### 1 | Explore Proportions By Mixing Paints

In this lesson, students work on a task about pink paint, which is made from a mixture of red paint and white paint.

How much red paint do you need to add to pink paint which is 60% red to turn it into pink paint which is 80% red?

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### 2 | Explore Probability by Experimenting with the Rolls of Dice

People frequently vastly overestimate or underestimate many everyday risks, which can lead to poor decision making.

One way to estimate the probability of something happening is to carry out an experiment and obtain some data on how many times it occurs out of a certain number of trials. The proportion of occurrences (relative frequency) gives an estimate of the probability, and this estimate generally improves the more trials that you do.

In this lesson plan by Colin Foster, students will carry out an experiment to estimate the probability that when three ordinary dice are thrown, they will show consecutive numbers. Students will explore how their estimates change as more and more throws are included, leading to an appreciation of the law of large numbers.

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### 3 | Use Puzzles to Show Students what Changes in Percentages Really Mean

Percentages are commonplace in everyday life, but can be sometimes be confusing and difficult to interpret. In this lesson plan by Colin Foster, students get to grips with a series of puzzles aimed at showing what changes in percentage actually mean…

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### 4 | Can Students Carry Out A Probability Experiment And Explain The Results?

If you choose which student will answer a question by drawing lolly sticks bearing their names from a cup, how many questions would need to be asked before everyone has a chance to answer? That’s the subject of this KS3 maths lesson that sets up a problem solving activity on probability.

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### 5 | Improve Mathematical Reasoning with a Real-Life Problem: How Many Sweets Can I Buy?

Working out how many large and small bags of doughnuts to buy so as to get your treats as cheaply as possible involves students in some careful mathematical reasoning.

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### 6 | Help Learners Understand Adding And Subtracting Directed Numbers By Linking To The Topic Of Electricity

Lots of learners have heard that ‘two minuses make a plus’ but may not apply this rule correctly or have much sense of when or why it is the case. In this lesson, links are made to the science topic of electricity to help learners understand adding and subtracting directed (positive and negative) numbers.

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### 7 | Gain Essential Skills in Simplifying Algebraic Expressions without Repetitious Drills

‘Simplifying Expressions’ lesson plan and accompanying task sheet (originally published in volume 5, issue 2 of *Teach Secondary* magazine).

Exercises are not the only way to improve students’ fluency in mathematical procedures – sometimes the very same skills can be developed in a more exploratory fashion.

Expanding brackets and collecting like terms to simplify expressions are perhaps not the most exciting of mathematical topics, but fluency in these skills is needed in order to solve equations and engage with more stimulating mathematical problem solving. Unless students take the time to master these skills, they will be disadvantaged later – but how can we avoid lessons on algebraic manipulation descending into endless mindless drill and practice?

One way is to engage students in devising expressions that will simplify to produce a given result – in this lesson, the expression ‘5x + 8y’. Restricting the possible expressions that can be combined to make this to five given linear binomials forces students to engage in some careful trial and improvement.

To find all the possible solutions, they will have to engage with negative coefficients, leading to plenty of opportunities for strengthening their skills in expanding and simplifying algebraic expressions.

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### 8 | Develop an Understanding of Numbers by Exploring What Makes Prime Numbers Special

Prime numbers are all around us, and they’re important, yet some are more important than others. Help your pupils discover how and why that is, using this lesson plan from Colin Foster.

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### 9 | Find The Centre Of Enlargement For A Given Shape

Students can often learn to perform a procedure successfully without much understanding of what they are doing – let alone why they are doing it! A good way to help them get a deeper understanding is to present them with an inverse task, where they have to try to do the opposite process.

This is usually harder and, unless they have learned it as a standard procedure, requires some careful thinking to unpick what they know. In the topic of enlargements of 2D shapes, students are normally asked to enlarge a given shape with a given scale factor about a given centre of enlargement, or to describe in these terms an enlargement that has already been drawn.

Sometimes they are asked to find the centre of enlargement or scale factor for a given enlargement.

In this lesson, students are asked to decide where the centre of enlargement can go for a given shape with a scale factor of 3 enlargement so that no part of the image will go off the edge of the grid. Solving this problem involves students in intelligent trial and error as they experiment with various possible centres of enlargement in order to locate the locus of possible positions.

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### 10 | Practise Estimating By Converting One Million Pounds Into Coins

A million pounds sounds like a lot of cash, but exactly how much is it really? Converting a vague notion of ‘wealth’ or ‘riches’ into actual pound coins can be an extremely useful and engaging way of encouraging learners to carry out estimations involving large numbers, mass and volume.

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### 11 | Spend A Lesson Exploring One Arithmagon Problem

Learning mathematics is about making sense of mathematical situations. These puzzles provide an interesting context in which students can make predictions and conjectures and develop their powers of mathematical reasoning.

In this lesson students will investigate the relationship between numbers in a diagram and justify their reasoning.

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### 12 | Use This Trial-And-Error Activity To Practise Multiplication

Multiplication is a fundamental operation in mathematics which every student needs to understand and be comfortable with. Although nowadays we have computers and calculators to do routine calculations for us, we still need to understand what multiplication does and when to use it.

It is also important to have a sense for what size an answer should be, so that we can make useful estimates when accuracy is not essential and spot errors when using technology.

In this lesson, students have to try to make the maximum product possible by making two numbers from the digits 1 to 9 and multiplying them together. This task allows students to generate lots of practice through trial and error but also focuses their thinking on place value and the optimal positions for the highest-value digits.

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