The focus of all of the resources on this website is to promote conceptual understanding by starting with context first. This makes them ideal low threshold, high ceiling lessons. Please read the notes below on how to use them. These resources and this idea is new and untested so feedback is welcomed! Please visit the website for more info on how to use these resources. (Some resources are borrowed or adapted from other places - Credit where it's due)

The focus of all of the resources on this website is to promote conceptual understanding by starting with context first. This makes them ideal low threshold, high ceiling lessons. Please read the notes below on how to use them. These resources and this idea is new and untested so feedback is welcomed! Please visit the website for more info on how to use these resources. (Some resources are borrowed or adapted from other places - Credit where it's due)

The ‘Why’: Why does subtraction work the way it does?
This lesson starts with a worded problem about people’s heights. This is to introduce the idea of a real world application of subtraction as well as unit conversion and problem solving skills. Next is an introduction to inverse operations. This explained as being one of the most beautiful and simplistic ideas in all of mathematics; that anything that can be done one way can also be done in reverse. This is shown through bar models and fact families.
As an extension of the work on addition, mental subtraction techniques such as partitioning and compensating are covered as an ‘I do, you do’ followed by some practice questions of each. Then, move on to written subtraction. This is one of very few slides where the answers haven’t been provided. However, a place value grid has been provided to talk through the answers and techniques. There is then some practice (with grids) that does include the answers and some problem solving questions as extension.
There is then a slide addressing a misconception that is often built in by teachers; that you “can’t take 3 from 2”. This isn’t strictly speaking true. If you take 3 from 2, you get negative 1. There is then an example of how, if you are secure in your understanding of place value, you can subtract using these numbers.
There is then explanation and practice for subtracting decimals. Although this will have been modelled earlier, this will be the students first chance to practice. Again, the first screen is an “I do, you do” (without answers) and then practice (with answers). There is then an example of how bar models and fact families can be used to solve algebraic expressions and lastly, some problem solving tasks using algebraic skills.
Activities included:
Worded height starter
The beauty of inverse functions
Partitioning and compensating mental subtraction
Written subtraction practice
Problem solving
Place value subtraction
Subtracting decimals practice
Using bar models for algebra
Algebra problem solving

The ‘Why’: Why does multiplication work the way it does?
This lesson starts with an exercise designed to get students to recognize that multiplication is a way of simplifying repeated addition. It is then followed by a refresher of the commutative and associative laws and why they work.
Similar to the addition and subtraction lessons, this is then split into mental and written techniques. Mental techniques covered include; doubles, using 10’s, hand tricks and partitioning. Students should be reminded that, whilst they may be able to answer quickly and another way, these are good techniques for them to have in their back pockets. The aim is to give a technique for as many time tables as possible. Each slide features an “I do, you do” example, followed by a time trail to see how many students can complete in one minute.
Following this is the written techniques that includes: bar models, grid method, lattice method and lines. Some of these descriptions are slight rewording of popular techniques. Grid method is the method by which the place value of each number is split and put in a table. The lattice method, is similar but features diagonal lines to give two digit answers followed by diagonal addition to give each place value. Lastly, line is the Japanese method of drawing a single line to represent each place value, followed by another line across is and then counting the points at which they cross.
The deliberate decision was made to not include column method as it will likely have been covered in primary and often leads students to an incorrect answer.
Towards the end of the lesson, there are some techniques for decimal multiplication including using similar sums and estimating followed by practice using any method. At the end is a mix of problem solving tasks including worded and spot the mistake problems.
Activities included:
Repeated addition starter
Commutative and associative law refresher
Mental multiplication methods
Written multiplication methods
Decimal Multiplication
Estimating and Using Similar Sums
Mixed Problem solving

The ‘Why’: Why do we use the current number system?
Because Place value is taught to Primary students, many come in to lessons with a working understanding of ‘what’ place value is and ‘how’ it works.
What is often not made clear, is the motivations behind it.
The early part of this lesson gets students to understand that numbers (as we think of them today) are in fact symbols that represent a value and that many other systems existed before this.
It then gets students to understand why it would be inconvenient to have a new symbol for every single number and how handy the positional notation system is.
Some students will go on to ask “Why do we count in tens?”
This leads nicely into talking about different bases and binary as extension.
Activities included:
Number Symbols from the past
Counting systems throughout history
Representing Number activity
Design your own Number system
Where our symbols came from
The History of 10
Positional Notation activity
Problem Solving Questions
Different Bases
Counting in Binary

Much of this lesson references the idea of thinking about division in terms of multiplication. As such, the lesson starts with an exercise designed to provide students with a complete set of multiplication grids. You will need to print slide 1 for students and hand them to them on the way in to lessons. The answers are on slide 2.
Then follows a true or false exercise designed to refresher students understanding of division. Do this using thumbs up or thumbs down across the room (They can always do in the middle if they aren’t sure). The next activity gets students to think of division in terms of worded sentences e.g. How many 5’s are there in 15? followed by a look at fact families. This is to get students to remember and understand the inverse relationship between multiplication and division.
Similar to the other arithmetic lessons, there are then mental and written methods of division. The mental methods of division are a series of divisibility tests and what to look for to see if a number will divide to give an integer answer. Provide students with a copy of the green grid on slide 7 and fill in the rules as they go along. It’s fun to do the number sort activities at the board with some board pens. When they have all the rules, they should attempt to complete the orange grid on slide 13. Bonus points for any students who can recognize that all the divisions can be completed but some will give a decimal answer.
To lead in to the written division techniques, first is a reminder of some of the literacy such as dividend, quotient and divisor and a visual demonstration of how division works as a method of grouping. There is then an “I do, you do” section to teach bus stop method. Most students should have seen this before. There is then a differentiated challenge. Students should challenge themselves to get as far as they can.
The next section is about dividing decimals including giving decimal answers, dividing a decimal by an integer and giving recurring decimal answers and some practice on these skills. A trickier extension is to ask students to explain how to divide by a decimal. This slide includes a visual explanation of why it works and some practice.
Lastly, there is some problem solving questions and a division dot to dot. Students will need a copy of slide 28 and 29. Students should start at an underlined question. They then need to join the question number to it’s answer. The answer then becomes the next question number until they reach a dead end. They should then start at the next underlined number.
Activities included:
Timetable grid starter
Division True or False
Division as a sentence
Mental Divisibility tests
Division Literacy
Written division explanation & practice
Mixed decimal division
Dividing by a decimal
Problem Solving
Division Dot to Dot

The ‘Why’: Why do we need to be able to add?
This lesson starts with students creating a spider diagram on what they think numbers are. Encourage them to think about what you can do with numbers? Some suggestions are included which could be revealed if students are struggling and prompt thoughts in other directions.
The term “Gut, data, gut” is used and is taken from a concept used in Marketing. It suggests that whenever money needs to be spent, you will have a rough idea (a gut instinct if you will) about how much something should cost. You then go seeking data to prove that and then realign this with your gut decision making. The example included is a simple scenario involving a shop.
Students will have some instincts about how to do mental addition. It is still important for them to understand the different techniques. There is a prompt to encourage this in students.
The final activity shows a total bill for four friends who went to lunch and ask them to check their gut feeling about how much they are being asked to pay.
Activities included:
What are numbers?
Gut, Data, Gut example
Commutative law Activity
Mental Addition Techniques
Associative law activity
Written Method Practice
Problem Solving
Estimating
Lunch at a café
Cryptarithms

The ‘Why’: Why do we count a value of less than zero?
This lesson starts by introducing the idea of a bank statement with money going and out of an account in different ways. At one point, a standing order for £100 comes out when only £99 exists in the account. It may be worth explaining what a standing order is although some will understand this implicitly. Debt is in fact the origin of negative numbers which is why the lesson starts here.
It then goes on to use other real life examples including temperature and moods.
Some mastery tasks are included in this from the White rose SOW including the number line and problem solving activities.
Activities included:
Bank Statement Starter
Temperature Explanation
Number Lines Activity
Temperature around the world
Adding & Subtracting Negative numbers
Mini Whiteboard Activity
Moods
Walking in a line to Multiply

The ‘Why’: Why do we count in 10’s?
This lesson builds on the understanding of Place value and includes a recap of this if the first place value lesson wasn’t used.
When asking students, “Why do we count in tens?” the suggestions around the room are often “Because we do” or “Because that’s the system that makes sense”. Students are often surprised to learn that it is likely due to the convenience of having 10 fingers.
Showing the pattern that leads to anything to the power of 1 and 0 also allows students to understand that this pattern goes on in both directions forever.
Once there is a good understanding of negative powers of 10, a task framing the usefulness of this to Motorsport lap times is included as extension. There is also a short introduction to standard form which students often see on their calculators.
Activities included:
Pocket Money Starter
The History of Number Systems
Place Value Recap
Counting in Tens
Definition of Powers
Multiplying by Powers of 10
Dividing by Powers of 10
Negative Powers
Standard Form
Motorsport