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

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‘Making Connections’ Series