When factorising algebraic expressions with powers students often struggle to identify the highest common factor when it involves an algebraic term. For example, factorising 3h + 12 as 3(h + 4) is attempted correctly much more often than factorising 3h2 + 12h as 3h(h + 4). In this lesson students learn how to identify the highest common factor of expressions that include algebraic terms.
The starter activity helps students to understand factors are not restricted to numerical terms but could also include letters (or unknowns).
To help with this I have included some possible factor pairs of 36x2y. When I first taught this one student made an interesting point of including 72x2 and 1/2 y as a factor pair. Algebraically this works but we would not typically include fractions or decimals when finding the factor pairs of a number.
I ask students to work on mini-whiteboards to write at least 5 factor pairs of the term 40ab2c. For the purpose of this activity factor pairs involving fractions and decimals are encouraged as this comes up later in the lesson. A couple of the higher ability students include factor pairs such as 40abd and bcd-1 which I applaud.
When factorising terms involving powers students need to understand the basic rules of indices. As we work through the first couple of examples I write out each term as a product of the highest common factor. For example,
r3t + rt2
r3t = rt × r2
rt2 = rt × t
rt(r2 + t)
6w2y – 8wy2
6w2y = 2wy × 3w
– 8wy2 = 2wy × - 4y
2wy(3w – 4y)
8u3c2 – 20u2c
8u3c2 = 4u2c × 2uc
– 20u2c = 4u2c × -5
4u2c (2uc – 5)
I work through the first questions in this way and ask the class to attempt the next two on mini-whiteboards with similar working out. After this students work in their exercise book to match the equivalent expressions.
As students work independently through the questions on the third slide I challenge the most able to factorise expressions similar to those in the extension. I remember a similar question appearing in the final GCSE paper a couple of years back. The examiner's report noted most students didn't have a clue.
To factorise 16(f + d)2 + 8f + 8d most students only recognise 8 as a common factor. We discuss the need to factorise 8f + 8d and rewrite the expression as 16(f + d)2 + 8(f + d). It is now easier to see 8 and (f + d) are both common factors. 16(f + d)2 + 8f + 8d factorises to 8(f + d) (2f + 2d + 1).
In the plenary we investigate factorising the algebraic area of a rectangle to find possible perimeters. I ask everyone to include taking out the highest common factor for one of their solutions.
The plenary takes about 8 minutes with students working in pairs on mini-whiteboards.
For the less able students I emphasise the need to find the highest common denominator as they quite often only partially factorise by taking out the numerical factor.
For more able students I include problems involving negative powers such as 18x-2y + 27xy-2 = 9xy(2x-3 + 3y-3) This requires a greater understanding of the rules of indices.
While I was teaching a higher GCSE class about Reflections, Rotations and Translations I wanted to explore extending transformations beyond shapes on a grid to include transforming straight line graphs.
About forty minutes into the lesson on reflections the majority of the students were quietly working their way through the activities. The class were well behaved, attentive and on task. However, were they being challenged? I suspected not.
We had recently covered plotting and deriving the equation of straight line graphs. This proved a challenging topic to the majority of the class as it always does. Linking it to transformations as a plenary activity seemed the perfect mix of challenge and consolidation the students needed.
I presented the straight line graph y = 2x + 3 on a blank pair of axes and asked the students to sketch it on their mini-whiteboards. I set two challenges.
Challenge One: Draw a reflection of the line y = 2x + 3 in the line y = 1.
Challenge Two: Derive the equation of the reflected line in the form y = mx + c.
When feeding back to the class we discussed the need to include the intercept on the sketched graph as this provided a reference point for the reflection.
Some students wrote y = -x/2 + 3 as they confused the gradient of the image as being perpendicular to the object line. We discussed how each point on the image line must be the same distance from the mirror line to the object line.
Once I was confident all students could perform and describe a rotation of a shape on a grid we extended it to include a straight line graphs.
Challenge One: Draw a rotation of 90° clockwise about the point (0, 3) of the line y = 2x + 3.
Challenge Two: Derive the equation of the rotated line in the form y = mx + c.
All students had correctly marked the intercept value and used it as the center of rotation. It was pleasing to hear a number of students discuss a rotation of 90° anti-clockwise would result in the same rotation as clockwise. We discussed why this is true.
A few students wrote the gradient as -2 as they remembered what happened the previous lesson when reflecting the line. We discussed the relationship between perpendicular gradients as the negative reciprocal. I remember the class found this concept difficult when I first taught it.
Translating shapes using a vector is often the easiest transformation for students to perform and describe. However, translating a linear graph and deriving the new equation proved to be the most difficult activity out of the three.
Challenge One: Draw a translation of the line y = 2x + 3 using the vector .
Challenge Two: Derive the equation of the translated line in the form y = mx + c.
When performing the translation some students needed to label the numbers on each axis to translate the line as a whole. We discussed the benefit of translating a point on the line rather than the line itself. The easiest point to translate was the intercept. Once the intercept has been translated it proved intuitive for the image to be parallel to the object. All students realised this meant the two lines had the same gradient.
While all the class found the gradient of the new line only a quarter had sufficient algebra skills to find the intercept. Using the translated intercept coordinate of (2, 2) we discussed x = 2 when y = 2. Substituting these values into the equation y = 2x + c gave 2 = 4 + c. The new intercept was therefore -2.
The point of extending transformations beyond shapes on a grid was not to teach the students something new but rather connect their understanding of transforming shapes on a grid, a topic they found relatively easy, to straight line graphs, a topic they found difficult.
Each activity was used a plenary of approximately 15 minutes with students working in pairs if they wanted to and on mini-whiteboards.
Most students were able to transform the lines but some did find deriving the equation of the images difficult . I think it was important for the students to explore extending transformations beyond shapes on a grid to develop their algebra skills and recap finding the equation of straight line graphs.
Here is a list of what I consider to be the best secondary school maths websites that I use every week.
UK Maths Challenge is the best high school maths website out there, second to mr-mathematics.com. The Questions and Solutions page provide a fantastic range of questions ideal for engaging and challenging any student. I regularly use these questions in lesson and for homework as I like how they promote problem solving and teamwork. The multiple choice answers are great at bringin common misconceptions. The team challenge resources are excellent for gifted and talented classes and fun for end of term lessons.
Geogebra is a fantastic website for teaching mathematics conceptually using an interactive whiteboard. Topics such as: transformations, graphs, angle properties and even data analysis can all be demonstrated using Geogebra. If you can’t find a ready-made applet it is fairly simple to create your own. I’ve made 58 applets ranging from trigonometric graphs to identifying parts of a circle.
This website has a good range of homework tasks for key stage 3 classes. Every worksheet includes a skill, challenge, literacy, research and memory section. I use a worksheet once a week for my key stage 3 classes as most of the topics at this level are covered. There isn’t as much at GCSE (especially the higher tier) but new topics are added on a regular basis.
I like how the homeworks are laid out but I often need to print them as pdfs so I can upload them for the students. Some sheets include an answer page which helps with marking. It would be better if all worksheets included answers.
Excellent website for helping students improve their weeknesses. I use the topic based practise papers as part of the revision process within my department. The expectation is for students to complete two topic papers every week from January to the summer exam. That’s a lot of printing but most kids will print out the papers at home or read the questions on the computer. Having answers and linked video demonstrations available help to reduce my workload and make the students more independent.
As we approach the start of a new school year I thought I would share some tips on behaviour management in a mathematics lesson. These are things that I have picked up over the years and have worked well for me. I am sure there are opposing viewpoints and you may find some of these tips work for you and some don’t.
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. An engaging and challenging lesson will do more to create a positive learning environment than any behaviour policy.
That said there are a couple of things you can do to set up an enjoyable lesson.
Get to know your student’s names as quickly as possible. Writing a seating plan is crucial to getting to know your students. How you arrange the plan depends on your teaching style and the needs of your class. Here are some common seating plans for a mathematics lesson.
If you have never taught the class before or don’t know the students ask your mentor or a colleague to help you. They will be able to advise who works well with who and who to avoid sitting next to each other. You may also need to allow space for a teaching assistant.
Some students may forget where they sit so make sure to check everyone is in the right place at the start of the lesson. Use your seating plan to address students by name as often as you can. This makes the students feel valued and helps them to understand the importance of sitting where they should.
Follow the policy
Consistency and reasonableness is key when teaching. Speak to your mentor about the school behaviour policy and get his/her advice on how to implement it in your lessons. What works well for them and could work for you? When the students understand you know the school rules they are less likely to challenge you on them.
The best behaviour policies are centred on praise and earning points for their House. Students should feel they have to work hard to earn your praise but when they do you acknowledge their hard work consistently and fairly. It’s up to you to define what hard work is in your lessons. The students will soon pick up what they have to do to earn your praise. Research shows the most effective praise vs sanction ratio for motivating people is about 6 : 1.
Watch your students learn
We all learn maths by doing maths. This is not a complicated idea but it is so often overlooked by too much directed teaching at the front of the class. We have all done it. As teachers we want to make sure the students have understood what they need to.
Look for ways to check their understanding at the start of the lesson. Mini-whiteboards, traffic lights and class discussions are all effective assessment for learning strategies to help you interact with students. The student planner often comes with a built in mini-whiteboard and set of traffic lights.
Knowing who can do what in the first 15 minutes of a lesson means you can focus your attention for the next 5 minutes on helping those who need it. You then have the rest of the lesson to watch your students learn so you can make the most of the plenary later on.
As I mentioned at the start of this blog these are things that have worked for me when it comes to behaviour management in a mathematics lesson. You will have your own teaching style. As is so often with teaching there is no one perfect answer, especially to something as complex as behaviour management. Please do leave a comment to share what works well for you.
As we prepare to say goodbye to our exam classes I thought it would be helpful to share some ideas about planning ahead in the summer term with your maths department.
With all the revision work that has been going on over the past few months you probably have loads of past papers and worksheets hanging around. Use this time to file reusable material and recycle when needed. A tidy classroom is key to keeping on top of your workload and lesson preparation. It also provides structure and space for the students. I like to organise my room like this… Here’s another blog about how one teacher has set up his maths classroom.
If you have a maths faculty office use this time to tidy it and perhaps reorganise. Do this as a team building activity. It will remind teachers of all the resources you have and provide a nicer place to work and be social. Inform your site staff you’ll be needing their help to move
As we go through the school year faculty meetings are often taken up with imposed administrative tasks and intervention planning so little time is available for forward planning. Now is the time to review your schemes of work for next year. This will help share best practise, reduce future workload and develop teachers understanding of pedagogy. Here’s a blog about some key ingredients for a maths scheme of work.
With your staff’s and school support now may be the time to review your department’s timetable. Use some time to team teach and share ideas when lesson planning. If you know a colleague has a challenging class, this can be of enormous support. Team teaching can show the kids the maths department are a team and are united in their expectations. Here’s an interesting blog about how to make team teaching work.
Secondary school teachers tend to underestimate what new year 7 students are capable of when they join in September. This results in students working at level below what they working at in year 6. Arrange to meet with the Year 6 maths teacher to discuss the topics in your scheme of work up to Christmas. They will have great insight into what students can already do – often much more than we secondary schools expect. The key stage 2 curriculum can be found here.
I’m a big fan of working with other secondary schools. Take the time to look at other maths department’s schemes of work, resources, and revision materials. I have learned so much by seeing what other schools do. I like to find out how the following:
How does your department make use of the gained time? Please do share some ideas using the comment form below.
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 cm2. What could the dimensions be?”
“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.
|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.
|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.