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?”

01 Race Times

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?

02 Tables

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?”

03 Using frequency

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?

04 Using frequency density

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.

05 Density of atoms

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.

06 Top or bottom

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!

07 Sketch bars

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.

08 Estimate Frequencies 1

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?”

09 Estimate Frequencies 2

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?

10 Estimate Frequencies 3

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!

11 Density and FD Table

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.

12 Formulae

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:

Previous blog posts:

‘Planning Topics’ Series:

‘What I Learned From Shanghai’ Series:

‘Making Connections’ Series

Twists and Turns with Straight Line Graphs

Feature Image

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?

01 Explicit y-intercepts

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.

Close to the y-axis

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.

Further from the y-axis

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.

021 Working

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.

03 step back to the y-intercept

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

04 Can't Step Back

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.

05 Explicit Gradient

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).

06 Difference in y shared between the difference in x

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.

061 Fractional and negative gradients

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.

07 y-intercept and a point

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?

08 Graph given

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.

09 Graph with different scales

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.

10 Two points given

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.

11 Algebra Coordinates

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.

12 Points off 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.

13 Y-intercept 12.5

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

14 Y-intercept 12.75

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:

00 The Question

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:

0. Function Machine

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

1. Function Machine

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

2. Function machines

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:

3. Function machines

And here’s the second:

4. Function machines

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!

4.5. Function Machine

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…

5. Multipliers Within

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:

6. Mutlipliers Within

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

7. Mutlipliers Within

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:
8. Mutlipliers Between

This could be generalised to the following:

9. Mutlipliers Between

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

10. Mutlipliers Between

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:

11. Fractions

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

Therefore, if 4:3 is equivalent to x:y then we can equate the fractions for the first part of each ratio:13. Fractions

Or we could equate the fractions for the second part of each ratio:14. Fractions

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:

15. Making them equal

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.

16. Making them equal

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

17. Making them equal

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.
18. X-Direct

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.

01 Thoughtless Practice 1

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).

01 Thoughtless Practice 2

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.

01 Thoughtless Practice 3

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.

01 Thoughtless Practice 4

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.

01 Thoughtless Practice 5

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.

01 Thoughtless Practice 7

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.

02 Perpendicular 1

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.

02 Perpendicular 3

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.

02 Perpendicular 2

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.

02 Perpendicular 4

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.

02 Perpendicular 5

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

02 Perpendicular 6

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.

02 Perpendicular 7

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.

02 Perpendicular 8

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.

03 Slant 1

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).

03 Slant 2

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.

03 Slant 3

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!

04 Match Shapes

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.

0 Equations

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.

0 Triangles

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

0 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:

1 Radius and 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:

2 Position of Radius and Dimeter

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.

Shaded Regions

3. Questions that don’t provide diagrams:

2.5 Worded descriptions

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:

3 More than one measurement

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:

4 Work out the radius or diameter

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:

5 Measure the radius or diameter

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:

6 Covered over circles

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:

7 Calculate the radius or diamter first

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

Thinking About Corresponding Angles

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.

1 Old Resource

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.

2 These angles are -

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?”

4 Where is its corresponding angle

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.

5 Three letter notation

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:

3 These angles are (parallel) -

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.”

6 Parallel or Not

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

A Shanghai Lesson on Adding Negative Numbers

Three years after visiting Shanghai, what has stuck with me? (Part 3)

In 2015, my colleague (@Carohami) and I took part in an England-China teacher exchange programme, organised by the Maths Hubs.  In this series of posts, I reflect on some of the things that have stuck with me over the last three years.

***

The second leg of the England-China exchange involved two teachers from Shanghai visiting England for a month and teaching some of our classes in Halifax.   In this post, I’d like to present an account of what happened during a lesson with a Year 7 class on adding negative numbers.

Account of the Lesson

The lesson started off with a brief demonstration of representing single numbers as arrows on a number line.  Positive numbers travel to the right; negative numbers travel to the left.

1 Representing Numbers

That done, the teacher and students used this representation to explore addition of two numbers.  Each addition in this set of introductory questions used the numbers 5 and 3, while incorporating every combination of positives and negatives.  The students drafted ideas on mini whiteboards (this was the first time the teacher had used mini whiteboards) and the teacher drew a copy of each diagram on the main board.

2 Eight Additions

The teacher was keen that drawing diagrams didn’t become the method that students used to solve additions with negative numbers.  So the teacher then used the diagrams from the board to highlight key features and derive generalisations.  The aim was for students to know and understand rules that would help them to calculate additions quickly in their heads.

The first key feature discussed was how sometimes both arrows pointed in the same direction and made one long path, while other times they pointed in opposite directions and overlapped.  This led to the realisation that when the two numbers in the addition have the same sign, the two arrows will continue on from one another.  Therefore, when the two numbers have the same signs then the solution to the addition will contain the sum of those two numbers (putting the negative and positive aside for a minute). A more precise way to describe this could be that the absolute value of the solution will be equal to the sum of the absolute values of each number – but this terminology wasn’t used in this Y7 lesson.

3 Same SignsConversely, if the two numbers in the addition have different signs to each other then the arrows overlap.  Therefore, the end result leaves you with the difference between the lengths of the two arrows.  This means that when the signs in the addition are different, the solution will contain the difference between the two numbers.

4 Different Signs

Next: how to decide whether the total will be positive or negative?  They started by looking at cases where the answer they got was positive. It seemed fairly obvious that if both numbers in the addition are positive then the total would be positive.  But how about if one is positive and the other is negative?

5 Positive Answer

In both of these pictures, they could see that the answers they got were positive because the longest arrows were travelling in the positive direction.  Then onto cases where the total was negative…

6 Negative Answer

These pictures show that the answers they got were negative because the longest arrows were negative.  This led to the conclusion that the total from an addition of two numbers will always share the same as the number with the greatest absolute value.  This all got summarised into three rules for working mentally:

  • Same signs: use the sum
  • Different sign: use the difference
  • The solution will have the same sign as the number with the greatest [absolute] value.

7 Rules

Once these rules were established and seemingly understood by the students, the remaining time of the lesson was given over to practising adding with negative numbers abstractly.  The first few questions were answered by the class one at a time on mini-whiteboards; the rest were done independently in books.

My Reflections

What struck many of us who saw this lesson was the strong emphasis on using tricks or rules to calculate abstractly.  The very first time I taught negative numbers, I taught it completely abstractly using rules and found that students kept forgetting the rules or would get them mixed up.  The following year, I endeavoured to teach the concept for deeper understanding by using number lines (although I tended to represent calculations on the number line as ‘starting number’ and ‘movement’).  This time, students appeared to understand what was going on, but they ended up relying on the number lines so much that they struggled to break away from them.  The number line ended up becoming their method to calculate with negative numbers.  This made performing mental calculations troublesome.

What I found interesting about the Shanghai teacher’s lesson was the relationship between diagrams and mental methods.  The purpose of drawing diagrams was never to get answers.  They were not used as a method; they were used to illustrate a method.  It was clear to the class that the end goal was to be able to calculate addition with negative numbers mentally and the diagrams were a vehicle to get them there.

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: A Shanghai Lesson on Adding Negative Numbers

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

Previous blog posts:

This series, ‘What I Learned From Shanghai’:

‘Making Connections’ Series:

‘Planning Topics’ Series:

Do Sweat The Small Stuff!

Three years after visiting Shanghai, what has stuck with me? (Part 2)

In 2015, my colleague (@Carohami) and I took part in an England-China teacher exchange programme, organised by the Maths Hubs.  In this series of posts, I reflect on some of the things that have stuck with me over the last three years.

***

In some ways, planning a mathematics lesson is a bit like staring at a fractal.

Fractal Triangle 2The mathematics curriculum is often separated into large sections, such as Number, Data, Algebra etc.  Each section tends to be split into subsections that might be referred to as topics, chapters or units of work: introduction to fractions, equivalence, four operations with fractions, and so on.  Within each unit students learn a set of skills, concepts and procedures, such as how to add fractions, how to multiply them, etc.  When teachers plan to teach each of these skills, we often break them down into a series of small steps (e.g. learning to add fractions might be broken down into adding like fractions, then related fractions, then unrelated fractions, mixed numbers and so on).  And then within each small step there are lots of even finer points.  When adding related fractions, how do you decide which fraction to change? Once you’ve added the fractions, how do you spot if the total is a whole number?

This fractal analogy is far from perfect! But it aims to emphasise those tiny parts of mathematics that can be hard to spot unless you deliberately focus on them.  It was these fine details that I reckon overlooked on too many occasions at the start of my teaching career.  I probably still miss a fair few now.

Sometimes I would presume that students would just pick up the fine details along the way, while learning bigger things.  For example, I presumed that students would just pick up what it means for two algebraic terms to be ‘like’ while they are busy ‘collecting like terms’.  

Collecting Like TermsOr they would learn how identify which angles in isosceles triangles are equal while solving missing angle problems.  

Isoscelese Triangle

Even when I did cover these points, on reflection, I don’t think I gave them as much time as they needed.

Other times, fine details would crop up sporadically and cause problems while students were working through an exercise.  For example, when students were practising to rearrange formulae and a question such as the one below came up, students assumed that they had made a mistake because their answer didn’t match the textbook’s answer. 

Formula

When I tried to reassure them that they were right, some would then ask, “So are there two correct answers then?”  I would then frantically try to convince them that both fractions were identical.

There were even small details that I was completely oblivious to.  Throughout my own GCSEs, A-levels, university degree and first few years of teaching, I thought that alternate angles and corresponding angles only existed between parallel lines. Reassuringly, I wasn’t alone with this!  It was only when I read an article by Huang and Leung (link at the bottom) that I learned that the lines don’t have to be parallel for the angles to be corresponding; parallel lines is just the special case when corresponding angles are equal.

Corresponding Angles

One thing struck me about the lessons I saw in Shanghai was how they paid special attention to little things.  On numerous occasions, the teacher would dedicate part of a lesson to discussing a tiny, tiny aspect of a topic and then get the students to practise that point in isolation before applying it to bigger questions.  These moments would only take up a few minutes of the lesson, but they were enough to prepare students for the more complex problems ahead. They taught each detail explicitly!

Here are some examples from the lessons we visited:

1. By spending enough time on these points:

(m-n) = –(n-m)     and      –(m-n) = (n-m)

students were better prepared to deal with situations like this:
Simplify brackets

2. In a lesson on multiplying terms with indices, the teacher started by asking the following:
Base and Power

3. There was an exercise that solely practised applying powers to negatives:
True of false

4. By tackling points 2 and 3 in advance,  students were equipped to tackle these questions:
Indices

5. Once students were confident on all of the previous points, this next question became straight forward:

Bracket Indices 2

There are so many tiny points like these in mathematics that addressing each of them separately would completely atomize the curriculum.  I’m not yet sure about how wise that would be.  Is it necessary for every single student to cover every single tiny point explicitly like this? Probably not. I’m not sure it’s even possible in the available time.  Then again, I find that students’ misconceptions are often caused by misunderstanding one of these tiny aspects of the topic.

If there’s one thing I’m certain about, it’s that the trip to Shanghai highlighted the importance of the little things.  On reflection, the reason why I probably didn’t think so much about these little things was because, as an experience mathematician, I didn’t need to think about them – the more experienced we get with something, the more we automate and the less we need to think about what we’re doing.  But my students are not expert mathematicians; they’re learning.  So the Shanghai exchange has encouraged me to pay more attention to the fine details and think carefully about if, when and how I should address them.

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

Previous blog posts:

This series, ‘What I Learned From Shanghai’:

‘Making Connections’ Series:

‘Planning Topics’ Series:

Not As Complicated As It Looks…

Three years after visiting Shanghai, what has stuck with me? (Part 1)

In 2015, my colleague (@Carohami) and I took part in an England-China teacher exchange programme, organised by the Maths Hubs. We were incredibly lucky to go and saw some fantastic maths lessons! And while we can’t replicate everything from Shanghai schools (and wouldn’t want to), there were plenty of things that seemed appropriate for any maths teacher to take away.  After all, we don’t watch other teachers to emulate them; we watch them to learn.

Overall, the trip provided lots of food for thought and inspiration to try out new ideas.  In this next series of posts, I’d like to reflect on some of the things that have stuck with me over the last three years.

***

I think that I may have been a little bit mean in a recent lesson.  The whole class were sat eagerly with their pens held ready, when they heard me say…

“I’m going to display a question on the board.

You’ll all know how to solve it, but the numbers are quite big.

But still, I know you’ll be able to do it!

I wonder who will be the first to show me the correct answer?

Are you all ready?

Set… Go!”

Simplify Fraction

While nearly all the students had their heads down, frantically trying to divide 248 by 82, one student held up the answer 124 after just a couple of seconds.  Initially, the rest of the class were flabbergasted that someone could calculate it so quickly.  But then a ripple of “Oh yeah!” noises spread around the room as each student noticed that the fraction simplified to a half.

Maybe it was cruel of me to lure them into such a trap.  But it was to make the point that it can be unwise to jump into calculations too quickly, and that sometimes the first method that comes to mind isn’t always the most straight forward.

Using cheeky questions like the one above was one thing that I took from my trip to Shanghai – in fact it is probably the thing that has stuck with me the most.  In lots of the lessons we visited, the students were presented with questions that looked more complicated than they really were.  For example, during a lesson on simplifying surds, the teacher posed the following two questions:

Surds

These questions may look quite complicated on the surface, but the numbers in them have been chosen so that they work out fairly nice once you get going.

Surds

Here is another one that I saw during a lesson after collecting like terms and substitution:

Subsitution

Each of these questions presented an initial shock (or at least I was shocked; the students seemed unfazed), a feeling of surprise once you got started and then a strange sense of satisfaction with the end solution.

My favourite questions were ones that led to discussions about comparing different methods.  Usually with these, simply number crunching your way through the calculations would make for hard work.  But when students pause, think and apply some reasoning, they may find an easier route to the solution.  For example, one of the NCETM’s pre-trip seminars presented a question a bit like this:

Decimals

On the surface, this could look like a question that requires multiplying decimals together.  However, there is an easier way:

Decimals 2.PNG

A similar idea could be used with questions on finding fractions of a quantity:

Fractions of 13

It turns out that I didn’t really need to go all the way to Shanghai to see questions like these; there were plenty being used in England.  However, the trip drew my attention to them (amongst other things), allowed me to see how students in these Shanghai schools tackled them and inspired me to start using them with my own classes.

Sometimes I’ve used these towards the end of a lesson, for the whole class to answer on mini-whiteboards.  But more than anything, I’ve found it helpful to incorporate them into practice exercises. Once students have practised something enough times that they don’t need to think about it as much, a couple of these types of questions can get them thinking again.

Over the last couple of years, I’ve been building bit of a collection of these (mostly by poking my head into classrooms or looking in textbooks for inspiration).  Here are a few more that I’d like to share:

Improper Fraction

That fourth method seemed a bit odd to my students when they first saw it, but it helped them with the next couple of questions.Fractions of 2 and 37

Super Fractions

The next group of questions share a similar theme of ‘easier if you simplify first‘.

For a lesson on substituting into expressions:
Subsitution after simplifying

For a lesson on fractional indices:

indices 1

When students are applying many rules of indices:

Indices 2

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:‘Less Complicated Than It Looks’

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

Previous blog posts:

This series, ‘What I Learned From Shanghai’:

‘Making Connections’ Series:

‘Planning Topics’ Series: