Grade 5

Line graphs

A bean plant is measured every Saturday. How can one picture show how its height changed, and still let you read the height at any week?

You need

Data tables, tallies and frequency tablesGrade 1 The number lineGrades 1 to 2

Where you are 12 in Data displays

Statistics7

Data displays12

Bar graphs and pictographs4 Dot plots3

Line graphs

Histograms4 Reading a graph: titles, labels and units ▾ Show all 12 (7 more)

When you reach for it

Xiu planted a bean seed and measured the plant with a ruler every Saturday for eight weeks. Yoshiro weighed his puppy, Pepper, once a month as she grew. Each of them ended up with a list of pairs: a time, and the number measured at that time. That is a data table, like the tables you already know, except that each row pairs a time with a measurement instead of a category with a count.

A line graph turns such a table into a picture. Time runs along the bottom, the measurement goes up the side, each reading becomes a point, and the points are joined in time order. You reach for a line graph when the question is about change over time. How tall was the plant at week 5? When did it reach 14 centimeters? When did it grow the most? Did the puppy ever lose weight? Which of two puppies was heavier, and when did that change? A table holds the same numbers, but the line shows the shape of the change at a glance.

This lesson takes three days.

Part 1: how a line graph is built, reading its scales, reading the value at a given time, and finding the times a value was reached.
Part 2: where the line rises, falls or stays flat, how big each change is, and why a reading between two points is only an estimate.
Part 3: double line graphs, which put two quantities on the same axes, and making a line graph from a table.

The procedure

Part 1: reading a value, and finding a time

How the graph is built. A line graph is two number lines that meet at a corner. The one across the bottom is the time axis: weeks, months or days, evenly spaced from left to right. The one up the side is the value axis: it starts at 0 at the corner and counts up in equal steps. Each axis carries a label that says what it measures and in what unit, and the graph carries a title.

Each reading in the table becomes one point. If Xiu’s notebook says that at week 2 the plant was 4 cm tall, the point goes straight above the 2 on the time axis, at the height of the 4 on the value axis. When every reading has its point, the points are joined by straight pieces, called segments, in order of time from left to right.

Figure 1Xiu’s bean plant. The dashed path reads the height at week 5, which is 18 cm.

Reading the scale. Before reading any point, find out what one space on each axis stands for. On Xiu’s graph every line on the value axis is labeled, 0, 2, 4, 6 and so on, so each space is 2 cm. Often only some of the lines are labeled, to keep the side uncluttered. Then take two neighboring labels, subtract, and divide by the number of spaces between them. If the labels go 0, 20, 40 and there are 4 spaces between neighbors, each space stands for 20 \div 4 = 5. This works because the axis is a number line: its spaces are all equal, so they share the gap between two labels equally.

Reading the value at a given time. Find the time on the bottom axis and go straight up until you reach the line’s point. From the point go straight across to the value axis and read the scale there. The dashed path on Xiu’s graph does this for week 5: up from the 5, across to the 18. So at week 5 the plant was 18 cm tall. Going straight up and then straight across is right because the point was placed by exactly those two moves, right to the time and up to the value, so undoing them gives back the reading.

Finding the times a value was reached. This is the reverse trip. Start at the value on the side and go straight across, looking for points at that height. From each point you meet, go straight down to the time axis and read the time. There may be one such point, several, or none at all.

Part 2: rising, falling, flat, and between the points

Each segment joins two readings that are next to each other in time. Reading a segment from left to right:

if it goes up, the value rose between those two times;
if it goes down, the value fell;
if it is flat, the two readings were the same.

The size of a change is found by subtraction. For a rise, the later value minus the earlier value is how much it grew. For a fall, the earlier value minus the later value is how much it dropped.

On Xiu’s graph every segment rises except the last one, from week 7 to week 8, which is flat: 22 cm at both readings. Here are the weekly changes.

Weeks	Change (cm)
0 to 1	2
1 to 2	2
2 to 3	4
3 to 4	6
4 to 5	4
5 to 6	2
6 to 7	2
7 to 8	0

The biggest rise, 6 cm, is from week 3 to week 4, and it is also the steepest segment on the graph. That is no accident. Every segment covers the same distance across, one week, so the segment that climbs the farthest in that distance is the steepest one. When the readings are equally spaced in time, the steepest upward segment shows the biggest rise, and you can find it by eye and then confirm it by subtracting.

The changes also add up to the whole story: 2 + 2 + 4 + 6 + 4 + 2 + 2 + 0 = 22, which is the height at week 8 minus the height at week 0. Adding every step of a trip gives the whole trip.

Between the points. Xiu measured only on Saturdays. The graph joins her readings with straight segments, but she did not watch the plant during the week. A straight segment is what the line would look like if the plant grew by the same amount every day of that week. If it did, the segment gives the right height on any day in between. If it grew faster early in the week, or grew in a rush at the end, the segment is wrong there. So a value read between two points is an estimate, a careful guess, and should be said that way: “about 11 cm”, not “11 cm”. Only the points themselves are measurements.

Part 3: two lines on one graph, and making a line graph

A double line graph shows two quantities measured at the same times, drawn on the same axes. Each one gets its own line, drawn differently (solid and dashed, dots and squares), and a key says which line is which. Each line is read exactly as in Parts 1 and 2. What is new is comparing them.

At one time, the higher point is the greater value, and subtracting gives how much greater.
Where the two points sit at the same height at a reading, the two values were equal.
Where one line goes from below the other to above it, the two quantities have swapped order.

These comparisons can be made by looking because both lines share one value axis. A point that is higher on the page really is the larger number, which would not be true if each line had its own scale.

Making a line graph from a table. Each choice in making the graph comes from what the graph has to do when it is read.

The time axis gets one mark for each reading, in order, with equal spaces for equal steps of time, so that steepness means the same thing everywhere. The value axis starts at 0 and goes a little past the greatest value, so every point fits. Its step is chosen so the values land on lines, or close enough to read. Then the scale, the axis labels with their units, and a title, so a reader knows what is being measured. Last, each reading is placed as a point by the two moves of Part 1, and the points are joined in time order.

Worked examples

Examples 1 to 3 belong to Part 1, Examples 4 and 5 to Part 2, and Examples 6 and 7 to Part 3.

Example 1

Reading heights off the graph

Use Xiu’s graph. How tall was the plant at week 3? At week 6?

Solution

Each line on the value axis is labeled, so each space is 2 cm.

For week 3, go straight up from the 3 on the time axis to the point, then straight across to the side. The point sits on the line labeled 8. The plant was 8 cm tall at week 3.

For week 6, the same two moves land on the line labeled 20. The plant was 20 cm tall at week 6.

Example 2

When was the plant this tall?

Use Xiu’s graph. At which measured weeks was the plant 14 cm tall? At which was it 22 cm tall?

Solution

This is the reverse trip: start on the value axis, go across, then down.

From 14 on the side, go straight across. The first point met is above week 4, and no other point sits at 14. The plant was 14 cm tall at week 4.

From 22, going across meets two points, one above week 7 and one above week 8, with the flat segment between them. So the plant was 22 cm tall at week 7 and at week 8. A value can be reached at more than one reading, and the reverse trip has to look all the way across to find every one.

Example 3

A scale that counts by twos

Use the graph of Pepper’s weight. How much did Pepper weigh at month 4? At month 2? At which month did she weigh 24 pounds?

Figure 2Pepper’s weight. Only every fifth line is labeled, so each space has to be worked out.

Solution

First the scale. The value axis is labeled 0, 10, 20, 30, and between two neighboring labels there are 5 spaces. So each space stands for

10 \div 5 = 2 \text{ pounds}

Weight is written in pounds, shortened to lb on the graph.

At month 4, go up to the point and across. It sits one line below the 20, so it is 20 - 2 = 18. Pepper weighed 18 lb at month 4.

At month 2, the point sits three lines above 0. Three spaces of 2 make 3 \times 2 = 6. Pepper weighed 6 lb at month 2.

For 24 lb, find that height first: two lines above the 20, since 20 + 2 + 2 = 24. Going across from there meets one point, above month 7. Pepper weighed 24 lb at month 7.

Example 4

How Pepper’s weight changed

Use Pepper’s graph. Between which months did her weight rise, and between which did it fall? What was the biggest rise? How much did she gain in all from month 2 to month 8?

Solution

Read every point the way Example 3 did: 6, 10, 18, 16, 20, 24, 26 lb at months 2 to 8. Then subtract the neighbors.

Months	Change (lb)
2 to 3	rise 4
3 to 4	rise 8
4 to 5	fall 2
5 to 6	rise 4
6 to 7	rise 4
7 to 8	rise 2

Each rise is later minus earlier, such as 18 - 10 = 8. The fall is earlier minus later: 18 - 16 = 2. The only segment that goes down is from month 4 to month 5; Yoshiro remembers that Pepper was sick that month. Every other segment goes up, and none is flat.

The biggest rise is 8 lb, from month 3 to month 4. On the graph it is the steepest segment, which is what Part 2 says to expect, because each segment spans one month.

In all, Pepper went from 6 lb to 26 lb, a gain of 26 - 6 = 20 lb. Check with the steps: the rises add to 4 + 8 + 4 + 4 + 2 = 22, and taking away the fall of 2 leaves 22 - 2 = 20.

Example 5

Between Saturdays

Use Xiu’s graph. Estimate the plant’s height halfway between week 3 and week 4. Then say when the plant first reached 10 cm.

Solution

The segment from week 3 to week 4 joins 8 cm to 14 cm, a rise of 14 - 8 = 6 cm. Halfway along the segment, the line has made half of that rise:

8 + 6 \div 2 = 8 + 3 = 11

So the plant was about 11 cm tall halfway through that week. This is an estimate. It is exactly right only if the plant grew the same amount each day of that week, and Xiu did not measure in between.

No reading is 10 cm. The last reading below 10 is week 3, at 8 cm, and the next one is week 4, at 14 cm. The plant grows taller without shrinking, so it reached 10 cm sometime between week 3 and week 4. The graph cannot say exactly when. If the growth was steady, 10 cm is 2 of the week’s 6 cm, about a third of the way through the week, but that too is an estimate.

Example 6

Two puppies

Zhiwei weighed his puppy, Mochi, at the same ages as Pepper. Use the double line graph. Which puppy was heavier at month 3, and by how much? When did they weigh the same? At which months was Mochi heavier? Which puppy gained more from month 2 to month 8?

Figure 3Two puppies weighed at the same ages. The key says which line is which.

Solution

The key says Pepper is the solid line with dots and Mochi the dashed line with squares. Both use the same scale as before, 2 lb a space.

At month 3, Pepper’s dot is at 10 and Mochi’s square at 8. Pepper was heavier, by 10 - 8 = 2 lb.

At month 5, Pepper’s dot sits inside Mochi’s square: both are at 16 lb. That is the only reading where they are equal.

Before month 5 Mochi’s line is below Pepper’s. After it, Mochi’s line is above: at months 6, 7 and 8 Mochi weighed 22, 26 and 30 lb, against Pepper’s 20, 24 and 26. So Mochi was heavier at months 6, 7 and 8.

Pepper gained 26 - 6 = 20 lb. Mochi gained 30 - 4 = 26 lb. Mochi gained more, by 26 - 20 = 6 lb. Mochi started lighter and ended heavier, which is the crossing the graph shows.

Example 7

Making a line graph from Zhiwei’s table

Zhiwei also grew a bean plant and measured it each week.

Week	1	2	3	4	5	6
Height (cm)	1	3	6	11	15	16

Make a line graph of the table, and use it to find the week with the biggest rise.

Solution

The readings are one week apart, so the time axis gets six marks, 1 to 6, with equal spaces.

The greatest height is 16 cm. The value axis starts at 0 and goes to 18, a little past 16, so the top point is not squeezed against the edge. Several heights are odd numbers, 1, 3, 11, 15, so a step of 2 would leave them between lines. A line for every 1 cm puts every reading on a line. Labeling all 19 lines would crowd the side, so the labels go on every other line, 0, 2, 4 and so on.

Then the labels: “week” across, “height (cm)” up the side, and a title. Each point goes straight above its week, at its height, and the points are joined in order.

Figure 4The graph made from Zhiwei’s table in Example 7.

The weekly rises are 3 - 1 = 2, 6 - 3 = 3, 11 - 6 = 5, 15 - 11 = 4 and 16 - 15 = 1. The biggest, 5 cm, is from week 3 to week 4, and it is the steepest segment on the graph. The last segment is nearly flat: the plant grew only 1 cm in its last week.

Boundaries

Only the points are measurements. The segments join readings. A value between two readings is an estimate, exact only if the quantity changed steadily between them. A flat segment says the two readings were equal, not that nothing changed in between: a puppy could gain a pound and lose it again within the month.
The graph says nothing outside its readings. Before the first point and after the last, the line does not tell you anything.
Equal time steps need equal spaces. If the readings are not equally spaced in time, each point has to be placed by its actual time. Squeezing uneven times into even spaces changes the steepness and the story, one of the Misleading graphs.
Steepness compares changes only over equal times. A segment covering two weeks can be less steep than a one-week segment and still show a bigger change.
The value axis starts at 0 when the graph is used to compare sizes. A value axis that starts higher makes small differences look large.
A double line graph needs one scale and the same times for both lines. Two different scales on one picture make the heights impossible to compare.
A line graph is for things measured over time, or along some other order. For separate categories, such as favorite fruits, there is nothing between the categories to join, and a bar graph is the right picture.

Where this goes next

Every point on a line graph is a pair of numbers, a time and a value, which is exactly an ordered pair on a coordinate plane. In Two number patterns side by side and graphing their pairs, the pairs come from two number rules instead of a ruler and a scale.

Later on, Reading values off a graph reads an input and an output off the graph of a function, the same two moves as in Part 1. The steepness of a segment becomes a number, the Rate of change, found by dividing the rise by the run. In Graphs that tell a story, a graph with no numbers at all tells a story by its shape alone. Continuous graphs, jumps and holes asks the question Part 2 opened: when does it make sense to join the points, and when should they stay separate?

Exercises

Part 1: reading values and times

Exercises 1 to 16 use the graph of Yoshiro’s bean plant.

Figure 5Yoshiro’s bean plant, for Exercises 1 to 16.

The value axis is labeled 0, 4, 8, 12, 16. How many centimeters does each space stand for?
How tall was Yoshiro’s plant at week 2?
How tall was it at week 7?
How tall was it at week 5?
At which measured weeks was the plant 8 cm tall?
At which measured week was the plant 10 cm tall?
Was the plant ever 12 cm tall at one of Yoshiro’s readings?
A value axis is labeled 0, 50, 100, 150, with 5 equal spaces between neighboring labels. What does each space stand for? What value does a point have if it sits 3 spaces above the 50?

Part 2: changes, and between the points

Between which two weeks did Yoshiro’s plant get shorter, and by how much? (A rabbit nibbled the top.)
Between which two weeks did the line stay flat? What does that say about the two readings?
How much did the plant grow from week 1 to week 3?
What was the biggest rise between two neighboring readings, and between which weeks?
Estimate the height halfway between week 6 and week 7. Why is your answer only an estimate?
Add up every rise on Yoshiro’s graph, then take away the fall. Compare the result with the height at week 7, and explain why they agree.
A puppy weighed 9 lb at month 2, 14 lb at month 3 and 17 lb at month 4. Without drawing, say which of the two segments on its line graph is steeper, and why.
Between which two neighboring readings did Yoshiro’s plant first pass 5 cm?

Part 3: double line graphs and making a graph

Exercises 17 to 21 use the graph of two bean plants, one grown in the sun and one in the shade.

Figure 6Two bean plants, one grown in the sun and one in the shade, for Exercises 17 to 21.

How tall was each plant at week 3? How many times as tall was the plant in the sun?
How much taller was the plant in the sun than the plant in the shade at week 6?
At which week was the plant in the sun first more than 5 cm taller than the plant in the shade?
Find the biggest weekly rise for each plant, and the weeks it happened.
The plant in the shade was 9 cm tall at week 4. Between which two readings did the plant in the sun pass 9 cm?
Xiu weighed her puppy, Biscuit, every month.

Month	2	3	4	5	6
Weight (lb)	5	9	14	17	19

Describe the line graph you would draw: the scale on each axis and where the points go. Then find the biggest monthly rise.

Reasoning

Why are the points on a line graph joined in order of time? Why are the bars of a bar graph of favorite fruits not joined the same way?
On a double line graph, puppy A weighs 10 lb and puppy B 12 lb at month 3. At month 4, A weighs 15 lb and B 13 lb. What do the two lines do between month 3 and month 4? Can you say on which day the two puppies weighed the same?
A puppy’s line graph is flat from month 6 to month 7. Zhiwei says the puppy weighed the same every day of that month. Is he right? Explain.

Answers

2 cm. Neighboring labels differ by 4, with 2 spaces between them: 4 \div 2 = 2.
6 cm. The point sits one space above the 4: 4 + 2 = 6.
14 cm. The point sits one space below the 16: 16 - 2 = 14.
4 cm. The point is on the line labeled 4.
Weeks 3 and 4. Going across from 8 meets two points, joined by a flat segment.
Week 6. Going across from 10, halfway between the 8 and 12 labels, meets only the point above week 6.
No. Going across from 12 meets no point. The line passes 12 between week 6 (10 cm) and week 7 (14 cm), but no reading was taken there.
Each space is 10, since 50 \div 5 = 10. Three spaces above 50 is 50 + 3 \times 10 = 80.
From week 4 to week 5, it got 4 cm shorter: 8 - 4 = 4.
From week 3 to week 4. The two readings were equal, 8 cm both times.
6 cm: 8 - 2 = 6.
6 cm, from week 5 to week 6: 10 - 4 = 6. The other rises are 2, 4, 2 and 4, and it is the steepest segment.
About 12 cm: the segment rises from 10 to 14, and half of the rise of 4 is 2, so 10 + 2 = 12. It is an estimate because no reading was taken there; the plant may not have grown steadily during that week.
The rises are 2, 4, 2, 6, 4, which add to 18, and the fall is 4, so 18 - 4 = 14. The height at week 7 is 14 cm too, and it started at 0. Adding every step, up and down, gives the whole change from the first reading to the last.
The first segment is steeper. It rises 14 - 9 = 5 lb, the second 17 - 14 = 3 lb, and each covers one month across, so the bigger rise climbs more steeply.
Between week 1 (2 cm) and week 2 (6 cm).
In the sun 12 cm, in the shade 6 cm. The plant in the sun was 2 times as tall: 12 \div 6 = 2.
9 cm: 21 - 12 = 9.
Week 3. The differences are 0, 1, 3, 6 at weeks 0 to 3, and 6 is the first one more than 5.
In the sun: 5 cm, from week 2 to week 3 (12 - 7 = 5). In the shade: 3 cm, from week 3 to week 4 (9 - 6 = 3).
Between week 2 (7 cm) and week 3 (12 cm). So the plant in the sun was past 9 cm before week 3, while the plant in the shade was still only 6 cm at week 3.
Time axis: months 2 to 6, equally spaced. Value axis: 0 to 20 lb with a line for every 1 lb, labeled every 5, because the weights include odd numbers. Points above months 2 to 6 at 5, 9, 14, 17, 19, joined in order. The rises are 4, 5, 3, 2, so the biggest is 5 lb, from month 3 to month 4.
The readings are moments in one quantity’s life, in order, and the segment from one to the next shows how the value moved from one reading to the next. Favorite fruits are separate categories with no order and nothing in between, so a line from apples to bananas would show a change that does not exist.
A’s line starts below B’s and ends above it, so the lines cross between month 3 and month 4. The two puppies weighed the same at some moment in that month, but the graph cannot say which day: there are no readings in between.
Not necessarily. The flat segment only says the two readings, at month 6 and month 7, were equal. In between, the puppy could have gained a little and lost it again. Only steady weight every day would make Zhiwei right, and the graph cannot show that.