Mr Mathematics Blog

How to calculate speed using ratio notation

February 17, 2017

Whenever I teach how to calculate speed as a measure of distance and time I either use the formula or the triangle method.  In my experience most students are know about the triangle method from their science lessons.  For this reason I would have expected speed to appear either within the algebra or shape and measures strands of the new syllabus.   Calculating speed and other compound measures is now in the ratio, proportion and rates of change strand of the key stage 3 syllabus.  So I wondered if there was a different, more conceptual way of teaching how to calculate speed using ratio notation.  I discovered there was.

Using the formula

17

Using the triangle method16

Writing speed as a ratio of distance to time

How to calculate speed using ratio notation

If we think of speed as a measure of distance covered per unit of time the ratio becomes simple and links nicely to writing ratios in the form 1 : n.  In the starter students are asked to match a two part ratio with its equivalent ratio given in the form 1 : n.  As this is prerequisite knowledge for the remainder of the lesson I have the class attempt this on mini-whiteboards with the multipliers clearly shown as part of their working.

How to calculate speed using ratio notation

In the development phase we discuss speed as the change in distance for a fixed period of time.  When the distance is given in metres the unit of time is per second and for kilometres or miles it is per hour.  Therefore, speed can be written as the ratio of distance to unit of time.

How to calculate speed using ratio notation
How to calculate speed using ratio notation

Once students are able to calculate a speed we move on to finding the time or distance using the same ratio notation.

How to calculate speed using ratio notation

How to calculate speed using ratio notation

My motivation for teaching this approach of how to calculate speed using ratio notation was mostly out of interest.  I didn’t really expect it to change my practise but as I watched the students work through the questions it became apparent how comfortable they were using ratio notation compared to rearranging formulae or using the triangle.  Teaching speed in this way gave ratio a practical context and reinforced their understanding of equivalence and proportional reasoning.

Is this the preferred method for teaching speed?

As I write this I do wonder whether most teachers have been using this method for a while and I’m preaching to the choir.  Having been a teacher for 15 years I’m always delighted to find new and interesting ways of teaching things especially when they fit so nicely with a conceptual way of learning mathematics.

 

nth term of a geometric sequence

January 28, 2017

GCSE foundation and higher maths students are now expected to find the nth term of a geometric sequence.

When I teach the nth term of geometric sequences I ask the class to work in pairs to categorise a range of sequences into two groups and present their solutions on mini-whiteboards. The students can decide what the categories are based on how they think the sequences are different.

nth term of a geometric sequence

Arithmetic and geometric sequences

I think it’s important for students to discover for themselves how a geometric sequence differs from an arithmetic sequence. Students are encouraged to use a calculator to aid their calculations. When you consider what is happening to the sequences on a term to term basis this is actually quite a simple concept and one that provokes quite a bit of class discussion.

Finding the common ratio

Moving on to the development phase I introduce the common ratio by considering the term-to-term rule. For the sequence 2, 4, 8, 16, 32, 64 the common ratio is 2. This means the following term is double the previous term. We begin to generalise this on a term to term basis taking ‘a’ as the first term, a, 2a, 4a, 8a, 16a, 32a.

nth term of a geometric sequence

nth term of a geometric sequence

Now we need to find the formula for the coefficient of a. Rather than telling the class the formula I challenge them to derive it independently. The majority of the class know to raise 2 to a power. A common mistake is to raise 2 to the power of n. We discuss what this sequence would look like (2, 4, 8, 16, 32, 64) and after another minute or so most the students arrive at a × rn-1 where a is the first term of the sequence, r is the common ratio and n is the position number.
I find that once students have found the nth term for a geometric sequence they are much more likely to remember it and be able to apply it in the future.

Differentiating geometric sequences for the most able

As we work through the remaining questions the common ratio changes from an integer to a fraction. More able students are challenged to find the first term of a sequence when given two other non-consecutive terms.

Real life problem with geometric sequences

nth term of a geometric sequence

In the plenary, the class are challenged to apply finding the nth term of a geometric sequence to compound percentage changes. I think it is really important students appreciate the practical, real life aspect of geometric sequences and compound interest links really nicely with this topic. I tend to allow about ten minutes for this question and have a student demonstrate the solution to their peers to feedback at the end.

Area of rectangles for a mixed ability maths class

January 5, 2017

Finding the area of a rectangle is such a key skill in mathematics as it leads on to many other aspects of shape, number, algebra and even handling data.  In this blog I’ll take you through how I teach the area of rectangles for a mixed ability maths class in Year 7.

Difference between perimeter and area

Area of rectangles for a mixed ability maths class

A common misconception for Year 7s is to confuse the area of a rectangle with its perimeter.  The starter addresses this by challenging students to find the perimeter of a star, regular octagon and pentagon and hexagon.

Students are typically able to find the perimeter as a product of the number of sides and side length for the three regular shapes.  Less able students may find the perimeter by long addition.  Some forget to find the two missing lengths in the blue hexagon and write its perimeter incorrectly as 40 cm.

Area of rectangles for a mixed ability maths class

Area of rectangles for a mixed ability maths class

To phase in the main part of the lesson I highlight the difference between perimeter and area.  I do this by counting the number of squares inside the rectangle.  The majority of Year 7s know this from primary school.

As we progress, I ask the students to sketch a rectangle on their mini-whiteboard (1 whiteboard per pair so they have to work together to aid peer support).  I pose two questions one for the lower and core ability and one for the most able.

Lower and Core Ability

“A rectangle has a fixed area of 24 cm2.  What could the dimensions be?”

Area of rectangles for a mixed ability maths class

More able

“A rectangle has a fixed perimeter of 36 cm.  What could the different areas be?”

Area of rectangles for a mixed ability maths class

Rather than asking students to repetitively find the area as a product of its two sides I challenge students to find a missing length when given its area or to find both the length and width when given area and perimeter.  See the table below.

LengthWidthAreaPerimeter
12 cm8 cm
9 mm12 mm
6 in30 in^2
15 m46 m
11 cm7 cm^2
25 m36 m^2

To add further challenge for the most able I pose similar questions with algebraic dimensions.

Find the missing dimensions for these rectangles.  All lengths are in cm.

LengthWidthAreaPerimeter
av
2x5r
55(c + 15)
2c + y10c + 2y
4ff^2 - 1
1820 - 2c - 4c^2

Ambitious?  Yes, especially for Year 7 students.  But it constantly surprises me how much students can understand when expectations are high.

Assessing progress and leading on to composite areas
Area of rectangles for a mixed ability maths class

To wrap up this lesson and lead into the next on compound areas the plenary challenges students to find the area of a composite rectilinear shape.  I remind some students to find the missing lengths.  Others need some help seeing the composite shape as the sum or difference of two rectangles.

Either way, by the end of the lesson students are much more confident to solve problems involving the area of a rectangle.

 

Best of 2016 Mr Mathematics Blog

December 28, 2016

Best of 2016 Mr Mathematics Blog

I've loved creating content for the Mr Mathematics blog this year and am always interested to see what you all read, shared, and commented on the most. Take a browse through the best performing posts below and we'll see you in 2017.


Best of 2016 Mr Mathematics Blog

Behaviour Management in a Mathematics Lesson

Engaging students in our lessons so their behaviour contributes to learning is often a top priority for new teachers. Try not to think of behaviour management as separate to teaching but rather a direct result of it.

Best of 2016 Mr Mathematics Blog

How to Visualise Dividing with Fractions

To help students visualise what is happening when we divide one fraction by another I do two things.  First I keep the fractions simple and second I use proportions of a circle because it is much easier for students to see one circle as a whole.

Best of 2016 Mr Mathematics Blog

Setting up your Maths Classroom

If you’re new to teaching mathematics or have recently been given a new teaching room this blog is designed to help you in setting up your maths classroom so you have more time to focus on the teaching.

Best of 2016 Mr Mathematics Blog

Geometrical Reasoning in Key Stage 3

We start the topic learning how to measure and draw acute and obtuse angles with a 180° protractor. Students have lots of practice to learn how to position the protractor correctly.  Understanding the types of angles is also key.

Best of 2016 Mr Mathematics Blog

Homework ideas for Mathematics Lessons

These four ideas are some homework tasks that have worked well for me in the past.  I find getting the students to rely on each other for completing homework often works best for ensuring accountability.

Best of 2016 Mr Mathematics Blog

Plan and Elevation Drawings of 3D shapes

An elevation drawing is the view that you would see in real life as you stand looking at either the front or side of the solid.  The plan is what you would see if you were looking directly down.   A set of elevation and plan drawings gives you the chance to see all of the object from the multiple viewpoints.

Plan and Elevation Drawings of 3D shapes

December 2, 2016

Being able to construct plan and elevation drawings of 3D shapes is a key skill which often leads on to topics such as nets, isometric drawings, volume and surface area.

An elevation drawing is the view that you would see in real life as you stand looking at either the front or side of the solid.  The plan is what you would see if you were looking directly down.   A set of elevation and plan drawings gives you the chance to see all of the object from the multiple viewpoints.

Prior knowledge of 3D shapes

To construct a set of elevation and plan drawings students need to know the properties of a 3D shape and how a solid can be presented on isometric paper.

The starter recaps both of these by asking students to arrange a set of cuboids.  The point of the activity is for students to discuss alternative methods of arrangement. By doing this is they remind themseleves of the various properties of a solid.  How they arrange the cuboids is left open.  Some choose to arrange by volume, surface area or area of cross-section.

Plan and elevation drawings of 3D shapes

To create a set of plan and elevation drawings of 3D shapes it is important to lay out the front, side and plan views so they align with each other.  The height of the front should align with the height of the side and the width of the plan should align to that of the front as you can see from the diagram.

Plan and Elevation Drawings of 3D shapes

When the students practise drawing the front and side elevations and plan view I ask them to draw sketches on mini-whiteboards rather than attempting accurate constructions.  This helps maintain the pace of the lesson as time is not wasted with handing out rulers, sharpeners, pencils and so on…

When students can accurately sketch the plan and elevations I hand out a collection of objects either bought from home or found in school.

Plan and Elevation Drawings of 3D shapes
Plan and Elevation Drawings of 3D shapes
t
Plan and Elevation Drawings of 3D shapes

I ask the students to construct the elevation and plan drawings as accurately as possible in their books.  Each table gets a different object and they pass it on to the next pair when finished.

Creating a solid from the elevation and plan drawings

In the plenary the class are challenged to sketch a multi-coloured 3D object from its plan and elevation drawings.

Plan and Elevation Drawings of 3D shapes

I hand out a set of colours and isometric paper so students can present their work to me and each other at the end.   Students tend to really enjoy this plenary as it reverses what they have just learned.

Simultaneous Equations through Elimination

November 15, 2016

Being able to solve a pair of simultaneous equations through elimination is a key topic for GCSE students.   With the new criteria focusing more on problem solving and application of knowledge it is much less obvious for students when simultaneous equations is being assessed.

It’s important students are exposed to the types of questions they are likely to be asked in exams from as early as possible.   Plus, the problem solving questions are much more fun and rewarding to solve.

Setting up equations in the starter

Before students work with simultaneous equations they need to be confident with setting up and solving equations with one unknown.  This is why I use the question below as my starter.

simultaneous equations through elimination

Because the identical rectangles overlap by 7 cm students find it difficult to solve this intuitively and are therefore more likely to form some kind of equation as part of their working.  Generally, writing the equation  proves more challenging than solving it.

simultaneous equations through elimination

Introducing solving simultaneous equations through elimination

To transition into the development phase I ask the class how simultaneous equations are different to the ones they have seen previously.  ‘Because there are two unknowns’, eventually follows.

This leads nicely on to the key point of, for every unknown we need an equation.  So for two unknowns, x and y we need two equations involving both x and y.

We discuss coefficients and how to eliminate an unknown by either adding or subtracting the equations when coefficients are equal.  Students often find it difficult knowing when to add or subtract the equations to eliminate an unknown.  A quick recap of adding and subtracting with negatives and collecting like terms follows.

I work through the questions a) and b) as shown in the video and ask the class to work in pairs on question c).   Once they have shown me their working out on mini-whiteboards I feedback so students can attempt question d).

At this point we’re about 20 minutes into the lesson and the students can now start working independently either in pairs or individually.

simultaneous equations through elimination

I gather those who need a little more help and use the interactive Excel file to work through a couple more questions.

Differentiation for the most able

While the class are working through the problems I challenge the more able students to find an alternative method of solving the pair of simultaneous equations.  I encourage using the substitution method and ask the students to look out which method would be most suitable for different questions.

Assessing progress in the plenary

About 10 minutes before the end of the lesson we move on to the plenary.  This is an opportunity to introduce students to the type of question they are likely to be asked in their exams.  It always surprises me how intuitively some can derive the necessary equations.

simultaneous equations through elimination

Whenever possible I ask a student, or a pair, to demonstrate how they tackled the problem at the front of the class.  Those who have struggled are much more likely to engage with their peer at this point than having to listen to me again.

 

Dividing with mixed numbers

November 8, 2016

Dividing with mixed numbers and top-heavy fractions is one of those lessons where students have to combine a lot of topics they have learned in previous lessons.  Equivalent fractions, mixed numbers, reciprocals and multiplying with fractions are all involved when dividing with mixed numbers.

Division with fractions using the visual method

The start of the lesson recaps division with fractions using the visual method which they learned the previously.  More able students are normally comfortable using the written method and realise the 6/8 can be simplified to 3/4.  The middle and lower ability students benefit from working on the diagram as this helps them visualise what is going on.   This starter typically takes about 10 minutes once I have finished checking the student's mini-whiteboards and feeding back.Dividing with mixed numbers and top-heavy fractions

Dividing with mixed numbers and top-heavy fractions

In the development phase we stick with the visual method for dividing with fractions but this time we use one whole circle and 3/4 of another.   I find that once students are able to visualise dividing with normal fractions, dividing with mixed numbers and top-heavy fractions is much easier.  To do this we work out how many eighths go into one and three quarters by counting the sectors then follow it up using the written method by converting the 1 3/4 to 7/4.

2

As we progress through the questions the majority of students move away from the visual method preferring instead
to use the written method.  It really is fantastic watching the class put together so many aspects of their work on fractions to solve a single problem.

Differentiation with dividing fractions and mixed numbers

On the worksheet students are free to choose (within reason) their level of questions.  Less able students use the diagrams to help them use the visual method.  The core ability using the written method to attempt questions similar to those on the presentation.  More able students apply their learning to the volume and lengths of cuboids.  There's a challenge at the end where students combine multiplication and division of fractions and mixed numbers.

Dividing with Mixed Numbers and Top-Heavy Fractions

 

Wrapping up

At the end of the lesson students match up a division with its solution.  Questions range from dividing a mixed number or top-heavy fraction by an integer, ordinary fraction or another mixed number.  The plenary takes around 8 to 12 minutes with students working on their mini-whiteboards so I can assess their progress and feedback.

4

 

How to Visualise Dividing with Fractions

October 28, 2016

Students first learn how to divide with fractions in year 8.  In the past I’ve struggled with teaching how to visualise dividing with fractions in a way that students gain a conceptual understanding, especially when the written method is a relatively simple procedure.

Dividing with integers is much easier to understand on a conceptual level, for instance 12 ÷ 4 can be visualised as 12 split into four equal parts.

How to Visualise Dividing with Fractions

However, questions like 2/3 ÷ 1/2 or, how many halves go into two thirds, is much more difficult for students to visualise.  Yet, when students are studying GCSE maths, we expect them to be confident and competent with both kinds of division.

How to visualise dividing with fractions

To help students visualise what is happening when we divide one fraction by another I do two things.  First I keep the fractions simple and second I use proportions of a circle because it is much easier for students to see one circle as a whole.

To keep the fractions simple I use those which are easy to visualise, such as halves, thirds and quarters or those where the denominators have more factors, such as eighths and twelfths.  I would avoid using fifths as 5 is a prime, so has only two factors.

Scaffolding the questions

I set the scene by using a circle to represent the whole.  I then split into a number of sectors, relevant to the question in hand.  For the example below, 3/4 ÷ 2, I’ve split a whole circle into eighths and display the proportion of the circle that corresponds to the numerator. Students can see 3/8s make up one half of the three quarter circle.

For the third question I find it is helpful to phrase 3/4 ÷ 1/8 as ‘how many eighths go into three quarters?’  It’s much easier to count the number of eighths within 3/4 than it is to try to calculate it using arithmetic.

3/4÷ 2 = 3/8

How to Visualise Dividing with Fractions

3/4 ÷ 1/4 = 3

How to Visualise Dividing with Fractions

3/4 ÷ 1/8 = 6

How to Visualise Dividing with Fractions

Teaching how to divide with fractions

How to divide with fractions

Assessing understanding

For the next series of questions I ask the students to sketch a circle split into 6 equal sectors on their mini-whiteboard and to wipe off one third.  Now we have two thirds of a circle split into four equal sectors.

I ask the class to attempt 2/3 ÷ 4 on their mini whiteboard.  Most students present 1/6 as their answer.  Some present 1/4 with the argument the shape has been split into four equal sectors.  We discuss that while their argument has some validity we need to look at 2/3 of the whole circle and the whole circle is split into sixths.   We try the next question.

2/3 ÷ 1/2 is attempted successfully by the vast majority of students.  Those who do have some difficulty are helped by their friend.

2/3 ÷ 4 = 1/6

How to Visualise Dividing with Fractions

2/3 ÷ 1/3 = 2

How to Visualise Dividing with Fractions

2/3 ÷ 1/2 = 4/3

How to Visualise Dividing with Fractions

Here’s the tricky bit

Most of the class found 2/3 ÷ 1/2 = 4/3 difficult to visualise.  You can see within the 2/3 there is a complete half and additional 1/3 of the second half.  A common misconception here was to calculate 1/2 + 1/3 which is 5/6.

Introducing the written method

The idea behind this approach was to help students visualise dividing with fractions.  The written method follows on from this as a natural progression of their understanding.  To link the two approaches together we work through the previous questions using the written method so students can see how the two methods arrive at the same answer, thus consolidating each other.

Finding the mean average from a frequency table

October 4, 2016

In this blog I will discuss finding the mean average from a frequency table.  Students need to understand why the average is the total of the data divided by the sample size.  I like to use multi-link cubes to demonstrate this concept.
Finding the mean average from a frequency table

Finding the mean average from a frequency table

The first step when finding the mean average of data in a frequency table is to calculate the sum of the data.  This begins with a discussion of what the total shoe size would be if all the students placed their feet in a line?  By working through the problem this way it becomes intuitive to find the product of the shoe size and frequency.

Typically I would demonstrate how to calculate the boy’s shoe size and ask the class to find the girl’s.  I find this example works well as most students would expect boys to have a larger shoe size so this backs that up.

If the class is large enough I would put a student in charge of collecting shoe sizes for boys and girls and repeat the question using primary data.  It’s always funny when boys are proud to have largest feet in the class.

Assessing progress and feeding back

Moving on from this I ask the class to work through the question on the third slide using mini-whiteboards.  I use mini-whiteboards for this as there is quite a bit of working out involved and it helps the students keep track of their method.  It also helps me to assess the progress so I can feedback.

Finding the mean average from a frequency table

Finding the mean average from a frequency table Plenary

The plenary takes between 10 to 15 minutes.  I think it’s important to leave plenty of time for this as some students can be a little overwhelmed with making sense of the question because data is normally presented to them in a table.  In this question however they have to read the frequencies of a bar chart.  To save time I would print a copy of the bar chart as a handout.  Students sitting at the back of the classroom can struggle to read the correct frequencies of the bar chat.

Next lesson

I find students need at least a couple of lessons interpreting data from frequency table so the following lesson  is normally about finding a combination of the median, mode, mean and range from frequency tables.

Do you have any nice examples for teaching the mean average or how to work with frequency tables?  Please leave a comment to share your ideas.

 

5 ways how to connect factor pairs to other areas of maths

September 20, 2016

How to connect factor pairs to other areas of maths

Knowing how to find the factors of a number enables students to connect with topics such as prime, square and perfect numbers.  You can also link factors to area, perimeter and algebra.  I am sure there are many more.  The highest common factor for a pair of numbers has obvious applications in fractions and solving real life problems.

In this blog I want to show you how I connect to and introduce a range of topics through factors both in class and as homework activities.

Prime Numbers

At the start of a lesson on primes I challenge students to think of at least 10 numbers that have exactly two factors.    I avoid phrasing it as having a factor of 1 and itself as this includes the number 1 which is not prime.  A common misconception when listing primes is to include 9, 21 and 27.  By working out the primes this way students are less likely to fall into this trap.

How to connect factor pairs to other areas of maths

Square Numbers

A nice homework activity is to have students list the first 12 numbers that have an odd number of factors.  This way they consolidate their learning of factors and prepare for the next lesson on square numbers.  An extension activity could be to have students explain why these numbers have an odd number of factors.

Perfect Numbers

Wikipedia defines a Perfect Number as a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself.   Finding perfect numbers is my favourite plenary when teaching factor pairs as students use a trial and error to find the perfect number thus embedding factor pairs.  This activity works well for all abilities because it is really easy to differentiate.  I ask less able students to find a perfect number between 1 and 10, answer is 6.  More able students have to find one between 1 and 30, answer 28.

Area and Perimeter

To consolidate factor pairs at the start of the following lesson I present a rectangle with a fixed area.  Students can earn a House Point for finding the dimensions of a rectangle with the i) the minimum and ii) the maximum perimeter for a rectangle with a fixed area of 24cm2. More able students will consider decimal dimensions, such as 0.1 cm by 240 cm to create a perimeter of 480.2cm.  Very few students have ever found the smallest perimeter using the square root of 24cm2.

How to connect factor pairs to other areas of maths

 

Algebraic Products

A nice plenary when teaching algebraic products is to find factor pairs for an algebraic expression, e.g., the factor pairs of 30x2y.  Students will typically list 30 and x2y or 30x2 and y but forget factor pairs such as 5x2 and 6y.  A further extension to this could be to have a rectangle with area 30x2y and students investigate different perimeters.