Choosing the right data display: a week of ready-to-teach maths

Five lessons for Year 9 Statistics. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus 12 variable cards, 12 situation cards, a mini-check and every answer with its working.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Print the two cut-out sheets, one copy per pair: sheet 1 has the variable cards (Day 1), sheet 2 the situation cards (Day 5). Cut them before Day 1.
4. Each student needs a ruler and a pencil; on Day 2 each student also needs a protractor and a calculator.
5. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.

The idea in plain words

The right display depends on two things: the type of data and the question it must answer. Categorical data (named groups) suits a bar chart, or a pie chart when the groups share one whole. Numerical data (counts or measurements) suits a dot plot, a histogram or a box plot. Two numerical variables measured together suit a scatter plot, and a variable that changes over time suits a line graph. A good choice can be justified in one sentence that names the data and the purpose.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.

How we write these answers

A class such as 160–165 holds the values from 160 up to but not including 165, so a height of 165 cm goes in 165–170. A sector angle is the frequency divided by the total, times 360°. An association is described by its direction (upward, downward or none) and its strength.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Day 1 classifies the data, which decides everything after it; Day 2 shows categories; Day 3 shows numerical data; Day 4 shows two variables at once, as a relationship or as change over time; and Day 5 puts the choice into words. It builds on the box plots and comparisons of the last unit and on reading graphs critically earlier in Year 9; it prepares planning a full statistical investigation.

Materials for the week (one trip)

• From the classroom: rulers, pencils, protractors and calculators for Day 2, scissors, envelopes, this pack printed.
• Cut-out sheet 1, one per pair: 12 variable cards to sort (Day 1).
• Cut-out sheet 2, one per pair: 12 situation cards to sort (Day 5).
• If you have them: a spreadsheet that draws charts, to redraw a hand-drawn display in seconds. No other equipment to buy.

Words for the word wall

categorical, nominal, ordinal, numerical, discrete, continuous, bar chart, pie chart, sector, histogram, class, dot plot, box plot, scatter plot, association, line graph, trend, seasonal, justify

Dear families

This week in maths, Year 9 is choosing the right graph for a set of data and explaining the choice, and reading graphs carefully for what they do and do not show.

Try this at home

• Find a graph in the news, a weather app or a sports page. What kind of data does it show? Is it the best kind of graph for that data?
• Ask: if you wrote down the temperature every hour for a day, would you draw a bar chart or a line graph? Why?
• Ask: a pie chart of everyone’s favourite sports adds up to more than 100%. What could have gone wrong?

My graph week

Fill one row a day. Write a graph you found, the data it shows, and whether it was a good choice.

Categorical or numerical?

Every choice of display starts with the type of data. Categorical data sorts things into named groups: nominal groups have no order (country of birth), ordinal groups do (shirt sizes). Numerical data is numbers you can do arithmetic with: discrete data is counted, continuous data is measured and can take any value in a range. Numbers used as labels, such as postcodes, are categorical.

We are learning to

• classify a variable as categorical (nominal or ordinal) or numerical (discrete or continuous),
• rewrite a survey question so it collects a different type of data,
• name the displays that suit each type of data.

Success criteria

• I can sort the twelve variable cards and explain the tricky ones.
• I can list the displays for categorical data and for numerical data.

You need

The worksheet; the 12 variable cards (cut-out sheet 1) per pair.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the sort; start the second with Part B.

Watch for these ideas

• Calling anything written in digits numerical: postcodes, phone numbers, jersey numbers.
• Calling a rounded measurement discrete: height in whole centimetres is still measured.
• Forgetting that ordinal groups have an order that a display should keep.

Answers

• Part A: the card key is with the mini-check answers.
• Part B: 1. categorical (nominal); 2. categorical (ordinal); 3. categorical (nominal). Rewrites, for example, ask how many pets live in your home (discrete); how many minutes the trip takes (continuous); how many hours you slept (continuous).
• Part C: categorical data, one variable: bar chart; pie chart when the groups share one whole; numerical data, one variable: dot plot, stem-and-leaf plot, histogram or box plot; two numerical variables measured together: scatter plot; a numerical variable over time: line graph.
• Part D: for example, a phone number looks numerical but is a label; the number of letters in your first name sounds like a name but is a count (discrete).

Categorical or numerical?

A. Sort the variable cards

Write each card’s letter in its pile.

B. What type of data?

Say what type of data each question collects. Then rewrite it so that it collects numerical data.

C. Which displays suit?

D. Stretch: make two tricky cards

Write a variable that looks numerical but is categorical, and one that sounds categorical but is numerical. Explain each in a sentence.

Displays for categories

A bar chart shows categorical data as separate bars with gaps between them. Nominal categories can go in any order, so sorting the bars from most to least popular, with “Other” last, makes the chart easiest to read; ordinal categories keep their own order. A pie chart shows how the categories share one whole: each sector angle is the frequency divided by the total, times 360°.

We are learning to

• draw a bar chart of categorical data in a helpful order,
• work out the sector angles of a pie chart and draw it,
• decide when a pie chart cannot be used.

Success criteria

• I can draw a bar chart with gaps, sorted by frequency, with “Other” last.
• I can find each sector angle and check that the angles add to 360°.

You need

The worksheet; a ruler, a protractor and a calculator each.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with the pie chart.

Watch for these ideas

• Drawing the bars touching, as if the categories sat on a number line.
• Sorting ordinal categories, such as sizes or ratings, by frequency and losing their order.
• Using a pie chart when people could give more than one answer.

Answers

• Part A: bars with gaps, in the order Sandwich 9, Wrap 6, Sushi 5, Pasta 4, Salad 3, Other 3.
• Part B: Pasta 4 ÷ 30 × 360° = 48°; Sandwich 9 ÷ 30 × 360° = 108°; Salad 3 ÷ 30 × 360° = 36°; Sushi 5 ÷ 30 × 360° = 60°; Wrap 6 ÷ 30 × 360° = 72°; Other 3 ÷ 30 × 360° = 36°; total 360°.
• Part C: keep S, M, L, XL: the sizes have an order (ordinal). Sorted by frequency (M, L, S, XL) the chart hides that sales rise to a peak at M and fall away on both sides.
• Part D: the counts add to 46, more than the 30 students, because many ticked more than one sport. The parts do not share one whole, so use a bar chart.

Displays for categories

Favourite lunch of 30 students, as collected: Pasta 4, Sandwich 9, Salad 3, Sushi 5, Wrap 6, Other 3.

A. A bar chart, most popular first

Label each bar underneath. Leave a gap between the bars; put “Other” last.

B. A pie chart

C. Keep the order?

Shirts sold in a week: S 14, M 26, L 22, XL 8. A shop drew its bars from tallest to shortest: M, L, S, XL. What does that order hide? Which order is better, and why?

D. Stretch: when a pie chart fails

30 students ticked every sport they play: Football 14, Netball 11, Swimming 12, Basketball 9. Why can these counts not go in a pie chart? Which display should be used?

Displays for numbers

Numerical data sits on a number line. A histogram groups the values into classes of equal width and draws a bar for each class with no gaps, because the classes meet: a height of 165 cm goes in the class 165–170. A dot plot shows every value as a dot above a number line, which suits a small set of whole numbers. Gaps mean categories; touching bars mean numbers.

We are learning to

• group numerical data into classes and draw a histogram,
• draw a dot plot of a small set of whole numbers,
• read the mode and the median class from a display.

Success criteria

• I can draw a histogram with equal classes and touching bars.
• I can say when a dot plot shows more than a histogram.

You need

The worksheet; a ruler each.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with Part B.

Watch for these ideas

• Leaving gaps between the bars of a histogram, or making classes of different widths.
• Putting a value that sits on a class edge into the lower class.
• Grouping a dozen whole-number values into a histogram, where a dot plot keeps every value.

Answers

• Part A: 145–150: 1, 150–155: 2, 155–160: 6, 160–165: 8, 165–170: 6, 170–175: 4, 175–180: 2, 180–185: 1; the bars touch.
• Part B: 0 goals: 3, 1 goals: 5, 2 goals: 7, 3 goals: 3, 4 goals: 1, 5 goals: 1 dots. The mode is 2 goals; 5 games had 3 or more goals.
• Part C: the median lies between the 15th and 16th heights. Counting up the classes (1, 3, 9, 17, 23, 27, 29, 30), both are in 160–165. The goals take only six whole-number values, so a dot plot shows every game and loses nothing.
• Part D: 140–150: 1, 150–160: 8, 160–170: 14, 170–180: 6, 180–190: 1. Still one peak, but the wider classes hide where the heights bunch: the 5 cm classes show more detail.

Displays for numbers

Heights (cm) of 30 Year 9 students: 152, 168, 160, 171, 158, 163, 175, 149, 166, 162, 157, 170, 164, 181, 159, 167, 155, 173, 161, 165, 153, 169, 178, 162, 156, 166, 160, 172, 164, 158.

A. A histogram

Count the heights in each class, then draw the histogram. A class such as 160–165 holds 160 up to but not including 165.

B. A dot plot

Goals scored by a team in 20 games: 2, 0, 3, 1, 2, 4, 1, 2, 0, 3, 2, 1, 5, 2, 1, 3, 2, 0, 1, 2.

C. Read the displays

Which class holds the median height? Why does a dot plot suit the goals better than a histogram?

D. Stretch: wider classes

Group the heights in 10 cm classes starting at 140 and sketch that histogram. What does it show, and what does it hide?

Two variables

Some questions are about two variables at once. When both are measured on each person, a scatter plot shows each pair as a point, and the cloud of points shows the association: upward, downward or none, and how strong. When one variable is time, a line graph joins the readings in time order, and the line shows the change from one reading to the next. The points of a scatter plot are never joined: the order of the people means nothing.

We are learning to

• plot a scatter plot of two numerical variables and describe the association,
• draw a line graph of a variable over time and describe the pattern,
• explain why the points of a line graph are joined and those of a scatter plot are not.

Success criteria

• I can describe an association by its direction and its strength.
• I can read the highest and lowest points of a line graph and name the pattern.

You need

The worksheet; a ruler each.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part B; start the second with the line graph picture.

Watch for these ideas

• Joining the points of a scatter plot in the order of the table.
• Reading “upward” as “every point is higher than the one before”.
• Saying one variable causes the other just because the two are associated.

Answers

• Part A: a strong upward association: students with a longer arm span tend to be taller.
• Part B: 7 of the 12 points lie above the line (taller than their arm span), 5 below and none on it. The points hug the line, so arm span is close to height.
• Part C: busiest in January (9.5 thousand), quietest in July (0.8 thousand): a seasonal pattern, high in summer and low in winter.
• Part D: the months follow each other in time, so the line shows how the visitors change from month to month. The students in the scatter plot have no order, so a line joining them would show nothing.

Two variables

A. A scatter plot

Plot each student as one point: arm span across, height up. Do not join the points.

Describe the association (direction and strength): ____________________________________________

B. Use the plot

Rule the line height = arm span through (150, 150) and (180, 180). How many students are taller than their arm span? ____________

C. A line graph

A council pool’s visitors each month (thousands). Plot them and join the points in order.

When was the pool busiest, and when quietest? Describe the pattern: ___________________________________

D. Stretch: to join or not

Why are the points of the line graph joined, but not the points of the scatter plot?

Choose and justify

Choosing a display is an argument: the type of data and the question it must answer decide the display, and a good justification names both. A line graph that joins unordered categories suggests a trend that is not there; a pie chart needs parts of one whole; and a scatter plot shows a relationship but not its cause.

We are learning to

• choose a display for a situation from the type of data and the question,
• justify the choice in one sentence,
• find and fix a display that does not suit its data.

Success criteria

• I can sort the situation cards and name the best display for each.
• I can write a justification that names the type of data and the purpose.

You need

The worksheet; the 12 situation cards (cut-out sheet 2) per pair; the unit’s self-check quiz.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the sort; start the second with Part B and the quiz.

Watch for these ideas

• Choosing a display because it looks good rather than because it suits the data.
• Justifying with the type of data alone, without the question the display must answer.
• Reading an association as a cause.

Answers

• Part A: the card key, with the best display for each card, is with the mini-check answers.
• Part B: for example, “The heights are numerical and continuous, and the question is how they are spread, so a histogram with touching bars.”
• Part C: the sports are categories with no order, so the line suggests a change from one sport to the next that means nothing. Draw a bar chart with gaps.
• Part D: no. Both sell more in hot weather: a third variable, the temperature, explains the pattern. An association does not show a cause.

Choose and justify

A. Sort the situation cards

Write each card’s letter in the pile of displays that suits it.

B. Justify three choices

Choose one card from each pile. Name the best display and justify it: the type of data, and the question the display answers.

C. Fix the mismatch

D. Stretch: association or cause?

Over a year, a corner shop’s sales of ice blocks and of sunscreen rise and fall together. Does buying sunscreen make people buy ice blocks? Explain.

Variable cards

Day 1, one set per pair: 12 cards. Each names something that could be recorded about a person or a thing. Sort them: categorical; numerical: discrete; numerical: continuous.

Teacher note: the key is with the mini-check answers, so this sheet can go to pairs whole.

Situation cards

Day 5, one set per pair: 12 cards. Each describes some data and what we want to see in it. Sort them by the displays that suit: histogram, dot plot or box plot; scatter plot or line graph; bar chart or pie chart.

Teacher note: the key, with the best display for each card, is with the mini-check answers, so this sheet can go to pairs whole.

What we know: Choosing the right data display

Work on your own. Show your working.

1. A football team’s players wear the numbers 1 to 23. Is jersey number categorical or numerical? Explain. ______________________
2. Is the number of cars passing the school gate in an hour discrete or continuous? ______________________
3. In a survey of 40 people, 14 walk to work. Find the sector angle for “walk” on a pie chart. ______________________
4. A survey’s answers are poor, fair, good, excellent. In what order should the bars of its bar chart go? Why? ______________________
5. Which display, a bar chart or a histogram, has touching bars? Why? ______________________
6. Which display shows whether hours of sleep and reaction time are related? ______________________
7. Which display shows the weekly price of petrol over a year? ______________________
8. On a scatter plot, the points fall from left to right. Describe the association. ______________________

Answers and marking guide

Answers with working

1. Categorical: the numbers are labels; adding or averaging them means nothing.
2. Discrete: the cars are counted.
3. 14 ÷ 40 × 360° = 126°.
4. In the order of the scale (poor, fair, good, excellent): the categories are ordinal.
5. A histogram: its axis is a number line and the classes meet, so the bars touch.
6. A scatter plot: two numerical variables measured on each person.
7. A line graph: the price changes over time, week by week in order.
8. Downward (negative): as one variable increases, the other tends to decrease.

A quick three-level guide

Eight questions, four ideas. A student at standard classifies data and chooses and reads a display that suits it.

Card keys

Variable cards (Day 1): categorical: A, D, F, K, L; numerical: discrete: C, E, H, J; numerical: continuous: B, G, I. D: digits used as a label: adding postcodes means nothing; K: ordinal: the sizes have an order.

Situation cards (Day 5): histogram, dot plot or box plot: A (histogram), D (dot plot), F (parallel box plots), G (dot plot), L (histogram); scatter plot or line graph: C (line graph), H (scatter plot), J (line graph); bar chart or pie chart: B (bar chart), E (bar chart), I (pie chart), K (bar chart).

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: types of data, displays for categories, displays for numbers, two variables, and choosing.