### 1 | Teach Highest Common Factor and Lowest Common Multiple Together to Show the Connection Between Them

‘The What Factor?’ takes the ideas of highest common factor and lowest common multiple, which are often taught separately, and deals with them together to help learners to appreciate the connections between them.

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### 2 | Use Algebra To Find Out Why A Number Trick Works

In this lesson, algebra is used to make sense of a number trick that the teacher performs.

Students are asked to explore the mechanics, experiment with some numbers, and then use algebra to try to probe exactly how and why the trick works.

It’s easy to check the rule on a few numerical examples by working them out, but a short piece of algebra helps students to see what is going on under the surface.

This lesson provides an opportunity to work on expanding brackets and simplifying algebraic expressions, as well as making sense of the final answer in terms of the place value of the original numbers.

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### 3 | Calculating Rate and Time by Exploring How Long it Takes Multiple People to Paint a Wall

This lesson is about calculating how long it will take to paint a wall if several people work at it at the same time. Will it take less time with more people helping? And if so then how much less?

Students need to make assumptions that are realistic to the real-world situation, but they also have to make assumptions that will make the calculations easy to do and give an answer that is close enough for practical purposes. Pupils often find calculations involving rates difficult. They may be tempted to add up the times rather than the rates, and this provides an opportunity for discussion and for deciding whether or not different possible answers obtained are reasonable.

This download contains one lesson plan guide and one accompanying task sheet.

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### 4 | Explore Probability by Playing Dice Games

In many everyday scenarios, we cannot calculate a theoretical probability of something happening.

Even some simple-sounding games involving dice are complicated to analyse and the easiest way to estimate the probability of a win is often to run a simulation on a computer.

In this lesson, students are asked to estimate the probability of winning a game in which they throw an ordinary dice and move a counter that many spaces forwards.

To win, they must ‘hit ten’ by landing on the tenth square from the start. This game is too complicated for students to work out the probability of winning theoretically, so they must estimate it by playing the game several times and seeing how often they win. This enables them to calculate how much money they might be willing to pay to play the game, if the prize for winning is £1.

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### 5 | Embed Adding and Subtracting Fractions into a Richer Problem to get the Technique to Stick

They’ve been taught the technique a dozen times or more…but through embedding it within a richer problem, ‘adding and subtracting fractions’ might finally make sense to students.

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### 6 | Link Maths To Real Life In This Problem Solving Lesson About Money

Mathematical questions about a photo in a carpark gives a real-life context to this maths lesson. Students will devise their own investigations about the photo that will involve money, algebra, and finding patterns. Logical thinking and careful arithmetic are also required in this KS4 lesson.

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### 7 | Quadratic Equations Bring Together A Host Of Different Mathematical Ideas And Techniques

Quadratic expressions are considerably more complicated to work with than linear expressions, and students often find them hard to handle.

Students may carry out factorising of quadratics by applying poorly-understood procedures that make little mathematical sense to them, and this can be especially so when it comes to non-monic quadratics (those where the coefficient of x2 is not 1).

In this lesson, students approach factorising non-monic quadratics by trying to find factorisations which will expand and simplify to produce a quadratic expression of a specified form.

This entails lots of useful practice at expanding pairs of brackets and collecting like terms, and also gives opportunity for students to unpick what is going on, so as to gain insight into how the inverse process of factorising works.

By working backwards in this way to obtain the necessary factors, students build a deeper understanding of factorising quadratics.

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### 8 | Help Students Understand When One Integer Will Divide Exactly Into Another

Understanding when one integer will divide exactly into another helps students to ‘make friends’ with the integers and feel more secure when carrying out calculations.

Even when students are using calculators, this sort of familiarity with factors enables students to estimate the sizes of answers (and such things as whether an integer should be obtained or not).

All of this provides valuable checks on the work that they are doing and can be more intellectually stimulating than merely following standard algorithms to get an answer.

In this lesson, students have the opportunity to experiment with producing ‘nice’ answers like 123 456 from multiplying integers.

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### 9 | Let Students Construct Their Own Quadratic Equations to Boost Mathematical Fluency

‘Connected Quadratics’ helps students become confident and fluent with techniques such as factorising quadratics by offering a scenario in which they need to construct their own quadratic equations in order to solve a bigger problem…

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### 10 | Shading Graphical Linear Inequalities by Locating Mystery Pairs of Coordinates

‘What’s the Point?’ teaches shading graphical inequalities, but not as a sequence of instructions to follow, as it is often done. Instead, asking learners to find a ‘mystery point’ can bring the process to life…

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### 11 | Show Students the Importance of Histograms by Illustrating the Limitations of Bar Charts

If your maths students are struggling to see the point of why we have histograms, this lesson devised by Colin Foster will quickly highlight the limitations of the bar charts they learnt about in primary school.

Students here are tasked with devising bar charts in a deliberately misleading way, by splitting data concerning visitor numbers at a sports centre into different categories. However, when they use the same data to plot histograms, it will soon become clear that their ability to show frequency density helps to communicate an accurate sense of what the data distribution really is.

**Click here to get this free lesson plan.**

### 12 | Use Flowcharts To Set Up Rules About Iterative Operations

Flowcharts are a highly visual way of communicating information and are used in this lesson to set up rules about operations that will be performed iteratively. Students are invited to explore what happens when they put different numbers into a given flowchart, and later to experiment with varying the features of the flowchart themselves. They will encounter some converging sequences and use algebra to predict and explain the outcomes.

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### 13 | Use A Problem-Based Approach To Recap Adding And Subtracting Fractions

Developing fluency with a particular mathematical technique to the point where students no longer need to think about the details can give them a powerful sense of mastery. This enables them to tackle more demanding mathematical problems that rely on that process and contributes to a wider confidence in the subject.

Adding and subtracting fractions is a procedure which students often find it very difficult to master, despite sometimes having been taught it every year from upper primary school to the sixth form. They’ll do it for a few lessons but then quickly forget it again for another year – particularly in this age of calculators and computers.

When revisiting this topic at Key Stage 4, therefore, it can bring up powerful memories of failure, so it is important to address the area without it feeling like an exact repetition of what they have done before.

One way to make the practice of a technique less tedious than a page of exercises is to embed it in a bigger problem which students are trying to solve. Having their attention on a grander purpose helps them to internalise the method to the point where they don’t have to think about it any more, which is always the goal of mathematical fluency.

A problem-based approach also means that the answers obtained have some significance for the wider problem, so students are more inclined to check their answers and notice if any of them doesn’t seem to be about the right size.

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### 14 | Make Sense Of The Mean And Find Efficient Ways To Calculate It

The language of averages is used all the time in the world around us – and budding mathematicians need to understand exactly what such terms represent.

Many students will know how to calculate the mean: they add up all the numbers and divide by how many there are. However, it is common to find that students are less sure about exactly what the value that they have calculated actually means.

This lesson helps to make sense of the meaning of the mean as well as exploring efficient ways to calculate it.

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### 15 | Use Ratio to Work out How to Carry Enough Food to Stay Alive in the Desert

Working out how to carry enough food to stay alive in a challenging environment poses interesting problems.

In this lesson, students use their knowledge of ratio to make sense of a real-world scenario in which enough food must be carried into the desert to allow a group of people to survive for a certain length of time.

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