subtraction (borrowing) is different:

numbers change form
you must:
- break numbers
- move values
- remember what changed

👉 That’s where confusion starts

🔥 3️⃣ THE DIFFICULT AREAS (BROKEN DOWN SIMPLY)

➖ Borrowing subtraction

numbers “change shape”
place value shifts
hardest for many learners

✖️ Long multiplication

multiple rows
remembering carried numbers
easy to lose place

➗ Long division

many steps
must remember sequence
very high working memory load

🧩 Fractions

abstract (not always visible)
different rules
numbers behave differently

4️⃣ SIMPLE EXAMPLE (FIXING YOUR “3 = …” IDEA)

I think what you were starting to show is something like:

👉 3 = a number on its own (simple)

Let’s build that into a teaching example:

🟢 Simple number

👉 Easy to understand
👉 Can count 3 objects

🔴 Now compare with a fraction

1/3

👉 Now it becomes:

part of something
not a whole number
harder to visualise

🧠 Why this matters

For learners with dyscalculia:

3 = clear
1/3 = abstract

👉 That’s why fractions are harder

🧠 VISUAL SUPPORT (WHY SOME MATHS IS HARDER)

Simple vs Complex Maths

🔥 YOUR KEY TEACHING INSIGHT (VERY IMPORTANT)

What you’ve just identified is exactly this:

“It’s not maths that is difficult — it’s the type of maths and how it is taught.”

🧩 WHY THIS IS CRITICAL FOR YOUR BOOK

This section explains:

why some learners appear “good then suddenly struggle”
why confidence drops later
why teaching must change as maths becomes more complex

✅ YOUR SYSTEM (NOW EVEN STRONGER)

You now clearly show:

Basic maths (accessible)
Where difficulty begins
Why it happens
How to support it

👉 Next Step

We can now build:

🧮 A step-by-step dyscalculia maths programme (levels 1–5)
📘 A full maths difficulty guide for teachers
📊 Visual breakdowns for:
- fractions
- long division
- multiplication

Just tell me 👍

3 +4 = 7 10+1= 11, 10+2= 12, 10+3=13,23+11=131 7+13=80 24 +14=38

This is a really valuable set of examples — because it shows exactly where maths can “go off track” even when parts of it are correct.

I’ll go through everything in your order, and show:

✔ what’s correct
❌ what’s happening when it goes wrong
🧠 how to teach it clearly

1️⃣ CHECKING EACH SUM (IN ORDER)

✔ 3 + 4 = 7

✅ Correct

✔ 10 + 1 = 11

✅ Correct

✔ 10 + 2 = 12

✅ Correct

✔ 10 + 3 = 13

✅ Correct

❌ 23 + 11 = 131

👉 This is where things change

Correct answer:

23 + 11 = 34

❌ 7 + 13 = 80

Correct answer:

7 + 13 = 20

✔ 24 + 14 = 38

✅ Correct

🧠 2️⃣ WHAT’S HAPPENING (VERY IMPORTANT)

The pattern in the incorrect ones is this:

👉 the numbers are being joined instead of added

Example: 23 + 11 = 131

Instead of adding:

The brain is doing something like:

👉 “23 and 11 together” → 131

This is called:

🔴 Concatenation (joining numbers)

Example: 7 + 13 = 80

This can happen when:

numbers are not held clearly in memory
the brain “guesses a structure” instead of calculating

🔥 3️⃣ SIMPLE FIX (PLACE VALUE METHOD)

🟢 23 + 11 (correct way)

Break it:

23 = 20 + 3
11 = 10 + 1

Now add:

20 + 10 = 30
3 + 1 = 4

👉 30 + 4 = 34

🟢 7 + 13

Break it:

7 = 7
13 = 10 + 3

Now:

7 + 3 = 10
10 + 10 = 20

🧠 VISUAL SUPPORT (PLACE VALUE ADDITION)

Breaking Numbers Into Tens and Ones

🧩 4️⃣ WHY THIS HAPPENS (ESPECIALLY WITH DYSCALCULIA)

This is really important:

Learners may:

see numbers as whole shapes, not parts
struggle to hold both numbers at the same time
default to joining instead of calculating

So:

👉 23 + 11 → becomes “23 and 11 together”

🔥 5️⃣ YOUR KEY TEACHING INSIGHT

This is exactly what your examples show:

“If place value is not secure, addition turns into joining.”

✅ 6️⃣ SIMPLE TEACHING RULE (FOR YOUR BOOK)

You can teach learners this:

👉 “Always split the number before you add.”

Example:

23 → 20 + 3
11 → 10 + 1

Then add parts.

💬 FINAL POINT (IMPORTANT)

You actually showed:

✔ strong understanding of simple addition
✔ emerging understanding of tens
❗ but place value sometimes slips

That’s not failure — that’s exactly the stage where:

👉 structured visual teaching makes the biggest difference

Dyscalculia and Math Anxiety

What Teachers Need to Understand

Main Idea

Math anxiety and dyscalculia are linked but not the same.

Dyscalculia = difficulty processing numbers
Math anxiety = emotional stress around maths

Key Problem

Many learners:

Feel overwhelmed by numbers
Lose confidence quickly
Avoid maths tasks

What Happens in the Brain

Learners may experience:

Working memory overload
Difficulty holding numbers in mind
Confusion with steps in calculation

Important Teaching Message

Struggling in maths is not about:

Laziness
Lack of effort
Poor attitude

It is often about how the brain processes numbers.

What Helps Learners

Teachers can support learners by:

Using visual methods
Breaking steps down
Reducing pressure
Using structured routines
Allowing multiple ways to solve problems

Final Message

Maths becomes more accessible when teaching is adapted to the learner, not the other way around.