# Mr Mathematics Blog

## Setting up simultaneous equations

October 13, 2020

I created this question to challenge the students in my higher ability Year 11 class. The question connects simultaneous equations to density, volume, surface area and rearranging formula.

## Setting up simultaneous equations

Each individual topic is fairly straight forward on its own. The challenge is to connect them together within a single problem. To begin with, I posed the question to the class and asked the students to discuss in pairs or small groups a strategy for solving the problem.

After a few minutes we decided to work the problem backwards. To work out the surface area we need the radius. We work out the radius we can use the volume. To find the volume we need to work out the value of y. To work out y we need to set up and solve a pair of simultaneous equations.

At this point about half the class began working independently either on their own or in pairs to set up the two equations. The other half needed sone help setting up the two equations with x and y. I prompted them to consider the mass of spheres A and B and the volume of sphere C.

When the students had set up the simultaneous equations almost all the class were able to complete the problem with no further help.

I wanted this problem to emphasise the need for developing a strategy to solving a problem. By working the problem backwards students were able to identify the starting point and how it would lead on further workings.

## Teach this lesson

My name is Jonathan Robinson and I passionate about teaching mathematics. I am currently Head of Maths in the South East of England and have been teaching for over 15 years. I am proud to have helped teachers all over the world to continue to engage and inspire their students with my lessons.

October 6, 2020

## Modelling Direct Proportion Learning Objectives:

By the end of the lesson:

• All students should be able to use the constant of proportionality to describe how two measurements are in direct proportion.
• Most students should be able to model two units that are in direct proportion using the constant of proportionality.
• Some students should be able identify whether two units are in direct proportion using the constant of proportionality.

## Starter/Introduction

The starter is used to introduce the term direct proportion and the symbol α.  I ask the class to work in pairs to match together measurements that increase or decrease at the same rate.  This takes about five minutes before I ask students to present their work on mini whiteboards.

When feeding back to the class the matched pairs are shown using the proportionality symbol, for instance, Test Score ∝ Time spent revising.

## Modelling Direct Proportion

I explain in this lesson we are going to derive a formula to model two measure two measurements that are in direct proportion.  The model will involve a value we call the Constant of Proportionality (k).  The value of k describes the rate at which two measurements increase or decrease together.  We will use this model to calculate one value when the other measurement is known.

As you can see in the video below, I work through the first two questions with the class and ask them to attempt the third question on mini-whiteboards so I can assess their understanding.

I use the interactive Excel file to demonstrate additional examples if they are needed. You can download this question and answer generator here.

When the class are ready, I ask them to work independently through the questions on the third slide and then the worksheet.

## Plenary

The plenary challenges students to apply what they have learned to a real-life situation.  After a few minutes if students are struggling to make progress, I help them to set up the formula. This way they can still attempt parts b and c. The extension is to rearrange the formula to calculate the weight when given the extension.

My name is Jonathan Robinson and I passionate about teaching mathematics. I am currently Head of Maths in the South East of England and have been teaching for over 15 years. I am proud to have helped teachers all over the world to continue to engage and inspire their students with my lessons.

## If you like this, you may also like …

### How to Write 3 Part Ratios

How to teach writing 3 part ratios.

## Drawing Histograms with Unequal Class Widths

September 28, 2020

Higher GCSE maths students are required to plot and interpret histograms with unequal class widths.  Drawing histograms with unequal class widths are very common in GCSE maths papers.

## Histograms versus bar charts

Histograms look like bar charts but have important differences.

• Histograms use a continuous horizontal scale which means the bars touch so the difference between them is zero.
• The frequency of the data is measured by area not height.

## How to calculate the height of the bar (Frequency Density)

When grouping continuous data, it may be necessary to have different class widths if the data are not equally spread out.  When class widths are not equal frequency density becomes the vertical axes. ## Working with grouped data I start the lesson by asking students to find a probability from a set of data in a grouped frequency table.  I use this example to help students recap inequality notation and to discuss whether this is an exact or estimated probability.  It is important to remind everyone that grouping helps to organise large samples of data but there is a trade off with accuracy when interpreting the results.

## Drawing histograms with unequal class widths As we move on to the main part of the lesson I ask the class to think about how the table in the second slide is different to that in the starter.  Students quickly pick up the class widths are no longer equal.

## Representing data with unequal class widths

We discuss it may be necessary to use unequal class widths for data that are unevenly distributed and when we do so frequency is measured as the product of frequency density (vertical scale) and class width (horizontal scale).

When drawing histograms for Higher GCSE maths students are provided with the class widths as part of the question and asked to find the frequency density.

I work through the first example with the class plotting the histogram as we complete the table.

## Completing a table and histogram with unequal class intervals

In the plenary students are challenged to complete a table and histogram by working out the scale of the frequency density axis.  This task frequently appears in exam papers.  I provide the students with a print-out of this slide, so no time is wasted copying it into books.  Examiner reports state that those who understand frequency as measured by the area of a bar often go on to achieve full marks.

Here is an extract of an Edexcel examiner report for a similar question.

‘In general candidates appear not to be aware that the area of the bars of a histogram are the frequencies, evidenced by a lack of frequency density calculations.’

To ensure all students have enough time to complete the plenary I ask those who have finished first to additionally estimate the number of flowers that grew to between 5 and 8 cm tall.   The most able students calculate this as a compound area of the bars on the histogram.

## Interpreting histograms with unequal class widths

In the next lesson students practice finding the frequencies from histograms to calculate an estimate of the mean.  As learning progresses we move on to using interpolation to estimate the median average from a histogram.

## Teach this lesson

My name is Jonathan Robinson and I passionate about teaching mathematics. I am currently Head of Maths in the South East of England and have been teaching for over 15 years. I am proud to have helped teachers all over the world to continue to engage and inspire their students with my lessons.

## If you found this helpful you may like …

### Comparing Datasets using the Mean and Range

In my experience, students, in general, find the concept of a mean straightforward to calculate and understand. However, the mean alone does not provide a complete picture of a set of data. To achieve this, a measure of spread is also required.

## Trigonometric Identities Sin, Cos and Tan

September 15, 2020

In my experience the best way to differentiate teaching the sin, cos and tan trigonometric identities is through discovery.

I begin the lesson by explaining we are going to discover some relationships between the three sides of a right-angled triangle and the interior angles.  We discuss theta (θ) as the name of the angle we will be working with and I explain the adjacent and opposite sides of the triangle depend on the position of theta.  The hypotenuse side is opposite the right-angle and is always the longest side.

## Introduction to Trigonometry

To begin the discovery, I ask students to draw a right-angled triangle where the adjacent is 5 cm and theta is 30°.   Students then measure the length of the opposite and hypotenuse sides in millimetres.  After checking their progress, I feedback by asking students to share their measurements of the two sides.  This helps to identify mistakes early on.

To save time I give the students a handout the table above to stick in their books.  Next, I ask the class to enter their results for the 30° triangle in the orange rows.  I demonstrate how to calculate the values for the blue and green rows for the same triangle using a calculator.  All these results are entered in the table.

Students work independently for about the next 15 minutes to construct the triangles.  They measure the sides of each triangle and calculate the necessary values.  I remind everyone to enter the results in the table as they go.

## Discovering Sin, Cos, Tan

When most of the class have got halfway through completing the table, I ask students to take a minute and try to identify any patterns within their results.  At this stage I do not provide any further prompts, so the task is as open as possible. I ask students to share any patterns they have noticed with the person next to them.

As students continue with the triangles, I encourage them to consider whether future values fall in line with their pattern or contradict it.  If future results contradict their patter I encourage students to reassess.

For those who have not found a pattern I encourage them to check their measurements and whether the data is recorded correctly in the table.

## Trigonometric Identities

Students work at different paces, so I encourage peer support.  For instance, one student records the results when theta is 30°, 50° and 70° and the other records the results for remaining angles.  Either way, every student is expected to have a completed set of results in their exercise book.

About 35 minutes into the lesson most students have completed the table of results.  I now stop the class and ask them to discuss with their peers any patterns they have noticed.

After their discussion I ask students to present their patterns in the form of equations to me on mini whiteboards.  About three quarters of the class present their patterns as sinθ = opp/hyp, cosθ = adj/hyp and tanθ = opp/adj.  The remaining quarter describe the same relationships in words.

## Applying Trigonometric Identities to Solve Problems

As part of an extended plenary I demonstrate how to use the three equations to calculate unknown lengths in these right-angled triangles.  You can watch a video of this demonstration here.

Typically, I work through questions a and b then ask students to attempt c and d in their books before I feedback.

In the final part of the lesson I ask the students to solve this problem on mini whiteboards.  They are free to work on their own or with a peer for support.  Again, having checked their progress I feedback on the main board.

## Future lessons on Trigonometric Identities

In the next lesson, students learn more about using trigonometric identities to calculate unknown angles in a triangle before moving on to lengths.  You can view the medium term plan here.

## Teach this lesson

My name is Jonathan Robinson and I passionate about teaching mathematics. I am currently Head of Maths in the South East of England and have been teaching for over 15 years. I am proud to have helped teachers all over the world to continue to engage and inspire their students with my lessons.

## If you liked this you may also like …

### Pythagoras Theorem in 3D Shapes

How to teach applying Pythagoras’ Theorem to 3D shapes.

## Calculating a Reverse Percentage

September 3, 2020

Questions that involve calculating a reverse percentage are difficult for two reasons: students do not always understand they are required to work out the original amount after a percentage change and the written method involves multiple lines of working which, without a clear writing frame, can be confusing.

Here are two examples from exam papers on calculating a reverse percentage.

Question 1

The normal price of a television is reduced by 20% in a sale.

The sale price of the television is £360

Work out the normal price of the television.

The common misconception was to incorrectly add 20% on to the sale price so £432 is seen as an incorrect answer.

Question 2

20% VAT is added to the price of the laptop.
Anita then has to pay a total of £400.

What is the price of the laptop with no VAT added?

The most common mistake was to use £400 as 100% instead of 120%, with students working out 20% of 400 and subtracting to get £320.

Some thought that as £400 was 120% they had to find 80% of £400 to get back to the original value.

## 100% represents the original value.

To overcome these difficulties students, need to understand the original amount, before any percentage change, is represented as 100% .  They also need to have a clear model that draws on prior learning to break down the problem.

## Calculating a Percentage Change

To address these misconceptions, the lesson starts by reviewing how to calculate an amount after a percentage change using a multiplier.  This is because, to calculate the multiplier students must have started at 100% as the original value.  I ask students to work on whiteboards so I can check this when feeding back.

## Using ratios to model reverse percentages

To calculate the original amount after a percentage change I model the percentage and amount using equivalent ratios.  As you can see in this video.

If students needed more practice I use the following Interactive Excel File to randomly generate more questions and solutions, which you can download by clicking on the image.

## Real Life Reverse Percentage Problems

When the class can model calculating a reverse percentage we move on to solving more worded, real-life problems. These are included in the Interactive Excel File.

Later, as learning progresses students work independently through the questions on the third slide and then through the worksheet.

## Teach this lesson

My name is Jonathan Robinson and I passionate about teaching mathematics. I am currently Head of Maths in the South East of England and have been teaching for over 15 years. I am proud to have helped teachers all over the world to continue to engage and inspire their students with my lessons.

### Efficient Calculator Methods for Percentages Changes

Using efficient calculator methods for percentages changes can be a tricky experience for students. Sometimes because the numerical methods involved can overwhelm the visual concept but also because students often see two distinct methods.

## Comparing Datasets using the Mean and Range

August 25, 2020

In my experience, students, in general, find the concept of a mean straightforward to calculate and understand. However, the mean alone does not provide a complete picture of a set of data. To achieve this, a measure of spread is also required. The range is the simplest measure that can be used for this. Not only can a range be used to describe a dataset, in simple examples, and in combination with other information, it can also be used to calculate missing pieces of data.

In this blog I discuss when comparing datasets using the mean and range interpreting the statistics is just as important as calculating them.

#### The importance of the range when comparing datasets

Students are taught to describe both the mean and range of datasets.  However, the importance of what we can learn about the data from the range is poorly grasped.  Often, the range is quoted without any context-based interpretation. This means students are losing out on marks in exams and are insufficiently prepared for the context-based answers as they progress in the statistics discipline.

#### Recapping the Mean Average and Range

I start the lesson with a reminder of the mean and range of datasets.  I show how the range can be used to calculate the unknown value in a small data set. My aim is to plant the idea that the range is an effective tool used to gain a better understanding of a data set.

#### Comparing Datasets using the Mean and Range

At this point in the lesson, students are confident using the mean and range to work out missing data.

We turn our attention to comparing two datasets. Students are asked to calculate the mean and range for each data set from the raw data presented.

At this point, I ask the class to compare the two datasets on their whiteboard.  Most students describe one set of data as having a smaller or larger mean than the other. Very few students make any comment regarding the range.

#### Context-Based Interpretations using the Mean and Range

I provide a written framework to help the students write a comparison.  This encourages the class to think about the differences in the mean and range in the context of the data.

As we move on to the third slide students are ready to practise comparing datasets through a variety of scaffolded questions.

I encourage students to adapt the framework for each example by drawing appropriate conclusions about the within the context of the data.  I hope this framework can make the narrative context-based style of interpretation both second nature and less intimidating to students.

#### Comparing and Summarising Data Additional Resources

Comparing datasets using the mean and range is the third lesson in the comparing and summarising data scheme of work. In future lessons students progress onto representing and interpreting datasets using stem and leaf diagrams.

#### Teach this lesson

My name is Jonathan Robinson and I passionate about teaching mathematics. I am currently Head of Maths in the South East of England and have been teaching for over 15 years. I am proud to have helped teachers all over the world to continue to engage and inspire their students with my lessons.

### Finding the mean average from a frequency table

Understanding the concept of the mean using multi-link cubes is key to finding the mean from a data in a frequency table.

## How to Solve Quadratics by Factorising

August 24, 2020

The method of how to solve quadratics by factorising is now part of the foundational knowledge students aiming for higher exam grades are expected to have.   Here is an example of such a question.

Solve x2 + 7x – 18 = 0

In my experience of teaching and marking exam papers students often struggle with solving equations by factorisation.  Common incorrect attempts include trying to manipulate the equation using the balance method or using a method of trial and improvement.  At best, both attempts only lead to one solution and are therefore not awarded any marks as no creditable attempt to factorise has been made.

In this blog I show how to solve quadratics by factorising using scaffolded questions and mini whiteboards to check student’s understanding throughout a lesson.

At the start of the lesson students revise how to factorise a quadratic into two brackets.  The purpose of this is to remind the class that some quadratics can be written as a product of two brackets.  I encourage students who have forgotten how to factorise a quadratic to multiply out the brackets on the right.

When all the expressions are matched, they should look for a pattern between the numbers in each bracket and the numerical terms in the matching quadratic.  The pattern being the constant term is their product and the coefficient of x their sum.

## How to Solve Quadratics by Factorising

To solve a quadratic, we discuss if it can be represented as a product of two brackets. When each bracket equals zero the quadratic itself equals zero.  In this video I demonstrate the method further.

Having demonstrated a few examples I ask the class to solve x2 + 11x + 24 = 0 and show me their working on mini whiteboards.  It is pleasing that all students have correctly factorised the quadratic into (x + 3)(x + 8) = 0 but some present 3 and 8 as their solutions for x.

To feedback we discuss how each bracket must equal zero for the overall quadratic to be zero.  For example, x + 3 = 0, x = -3 and x + 8 = 0, x = -8.  I demonstrate these are the correct solutions by substituting the values -3 and -8 back into the original quadratic.

When working through the following questions on mini whiteboards students experience some difficulty in dealing with combinations of negative and positive factors of the constant term.

x2 + 3x – 4 = 0

x2 – 11x + 28 = 0

However, this is more to do with needing greater practice with factorisation than it does understanding how to solve a quadratic.

Solving quadratics with a zero term, such as:

x2 – 9 = 0

x2 – 3x = 0

also proved challenging for some students.  To address this, I ask the students to consider the value of the term that is missing.  In the case of x2 – 9 = 0 the b term is 0.  Therefore, we need two numbers that multiply to make -9 but have a sum of 0.  For x2 – 3x = 0 we need two numbers that have a product of 0 but a sum of -3.  About 25 minutes into the lesson students are ready to work independently through the questions below.

## Using Factorisation to Solve Real-Life Problems

In the final part of the lesson students apply their learning to model a real-life problem as shown.

To help the class set up the quadratic equation I encourage students to consider the height of the ball from the ground at time t = 0 and the value of t when the height is 96 m above the ground.

About 5 minutes later, when the class have had time to sketch out some ideas, I provide further prompts for those who are struggling.  These include how to make the quadratic equal to zero and to factorise out the 2 from 2t­2 – 28t – 96 = 0 to make t2 – 14t + 48 = 0.

Having factorised and solved the equation some of the students are not sure whether the time is 6 or 8 seconds.  My response is to sketch the graph of the changing height of the ball as a function of time.  The resulting parabola helps all students understand the ball passes 96 m travelling upwards after 6 seconds and again travelling downwards at 8 seconds.

## Next Steps

I recently taught this to a Higher GCSE class as the first lesson on solving quadratic equations.  The next lesson is to solve quadratics where the coefficient of x2 is no longer one.  If I had taught this to a Foundation class, the next lesson would be to model and solve real life problems through factorising quadratics.`

## Teach this lesson

My name is Jonathan Robinson and I passionate about teaching mathematics. I am currently Head of Maths in the South East of England and have been teaching for over 15 years. I am proud to have helped teachers all over the world to continue to engage and inspire their students with my lessons.

### Calculating Instantaneous Rates of Change

To introduce how to calculate an instantaneous rate of change on a curve we discuss how the steepness of the graph changes depending on the x value. I like to use a Geogebra to demonstrate how the gradient of the tangent changes along the curve.

## Area of Compound Shapes

August 23, 2020

To find the area of compound shapes students need to understand what the word compound means.  Therefore, I ask students to discuss in pairs a definition for the word compound and to extend it to include the shapes below.  As a result of their learning in science students agree that a compound can be defined as a mixture of elements.  In the context of today’s lesson, it is one shape formed by a mixture of rectangles.

## Area of Compound Shapes

I present the shapes on the board to the class and ask them to consider different mixtures of rectangles that will create the pink compound shape.

All the students showed they could create the pink composite shape by adding two smaller rectangles together.  However, the majority of the class were unable to create it as the difference of two rectangles.

It was pleasing, that all students made a clear attempt to split the composite shape up into rectangles.   Examiner reports often explain candidates who do split up the composite shapes are far more likely to arrive at the correct solution.

As we move on to question B I ask students to calculate the composite area using the three methods we discussed and to circle the method they find easiest.  Most students prefer to add the two rectangles together rather than find the difference of them.

I ask the class to attempt question C using their preferred method.  Students who finish early are to choose a different method to check their solution.  I am pleased that about half the class chose to calculate the green area as a sum of two rectangles and the other half choosing to find it as the difference.

At this point students work independently through questions on the third slide using which ever method they prefer.  If they run into a problem, I encourage them to consider a second approach before they ask for help.

## Area of Compound Shapes Plenary

The plenary takes about 12 minutes.  When I present the problem I encourage students to think carefully about the individual rectangles that make up the shape.

After a few minutes it is clear some students are struggling to progress.  For these students I take a few minutes to discuss how the blue region is formed, as shown below.

The next level of support, for students who need it, includes breaking the middle shape down into individual rectangles and their dimensions.

## Teach this lesson

My name is Jonathan Robinson and I passionate about teaching mathematics. I am currently Head of Maths in the South East of England and have been teaching for over 15 years. I am proud to have helped teachers all over the world to continue to engage and inspire their students with my lessons.

### Area of Rectangles for Mixed Ability Key Stage 3

Read more about teaching shape and space in this blog on teaching the area of a rectangle to a mixed abilty Year 7 class.

## Problem Solving Maths Lessons

August 9, 2020

### Problem solving is at the heart of engaging and inspiring mathematics lessons.

Last year my faculty went through two OFSTED inspections.  A trial inspection with three HMIs and a real inspection shorty before lockdown.  The trial inspection showed all students have a love of maths. Later, the real inspection, in the words of my headteacher, simply blew the lead inspector away.

In each inspection we showed how we teach the curriculum through problem solving.  Shortly after blogging about the inspections I received hundreds of messages from educators seeking advice about how they can embed such an approach.  My answer is simple, make problem solving questions a feature of every lesson for students of all abilities.

To help with this I would like to share with four new problem solving lessons. These range from applying the rules of arithmetic to solving real life problems with compound interest.  Every lesson comes with a worksheet and tutorial video that model the solutions.  More problem solving lessons will be added more soon.

## Problem Solving Mathematics Lessons

#### Perimeter and Area

Students are challenged to solve problems involving perimeter and area of rectangles and triangles.

Go teach it

#### Averages

Students are challenged to apply their understanding of the mean, mode, median and range to solve problems involving datasets.

Go teach it

#### Written Methods

Students are challenged to apply the rules of arithmetic to a series of real-life, functional problems.

Go teach it

#### Compound Percentages

Students are challenged to apply compound percentage changes to calculate an original amount after a percentage change, a percentage rate and compare real-life investments.

Go teach it

My name is Jonathan Robinson and I passionate about teaching mathematics. I am currently Head of Maths in the South East of England and have been teaching for over 15 years. I am proud to have helped teachers all over the world to continue to engage and inspire their students with my lessons.

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## Teaching in a Bubble

July 28, 2020

To prepare for teaching in a bubble when students return to school in September, I thought it would be helpful to share some ideas on how my department will adjust to the new restrictions associated with year group bubbles.

This includes how we:

1. make the most of formative assessments to gauge student’s understanding
2. mark student’s books to provide meaningful feedback
3. and manage equipment including exercise and textbooks

## 1.  Making the Most of Formative Assessments

In the past if a student had not fully understood what was being taught at the front of the classroom, they could seek a more detailed and private explanation from the teacher at their desk.  However, as teachers now need to maintain 2 metres distance from their students and cannot walk around the classroom this is no longer possible.   Therefore, the formative assessments we make during the initial teaching phase need to ensure students are ready to work independently.

We can formatively assess a student’s understanding in the following ways:

##### Using mini whiteboards

When the teacher has posed a question to the class to check their understanding students present their method and solution on a mini whiteboard.  I ask students to show me their whiteboards at the same time.  I do not comment on individual whiteboards, so students do not risk embarrassment.  Students can either work in pairs on a single whiteboard or individually.  If they do decide to work in pairs, they must both agree with what is written.  I look at everyone’s whiteboard and feedback any misconceptions by modelling the solution on the board.  You can learn more about using mini whiteboards in this blog.

##### Traffic Lights/ Thumbs Up, Middle, Down / Fingers Up Scale

Students can signal to the teacher their level of understanding using either a set of traffic lights, (which are often included in student planners) using thumbs up/middle/down or holding up 1 finger for poor understanding to 5 fingers for great understanding.

When the teacher has worked through some examples on the board the students could be asked to signal their understanding.  If two students are sharing a desk and one presents green the other presents amber or red the student who presented green can help their peer.

##### Prolonged Thinking Time

Research tells us the average time a teacher waits before calling on a student to respond is about one second.   By giving students longer to consider the problem, formulate an approach and present their solution we will gain a far more accurate assessment of their understanding.  To increase the thinking time, we could encourage students to work in pairs or to work collaboratively on the problem with a single mini whiteboard.

## 2.  Marking Student’s Work

Because student’s exercise books cannot be taken outside of the year group bubble, we will base our written feedback on half termly assessments.  These will cover all the topics covered over the half term and completed online for homework.

Teachers will analyse the results and focus their diagnostic comment on the topics where improvement is needed.  We will use the proforma below to write our personalised comment and give to students to stick in their book.  The class will then be given time to complete the response task in their book.

## 3.  Managing Equipment

To manage moving around the school it is important I carry as little as possible.  Therefore, students will need to have their own equipment.  This includes scientific calculator, 30 cm ruler, pen, pencil, ruler, protractor, eraser and pair of compasses.

On our return in September, I will write to parents with details about the necessary equipment and how it can be purchased on the high street, online and through the school.   You can download a template of the letter below and amend it for your own school.  In addition to the mathematical equipment students will also need their own mini whiteboard and dry wipe pen to use during lessons.  These will be provided by the school.