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Many problems involve three-dimensional objects or spaces. Pythagoras Theorem in 3D Shapes can be used as much with these problems as those in plane shapes.

The starter recaps applying Pythagoras Theorem as part of a larger problem involving the perimeter of a trapezium and square. The aim of this starter is for students to experience the range of problems that can be solved using the theorem. In this instance there are no obvious right-angled triangles yet the problem cannot be solved without Pythagoras.

When you work with 3D shapes it is important to look for and draw out right-angled triangles. Labelling the vertices of the triangle with the same label as those used in the 3D shape helps students to visualise it two dimensionally.

The base ABCG is in a horizontal plane and triangle ACD is in a vertical plane. First look at the right-angled triangle ABC and use Pythagoras’ Theorem to find length AC.

Next, look at triangle ACD. Label the length of AC that was found in triangle ABC, leave the answer in its most exact form as the root will be cancelled when squared. Use Pythagoras’ Theorem to work out length AD.

We work through the second problem as a class. I encourage students to draw the cuboid on one half of their mini-whiteboard and in the second half draw the right-angled triangles. Similar to the diagram above. It’s important to leave the working of the first example on the board to help students see the various stages.

The interactive Excel file can be used for additional practise before students work independently through the questions on the third slide and worksheet.

In the plenary students are challenged to find the total perpendicular height of a composite solid. In addition to this being a composite shape a new level of difficulty is added as students now have to use the hypotenuse to find a shorter side.

I like to set the problem shown below for homework after this lesson as it challenges students to link Pythagoras’ Theorem with volume of pyramids.

The diagram shows a pyramid.

ABCD is a square with lengths 8 cm.

The other faces of the pyramid are equilateral triangles with sides of length 8 cm.

Calculate the volume of the pyramid.

Drawing frequency trees for GCSE maths is a new topic and appears on both the higher and foundation curriculum. I’ve taught this lesson a couple of times, once to Year 10 and once to Year 11 and I have to say the kids really enjoy it.

Frequency trees can be confused with probability trees. Frequency trees show the actual frequency of different events They can show the same data as a two-way table but frequency trees are clearer because it shows the hierarchy of the frequencies. Probability trees show the probability of a combination of events.

I teach frequency trees after a lesson on two-way tables. By the end of the lesson I want all the students to create a frequency tree from a written description. Recapping two-way tables in the starter both consolidates student’s previous learning and helps them to understand the need to organise data in a clear and efficient way.

As we begin the main activity I provide the frequency tree template for the question. This helps students with poor literacy to break down the problem by highlighting a particular phrase and matching it to its position on the diagram. Each time I taught this lesson I found students had no difficulty with the numerical calculations but some did struggle to understand what part of the frequency tree they were calculating. By having the template already drawn students could use the built in hierarchy of the tree to read the text.

Once we had completed the frequency we discussed how to check our answers using the numbers at the end of the branches. If our tree was correct the frequencies would add up to the number at the start of the branch.

At this point we’re about 25 minutes into the lesson and students are ready to work independently through the worksheet. I let the students decide for themselves which question to start at. Those who had difficulties with the written description all decide to start with the two-way tables as they were already familiar with two-way tables from the starter and previous lesson.

The plenary challenges the students to create a frequency tree with 6 combinations. Initially I hid the frequency tree to see who could create it on their own. Some students found this quite difficult because the text only gives the total number of boys in the sample and they had to calculate the number of girls. Once they understood to include boys and girls in the tree the majority of students completed the problem fairly easily.

In the next lesson we go on to designing questionnaires and identifying bias.

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**

**Using the triangle method**

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.

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.

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 cm^{2}. What could the dimensions be?”

**More able**

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

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.

Length | Width | Area | Perimeter |
---|---|---|---|

12 cm | 8 cm | ||

9 mm | 12 mm | ||

6 in | 30 in^2 | ||

15 m | 46 m | ||

11 cm | 7 cm^2 | ||

25 m | 36 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.

Length | Width | Area | Perimeter |
---|---|---|---|

a | v | ||

2x | 5r | ||

5 | 5(c + 15) | ||

2c + y | 10c + 2y | ||

4f | f^2 - 1 | ||

18 | 20 - 2c - 4c^2 |

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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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.

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

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.

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.

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.

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.

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.

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

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.

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.

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.

Intersections – nrich activity

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

3/4 ÷ 1/4 = 3

3/4 ÷ 1/8 = 6

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

2/3 ÷ 1/3 = 2

2/3 ÷ 1/2 = 4/3

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.

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.