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.
Expressing One Number as a Percentage of Another
Solve Problems Involving Percentage Change
Finding the Original Value
Simple Interest in Financial Mathematics
Ratio, proportion and rates of change
Back in May 2017 maths teachers around the country eagerly awaited the first exam for the new GCSE Mathematics syllabus. Proving geometrical relationships using algebra featured at grade 9. In Paper 1 of Edexcel’s test paper the last question of the higher tier looked like this.
Edexcel wrote about student’s performance on this question in their Examiners Report.
“This question was set as a differentiator for those aiming for grade 9, so it was not unexpected to find that many candidates did not answer the question at all.”
It’s because of this I decided to blog about how I teach proving geometrical relationships to students aiming between grades 7 to 9 in GCSE maths.
Whenever I set this the consensus in the classroom is I have not given enough information as students try to find the value of individual angles to add them together. However, after a couple of minutes of perseverance most students recognise the triangle at the centre and its vertically opposite angles are the key to this problem.
The main part of the lesson starts with some simple proofs which the majority of the students would have seen before. It is important when working through these questions to emphasise the need for clear annotations on the diagrams that flow with the algebraic notation. In my experience students often lose track when deriving proofs as they involve multiple stages of working.
I work through the questions on the second slide asking the class for prompts as I go. The questions on the third slide are like those on the second so most students can begin working independently. I encourage peer support throughout the remainder of the lesson and for everyone to sketch the diagrams in their books.
The plenary takes about 15 minutes for all students to complete. The most able generally complete it in under 10 minutes. I ask those who finish early to provide peer support for any who are struggling.
The most common approach is, we know that angles BAC and BDC are equal due to angles in the same segment. Angles ABC and DCB are both 90° due to angle at the centre being double the angle at the circumference (or angle at the circumference of a semi-circle). BD is the same as AC as they are both diameters of the same circle. Angle, Angle, Side proves congruency in this case.
I think it is important to encourage students to derive proofs as early as possible and when they first appear with a unit. For instance, in Key Stage 3 students prove each of the interior, alternate and corresponding properties of parallel lines and why the angles within a triangle add up to 180. In Key stage 4 students prove the various formulae associated to non-right angled triangles, each of circle theorems and the quadratic equation.
Proving geometrical relationships with algebra is the second lesson in the Mathematical Proof unit of work. It follows Algebraic Proof where students learn how to prove various numerical properties and precedes Proof with Vectors where students prove whether lines are parallel.
Circles, cylinders and circular shapes follows on from area of 2D shapes and surface area of 3D cuboids and prisms which students study in Term 2 of Year 8.
In this unit students learn how to calculate the circumference and area of circles both as decimals and in terms of π. Learning progresses from 2D circles to finding the total surface area and volume of cylinders.
Higher GCSE maths students are required to plot and interpret histograms with unequal class widths. Drawing histograms with unequal class widths are very common in GCSE maths papers.
Histograms look like bar charts but have important differences.
When grouping continuous data, it may be necessary to have different class widths if the data are not equally spread out. When class widths are not equal frequency density becomes the vertical axes.
I start the lesson by asking students to find a probability from a set of data in a grouped frequency table. I use this example to help students recap inequality notation and to discuss whether this is an exact or estimated probability. It is important to remind everyone that grouping helps to organise large samples of data but there is a trade off with accuracy when interpreting the results.
As we move on to the main part of the lesson I ask the class to think about how the table in the second slide is different to that in the starter. Students quickly pick up the class widths are no longer equal.
We discuss it may be necessary to use unequal class widths for data that are unevenly distributed and when we do so frequency is measured as the product of frequency density (vertical scale) and class width (horizontal scale).
When drawing histograms for Higher GCSE maths students are provided with the class widths as part of the question and asked to find the frequency density.
I work through the first example with the class plotting the histogram as we complete the table.
In the plenary students are challenged to complete a table and histogram by working out the scale of the frequency density axis. This task frequently appears in exam papers. I provide the students with a print-out of this slide, so no time is wasted copying it into books. Examiner reports state that those who understand frequency as measured by the area of a bar often go on to achieve full marks.
Here is an extract of an Edexcel examiner report for a similar question.
‘In general candidates appear not to be aware that the area of the bars of a histogram are the frequencies, evidenced by a lack of frequency density calculations.’
To ensure all students have enough time to complete the plenary I ask those who have finished first to additionally estimate the number of flowers that grew to between 5 and 8 cm tall. The most able students calculate this as a compound area of the bars on the histogram.
In the next lesson students practice finding the frequencies from histograms to calculate an estimate of the mean. As learning progresses we move on to using interpolation to estimate the median average from a histogram.
To add and subtract with numbers in standard form students apply a range of skills and knowledge of different topics. They need to be equally confident converting large numbers to standard form as they are with writing small numbers from standard to ordinary form. Column subtraction and addition may seem basic skills, but they become more difficult when lots of zeros are added.
When I teach how to add and subtract with numbers in standard form I use the starter activity to make sure students know how to convert between ordinary and scientific numbers.
I ask them to arrange two sets of numbers in ascending order. I don’t mind if students choose to write all the numbers in scientific or ordinary form as both are relevant and necessary to the lesson.
Depending on the class I may use the first set as an example and ask the students to attempt the second set on mini-whiteboards. It is important to allow sufficient time for this as without it the rest of the lesson becomes very difficult.
There are two ways to add and subtract with numbers in standard form. The first is to write them both as the same power of ten and add or subtract the decimals.
To add the numbers in standard form both numbers need to written using the same power of ten.
I find this approach is best for more able students as it requires them to work with decimals rather than integers.
Here both numbers are converted to ordinary form to use the column method of addition or subtraction. This is a common approach as students are more comfortable working with integers than decimals.
Whenever these questions appear in exam papers they involve positive powers of ten but I like to include negative powers to add a greater level of challenge and interest. I also like to include a mixture of addition and subtraction questions. I do this because the challenge is not in the arithmetic but in the conversion between standard to ordinary form and including both consolidates this skill.
I work through the questions on the second slide with the class. After demonstrating a few solutions I ask the students to attempt a few on their mini-whiteboards. I leave them to decide whether to use the standard or ordinary form method. Typically, only the more able students attempt to add and subtract the numbers while in standard form.
After sufficient practice I ask the class to work through the questions on the third slide. I encourage all students to check their work on calculators rather than waiting for me to provide answers. I find this helps maintain the pace of the lesson as students get instant feedback.
I leave about 12 minutes for the plenary. I like this question as it brings together multiple skills. Students first must find the total land area of Asia, Africa and North America then subtract that from the global land area. I ask the class to attempt this on their mini-whiteboards so I can assess progress and feedback.
For additional challenge I provide the approximate populations of each continent so students can calculate population densities.
Addition and subtraction of numbers in standard form is a free lesson which you are welcome to download. Like all lessons available at Mr Mathematics it includes a detailed lesson plan, PowerPoint, Notebook, Flipchart and differentiated worksheet. Solutions are provided for the in the lesson plan and worksheet.
As a Head of Maths I understand the importance of a detailed, flexible and simple scheme of work. I designed the Key Stage 3 and GCSE schemes of work for maths teachers available at mr-mathematics.com to be just that. They are fully aligned with the current specifications and are suitable for study across all UK examination boards including Edexcel, AQA, WJEC and OCR.
The schemes provide a long and medium term plan for the entire Key Stage 3, Foundation and Higher GCSE courses. They emphasise helping teachers to create pace and challenge for every class as each lesson draws on and extends previous learning so progress is fluid and rapid.
Linked within the schemes are engaging, interactive and fully differentiated presentations that are designed to empower the teacher to enthuse their students. The presentations are available as a Smart Notebook, ActivInspire Flipchart and Microsoft PowerPoint. Every lesson comes with a plan to talk the teacher through the flow of the presentation and independent learning worksheet for consolidation. There are additional links to informative blogs, group work activities, YouTube videos and much more.
Every lesson and maths worksheet has been designed and used by myself. As well as a Head of Maths I am a certified teacher trainer and Master of Education specialising in Research in Mathematics Education. These schemes of work for maths teachers represent my intention to help improve mathematics education around the world.
Solving problems with non-right-angled triangles involves multiple areas of mathematics ranging from complex formulae to angles in a triangle and on a straight line.
As the GCSE mathematics curriculum increasingly challenges students to solve multiple step problems it is important for students to understand how to prove, apply and link together the various formulae associated to non-right-angled triangles.
There are three ways of teaching students how to derive the Sine, Cosine and Area rules.
I begin the topic of solving problems with non-right-angled triangles with the Sine Rule. At the start of the lesson students arrange a jumbled up derivation using right-angled trigonometry. In the main teaching phase we work through a series of problems involving missing angles and lengths. The plenary is more challenging as students need to apply various angle properties to have a matching angle and side.
After the Sine Rule we progress on to deriving and using the Cosine Rule to calculate unknown lengths. The start of the lesson is another jumbled up proof for the students to complete. When teaching the derivation most students are able to identify the a2 = b2 + c2 (-2bcCosA) as related to Pythagoras’ Theorem. The development phase teaches students how to substitute known values into the formula. As learning progresses students are challenged to combine the Sine and Cosine Rules within a single problem.
In this lesson students learn how to find an unknown angle in a triangle when all the lengths are known. We start with another jumbled up proof then quickly move on to sketching problems given as written descriptions. Using mini-whitboards to sketch their diagrams helps students to visualise the correct information.
This is the final lesson in the topic. Students apply both the Sine and Cosine rules to solve a range of problems involving the area of a triangle. At the start students learn how to find the area of a triangle. As learning progresses they use the area to calculate a missing angle or length. Exam questions often include the area of a triangle in the non-calculator paper as Sin 30 can be worked out exactly without the need for a calculator.
Students struggle solving larger problems that involve multiple formulae as they do not take the time to devise a strategy. I encourage students to take a minute or so to sketch a flowchart that will break the problem down into smaller, more manageable steps.
When the students have come up with a strategy we discuss how to identify which formula to use with the following prompts.
When a problem is given without a diagram students find it difficult to visualise what they are being asked to calculate which is why an accurate sketch is so important.
When I teach solving problems with non-right-angled triangles students are required to sketch the diagrams as part of their working. This helps them to accurately interpret the more wordy type problems. Understanding the correct notation is important. Angles are marked with capital letters. Opposite sides are marked with lower case letters.
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.