Mathematics Year 6: The histogram and the line graph
Mathematics, Year 6 (Portugal): The histogram shows how values grouped in classes are distributed; the line graph shows how a value changes over time. Both need a scale, a title, a source and a legend.
Two graphs for measured data
The histogram draws the table of classes: the horizontal axis carries the measured characteristic (kg), split into the classes, and each class has a bar whose height is the frequency. The bars touch, because one class starts where the previous one ends. The line graph is for values that change over time: time goes on the horizontal axis, the value on the vertical one, you mark one point per measurement and join the points. A plant measured each week: (0, 3.0), (1, 4.2) and (2, 6.2), in weeks and centimetres.
The bars touch and the line joins
Classes leave no gaps: between [3, 4) and [4, 5) no possible value is missing, so the bars of a histogram have no spaces. If you have made a stem-and-leaf diagram, notice this: the leaves of each stem form a bar lying on its side, and the histogram is the same picture, with the bars standing up and without the values. The line of a line graph joins the points because the value keeps changing between one measurement and the next. Scales have equal steps and say where they start, and every graph carries a title, a source (who measured and when) and a legend (what each axis measures, and the unit).
From the table to the histogram: 20 bags
Table: [2, 3) 1, [3, 4) 7, [4, 5) 4, [5, 6) 6, [6, 7) 2. Step 1: horizontal axis from 2 to 7 kg, 1 cm per class, five bars side by side. Step 2: the highest frequency is 7, so the vertical axis runs from 0 to 8 students, 1 cm per student, with equal steps. Step 3: bar heights: 1, 7, 4, 6 and 2 cm. Step 4: title “Weight of the class's bags”, source “weighing done by the class, 20 bags”, legends “Weight (kg)” and “Number of students”. Check: stem 3 of the stem-and-leaf has 7 leaves (2 4 5 6 7 8 9) and the bar [3, 4) is 7 cm high.
The line misleads when time is not to scale
A plant was measured in weeks 0, 1, 2, 4 and 8 and measured 3.0, 4.2, 6.2, 10.2 and 18.2 cm. If you put the five points equally far apart on the time axis, the increases between points are 1.2, 2.0, 4.0 and 8.0 cm, and the plant seems to grow faster and faster. But the time gaps were 1, 1, 2 and 4 weeks: per week, the increases are 1.2, 2.0, 2.0 and 2.0 cm. From week 1 to week 8 the plant always grows 2.0 cm per week. On the time axis, each week must take up the same space.
Screen time, temperature, heights and times
Line graphs are in the phone's screen time day by day, in the hour-by-hour temperature of a weather forecast, in how a mark or your savings in a piggy bank change. Histograms show up whenever there are many measured values: the heights of a class, the times of a swimming race, the ages of a group. Before you read them, look for three things: the title, the source and what each axis says, with its unit.
Test what you learned
In a histogram of the bag weights, which statement is true?
- a) The bars touch and the height of each is the number of values in that class.
- b) The bars have spaces between them, because each class is independent.
- c) The height of each bar is the weight of the bag.
- d) The order of the bars can be swapped without changing the picture.
Correct answer: a) The bars touch and the height of each is the number of values in that class. — Right: the classes follow one another with no gaps and the height measures the frequency, not the weight.
In the histogram of the 20 weights, the bars from [2, 3) to [6, 7) are 1, 7, 4, 6 and 2 cm high, with 1 cm per student. What percentage of the students have a bag of 5 kg or more?
- a) 30%
- b) 8%
- c) 20%
- d) 40%
Correct answer: d) 40% — Right: 6 + 2 = 8 students and 8/20 = 40/100 = 40%.
A plant measured 3.0, 4.2, 6.2, 10.2 and 18.2 cm in weeks 0, 1, 2, 4 and 8. What happens if the five points are placed equally far apart on the time axis?
- a) Nothing: the graph stays the same, because the values are the same.
- b) The plant seems to grow faster and faster, but it grows 2.0 cm per week from week 1 on.
- c) The plant seems to grow slower and slower.
Correct answer: b) The plant seems to grow faster and faster, but it grows 2.0 cm per week from week 1 on. — Right: the increases between points are 1.2, 2.0, 4.0 and 8.0 cm, but over 1, 1, 2 and 4 weeks, which is 2.0 cm per week.
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