There are three common methods for sharing an amount to a given ratio. Depending on the age group and ability range I am teaching I would choose one approach over the other two.
The three methods are:
In this blog I will demonstrate each of the three methods for the same problem.
Nikki and Gemma share £36 in the ratio 4 : 5.
Work out how much Nikki and Gemma each receive.
Change the shares for each person in to a fraction.
Nikki’s Share = 4/(4+5) = 4/9
Gemma’s Share = 5/(4+5) = 5/9
Calculate each fraction of the total amount.
Nikki receives £36 × 4/9 = £16. Gemma receives £35 × 5/9 = £20.
I use this approach when teaching more able students as it reinforces the link between ratio and proportion. The fractions are seen as proportions of the total amount.
The unitary method emphasises the need to find the value of one share by dividing the total amount by the total number of shares. This can be taught illustratively or with clear writing frames.
The illustrative approach represents each share as a box. Each box contains an equal proportion of the total amount. Illustrating the shares as boxes helps the younger and less able students to visualise the importance of finding one share and using that to split the amount correctly.
Unitary method using writing frames to find the value of a single share
Unitary Method using Writing Frames
Step 1: Find the total number of shares: 4 + 5 = 9.
Step 2: Find the value of one share: £36 ÷ 9 = £4 per share
Step 3: Multiply each part of the ratio by the value of one share.
Nikki = 4 shares × £4 = £16, Gemma = 5 shares × £4 = £20
The written unitary method is my most common approach for teaching how to share an amount to a ratio as it breaks the problem down into three intuitive stages.
The first column of the table uses the ratio given in the question. Subsequent columns are multiples of the first column. This method works well when the total shares is a factor of the amount.
I use this method for lower ability students and those in key stage 3. It reinforces multiples and patterns while providing a visual representation of the increasing shares.
To introduce plotting scatter graphs and understanding correlation I ask students to think about the relationships between different variables and to describe how they might be related.
Here’s my starter activity which students discuss in pairs then present to me on mini-whiteboards.
When the students have had time to discuss the matching pairs we talk about how each graph is likely to look if we were to plot eight typical samples.
As we begin the first example we discuss the type of relationship we expect to see when time spent reading is plotted against time spent watching TV for a sample of ten people. The consensus is the more time people spend reading the less time they are likely to spend watching TV. I ask the class to sketch on their mini-whiteboards what the scatter graph might look like if our hypothesis is correct.
For the first couple of examples I provide the scaled axes for the class. In later examples on plotting scatter graphs and understanding correlation I expect students to choose and draw their own axes on A4 graph paper with appropriate scaling.
When we have plotted the points, I introduce the term correlation as a means to describe the relationship between two variables. There are two types of correlation.
If two variables are not related the points will be scattered so no correlation is apparent.
A line of best fit can be used to clearly illustrate the directional trend of the data. The closer the points are to the line of best fit the stronger the correlation. We discuss the strength of the correlation as an indication of how closely two variables are related.
The line of best fit also helps to predict the value of one variable when the other is known. It is noted in several examiners reports by AQA and Edexcel that students are more likely to correctly estimate the value of a missing data point and identify anomalous data points if they use a line of best fit.
In my experience, there are three main misconceptions when drawing lines of best fit.
The line of best fit connects to the origin.
The line of best fit is drawn as a line segment connecting the extreme value to the origin.
The line of best fit passes through each of the points.
As an extended plenary I challenge the students to create a scatter graph based on their own hand and foot size. Before we collect any data, I ask the students to write down their own hypothesis at the top of the A4 graph paper.
This is a fun activity which requires the group to work together so everyone has everybody else’s data. Whenever I do this it amazes me which student steps up to take charge of organising everyone. I do my best to keep out of the way and let the students manage the data collection so it is fair and accurate.
In the next lesson we go on to discuss the limitation of using scatter graphs and correlation to identify causation. We consider examples such as the number of ice creams sold and drownings would correlate in the summer months as increased temperatures causes people to go swimming and eat ice cream. However, eating ice creams do not cause people to drown.
In recent examiner reports it is noted how important it is for students to understand the properties of a parabola when plotting quadratic graphs on Cartesian axes. Students who have a secure understanding of parabolas can use them to correct miscalculated values in their table of results and are more likely to attain full marks on this Grade 5 topic.
To help students draw a quadratic graph they need to understand the properties of a parabola. A parabola is a smooth, symmetrical curve with a clearly defined maximum or minimum turning point.
To demonstrate the shape of a parabola I stand at one end of the classroom and ask for a volunteer to stand at the other. I throw a whiteboard pen up in the air towards the volunteer and ask the students to sketch on their mini-whiteboards the path of the pen as it travels under the force of gravity.
Gravity is a force pushing the pen vertically downwards, which is working against the pen moving upwards. This decelerates the pen until zero velocity at the maximum point and then the pen accelerates back towards the ground. The horizontal motion is due to throwing the pen in that direction. Because gravity has a constant acceleration of 9.8 m/s2 the pen falls at the same rate as it climbed. This is what makes the path symmetrical.
When plotting quadratic graphs on Cartesian axes students are expected to complete a table of results to help calculate the y value for changing values of x. I like to include an additional row in the table for each calculation involved in the equation. The x and calculated y value form a series coordinate pairs which are plotted on a Cartesian grid. A common mistake when completing a table of results from a quadratic equation is to incorrectly calculate -32 as -9. In my experience this is more likely to happen if students use a calculator.
It is important for students to understand the table of results are used to generate a series of coordinate pairs involving x and y. A common mistake here is to complete the table but not know how to use it to plot the graph.
To help the pace of the lesson students are provided with a pair of correctly scaled axes. As learning progresses I expect them to choose their own scale and draw the grid on A4 graph paper.
After we have plotted the coordinates I share some tips for drawing the parabola which passes through the points. It is often easier to draw the parabola with your writing hand inside the curve. Turning the paper to draw the parabola is more comfortable than rotating their hand.
The parabola does not have a clearly defined minimum or maximum point.
A coordinate pair is either calculated or plotted incorrectly.
Drawing line segments between each coordinate pair suggest the relationship between them is linear which it is not.
The graph is does not pass through each of the coordinate pairs to form a clear defined and smooth parabola.
When students have completed plotting the first equation I ask them hold up their graphs for me to check. To feedback we have a short discussion about whether the graph they have drawn obeys the discussed properties of a parabola. The students who had made mistakes go back and correct them.
In the next lesson we progress on to using parabolas to solve quadratic equations graphically. Here, we talk about whether an equation has one to two solutions depending on where the linear graph cuts the quadratic. We also look at whether a quadratic and linear will have any solutions depending on the intercept value and gradient of the linear graph.
You can find this lesson in the Key Stage 3 Scheme of Work on Functions, Equations and Graphs.
To introduce teaching reciprocals of numbers and terms I begin the lesson explaining that everything has an opposite. The opposite of shutting a door is to open it. The opposite of saying hello is to say goodbye. Numbers have opposites too we call them reciprocals.
To start the lesson I say to the class,
“I want you to discuss what you think might be the opposite of 2 and you must explain why you think this.”
In your explanation try to use diagrams such as the place value table or a number line to back up your argument. I asked the class to present their reasoning on mini-whiteboards for me and other students to read.
The most common response at this point is to say negative two. We discuss how this could work on a number line.
Negative two is the opposite of positive two because it is the same magnitude of distance away from zero on the number line. This makes sense. Or does it?
If the reciprocal of a number is the same distance of that number from zero what is the reciprocal of zero? If zero represents ‘nothing’ or no place value how can it be positive? If zero cannot be positive, then how can the opposite of zero be negative?
I encourage students to think some more.
After a short time, we discuss how the number two can also be written as a fraction . The almost immediate response now is the opposite of two is one half because you can flip the fraction. This makes sense. I remind the class we thought we had it last time but got stuck on the opposite of zero.
If we take the reciprocal of a number to be the flipped fraction when the number is written over 1. Then the opposite of zero must be infinity. When you write zero as and you flip it to make the question now becomes how many zeros go into 1? Infinite zeros do. The reciprocal of zero is therefore infinity. More simply – the opposite of nothing (zero) is everything (infinity).
Now we understand the reciprocal of a number to be defined as one divided by that number we can look at more complex values such as finding the reciprocal of ordinary and top-heavy fractions, mixed numbers, decimals, powers and even algebraic expressions.
When teaching reciprocals of numbers and terms I know it would have been much easier for me, and possibly for the students too, to say the reciprocal of a number is 1 divided by that number. I could even have stated n × 1/ n = 1. Sometimes I will do this depending on the students I’m teaching. However, I have found though that when there is an opportunity to explore mathematics in this way students become engaged much quicker and are more likely to maintain their engagement for longer.
When I teach how to find the surface area of cylinders I like to add a constant level of challenge and enjoyment to the lesson. Rather than repetitively calculating the surface area of a cylinder I introduce more complex cylindrical shapes.
To find the surface area of a cylinder students need to understand which parts make up the net. To demonstrate how the curved surface makes a rectangle I roll up a sheet of A4 paper into a hollow cylinder and open it. This helps students to see how the circumference forms the length of the rectangle and the width is the height of the cylinder.
The area of the two circles which form the base and top of the cylinder are connected to the top and bottom of the rectangle.
When calculating the area of the net I leave the individual areas in terms of pi. This prevents any rounding up errors. The total surface area of the cylinder is found by adding the three individual areas together.
When I teach this I typically work through the Q1 (shown below) for the students.
The class and I will work through Question 2 together and they will attempt the Question 3 independently. After checking their mini-whiteboards I’ll address any misconceptions at the front.
At this point we have worked through three questions and are about 18 minutes into the lesson. To add more challenge, I ask the class to find the area of the semi-circular prism.
To give a little help we discuss which faces make up the surface area and how to find their dimensions.
After about 5 to 8 minutes most of the class have completed the process and confidently present their workings to me on mini-whiteboards. The only misconception was to find the length of the curved face as the full circumference rather than half of it.
The most common successful approach for this question was to consider the individual areas of each face without connecting them as a net. Some students attempted to create a net but found it difficult connecting the inside curved face.
In the plenary I challenge students to find the surface area of a composite solid involving two cylinders. I asked the class to do this in their books as there was not enough space on their mini-whiteboards.
By this stage in the lesson nearly all the class were comfortable leaving the areas in terms π.
This was a challenging and fun lesson for the students. By the end of the lesson all students could find the surface area of a cylinder in terms of pi and most could solve problems involving composite cylindrical shapes.
The next lesson in the Circles, Cylinders and Circular Shapes unit goes on to problem solving with the volume of cylinders.
Inspiring students to enjoy maths and feel the success that comes with attempting a difficult challenge is why I teach. The feeling of success is addictive. The more students experience it the more they want it and the further out of their comfort zone they are willing to go to get more of it. Teaching mathematics for a growth mindset is central to how I teach maths. It is always in the forefront of my mind when I plan a new lesson.
I was very grateful to be given the opportunity to share some ideas about how I teach mathematics and lead my department to develop a growth mindset in my students at #BCME9.
In my presentation I talk about the link between a student’s growth mindset and their self efficacy. I suggest a range of strategies for how a maths teacher can convince students they do have the skill set to attempt and solve problems they previously thought were beyond them.
The strategies I share in this blog are things that have worked well for me and I have seen work well in my department. I hope you find some ideas that you think could work well for you in teaching mathematics for a growth mindset.
When teaching for a growth mindset I challenge students to apply their learning to solve multi-step problems rather than repeating the same skill for similar questions.
Rather than repeatedly finding the area of a triangle using the formula, A = 1/2 bh I challenge students to find the area of composite shapes where finding the area of a triangle is only one step of a much larger problem.
When students are able to find the factors of a number we can develop their growth mindset by applying that skill to investigate the different perimeters for a rectangle with a fixed area. This could lead on to finding the smallest possible perimeter using irrational numbers or even the largest perimeter using decimals while exploring the concept of infinity.
I suggest when a student has learned how to solve an equation using trial and improvement they could apply it to solve much larger and more complex problems such as the example in the PowerPoint.
In 1976 Richard Skemp wrote a paper called Relational Understanding and Instrumental Understanding. Skemp describes Instrumental Understanding to be learning by rote. Relational Understanding is when a student can connect what they are currently learning to what they previously knew to be true. This type of understanding is more likely to develop a growth mindset as it constantly reinforces and extends their knowledge. Students are then more able to solve complex problems than a student who has an instrumental understanding.
To teach for a Relational Understanding I believe a lesson needs to be structured so students recall their prior learning at the start, extend it through the key learning objective in the main and apply the new skill in the plenary. I have blogged about flow of a mathematics lesson in more detail here.
The examples in the PowerPoint highlight a couple of activities that I use to get students talking about the topic at the start of the lesson. From listening to their conversations I can assess their prior knowledge which helps me decide the level of pitch for the beginning of the main phase. Read more about making the most of a mathematics starter here.
In the main phase I demonstrate how to progress from teaching the main learning objective to using it as a skill to solve a range of scaffolded questions. This approach helps to avoid plateauing so students remain engaged and constantly challenged.
The plenary provides an opportunity to assess the learning for each individual while challenging them to connect the new skill to other areas of mathematics. In this video I talk more about using a plenary in a mathematics lesson to assess progress and extend student’s learning.
To develop a student’s growth mindset they need to believe they have the intelligence and skill-set to solve a problem which they would not have previously attempted.
Comprehension of these questions must be accessible to all students if this is to be achieved. It is important therefore to present the question both visually and orally whenever possible.
Giving students some control over the amount of time they have to solve a problem helps to take the pressure off so they work calmly and maintain their focus.
As teachers we inherently want to help students with their work as and when they think they need it. I propose being less helpful. Encourage using peer support to help them get started with a challenging question.
If they do need help from the teacher they should be specific with what help they need and how it fits in their strategy for solving the problem. If a student believes they do not know where to start, get them to read the question to you or ask them what the solution may look like.
If you would like a student to share their working with the class give them prior notice so they can prepare in advance. They make want to rehearse presenting their method with a friend or another adult. I know I rehearsed my presentation at BCME9 several times!
When teaching mathematics for a growth mindset the resources we provide students with to answer the question is key to how successful students will feel.
I use mini-whiteboards in every lesson. I ask students to show me the whiteboards at the same time. I do not comment on individual whiteboards so no student feels embarrassed by what they have written. Students can either work in pairs on individually. If they do decide to work in pairs they must both agree with what is written.
When a student does share their work with the teacher it is important to value their processing rather than the final solution. A final answer can be ultimately wrong but the method could be valid and insightful. If we only ask students for their final answer we miss out so much of their method. Students can feel successful if they know their processing is correct even if their final answer is not.
I encourage students to be critical of each of other and to have a discussion when different solutions or methods arise. This encourages them to talk about the problem which is key to developing their mathematical reasoning. When I look at the student’s working on their whiteboard I will ask those with different approaches to discuss and share ideas.
Generic praise patronises. When teaching mathematics for a growth mindset be specific about what your awarding the praise for. This will show the students it has been earned and they deserve it. Awarding praise in this way will make students more likely to apply the same level of effort (or higher) next time.
Be clear and creative with your praise. What one student considers praise another may find humilating. For instance, one student may thrive on public praise whereas another could benefit from a discrete smile or nod.
Priase persverence and persistence not quick answers with little working. When a student refuses to progress on with a lesson because a previous problem is bothering them I will always give them the time to solve it. In the student’s mind this adds value to their processing which means they will be more likely to apply the same level of effort in future problems.
Here’s a video about a study on praise and growth mind-sets. I did not include in my presentation but I would like to share it here as the research findings echo the points above.
A written note from the teacher in a student’s book that directly comments on a piece of their work will add so much value to their processing and effort. Being recognised in this way will encourage greater efforts in the future.
In my department we display an Outstanding Mathematicians Board in the faculty corridor. Students know they will only be on the board if they take risks in their learning and have persevered both in their class and home work. Any student who gets their name on the board also recieves a letter home courtesy of the mathematics department.
There are lots of great websites with resources available to for teaching mathematics with a growth mindset.
There are over 450 lessons available for members on my own site https://mr-mathematics.com. The UK Maths Challenge site is excellent for providing challenging and indepth problems. Geogebra has a huge range of interactive animations that promote a relational understanding.
I hope you have found this blog useful and are able to take away some ideas to try for yourself. Please do leave a comment below to share how you go about teaching mathematics for a growth mindset.
To find the equation of straight line graphs students need to calculate the gradient using two pairs of coordinates and the intercept which is the y value of where it crosses the vertical axis. Examiner’s reports of past exam questions show students are more able to find the intercept of a straight line than they are at calculating the gradient. This lesson on interpreting straight line graphs is available here.
The starter recaps the learning in the previous lesson on using graphs to solve equations by interpreting the x and y values of a line. By adding the x and y values together students can see the equation of the line is x + y = 5.
As an extension I ask the class to imagine the line extended beyond the grid and to show me other coordinate pairs the line would pass through. We have a short discussion about coordinate pairs with decimal numbers and move on to the main teaching part.
To introduce the main part of the lesson we discuss how the steepness (gradient, M) and position (intercept, C) of where the line crosses the y axis define how the graph looks on a grid. Equations of linear graphs can be written in the form y = mx + c.
The gradient (M) can be calculated using two coordinates that lie along a line. The gradient is the change in vertical distance divided by the change in horizontal distance. This gives a measure of the steepness of the line.
The intercept (C) is the point where the line crosses the y axis. The intercept can also be defined algebraically as the value of y when x = 0.
I work through the first two examples on the second slide with the class and encourage the students to attempt the third question on their mini-whiteboards. If more practice is needed I use the Interactive Excel File to generate further questions. When they are ready students work through the questions on the third slide independently. All the answers are provided in the lesson plan.
The plenary extends the learning because the line now has a fractional gradient, negative intercept and students need to choose their own coordinates to use.
Rather than calculating the gradient they are asked to explain why it is 2/3. To do this they simply use the method as before leaving the answer as a fraction. I encourage students to use two sets of coordinates, so they can check their working. This also consolidates their method. The equation of the straight line is y = (2/3)x – 1.
The next lesson in the Functions, Graphs and Equations unit goes onto plotting quadratic graphs from a table of results where students explore the properties of parabolas.
Students learn how to generate and describe arithmetic and geometric sequences on a position-to-term basis. Learning progresses from plotting and reading coordinates in the first quadrant to describing geometric sequences using the nth term.
There are three common ways to organise data that fall into multiple sets: two-way tables, frequency diagrams and Venn diagrams. Having blogged about frequency diagrams before I thought I would write about how to draw a Venn Diagram to calculate probabilities.
This activity works well to review two-way tables from the previous lesson. I encourage students to use their calculators so little time is wasted on arithmetic. The probability questions are included to link this with the remainder of the lesson.
The box of the Venn Diagram contains the Universal Set which in this example is the 32 students of the maths class. Each of the ovals represent the A Level subject, Mathematics and Statistics. These are called sub-sets. Because a student can choose to study both Mathematics and Statistics the ovals overlap. This is called the Intersection. The area contained within the two ovals is called the Union.
We begin by writing the 6 students who choose both subjects in the intersection. There are 20 students who choose maths and 6 of them also choose statistics. This means 14 students must be the left most value. The same method tells you 8 students choose A Level Stats but not Stats and Maths. There are 4 students outside the Union who do not choose maths or stats at A Level.
After we have calculated the hidden numbers we work through the probability questions.
I ask the class to attempt the next problem in pairs on a single mini-whiteboard. I find having two students working together on a mini-whiteboard promotes discussion and peer support. Before they show me their Venn Diagrams I ask the class to think about how to check their working is correct. Most realise the numbers should add to the total sample of 100. This helps a couple of pairs rethink and correct their working.
We work through the probability questions one at time. Problems b) and c) prove simple for most students. In part d) only half the class realise the sample size has now changed from all the airplanes to only those departing from America. We discuss the importance of ‘Given that the plane was from America’ in reducing our sample size.
After 8 minutes all the students have drawn the Venn Diagram confidently and most have found the probability that the student plays piano and drums. The most able students have also found the probability the student plays the drums given they also play the guitar.
How to draw a Venn Diagram to calculate probabilities is the third lesson in the Probability, Outcomes and Venn Diagrams unit of work. It follows Calculating Probabilities from Two-Way Tables and precedes Understanding Set Notation.
Students learn how to find a percentage of an amount using calculator and non-calculator methods. As learning progresses they use decimal multipliers to find a percentage change and calculate a simple interest in financial mathematics.
Ratio, proportion and rates of change