# Using Numbers to Highlight Connections

This year I had the pleasure of contributing a chapter to a book edited by Ed Southall and published by The Mathematical Association called ‘If I Could Tell You One Thing‘. The chapter discusses a collection of situations where the numbers that are used in a worked example or practice question can obscure the reasoning behind the calculations or even fuel a misconception. However, numbers don’t just cause problems; they can also create opportunities! This blog post discusses some of the positive power that number choices can have in examples and questions! In particular, how carefully chosen numbers can help draw attention towards some wonderful connections that can be found within mathematics.

As a mathematics teacher, I am fascinated by deep structures that underpin mathematical ideas and the connections between different ideas. When looking at topics through this lens, numbers sometimes seem irrelevant and calculations can just be a distraction from what is going on at a conceptual level. It is arguably much easier for teachers than students to see mathematics in this way because we have already experienced the whole mathematics curriculum as a student and then several times again as a teacher. However, I tend to find that when students are looking at mathematics a lot of their attention to is focused on the numbers and calculations. So how can we use numbers and calculations to steer their attentions towards these deeper structures?

I’ve previously written a blog post about situations where questions from different topics can be solved using the same method – occasions where it’s almost like we have the same question dressed up in different disguises. An example of this is with equivalent fractions, equivalent ratios and other problems with proportion. The way that the questions and methods are laid out below helps highlight the similarities between them (even if they are are not necessarily the methods we might normally use). But the fact that the numbers are all the same in each question makes the connections even more obvious to anyone who is just focusing on getting answers – not only are the answers all the same, so are the calculations!

There are also occasions in mathematics where we are not quite doing the same thing but still find ourselves using similar processes in different topics. For example, we can simplify a ratio by finding for the highest common factor and dividing each part by it. We also find ourselves doing this when factorising an algebraic expression into a single bracket. The highest common factor plays a slightly different role in these two problems, both conceptually and in the final solution, but the mental methods are certainly similar.

Students could be prompted to notice the connections between these two problem types by solving one of each consecutively or seeing an example of each side-by-side on the board. If this was done using the two examples above then students might spot that they are doing similar calculations with each example. However, it’s still fairly subtle. Both the questions and solutions appear quite different in appearance and the main similarities are in the mental calculations, which they can not necessarily see on the page. Also, if students are used to completing exercises with unrelated questions (such as in mixed topic starters) then they might not think to look out for connections between them. However, if the two questions use the same numbers then it could help shine a brighter spotlight over the connections.

For students who are focusing mostly on the numbers and calculations they’re using, they might now find themselves not doing similar calculations but doing the exact same calculations in each. This might be enough to spark intrigue about why they’re getting a feeling of déjà vu. But if not then it leaves a nice opportunity for a teacher address it explicitly by asking questions such as, “Look at these two questions that have the same numbers. Why do their solutions also have the same numbers? What did we do in each case?” This connection could be exaggerated even further by using two questions like the one below.

However, it seems important in these cases for a teacher to also draw attention to the differences between the two concepts and processes. In particular highlight the differences in the role that the HCF plays in each question, just in case students then start to make mistakes like the one below.

There are some situations in mathematics where different topics appear completely unrelated on the surface but have some hidden connections when we look under the bonnet. Take histograms and density, which I have previously written about here. The two questions below could not appear more different if they tried…

However, these questions do share something common: at some point we find ourselves dividing the amount of “stuff” there is by the amount of space available for it. On the left we do this when calculating density by dividing mass by volume; on the right we do then when calculating frequency density by dividing frequency by class width. This might not be obvious to students, even if it’s explicitly explained to them. However, by presenting two questions together that are in a similar format and use the same numbers (such as below), it might help similarities to stand out a little more clearly. It’s arguably a little contrived, but its purpose is to shine a bright spotlight on the connections between these two contrasting topics.

But even when we have questions, answers and calculations that all appear completely different, there are still cases where our methods are driven by similar principles. Take for example calculating with fractions and calculating with standard form.

These questions look quite different and we’re not really doing similar mental calculations either. However, by looking at each of calculations from the first set side by side there are some connections to be found. In both topics, multiplication seems to have the most straight forward process: in each case we multiply the parts that correspond with each other and adjust/simply our product at the end if we need to.

Addition is quite straight forward when the “things” that we have are the same (i.e. the powers of 10 or the denominators). In these situations, we can just add together the other “things” (i.e. the decimals or numerators) and adjust/simply our solution at the end if necessary. Students also see this when simplifying simple algebraic expressions.

However, for both standard form and fractions, addition becomes trickier when the critical parts are not the same. In both cases, these problems can be solved by finding an equivalent addition where the parts do match, and then adding as before.

So students can think about how even though the methods and calculations are different, the ideas behind those methods are the same. They might conclude that the reason why these two types of addition problems are similar is because “that’s just how addition works: you can add when the two numbers are amounts of the same thing” which would be a helpful takeaway. However, we can shine a brighter light on why these two problem types are so similar by exploring pairs of questions like the ones below. In this case it’s potentially clearer for students to see why the methods are linked, because they are seeing two different ways of writing the same addition.

For one final example, I’d like to look at the connection between multiplying together a pair of two digit numbers and expanding a pair of binomials. This too can be highlighted through a careful choice of matching numbers.

To be honest, I’m not a particular fan of using the grid method as a final method for solving either of the questions above but more so as a stepping-stone to help students understand the reasoning behind more efficient methods. In this case, solving these two questions side-by-side using the same method demonstrates that despite the questions appearing different, they are structurally similar. It can also lead to some other interesting questions to reason with, such as “What value of x would make these two questions exactly the same?”

Also, it may also encourage creative thinking about other potential methods we could use (even if they are not the best). For example, if we can answer both of the questions above using the grid method, then what would it look like if we used the FOIL method that we normally use for expanding brackets to perform the calculation?

And what would it look like if we used the column method that we normally use to calculation long multiplication to expand the brackets?

I haven’t particularly warmed to the idea of using FOIL to calculate multiplications or column method to expand brackets (although I’d be interested if anyone does). But exploring methods that are normally associated with one problem-type to another can sometimes draw attention to aspects of each method that may have become automated or unnoticed. For example, expanding brackets using the column method emphasises the role that the ‘place holder zero’ plays in lining digits up correctly in long multiplication.

When I was an early career teacher, I used to regard the numbers I chose for worked examples and practice questions as fairly unimportant (so long as they didn’t make the calculations distractingly complicated or give decimals at unwanted times). However, as discussed in Chapter 18 of ‘If I Could Tell You One Thing’ and this post, careless number choices can lead to problems and careful number choices can lead to opportunities!

If you would like to use or share any of the images from this blog, feel free to use this Power Point to help you: Using Numbers to Highlight Connections

For other blog posts, please visit my contents page.

# Playing With Pythagoras and Trigonometry

During my first few years of teaching, I may have overused “ladder problems” for Pythagoras and trigonometry. I had other problem-solving questions involving splitting isosceles triangles in half, diagonals of rectangles, bearings, angles of elevation and other worded problems. But they were fairly predictable (especially if used during a lesson about Pythagoras or trigonometry) in that they all asked students to find a missing side or angle for a triangle that was somewhat hidden in the question. I used to find myself exhausting my collections of different setups for these sorts problems quickly. So I was keen to build up my repertoire of novel ways to disguise right-angled triangles for Pythagoras and trigonometry problems and also to find other interesting questions that might encourage students to think a little deeper about these two topics.

Since then, I’ve drawn inspiration from colleagues, textbooks, online chat and other resources to spark new ideas for getting students reasoning and problem-solving with Pythagoras and trigonometry in a variety of ways. I’d like to use this post to share a selection of my favourites. Some of them are sets of problems that focus around a particular theme (e.g. right-angled isosceles triangles), while others are ideas that just try to get students to consider the concepts in slightly different ways.

Applying The Formulae “Backwards”

In a previous post, I talked about working forwards and backwards through concepts and gave an example like the one below. In this case, rather than asking students to use Pythagoras’ Theorem to calculate the length of a missing side, the question asks them to decide whether the triangle contains any right angles.

This sort of problem could be extended further by asking students to determine whether the largest angle is acute, obtuse or a right-angle. I find this encourages students to think hard about the relationship between the lengths on the triangle and the structure of the formula in order to decide if the angle is acute or obtuse. For example, if the sum of the squares of the two shorter sides is greater than the square of the hypotenuse, an argument often breaks out about whether that means the largest angle is greater than 90 or less than 90.

Another situation where it can be interesting to “work backwards” in a similar way is with trigonometry. For example, rather than using sin to calculate a missing length or angle, a question could give the measurements for two sides and an angle then ask if the triangle is right-angled.

A Trojan Horse of Trigonometry

Craig Barton makes a great point in his book (2017, p.416) that suggests when students see problems that require Pythagoras’ Theorem in a lesson that is about Pythagoras, then they already know that there’s a good chance they’ll need to use Pythagoras’ Theorem before they even read the question. The same goes for other topics, such as trigonometry problems in a trigonometry lesson. The hardest thing about solving problems like these outside of their lessons is spotting the right-angled triangles. Therefore, when studying trigonometry, it can be useful to hold a few different types of trigonometry problems back for a later time and then sneak them into an unrelated lesson by stealth! That way, they are not already on the lookout for right-angled triangles and have to recognise the situation it for themselves.

For example, I slipped the question below into one of my starter activities with Year 10 a few weeks ago.

It had been a little while since we had done any trigonometry, so the students weren’t on the look out for right-angled triangles. Instead, this conversation happened…

Student: Can I just put “acute”?

Teacher: No, I’d like the size of the angle, please.

Student: Can I estimate?

Teacher: No, I’d like an accurate measurement given to one decimal place, please.

Student: Can I come to the board and measure it with a protractor?

Teacher: No, you don’t need to.

Student: Eh?

But once one student had spotted that it could be solved with trigonometry by visualising a vertical line like the one below, a ripple of realization made its way across the room.

This sort of idea can lead to a whole host of similar problems, like the ones below. In each case, you can make a right-angled triangle by looking for a point on the diagonal line that goes through the corner of one of the squares in the background. However, students need to then do further calculations to find the size of the red angle.

These problems can still be great to use in a lesson that is about trigonometry, as they demonstrate an interesting application of the formulae. And I’ve noticed that even when students know they need to use trigonometry, they still find the decision making around their route to the solution a little tricky. Sneaking them into an unrelated lesson, however, just adds a satisfying element of surprise.

Problems Using Right-Angled Isosceles Triangles and/or Regular Polygons

There is a lot of fun to be had with right-angled isosceles triangles. More than I think I initially appreciated. Even a basic question like the one below is interesting in its own right, because it can be solved with either Pythagoras or trigonometry based on the information given.

I often find students opt for using Pythagoras with the question above, rather than trigonometry. It’s the more obvious choice when the 45 degree angles aren’t labelled and my students have tended to find Pythagoras easier to use than trigonometry. However, when this question is turned around (like the one below), a method using Pythagoras becomes a little bit trickier than with the previous question, as students sometimes hesitate about how to get started. On the other hand, the methods with trigonometry seem just as straightforward as they did in the previous question.

Whether students use Pythagoras or trigonometry, their methods for dealing with right-angled isosceles triangles can be applied in many, many ways. For example, it allows them to solve problems involving diagonals of squares.

A problem like this can lead students to an alternative formula for calculating the area of a square: “area of square = 0.5d2“. This formula could also be discovered before students learn Pythagoras by drawing two diagonals to split the square into four congruent right-angled triangles.  However, I find the Pythagoras route neater!

Dealing with the diagonal of a square can also give some interesting opportunities for mathematical reasoning with Pythagoras’ Theorem and irrational numbers, such as with the question below.

Students often start this problem with trial and error by choosing a value for x and seeing if they get a whole number for y. But then when they start to work solely with the algebra that is given to them, they can deduce that y is equal to root2x, meaning that at least one of the two lengths must be irrational. This problem can also provide a nice segue into some history of mathematics and the story of Hippasus (an example of this story can be found on nrich).

The hypotenuse of a right-angled isosceles triangle could also disguise itself as a chord in a circle, like with the problem below. This question asks for the area of the circle, but alternatives could be to ask for the circumference, the length of the minor arc AB, area of the segment, and so on. Regardless of the end product, the start of the problem is still essentially the same as the second example in this section of the post.

The two radii and right-angle marker provide visual clues that the question involves a right-angled triangle. The entry into this problem could be made trickier by presenting the information without these things drawn, like in the example below. Students have to wrestle with the terminology a little first and deduce that the arc AB is one quarter of the circumference before they can be sure that the angle at the centre is 90 degrees.

A question like this could catch students out when the fraction given in one quarter rather than one third, like with the question below. Now, the angle at the centre is 72 degrees rather than 90 degrees, which prompts a change in strategy. However, seeing the word “quarter” in the question makes it very tempting to think that the arc is a quarter of the circumference, like in the problem above.

The hypotenuse of a right-angled isosceles triangle could also hide itself on the side of a regular octagon. Once again, solving this problem involves dealing with the same triangle as in the second example from this section, but visualising it can be a little tricky until students start drawing extra lines on the diagram.

Exploring regular polygons leads to a whole other exiting avenue for creating interesting Pythagoras and trigonometry problems (and not just ones with right-angled isosceles). For example, one tweak to the question above can make the problem require a different strategy.

The route to the solution is different but still has many options. The most direct way would be recall the interior angle size of a regular octagon and use the cosine rule.  However, if students haven’t met the cosine rule and have only learned right-angled trigonometry then they could split the shape into two congruent right-angled triangles and use half the interior angle. If students have only learned Pythagoras, then they could solve it by using the formula twice: once with GH as the hypotenuse and then again with HB as the hypotenuse.

There really is a whole Aladdin’s Cave to be of wonders to explore in problems that hide right-angled triangles in regular polygons.

The Two Small Areas Add Up to the Big Area

Another interesting avenue of Pythagoras problems can be started with the example below.

This problem is already a little bit quirky because it gives the lengths of the squares rather than the triangle itself. But it doesn’t take much for students to move those labels to the short sides of the triangle, calculate the length of the hypotenuse and then calculate the area of the biggest square. At least that is what I find most students want to do when they see this question (and then kick themselves afterwards for bothering to square root the 100). Alternatively, students who know their Pythagorean triples well will spot that the hypotenuse is 10 cm straight away and then square it to get the blue area.

However, this problem can become more thought provoking if we made the hypotenuse an irrational number and didn’t allow students to use a calculator.

What I like about this problem is that it requires students to stop thinking of Pythagoras’ Theorem as just a formula that you plug numbers into and get a number out; it encourages them to think more deeply about what the theorem tells us. It doesn’t just give us a relationship between three lengths, but also the relationship between the areas of three similar shapes. By looking at the problem through this lens, it becomes easier than students might initially think because they can just add up the two green areas to get the blue area.

Now there is a whole host of new questions that a class can explore with this message at the heart of the problem. For example, the last question could be made slightly more complex by making students work a little harder to get the areas of the smaller squares first, like below.

It can also be interesting to explore using shapes other than squares around a right-angled triangle.

Students who see this for the first time often solve it by substituting the lengths 16 cm and 12 cm into Pythagoras’ Theorem to calculate the length of the hypotenuse, half it and use that to calculate the area of the blue semicircle. They then have the nice surprise that they can also get the same answer by calculating the areas of the two green semicircles and adding them together.

Related to this are problems based on The Lunes of Alhazen, such as the one below. This comes with the delightful surprise at the end that the total green area is equal to the area of the triangle. The problem could be extended further by considering whether that result is just the case for this particular problem or if it can be proven to be true for all possible measurements on a right-angle triangle.

There are other problems that can be solved simply when students use Pythagoras’ Theorem to focus on relationships between areas rather lengths. For example, calculating the length of the hypotenuse in the problem below would be a nightmare at high school level!

The final fun thing I like to do with this idea is to break the shapes away from a right-angled triangle altogether. For instance, if the lengths for the three squares below satisfy the formula a2 + b2 = c2 while they are connected around a right-angled triangle…

Then those same lengths would continue to satisfy the formula if the three shapes were pulled apart…

So the areas of the two green squares still add up to the area of the blue square because the lengths satisfy the Pythagorean formula.

This means that if students are good at spotting Pythagorean Triples, then they could use this idea to solve problems that don’t even have a triangle in sight!

This next problem could be solved by calculating scale factors between two of the trapezia. But if it’s posed as a non-calculator question, then the calculations could get very fiddly! However, because they are similar shapes and the three lengths make a Pythagorean Triple, the smallest area is the difference between the other two areas.

We could also start to think 3D…

If students need a little more convincing about this last problem, then it could be explored by looking at nets for cubes around a right-angled triangle.

The limits to creating ir finding interesting problems using Pythagorean Triples is endless and there are so many ways to surprise students with their applications!

If you would like to use or share any of the images from this blog, feel free to use this Power Point to help you: Playing With Pythagoras and Trigonometry

For other blog posts, please visit my contents page.

One topic that I used to get very frustrated with teaching is ‘probability trees’.  It would frustrate me for the same reason that topics such as ‘angles in parallel lines’ (which I’ve previous written about here) would frustrate me: it’s relatively easy compared to other parts of the maths curriculum, but students seemed to struggle with it more than with difficult topics.  Even when I’ve taught top set Year 11, who could confidently solve non-linear simultaneous equations and complex trigonometry problems, I’d be surprised that probability trees would be one of their most requested topics for revision.

Looking back over my old teaching resources on probability trees, one possibility for these difficulties could be that I didn’t invest enough lesson time on the aspects of the concept that were most novel to them.   Take the question below for example.

Questions such as this one require students to:

• know how to write probabilities of single events as fractions;
• know how to multiply together fractions;
• know how to add together fractions;
• know how to structure a probability tree to represent a sequence of possible events;
• understand why we multiply the fractions together for combined events.

The first three bullet points are things that students (hopefully) already know before getting to this section of their programme of study.  This means that when students are introduced to probability trees, the most unfamiliar aspects of the topic are the last two bullet points.

Thinking back to those old lessons, a lot of my explanations would centre around solving whole probability tree problems from the get-go. While solving a problem on the board, I would call on students to give me the fractions that go on each line, multiply together each pair of fractions and add up the fractions that I needed to solve the problem.  The students would answer these questions so confidently that I’d think, “They’ve got this!” and set them off with independent practice.  I’d then always be surprised with how difficult they found solving probability tree questions by themselves, how many marks would be dropped in exams on them and how it would often be a common request for revision.

The biggest issue tended to be drawing the tree!  Knowing how to set them out, how many branches to use, how many sets of branches, how to arrange the labels, etc.  So even though students knew how to perform all the calculations they needed to solve a probability tree question, they made mistakes before getting to the point of calculatuon.  I think the biggest issue was that some of my students didn’t seem to understand what was going on when they were drawing probability trees and what the diagrams really meant.

Evidence of such misconceptions would occasionally present themselves in the way students set out their branches…

…the way they arranged the information on trees…

or their choices of fractions in problems that were worded like the one below.

In hindsight, I suspect that some of the students’ uncertainties and mistakes with this topic were because I tended to skim over explanations about how to set out probability trees and their meaning.  Instead, I jumped too quickly to calculations and solving complete problems.  This meant that more time was spent discussing the aspects of the problems that students were already familiar with (how to probabilities and calculate with fraction) than the aspects that were most novel to them.

This blog post shares a selection of things that have gradually been added to my lessons on this topic over the years.  Some of them support the introduction of probability trees by helping students think about their structure and meaning; some help extend and challenge students further within the topic. I don’t always use them all every year; it usually depends on how much support of extension I feel is appropriate. I’ve also discussed some of these things in conversion with Craig Barton on his podcast (available here).

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Laying Some Early Foundations

Students can start to be familiarised with some ideas relating to probability trees long before they get to the ‘probability tree’ section of the scheme of work.  For example, students may work with frequency trees in younger years to solve problems like the one below.

The numbers above were modelled roughly on the probability tree from the question below.  While the frequency tree looks at observed data for many people playing the games, the probability tree looks at the theoretical probabilities for each individual person playing the games.

Inspiration for this connection was taken was taken from a series of interesting articles on the Nrich website about transitioning from frequency trees to probability trees (available here).  They describe an empirical-to-theoretical approach for probability, where students go from exploring frequency to proportions to expectation.

Another thing that I’ve found useful, as a way provide students with some early exposure to probability trees, is to represent outcomes of single events with branches during lessons on basic probability.  These don’t necessarily change the way I teach probability to Key Stage 3 or affect how students use probabilities.  They are simply used to as a way to supplement explanations with illustrations on the board.  I find that if students are at least familiar with seeing outcomes and probabilities represented this way for single events, then they are already one step ahead when it comes to combining them to build probability trees.

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Thinking About How to Structure a Probability Tree

My go-to tool for introducing various ideas within this is to use scenarios where someone is drawing different coloured marbles from a bag.  While these ‘marbles-in-a-bag’ scenarios are fairly contrived, they provide a helpful stylised way to explore aspects of probability trees because they are easy to visualise and play out if necessary.  So, at each stage of concept development, I often begin with a marbles-in-a-bag scenario first and then later ask students to apply what they have learned to a more contextual scenario.

The first thing I tend to focus on is how to choose an appropriate tree structure to match a sequence of events.  This is without writing any probabilities for the time being.  For example, we might start by considering how to draw trees for the four scenarios below.

By comparing and contrasting these scenarios, students can be encouraged to consider the similarities and differences between the four probability trees.

It can be helpful for students to explicitly consider and discuss some of the key questions in their decision making. “How do we know how many branches to draw at each intersection? How do we know how many layers of branches to draw?”  Once students have understood the reasoning behind the structure for these four probability trees, they might then apply what they have learned to draw trees for other contexts.

I tend to find that Scenario C causes the biggest debate with students. They can sometimes be tempted to draw two layers of branches for ‘on time’ and ‘late’, with three branches at each intersection (one for each person).  This is similar to the mistake at the start of the post. This task provides a chance to address the issue before students meet it when solving complete problems.

One useful follow-up to this activity can sometimes be to take the contextual scenarios and map them to the marble scenarios.  “Why do the probability trees for ‘Marble Scenario 4’ and ‘Contextual Scenario A’ have the same structure?  Which of the marble scenarios has the same structure as Contextual Scenario C? How could I alter one of the contextual scenarios to make it have a tree like Marble Scenario 3?”

In all of the examples above, the conditions are consistent throughout the tree.  For example, if one intersection in a tree has three branches, then all intersections have three branches.  So, it can also be valuable to explore ways that the number of branches can vary within a tree.

In the scenario below, we are choosing marbles from two different bags.  Students can consider the different ways that we could structure our probability tree.

They might reason that the number of branches per intersection differs between layers is because the number of different options differs between picks.  This could also occur when drawing two marbles out of the same bag, when there is only one marble of a certain colour. For example:

In the scenario above, students can observe how the outcome of one event affects the number of possible outcomes in the next event.  This highlights that some branches can terminate before others.

Other contextual situations where branches terminate like this could be playing in a knockout tournament (e.g. there are four rounds of matches and you play until you lose) or in Monopoly where you have three attempts to roll a double to get out of jail without paying.

While all the examples above start by describing a scenario and then asking students to think about the tree, it can also be interesting to do this the other way around.  Students could be presented with a completed probability tree and then be asked to think about the scenario.

Once students are comfortable setting up trees of different structures, it doesn’t take much more effort for students to start labelling branches with probabilities.

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Thinking About Why We Multiply Probabilities

Another novel aspect of this topic is the idea of multiplying two or more probabilities together.  Students have most likely solved problems before where they needed to add probabilities together (i.e., when finding the probability of one thing happening or another) and hopefully understand why they should use addition.  But this could be the first time that they find themselves multiplying probabilities together and trying to understand why they should do so.

One way that I like to get this across is by referring to their previous experience with using two-way-tables as sample spaces.  For example, students could represent the scenario below by using both a probability tree and a table.

By comparing the two diagrams, students can consider how to find probabilities for combinations of events. Students tend to see clearly from the table that the fraction of BB is four twenty-fifths.  They might then spot that they can also get four twenty-fifths from the tree by multiplying the two branches together.  Digging a little deeper, they might reason that this works because the table shows that two-fifths of the outcomes have blue as the first pick (10 out of the 25 boxes) and then two-fifths of those also have a blue as the second pick (4 out of all 25 of the boxes).  Therefore, they are finding a fraction of a fraction, which can be done by multiplying the two fractions together. Once this rule has been established and checked then we can put aside the two-way table and focus on solving future problems using just probability trees.

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Reasoning and Problem Solving with Probabilities

Once students get to the point where they understand how to structure probability trees and can use them to calculate probabilities of combined events for both independent and dependent cases, it can be tempting for to think “job done!” and move on to the next topic.  This is certainly where my lesson resources used to end (and occasionally still do). In most cases, this often enough to enable students to solve most probability tree problems at GCSE.  However, there is so much scope to extend and apply probability trees within the GCSE syllabus.

One avenue of reasoning could be to explore how the order of events can affect probabilities depending on whether they are dependent or independent on each other.  Take the problem below, which asks students to consider two independent events.

Some students might be hesitant to set up the tree for a problem like this because the wording is a little vague.  They might be more used to scenarios where an event is repeated twice, or two different events happen in an obvious order (e.g., Monday and Tuesday).  However, the wording in the scenario above doesn’t give any clues about what order the boxes are chosen from.  This vagueness though provides opportunity to pose questions such as: “Does it matter which box Leyland chooses from first? Will the order affect the probabilities of the combined outcomes?”  It can be good to get students to explore this through verbal reasoning first, discuss how the fractions will be the same in either order and how multiplication is commutative.  These ideas can be confirmed by drawing probability trees for each order.

But then what if we made some adjustments to the problem to make it so that the probabilities for one event depended on the outcome of the other?  For example, there have been problems in GCSE over the last few years where a marble is taken from one bag and paced into another.

In the image below, the same problem is presented twice but with the order switched.  Similar questions could be posed again: “Does it matter which box Laura chooses from first? Will the order affect the probabilities of the combined outcomes?”

Once again, it can be interesting for students to explore this through verbal reasoning before drawing any trees.  They may recognise that the fractions will be different but could still have some doubts about if it matters.  “What if the fractions are different but they multiply and simplify to give the same products?”  Some students I’ve taught have managed to convince others that the order does matter by focusing on the denominators: in the left-hand scenario, the denominators will be 5×8=40; in the right-hand scenario, the denominators will be 7×6=42. Students predictions can be confirmed by drawing the probability tree for each case.

There are also plenty of opportunities to introduce other aspects of the mathematics curriculum into probability trees.  One way could be to include other topics within the context of a problem, such as in the question below.  While this scenario is fairly contrived, it requires students to draw upon their knowledge of area, perimeter and factor pairs in order to calculate the probabilities that go on each branch.

The context of the problem isn’t the only way to weave in other topics.  There is scope for students to exercise other maths skills even within more abstract probability tree questions.  Take the one below for example.  This could be solved numerically through trial and error or algebraically through equations.

However, what I particularly like about this problem is its scope for extension.  A follow-up problem could be to say, “When I wrote this problem, I chose to use the terms x, 2x, 2x, and 4x and it worked out quite nicely.  Would it have worked just as well if I had chosen a different set of four terms?  For example, what if I change this 4x to 3x?  Would the problem still work?”

Once students start working on this, they can find that it is unsolvable.  They might find values that work for some combined outcomes, but not work for all.  But what if I use the terms x, 4x, 4x and 16x instead? These do work!  This now seems ripe for turning into an open-ended investigation: “Find as many sets of four coefficients as you can that work in this problem.”

Students often start by choosing four terms, drawing the tree diagram and then trying to work out the fractions to determine if it’s possible – similar to the order that they solved the first of these problems in. However, they might then figure (or be prompted) that a better strategy could be to work in the reverse direction: choose a number of marbles for each colour, draw the tree and then work out the terms.  Once they get into the swing of doing this, they might start using a systematic approach to choosing the numbers of marbles, which reveal patterns in the coefficients.

By keeping the number of marbles for the first colour as 1 and changing the number of marbles for the other colour, students can generate the following sets of terms:

There are interesting patterns that students might explore here on a basic numerical level here.  They might notice that the middle two terms are always the same, the coefficient of the fourth term is always a square number and is the square of the second/third coefficient, all four coefficients always add up to the next square number that comes after the fourth coefficient, etc.

Students can dig deeper by changing the number of marbles for the first colour.  Before doing so, it could be good for them to predict which of their observations will remain true when there is more than one of each colour.  For example, when there are 2 marbles for the first colour and they vary the number of the second colour, they can get the following terms:

When there are three marbles for the first colour and the number of marbles for the second colour varies, they can get the following:

Once again, there are interesting things for students to explore on a numerical level.  The coefficients still add up to a square number (but not the next square number after the fourth coefficient anymore).  You can get the first and fourth coefficients by squaring the number of marbles of each colour.  You can get the second and third coefficient by multiplying together the number of marbles for each colour… and so on.

If this is all that students get out of this problem, then great! However, students may also make a link between this problem and another aspect of the mathematics curriculum: squaring binomials.  The patterns of coefficients in each set follow the same patterns as they do when they expand brackets for the square of a binomial.

For students who are planning to study A-level Maths, investigations like this can act as a ‘teaser trailer’ for ways that probability becomes more algebraic in the further study. For students aren’t planning to study A-level Maths, it can simply highlight a surprising connection between two very different-looking things that they have previously learned for GCSE.  But even if they don’t make the algebraic connections, the numerical reasoning can help spot patterns in probability trees with repeated independent events.  This might come in handy when they next solve a basic probability tree problem (such as the one below), because it provides them with some ‘checking mechanisms’ such as, “I should expect these numerators to be square numbers,” and, “I should expect these two numerators to be the same and also be product of the numbers of marbles for each colour.”

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If you would like to use or share any of the images from this blog, feel free to use this Power Point to help you: Thinking About Probability Trees

Previous blog posts:

‘Planning Topics’ Series:

‘What I Learned From Shanghai’ Series:

‘Making Connections’ Series

# This Way, That Way, Forwards and Backwards

For my daily commute, I take one route to get to work in the morning and a different route to get home in the afternoon. The shortest route is quicker in the morning but takes twice as long to get home because of traffic in the afternoon. I’ve taken these journeys so many times now that I can navigate my way with very little thought!

One day, I was driving to work later than usual and so decided to use my ‘afternoon route’ instead, to avoid the traffic. I’d only ever driven home this was before, so it meant I was navigating this route in the reverse direction to what I was used to. I thought it would be a doddle but was surprised at how tricky I found it! Despite driving down these roads so many times, I had to pay more attention to where I was going than I’d done for years! “I know I usually turn right onto this main road from a side street, but which side street is it?” It also made me realize that I didn’t know this part of town as well as I thought I did; I was just used to following the same sequence of turns each day. All this from the simple act of driving in the reverse direction to what I was used to!

How is this relevant to learning mathematics?

It’s a slightly tenuous analogy, but I also find that the same can be said for many aspects of learning mathematics. There are many topics where I often find myself asking students to work through problems in one direction far more often than the other. With enough practice, students would get to the point where they could carry out the correct processes in the correct order routinely with little thought or effort. However, the simple act of turning these problem around can encourage students to re-concentrate their efforts, deconstruct processes that they are familiar with or look at things in a different way. While this sometimes serves to increase challenge in a task, other times it can help students assimilate a more complete image of the concept they are studying.

For example, take the two questions below which require students to manipulate surds. I reckon I’ve given my classes far more practice on questions where they simplify surds (like the one on the left) than questions where they ‘unsimplify‘ surds (like the one on the right).

Even though ‘unsimplifying’ a surd might seem like a strange thing to do, it can still be useful from a concept learning point of view. There are two key things that I usually want my students to think about when simplifying surds, like in the question on the left: factor pairs of 45 that contain square numbers and remembering to square root the 9 when they take it out of the radicand. The question on the right also shines a light on these points but from a different angle: they have to square the 3 before they can bring it into the root, which means they multiply the 5 by a square number.

Working both ‘forwards’ and ‘backwards’ through a concept can facilitate a fuller inspection of it. This could even be done with the tiniest of tasks. For example, in a previous blog post I looked at some activities that students can do during lessons about corresponding angles. In one activity students were given two angles and asked to determine if they were corresponding; in another activity students were given one angle and asked to mark the angle that corresponds with it.

There are a lots of opportunities for students to work ‘forwards’ and ‘backwards’ through concepts. In some cases, it is because a process has an inverse that is either definable, easy to describe or taught explicitly as a separate learning objective. Examples of these include expanding brackets and factorising, converting numbers into and out of standard form, or drawing straight lines for equations and writing equations for straight lines.

When students learn a formula, it is typical for textbooks, worksheets and assessments to ask students to work with that formula in different ways by varying the information that is provided. Area tends to be a common one for this.

There are some topics where students need to know that certain conditions are met before they can perform a calculation (i.e. “if A is true, then B will work”). For example, they need to know a triangle has a right angle before they can use Pythagoras’ formula. These situations can also be turned around to ask students to perform calculations in order to evaluate the conditions (i.e. “if B works, then A is true”). E.g. students can substitute the three lengths on a triangle into Pythagoras’ formula to determine if it contains any right angles.

This idea can also be applied to similar shapes. A student can either use the fact that two shapes are similar to work out a missing length, or use the lengths on both shape to decide if they are similar.

It also works well for getting students to think about angle rules in different ways too…

With both Pythagoras and similar shapes, I usually teach students how to calculate missing lengths before teaching them how to check for right angles or similarity. However, this year I tried it the other way around for the first time. In other words, students first used Pythagoras’ Theorem to determine if a shape contained right angles before they used it to calculate missing sides. With similar shapes, they practised testing for similarity before they practised calculating missing sides. It seemed to work quite well but I think I need to try it again with another class before I decide which way I prefer.

Other opportunities to ‘work backwards’ also present themselves in situations where students learn to construct things, such as graphs. I’ve found that asking students to deconstruct these things helps them to think more carefully about their features and properties. This sometimes helps later on when students move on to interpreting graphs and making conclusions because they have already practised pulling raw data out of images.

Students can also be encouraged to work in the reverse direction to what they are used to by providing students with solutions to think about rather than (what would conventionally be seen as) the question. Sometimes this gets phrased as, “Here is the answer. What was the question?” But they can also be problems in their own right.

The quadratic equation example also benefits from becoming more open. This means that once a student has found the most straightforward equation (x2 + x – 6 = 0) they can then look for ways to manipulate it to find less obvious ones (e.g. x2 + x – 4 = 2; 2x2 + 2x – 12 = 0; -x2 – x + 6 = 0).

A slightly more contrived way to achieve a similar effect to the trigonometry example could be to provide both a question and its solution, but to conceal aspects of the question. This can encourage students to think about how aspects of the solution relate to aspects of the problem in a different way to what they might be used to.

The examples above don’t necessarily provide extra challenge but encourage students to think about concepts in multiple ways. However, sometimes working backwards can give additional opportunities to increase the difficulty of a task or include extra stages of working.

The cumulative frequency and scatter graph images were created using Desmos.

If you would like to use or share any of the images from this blog, feel free to use this Power Point to help you: This Way, That Way, Forwards and Backwards

Previous blog posts:

‘Planning Topics’ Series:

‘What I Learned From Shanghai’ Series:

‘Making Connections’ Series

# Filling in the Gaps With Histograms

This blog posts focuses on the sorts of histograms that students see in GCSE Mathematics: histograms with unequal class intervals and where the heights of the bars represent frequency density.  In particular, it aims to share a handful of tiny exercises that I’ve started to use at the start of a unit of work on histograms. Their intention is to give students a bit more of a ‘run up’ to the topic before they plot their first histogram and create more links with other parts of the curriculum.

I should probably start by saying that statistics has never been a either a favourite area of maths to teach or a strong point for me. The first time I taught histogram to high school students, I squeezed the full topic into one lesson which consisted of the following:

1) “Here is how to convert a grouped frequency table into a histogram.”

2) “And now here is how to convert a histogram into a grouped frequency table.”

In hindsight, this was not good! Many students forgot aspects of the topic pretty quickly and others got stumped by any question that asked them to do anything out of the ordinary with a histogram. This was probably because it was a very fast, procedural and surface-level way of teaching histograms and with very few links made to the students’ prior knowledge. Figuratively speaking, it positioned the topic on its own island, separate from anything familiar and with no map for how to get back to it.

I decided to start looking for two things for future lessons on histograms: more ways to make the statistics meaningful towards the end of the unit (e.g. applying what they’ve learned to larger and more realistic datasets) and more ways to break the topic down and prepare students at the start of the unit. For the rest of this blog post, I’d like to focus on the latter. Each year, I’ve found myself adding more and more tiny pieces to my lessons on histograms around around the introductory part of the unit. Some of them were included to address an issue that came up in the previous (e.g. “Why do we use unequal groups?” and other aim to highlight connections with other aspects of the curriculum (e.g. how frequency density in statistics relates to density in physics).

1.  Taking more time to explain why a table of data may contain unequal groups.

One year, when I was explaining how to calculate frequency density, one student asked, “Why use unequal groups in the first place?  If the class widths were equal then we wouldn’t have to use frequency density.”  She had a point!  Frustratingly, while many histograms I saw outside of school (e.g. for university) tended to use equal class-widths and frequency, GCSE questions required students to use unequal class widths and frequency density.  So I now remind myself to dedicate a bit of time at the start of the unit to exploring the advantages and disadvantages of using equal and unequal group sizes.

To do this, I’ve tried using something like this as a starting point.

I’d stress to the students that the graph is not an accurate graph and is just a rough sketch to give an idea of the sorts of race times. It is definitely not something they’re going to draw (but might plant some early seeds about normal distributions for later down the line).  But students could still use the graph to make predictions about what the raw data will look like. “I’m going to show you the list of actual race times in a minute.  But before I do, what numbers are you expecting?”  The key things for students to expect are the times to be roughly between 120 and 270 seconds, and for there to be lots numbers between 180 and 190 seconds.  So once this has been discussed, I’d display the raw data and ask, “Is this what you expected?”

Then onto looking at ways to sort this data.  One approach could be to ask students to sort this data for themselves and let them choose their own intervals for a grouped-frequency table.  It may give them first hand experience of some of the issues that arise when deciding what group sizes to use. However, if I want to take a more direct approach (e.g. if time is tight) then I could present a handful of prepared options for the students to examine instead.  For example, I might ask students to look at the tables below and discuss: “How did each person sort the data?  Why do you think they chose those groups?  What are the pros and cons of each table?

Once conversation starts to focus on Donna’s table with unequal groups, it may be worth showing what Donna’s data would look like represented using bars with if the heights showed frequency.  The first time I did this, I hadn’t shown them the curvy frequency graph at the start, so students thought that the graph below looked perfectly fine. “The last group has the highest frequency, so of it course it should have the biggest bar!” But on occasions where students had seen the frequency curve first, I’ve been able to ask “How does this compare to that rough sketch we saw at the start? Does it look like the same data? How are these bars misleading?”

Then, we could then look at the graph below instead. They might not know what frequency density is yet, but they can consider:  “Does this look more like the rough sketch from earlier?  Between this graph and the last one, which do you think is a better representation of the class’s race times?

I still don’t think that this introduction to histograms is great (e.g. equal class widths of 20 isn’t so bad for this data). So I’m still looking for better ways for the future.  But it aims to give a little bit more consideration to why data might be presented in unequal groups and why we need to bother learning about this thing called ‘frequency density’.

2. Getting students to think more deeply about the meaning of ‘frequency density’.

One thing that I’ve started to do a lot more is make clearer links between ‘frequency density’ and the students’ previous understanding of density from Science and maths.  So, I might start by getting students to think in general terms about density with an activity like the one below.  While the third question can’t really be answered without taking measurements, the purpose of it is to provoke discussion about the factors that effect density.

To help students transfer their knowledge of this kind of density to the idea of frequency density, I might then give them a set of questions like the ones below.  I’m not necessarily looking for students to calculate the frequency density of any of the groups, but to simply provide reasons why one group is denser than the other.  The numbers are chosen to make the questions comparable with the previous activity. It’s also worth noting that, unlike in the previous exercise, the third question can be answered accurately this time through proportional reasoning – like they would solve a ‘best buy’ style of problem.

3. Spend some time sketching histograms and estimating frequencies.

Let’s say the class have a general idea of what histograms look like and now know that the heights of the bars represent the frequency density rather than frequency. This next short activity is designed to encourage students to concentrate on using height to represent frequency density – still before they have learned to calculate frequency density with a formula.

Each table below can be displayed one at a time and students asked to roughly sketch what they think the histogram would look like.  On occasions when I’d previously used that wavy rough sketch from earlier, I’ve had to emphasise that I want them to draw bars.  Also, I’m not necessarily looking for them to draw the bars to scale yet; they just need to show which bars are taller than other bars. Although that’s not to stop them from using scales if they have figured out how to calculate frequency density!

Each question aims to highlight a different point about frequency density.  In the first one, the class-widths of the groups are all the same but the frequencies differ; in the second one, the frequencies are all the same but the class-widths differ.  The third one requires quite a bit of thought because both the class widths and frequencies differ.  But through proportional reasoning, students can come to the satisfying conclusion that all the bars are of the same height.

To challenge students a little more with this kind of thinking, the activity could be turned around.  Instead of giving them tables and asking them to sketch the histograms, we could give them histogram without scales and ask them to estimate frequencies.  These are open questions with infinitely many correct answers, so can be harder for the teacher to check.  But they aim to make students think carefully about the relationships between class width, frequency, frequency density and (most importantly) the heights of the bars.

One possibility for the one above could be to give the highest frequency to the group with the tallest bar, a slightly smaller frequency to the group with the next tallest bar, and so on.  However, students may spot they could also choose to give all groups the same frequency.  If not, then they could be guided towards this by asking, “Would the third bar still be the tallest if it didn’t have the greatest frequency?”  Or “If the frequency for the second group was 20,  what would be the smallest frequency that I could give to the third group?”

I’ve found that this second question (above) catches students out because it initially looks easier than it actually is.  A student might decide to give the first group the greatest frequency, the fourth the next highest frequency and so on, but still be wrong!  Students need to recognise that the frequency for the first group needs to be more than three times the frequency for the second group to ensure that the bar is taller. But then they also need to consider the other bars. The class discussion could talk about good strategies for approaching the problem: “What would be the easier way to start this problem? Should we start by choosing a frequency for the first group or a different group?

This one definitely requires some calculations from the students to ensure equal heights for all bars. It could still be done through proportional reasoning but could be a nice lead-in towards calculating frequency densities by using the formula.

The main purpose of the questions in this section is to encourage students to think hard about how the heights of the bars should represent frequency density rather than the frequency. A nice question to throw in during these activities could be, “What would happen if I take this table and doubled all of the frequencies?  Would the shape of the histogram change?”  This might lay some early foundations for questions like the one below, which they might later see in a GCSE paper or towards the end of this unit of work.

4. Compare the formulae for frequency density and density.

To highlight connections between the two formulae for calculating density and frequency density, I’ve tried displaying the following two questions on the board side-by-side.  The hope is that if students can see the similarities between the roles of mass and frequency, and between volume and class-width then they might be more likely to remember the formula for frequency density in the future.  I’ll try anything to stop them from using the midpoints!

Students have also occasionally found it helpful to see the new formula for frequency density along side the formula that they already knew for density.  It allowed them to compare the two and establish the general idea that they are dividing the amount of “stuff” they have by the amount of space available for it.

Another thing I like to do somewhere within a unit of work on histograms, is to give students a combination of questions like the three below. I might use this in a starter activity for the lesson after they’ve learned how to plot histograms and ask them “How do all of these questions relate to what we’ve been learning?”

Summary

The focus of this blog post has been to share some little things that I have tried adding into lessons at the very start of a unit of work on histograms. I don’t necessarily use all of them every year, but tend to find that using at least some of them give students a bit more of a gentle lead in to this topic before they start plotting and interpreting full histograms. None of the exercises take up an enormous amount of time (maybe a 5-10 minutes each) but aim to help students to think more deeply about the calculations and processes they are performing for the rest of the topic.

It is worth noting that all of the exercises above use small amounts of data and are very contrived! This makes the data less realistic and more abstract, but allows students to work on very specific ideas and key skills while they are first getting to grips with histograms. However, to make statistics learning meaningful, I also feel it is important for students to also experience using histograms (and other statistical graphs) for larger and more realistic data later on.

If you would like to use or share any of the images from this blog, feel free to use this Power Point to help you: Filling in Gaps With Histograms

If you found this post interesting, then I would also recommend the following:

• The graphs in this blog post were drawn using Autograph, which can be downloaded here on this website: https://completemaths.com/autograph
• Here are a couple of websites that have large datasets that can be used in schools (always looking for more):

Previous blog posts:

‘Planning Topics’ Series:

‘What I Learned From Shanghai’ Series:

‘Making Connections’ Series

# Twists and Turns with Straight Line Graphs

This blog post looks at straight line graphs for high school students.  In particular, it focuses on questions that ask students to “Write down the equation of the straight line that…” and then gives them some information to work with.  This often involves them finding the gradient and y-intercept and then writing an equation in the form y=mx+c.  But how in many different ways can the clues about the line appear?

For the majority of this post, I’m going to use examples where the equation of the line would be y=2x+3 and consider a selection of ways to present the information that leads to the equation.  This isn’t a sequence of questions that I would necessarily give to students, but it aims to highlight the variety of ways that the question can be posed.

Let’s first consider information that leads to the y-intercept.  In Examples 1-10, the information about the gradient is kept the same so that we can focus on the different ways of presenting details about the y-intercept.  How obvious or obscure can we make the y-intercept? And what different phrasings can we use?

In Examples 1-4, the y-intercept is given fairly explicitly; students don’t need to do any calculations to work it out.  However, even with these fairly straight forward cases the wording can differ in many ways.

In Examples 5-8, the wording is the same each time but the level of obscurity about the position of the y-intercept differs.  In each case, the y-intercept could be found by substituting the coordinate into the equation y=2x+C and then rearranging it to calculate the value of C.  However, this isn’t entirely necessary for Examples 5-7 because the coordinate given each time is very close to the y-axis.  Students may be able to work out the y-intercept for these mentally through mathematical reasoning with the gradient.

For instance, in Example 5 a student could think, “The coordinate (1,5) is one step to the right of the y-axis and the gradient is 2.  So to travel along this line to the y-axis, I would take two steps down and one step left.  That will be the coordinate (0,3).”  It’s not a very neat way of finding the y-intercept!  But it’s an approach that students could take if they haven’t yet learned the more formal method of substituting a coordinate into the partly formed equation.  It also exercises their understanding of the gradient too.

Example 7 is a little trickier because they need to think about taking two steps to the right and so take four steps up.  The calculations become more arduous to perform mentally when the point is further away from the y-axis (such as in Example 8) but can still be done with a combination of multiplication and subtraction.  But still these calculations would be the same as the ones they’d do if they used the substitution method instead (see below).  Either way, they would start by multiplying 17 by 2 and then would subtract that answer from 37.  Nonetheless, it’s probably a good indicator at this point that a slicker written method (such as substitution) might be preferable, especially before getting to questions with fractions and negatives.

A similar argument could be made for cases where this information is presented on a graph.  In Example 9, it’s fairly straight forward to visualise ‘stepping back’ to the y-axis from Point A.

However, this method would be tricky if the y-intercept wasn’t on the part of the graph that was visible.

Personally, I prefer to begin by giving students some questions where the point is near the y-axis so they can first think about how to find the y-intercept by taking steps towards it, both with and without diagrams.  Then when it comes to introducing the written substitution method, I might redo those same questions again so that the class can see how the two methods lead to the same answer and can hopefully reason why.

So now we’ve looked a handful of ways of changing the information about the y-intercept, let’s do the same for the gradient.  In Examples 11-18, the information about the y-intercept is kept constant while the gradient is presented in different ways.

Example 11 is the same as Example 1 from earlier, but we’re starting with it again because it gives the gradient very explicitly.  Example 12 still doesn’t require any calculations but the information about the gradient is not quite as direct. This kind of description though can provide a starting point and then scope for thinking about how to calculate the gradient between points that are more than one unit apart in the x-direction.  For instance, in Examples 13 and 14 students need to start thinking about sharing the vertical distance out amongst the horizontal distance.  Like with the stepping method from earlier, it’s a fairly loose way of thinking about something that can later lead to a more reliable formula (y2-y1/x2-x1).

These descriptions also become tricky when negative and fractional gradients are involved.  It can be very easy for someone to mistake the gradient for each question below as being 2 if they simply skim read the information.  Nonetheless, such questions can get students to pause and think carefully about what the gradient tells them about a line before they start calculating with the formula.

For the next set of examples, students are not given any information about the gradient but they are given a second coordinate so that they can work out the gradient for themselves.

Just like with Example 5 earlier, the first two provide points that are close to the y-intercept so students might be able to find the gradient through mentally reasoning.  “To get from (0,3) to (1,5) I’ve taken one step to the right and 2 steps up.  So the gradient must be 2.”  Once again, this becomes trickier when the two points are further away from each other (such as with Examples 17 and 18) and require multiple calculations.  Students can also get into big problems with this method when negative and fractional gradients are included.  Personally, I like to begin with questions like Examples 15 and 16 to get students to reason their way to the gradient and then use harder questions like Examples 17 and 18 to lead towards the desire for a more reliable formula.

So far we’ve looked at ways to vary the information about both the y-intercept and the gradient.  When we consider all the different combinations of these variants (plus others), along with situations where the gradient and/or the y-intercept are fractional and/or negative… the number of possibilities seem endless!  Also, in all the previous examples at least one out of the two parts of the equation is given very explicitly.  Let’s now look at situations where students have to work out both parts for themselves.  How many different ways can we mix this up?

In Example 19, students are not told either the y-intercept or the gradient but they are given the graph.  In this case, it’s fairly straight forward to obtain both pieces of information by visually inspecting the line.  This is particularly so because the gradient and intercept are both integers and the scales on both axis are 1 square for every 1.

However, this is trickier when the scales on the x- and y-axis differ from each other.  It’s all too easy for someone to mistake the gradient in Example 20 for being 1 if they are in the habit of just counting squares without considering the scale.

Examples 21-23 demand more abstracting thinking about the gradient and y-intercept than the previous two (unless students use the coordinates to draw the graph).  Even though all three examples are the same kind of question, the number of negatives used in each question differ.

We can make this even more abstract by including coordinates that contain algebraic expressions.  Here students can calculate the gradient by either thinking about how the positions of the second and third coordinates relate to each other (so long as they pay extra attention to whether the gradient is positive or negative), or by substituting the expressions into the gradient formula.

We could even present the question in a more problematic way like Example 26.  Here, students can see the line but they can’t see the on scales either of the axis.  They are told the coordinates of two points but neither of them are on the line.  To make matters worse, the gradient between those two points is not even the same as the gradient of the line.  However, they can use the points to work out the scale and then use that to find some coordinates on the line.

Take Example 27, for instance.  A question like this can encourage students to think about what information is required to form an equation for a straight line and how to obtain it.  With this question, I tend to notice that students find it easy to spot that the y-intercept is 12.5.  But explaining how they know for certain that it is 12.5 and not 12.4 or 12.6 requires them to think more carefully about the coordinates on either side of the y-axis.

This then gives an opportunity to explore what happens to the y-intercept when those points are moved from side to side.

The aim of this post has been to highlight the sheer variety of ways that students can be asked to find the equation of a straight line.  These examples are far from an exhaustive list and probably only scratch the surface.  On reflection, I feel that I may have occasionally skimmed over this topic too quickly in the past and could have explored it in far more depth.  There seems to be endless scope for getting students to reason and problem-solve with information about straight lines, and this post hasn’t even touched on using parallel and perpendicular lines!

If you would like to use or share any of the images from this blog, feel free to use this Power Point to help you: Twists and Turns with Straight Line Graphs

If you found this post interesting, then I would also recommend the following:

Previous blog posts:

‘Planning Topics’ Series:

‘What I Learned From Shanghai’ Series:

‘Making Connections’ Series

# Thinking About Ratios and Algebra

The ratio questions in the new GCSE Maths exams have drawn my attention to a particular skill: setting up equations and functions by using pairs of equivalent ratios.  To be honest, I had never explicitly taught students how to do this when studying for the old GCSEs.  However, knowing how to set up equations and functions can helpful for all sorts of ratio problems in the new GCSEs.  For some examples of these, see Jo Morgan’s blog post from December 2017.

So I’ve been looking at ways to teach this and been thinking about various ways that classes can approach this skill.  The process itself is pretty easy to perform (see point 6 at the end of this post), but in what ways can students make sense of the process?  What prior knowledge can students use to think with when first learning how it works?

For example:

This blog post discusses a handful of different ways that students can think about this task. Some of the ways are clunkier than others. Some of them share the same method but have slightly different reasoning behind them.  I’m not sure yet which ones are the best to use, or whether it depends more so on the prior knowledge students are most secure with.  But here they are…

1. Thinking about it with function machines

I think that out of all the approaches discussed in this post, this one is probably the most cumbersome! However, if students are already experienced with using function machines to make sense of expressions, equations and linear functions, then that same knowledge could be applied to this ratio skill.

For example, my Year 7 classes often work with function machine when learning about algebraic expressions, such as these:

And they may also see function machines in questions like this:

While this question has infinitely many solutions, here are two fairly straight forward responses:

These two questions involving function machines could be combined together in order to think about a way of writing x:y=4:3 as a linear function.  In this case, the task is to consider how to get from the first part of the ratio in 4:3 to the second part and then apply the same thing to x:y.  This could be done by using either of the solutions above.  Here’s the first one:

And here’s the second:

One student this year took it upon himself to do something a little different with this.  We had previously been practising writing ratios in the form 1:n and n:1, so he decided to start each question by writing the numerical ratio in the form 1:n before making a function machine.  He told me he did this so that he would only need to use one box for the function machine instead of two, which I thought was quite creative!

2. Thinking about the multipliers within the ratios

This is arguably just a more sophisticated version of the method above, but with a different way of thinking about it.  A class could work on the principle that if two ratios are equivalent, then the multipliers that get you from one part to another within the ratios will be the same.  For example, students may already be familiar with arguments like this for why the ratios 4:12 and 20:60 are equivalent…

We can see that the two ratio are equivalent because the second part is 3 times the first part in each.  The multiplier of 3 for each ratio comes from dividing the second part by the first.  This can be generalised to the following:

Students can think about the same principle to write x:y=4:3 as a linear function:

3. Thinking about the multipliers between the ratios

We can also take the same principle as in the previous approach and apply it in a different direction.  If two ratios are equivalent, then not only are the multiplier within the ratios equal but so are the multipliers between their corresponding parts.  For example:

This could be generalised to the following:

This can be used to write x:y=4:3 as a linear function in the following way:

So far, I’ve found that my students seem more comfortable using this approach than the previous one because it places both variables on the numerator before they start rearranging.  I also personally prefer it because it shows where the shortcut method comes from (at the end of this post) more clearly than when equating the multipliers within the ratios.

4. Thinking about each part as fractions of the whole.

Students may have previously learned to convert between ratios and fractions in various ways (there’s a nice activity for this on VariationTheory.com).   So a different approach towards making functions out of ratios could be to consider each part of a ratio as a fraction of the whole.   For example, students may have previously learned to do the following with numerical ratios:

They may have even learned to generalise this using algebraic ratios.

Therefore, if 4:3 is equivalent to x:y then we can equate the fractions for the first part of each ratio:

Or we could equate the fractions for the second part of each ratio:

An advantage for this approach is that students could do it one way and then check their answer by doing it the other way to see if they get the same solution.  A thing to be cautious about with this approach is that it demands slightly more advanced rearranging skills than either of the two methods before it, in order to deal with the two terms on the denominator.

5. Thinking about how to make the two parts equal

This approach strongly emphasises the difference between the meaning of a colon in a ratio and the meaning of a equals sign in an equation.  In x:y=4:3 students need to understand that x is not equal to y, because 4 is not equal to 3.  This can be illustrated using the following bar model:

Therefore students could consider, “How could we manipulate these parts so that they are equal to each other?”  In the case of the bar model, they are thinking about how to make the bars the same length.  They can do that by using three lots of x and four lots of y.

Without bar models, they might think about the same thing like this:

6. Using a shortcut

The quickest and easiest way to write x:y=4:3 as a function is probably to just multiply diagonally across them.

For me personally, this is what I ultimately want my students to do when they are making functions or equations to solve bigger problems.  While the other approaches aim to develop understanding through prior knowledge, this method seems more efficient to perform. However, I probably wouldn’t choose to start with this shortcut method because isn’t immediately obvious why we multiply across diagonally.  Without understanding how and why this method works, I also reckon it would be more tempting to make the mistake of multiplying the corresponding parts together to get 4x = 3y.  Therefore, my preference so far has been to start a class with one or more (or all) of the other approaches, use it to lead to the shortcut method and then use that from there.  Undoubtably, there are plenty of other ways to approach making functions and equations out of pairs of ratios, which I’ll continue to look out for in the future.

If you would like to use or share any of the images from this blog, feel free to use this Power Point to help you: Thinking About Ratios and Algebra

If you found this post interesting, then I would also recommend the following:

• As mentioned at the start, Jo Morgan (@mathsjem) has a blog post about different methods for two types of ratio problems in the new GCSE: ‘New GCSE: Ratio’
• I also really like this blog post by Miss Konstantine (@giftedHKO) with activities using function machines: MathsHKO: Function Machines

Previous blog posts:

This series, ‘Planning Topics’:

‘What I Learned From Shanghai’ Series:

‘Making Connections’ Series

# Thinking About Areas of Parallelograms

When I used to look for practice exercises to use in lessons, I mostly concentrated on finding sets of questions that matched the topic I was teaching.  For example, for lessons about areas of parallelograms, I would look for sets of questions about areas of parallelograms.  It seemed like a fairly straight forward thing to do! However, at the time, I didn’t appreciate how the designs of questions (or sequences of questions) can affect what students think about while they are working. Carefully considered questions can direct attention towards important specifics of a topic, while exercises that aren’t carefully considered can lead to students overlooking key points.

For example, let’s consider lessons that focus on finding areas of parallelograms. Once a class has reached the point where they have established that the formula is base x height, the teacher may then want to set them some practice questions on that skill.  What options are available?  What things might a teacher want students to pay attention to and think about while they are working?

Below is a practice exercise that I dug out of my resource box from a long time ago.  Looking back on this now, I’d argue that students would be able to complete this exercise while ignoring several important points that are pertinent to the purpose of the lesson.

The first thing that student might not pay much attention to is the instruction line at the top.  They only really need to read this once because every question requires the same thing.  So after reading it to begin the first question, they might not read the word “area” again for the remainder of the practice – meaning the exercise looks more like the picture below.  Arguably, this could result in students not thinking very much about how the process they are using is for calculating area (in oppose to perimeter, for instance).

Another thing that students might not consider is which lengths are appropriate for the formula and that they need to be perpendicular.  Why would they think about this when every question provides only the two numbers they need?  In this case, the students may only be attending to the details that remain in the picture below.

However, even these are surplus to requirements for this exercise.  Little attention needs to be paid towards the units because they are all cm.  Students can ignore the units and safely assume that all answers will be in square centimetres.

One could even go so far to argue that, with all questions being parallelograms, students don’t really need to pay much attention to what shape they’re dealing with either.  Every question requires the same process: multiply the base by the height (or in this case, just multiply the two numbers you see).  Therefore, students might not think very much about the association between the formula they are using and the shape they are using it for. From the students’ perspective, the most important thing for them to attend to in each question is the pair of numbers provided.

In a worst case scenario, what was intended to be practice on calculating areas of parallelograms may end up just being ‘times tables’ practice instead.

I’ve painted a fairly gloomy picture that is likely to be over the top and unrealistic!  However, despite the hyperbole, it aims to highlight the importance of carefully considering the purpose behind questions that are used for practice.  I’ve learned that simply selecting a set of questions that matches a topic might not match my intentions for practice.

So what things might I want students to think about while they are practising calculating areas of parallelograms?  And how might I direct their attention towards those things?

Getting students to think about how the lengths used in the formula need to be perpendicular:

A teacher might not want to go too complex with this too quickly and may want to first start students off with some fairly safe questions, like the ones below.  These questions don’t provide much opportunity for the students to go wrong, but gently guide students through some of the ways that the positioning of the lengths can vary.  After answering the first question, the second question aims to demonstrate how the height can be labelled outside of the parallelogram, while the third one shows that the measurements don’t always need to be horizontal and vertical.

Once students have got to grips with this, they might next turn their attention towards selecting appropriate lengths and rejecting lengths that are not useful for calculating the area.  This could be done by labelling each parallelogram with more than two lengths, like in the examples below.

While the first question eases students in by using a familiar orientation for the measurements (horizontal and vertical), the second question changes that up by requiring two lengths that appear diagonally on the page.  The third question provides students with four measurements to choose from and can be solved in more than one way.

However, it might be that a teacher wants to shift the emphasis of the task away from computation and more towards reasoning with what lengths can be used and what lengths should be rejected.  In this case, questions such as the ones below may serve that purpose.  I’ve found that students tend to find questions like the third one the most unsettling.  This is because they are not asked to calculate the actual area and can’t calculate it with the information provided.   Therefore, the requirements of the question lie firmly in explaining that the 9 cm and 11 cm aren’t perpendicular to each other.

A different way of encouraging students to think about finding perpendicular lengths could be by getting them to measure parallelograms for themselves, such as in the examples below.

I tend to see that students find the first two questions easier than the third one.  I think this is because they can measure the distance between the top and bottom side in the first question by laying their ruler vertically through the middle of the parallelogram.  Similar for the second one, but with a bit of turning.  However, the way that the third parallelogram slants prevents them from adjoining the top and bottom sides in the same way.

Therefore, they need to be a bit more creative with how they obtain the two perpendicular measurements.

Another interesting way to get students to think about the relationship between the base and the height is to get students to draw their own parallelograms that meet certain stipulations.  For example, the question below invites mistakes and potential discussion by its choice of numbers.

It could be tempting to approach this question by trying to find a factor pair of 24 that adds up to 22 (such as 3 and 8).  However, students have to remind themselves that areas of parallelograms aren’t calculated by multiplying the adjacent sides.  For more questions that get students to think about this distinction, see my recommendation at the end of the post.

Getting students to think about how the degree of slant doesn’t change the area:

One way to address this could be by comparing parallelograms that have the same area but look different because of how they slant, such as in the example below.  A variation for this could be for a teacher to first display the two parallelograms without the rulers, ask the class to discuss the question and vote, then display the rulers and ask the class discuss and vote again.

Working on this idea also provides opportunities to refer back to the relationship between parallelograms and rectangles (which may have been their starting point for learning the formula for parallelograms).

Having a good grasps on this matter makes questions such as the one below much easier.  Students sometimes go the long way round with questions like this, by first calculating the height of the rectangle (8 cm), transferring that to the height of the parallelogram and then multiplying 5 by 8 to get the area of the parallelogram.  While other students instantly recognise that the area of the two shapes are equal.

Getting students to think about the associations between each shape and its area formula:

Once students have learned how to calculate areas for a few shapes, I sometimes find that they then start to mix up the different formulae. They may not have had to think about formula selection so much during the practice exercises that just dealt with parallelograms alone.  Therefore, it can be helpful for them to practice recalling and choosing appropriate formulae for different shapes, so that they concentrate on associating each shape with its’ formula.

The picture below tries to exemplify this by providing a mixture of rectangles, squares, triangles and parallelograms in one practice exercise.  The first three questions aim to give students a bit of a “run up” by using the same orientations.  However, curve balls can be thrown to encourage students to concentrate harder on determining each shape and its’ formula.  For example, students can sometimes mistake a square like in the fourth question for a rhombus.  Also, the parallelogram in the last question is trying very hard to look like the triangle in the question before it!

If you would like to use or share any of the images from this blog, feel free to use this Power Point to help you: Thinking About Areas of Parallelograms

If you found this post interesting, then I would also recommend the following:

• I’ve promoted this website before, but I think Boss Maths has lots of great sets of questions, including practice exercises on areas of parallelograms.
• I also really like this set of questions by Don Steward, where the values for the area and perimeter are equal in each parallelogram and students have to find missing lengths: Equable Parallelograms

Previous blog posts:

This series, ‘Planning Topics’:

‘What I Learned From Shanghai’ Series:

‘Making Connections’ Series

# Thinking About Calculating Areas of Circles

It would be a fair and honest reflection to say that lots of resources I used during my first few years of teaching were quite repetitive. While sets of repetitive questions can have their purpose, they’re not always appropriate.  Since then, I’ve been making a more conscious effort to find or write exercises that at least mix up the ways that questions are presented and include more twists and challenges.

I’ve found this to be easier for some topics than others.  For example, even a sequence of fairly similar equations can be spiced up a little by simply switching round the orders of the terms.

For practice exercises on calculating areas of triangles, resource writers often mix things up by changing the orientations of the shapes or providing increasing amounts of information in each question.

However, one topic that I’ve struggled to do this for is areas of circles.

Let’s say that my class have got to the point where they have learned the formula for the area of a circle and now I want them to practise using it to calculate the areas of lots of whole circles.  How can I provide a variety of questions without veering off just yet into areas of semicircles, sectors or compound shapes?

I’ve been searching through lots of textbooks and online resources, looking for ideas for ways to do this.   I’d like to use the rest of this blog post to collate and share some of these ideas for questions on calculating areas of whole circles.  No doubt there are plenty more ideas to find, but here is a selection…

1. Varying whether students are given the length of the radius or diameter:

This seems the most obvious and common way to mix up circle area questions.  The main thing students need to ask themselves while working through these sorts of questions is, “Do I need to halve the number before substituting it into the formula or not?”  I quite like exercises that include even numbers for radii and odd numbers for diameters, like in the third and fourth examples.  If students aren’t thinking carefully about what they are doing, they might be tempted to halve the 6 cm because it’s even or doubt whether they should halve the 9 cm because it would give a decimal.

2. Varying the ways that the lengths are labelled:

I only came across examples like the third and fourth ones a couple of years ago.  Since using them, I’ve noticed that students seem to find these a little trickier than the first example because the length of each radius is not quite as obvious.  However, labelling lengths in such a way in the early stages could make it easier for students later on, when they begin calculating areas of shaded regions.

3. Questions that don’t provide diagrams:

The bottom two questions require students to think a little more carefully about the meaning of ‘radius’ and ‘diameter’ before deciding what numbers to substitute into the formula.  There were plenty of alternative worded descriptions for this kind of thing.

5. Questions that provide more than one length for students to choose from:

These questions can catch out students if they have got into the habit of just substituting what ever number they see (or half of it) into the formula.  They have to think a little more carefully about the fact that they need the length from the centre to the edge of the circle.

5. Questions where students need to work out the length of the radius themselves before they calculate the area:

I found a bunch of nice questions like this. Before students can calculate the area, they have to consider what information they need and how they can get it.

6. Questions where students need to measure the radius or diameter with a ruler before they calculate the area:

I’ve noticed that students tend to find the empty circle the most difficult because they instinctively try to measure the radius.  This is usually by attempting to guess where the centre is, but it can be very hard to do this accurately.  However, if they understand that the diameter is the furthest distance from one side of the circle to the other, then they could try to measure that instead.  For example, they could slide their ruler up the circle, watch the measurements initially increase and then decrease again as they pass the half way point, slide back to the greatest measurement and take that as the length of the diameter.

7. Questions where students need to measure the diameter with a ruler, but part of the circle has been covered:

These questions seem to back up point from the third example in the previous section.  In the second question, students can be sure that they are able to measure the furthest distance across the circle because it starts to curve back inwards again just before it’s covered.  However in the third example, the circle is still curving outwards at the point where it starts being covered.

8. Questions where students need to calculate the length of the radius before they calculate the area of the circle:

If you would like to use or share any of the images from this blog, feel free to use this Power Point to help you: Thinking About Calculating Areas of Circles

If you found this post interesting, then I would also recommend the following:

Previous blog posts:

This series, ‘Planning Topics’:

‘What I Learned From Shanghai’ Series:

‘Making Connections’ Series

Out of all the training sessions I’ve attended, two quotes in particular have stuck with me and affected how I approach lesson planning. These were from separate presentations but address a similar issue:

“Teachers here don’t plan what they want their students to do; they plan what they want students to think about.”

“Memory is the residue of thought. Or in other words, people remember what they think about.” (This presenter was talking about Daniel Willingham’s book).

Since hearing these, I’ve been looking back over my old lesson resources and considering “What are students likely to think about while answering these questions?” This next series of posts share some thoughts on this question.

***

One topic that bugged me as a teacher for a long time was ‘corresponding and alternate angles’!  It seems like it should be a fairly straight forward topic, so it frustrated me that students often dropped marks on it in exams.  They tended to be okay at working out missing angles, but failed to provide accurate reasons for their answers.  In the follow-up lessons after the exams, I’d find that the class remembered that certain angles on parallel lines were equal but had completely forgotten the words ‘corresponding’ and ‘alternate’  and what they meant.

So I had a dig around through my old resources and for each one considered “What would students be likely to think about during this activity?”  One of my earliest lesson files for introducing corresponding angles contained the set of questions below, which I had taken from the internet.  From what I remember, this exercise was fairly typical for what would come up when Googling “corresponding angles worksheet” at that time.

Looking back on this resource, I’m not sure it was the most appropriate set of questions for this lessons.  It’s not that the questions themselves are bad, they just didn’t suit my purpose for that lesson.  The new content that students were learning was the concept of corresponding angles, so I wanted students to think about the phrase ‘corresponding angles’ and its meaning.   But it is more likely that this exercise just got students to think “Is the answer the same as this number? Or do I need to subtract it from 180?”  While this might be useful later on, it wasn’t what I wanted at this point of introduction.

Since then, I’ve been looking for alternatives to such a numerically focused exercise.  In particular, I’ve tried to find activities that encourage students to think about the meaning of corresponding angles in lots of different ways.

For example, one task could be to present students with pairs of angles and ask them to consider “Are these angles corresponding or not?”  The questions below require students to compare examples with non-examples so that they can learn to discriminate between them.  While doing this, I would insist that students wrote the words ‘corresponding’ or ‘not corresponding’ rather than just putting ticks or crosses, so that they keep mentally pairing the word with its meaning.

Once they’ve practised evaluating pairs of angles, a follow-up task could approach the concept from the opposite direction.  The questions below provide students with one angle and then asks them to think “Which angles correspond with this one?”

Alternatively (or subsequently) students could be challenged with a slightly more open task.  Rather than focusing on just one angle at a time, like with the questions above, the next exercise requires students to consider “Which angles correspond with each other?”  My classes have found this task more difficult than the last one because it requires three letter notation, and because they don’t initially know how many pairs of corresponding angles they are looking for in each diagram.

In all of the exercises above, the questions have been restricted to looking at corresponding angles on parallel lines.  But the lines don’t have to be parallel.  For example, that first exercise could have looked like this:

I reckon that because the exams solely focus on corresponding angles around parallel lines, so did all the textbooks, Power Points and worksheets that I found.  So for a long time, I just presumed that the lines had to be parallel for angles to be defined as corresponding.  However, I’ve since learned that this is wrong and that parallel lines are just a special case were corresponding angles are equal to each other.

This opened up a new alternative for how to include numerical examples while still keeping the focus firmly on corresponding angles.  For example, rather providing students with a pair of parallel lines and asking them to find a missing angle, they could be provided with the angles and be asked to decide whether or not the lines are parallel.  The intention for the task below is to encourage students to think “If the lines are parallel then the corresponding angles will be equal.”

The key thing that I’ve learned from scrutinising my old resources in such a way is that questions requiring numerical answers can sometimes distract students from the main point of the lesson.  So when designing tasks, rather than thinking “What do I want students to work out?” in each question, it may be more appropriate for me to consider “What do I want students to think about?”

If you would like to use or share any of the images from this blog, feel free to use this Power Point to help you: Thinking About Corresponding Angles

If you found this post interesting, then I would also recommend the following:

Previous blog posts:

This series, ‘Planning Topics’:

‘What I Learned From Shanghai’ Series:

‘Making Connections’ Series