Reading scaled graphs and line plots
Scaled picture graphs, scaled bar graphs, and line plots of fraction measurements
About two to three lessons of 45 to 60 minutes
One picture can answer a hundred counting questions
Instead of counting a huge pile of survey tally marks one at a time, a graph turns the count into a shape you can read at a glance. This unit's first job is reading that shape correctly, using whatever scale the graph key gives you, because in Grade 3 one picture or one bar-square is worth more than 1.
Then we measure something ourselves, to the nearest quarter inch, and show a whole class's worth of measurements as dots stacked above a number line: a line plot. It looks different from a bar graph, but it answers the very same kind of question: what happened most, and what is the range of the data?
- A pictograph where one apple picture stands for 5 apples4 pictures means 4 x 5 = 20 apples, not 4
- A bar graph of favourite pets, scaled by 2sa bar reaching the 12 line means 12 votes, read from the gridlines
- A row of X's above 2 1/4 inches on a line ploteach X is one pencil measured at exactly that length
What students will be able to do
Students will read and interpret scaled picture graphs and bar graphs (including 'how many more/fewer' comparisons), and measure lengths to the nearest half and quarter inch and display that data on a line plot.
- I can read a scaled picture graph by checking the key first, then multiplying.
- I can read a value from a scaled bar graph using the gridlines.
- I can answer a 'how many more' or 'how many fewer' question from a graph.
- I can measure a length to the nearest quarter inch and mark it correctly on a line plot.
- I can answer a question using the data shown on a line plot.
Standards this unit teaches
- 3.MD.B.3Common Core (US)Scaled picture and bar graphs
Draw scaled picture graphs and bar graphs and solve problems using the data shown.
- 3.MD.B.4Common Core (US)Line plots with fractions
Measure lengths to the half and quarter inch and display the data on a line plot.
- AC9M3ST01Australian Curriculum v9 (ACARA)Many-to-one displays
Collect data on categories and discrete numbers, then represent it with many-to-one pictographs and column graphs and discuss the results.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Scaled graph
- a graph where one picture or one grid square stands for more than one item
- Key
- the part of a graph that tells you what one picture or one square is worth
- Bar graph
- a graph that uses the height or length of bars to show a count for each category
- Line plot
- a graph that shows data as X marks or dots stacked above a number line
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Scaled picture graphs
PictorialThe single most important habit for a scaled graph is checking the key before reading a single value. In earlier grades one picture was one item; from Grade 3 on, one picture can stand for 2, 5, or 10 items, and every count depends on multiplying by that scale.
A pictograph shows fruit picked, with each apple picture worth 5 apples. The apples row has 4 pictures. How many apples were picked?
- Read the key: 1 picture = 5 apples.
- Count the pictures in the row: 4.
- Multiply the picture count by the scale: 4 x 5 = 20.
Answer: 20 apples were picked.
- If the oranges row has 3 pictures at the same scale, how many oranges is that?
- Why can't you just count the pictures without checking the key?
2. Scaled bar graphs and comparisons
PictorialA scaled bar graph works the same way, except the scale is the jump between gridlines, not a picture. Read what number each gridline represents before reading any bar's value, then use subtraction for 'how many more' or 'how many fewer' questions.
A bar graph of pets owned, scaled in 2s, shows the Dogs bar reaching the 12 gridline and the Cats bar reaching the 8 gridline. How many more people have dogs than cats?
- Read each bar against the gridlines: Dogs = 12, Cats = 8.
- 'How many more' means subtract: 12 - 8 = 4.
Answer: 4 more people have dogs than cats.
- If the gridlines are scaled in 2s, what value does a bar sit on if it is exactly halfway between the 8 and 10 lines?
- How would you check that you have read the scale correctly before trusting any bar's value?
3. Line plots of fraction measurements
PictorialTo build a line plot, first measure each object to the nearest half or quarter inch, lining the ruler's zero up with the object's edge. Then draw a number line marked in quarters and put one X above the matching mark for every single measurement, stacking X's when a length repeats.
Eight pencils are measured to the nearest quarter inch: one 2 in, two 2 1/4 in, one 2 1/2 in, three 2 3/4 in, and one 3 in. How many pencils were measured in total, and what is the difference between the longest and shortest pencil?
- Add every X on the plot, one per pencil, regardless of length: 1 + 2 + 1 + 3 + 1 = 8.
- The longest pencil is 3 in and the shortest is 2 in.
- Difference: 3 - 2 = 1 inch.
Answer: 8 pencils were measured in total, and the longest is 1 inch longer than the shortest.
- How many pencils were exactly 2 3/4 inches long, and how do you read that straight from the plot?
- Why does counting every X give the total number of pencils, while counting the number of different marks does not?
Common misconceptions and how to address them
MisconceptionReading a scaled pictograph by counting the number of pictures only, ignoring the key, for example reading 4 pictures as 4 apples instead of 20.
Why it happens: In earlier grades one picture always meant one item, so students carry that rule forward without checking whether the scale has changed.
How to address it: Make 'check the key first' a non-negotiable first step, said out loud, before reading a single row of any pictograph.
MisconceptionReading a bar's value from the nearest labelled number rather than from the actual gridline scale, for example reading a bar as '5' when the gridlines jump in 2s and the bar actually sits on 6.
Why it happens: Students look for a familiar-looking number near the bar's top instead of counting gridlines from zero using the graph's own scale.
How to address it: Trace a finger along the gridlines from zero, counting up by the scale amount at every line, before reading any bar's height.
MisconceptionOn a line plot, counting the number of different lengths (columns) shown instead of the total number of data points (every X).
Why it happens: The columns are the most visually distinct feature of a line plot, so 'how many' gets answered from the columns rather than the individual marks.
How to address it: Have students physically point to and count every single X on the plot, out loud, one at a time, rather than counting stacks.
MisconceptionMeasuring a length and rounding it to the nearest whole inch out of habit, rather than reading it to the nearest quarter inch as the task requires.
Why it happens: Whole-number measuring is the more practised skill, so it is the default even after quarter-inch precision has been taught.
How to address it: Line the ruler's zero exactly on the object's edge, then ask which of the four quarter-inch ticks in that inch is closest before writing an answer.
Guided practice (with answers)
1. A pictograph key says 1 star = 10 books read. The row for Ms Lee's class has 3 stars. How many books did the class read?
Answer: 30 books, since 3 x 10 = 30.
2. A bar graph scaled in 5s shows the Football bar at 20 and the Basketball bar at 15. How many more votes did football get?
Answer: 5 more votes, since 20 - 15 = 5.
3. On a line plot with marks at 1, 1 1/4, 1 1/2 (each with one X except 1 1/4, which has three X's), how many measurements are shown in total?
Answer: 5 measurements: 1 + 3 + 1 = 5.
4. A pencil's tip lines up with the ruler's zero and its end lines up exactly halfway between the 4 inch and 4 1/4 inch marks. What length should be recorded to the nearest quarter inch?
Answer: 4 1/4 inches (or 4 inches, whichever quarter-inch tick is truly closer); the key habit is reading only the marked quarter-inch ticks, not estimating between them.
5. A pictograph key says 1 picture = 5 pets. If 35 pets were counted for 'dogs', how many pictures should be drawn?
Answer: 7 pictures, since 35 ÷ 5 = 7.
Independent practice worksheets
Set the matching ChalkBee worksheets for independent work. The answer keys are computed in code, so they are never wrong. Start with picture and bar graphs, then move to line plots once measuring to the quarter inch is secure.
Differentiation
- Start with a scale of 2 before moving to 5 or 10, and keep the key visible and highlighted on every graph.
- For line plots, pre-mark the number line in quarters so students only place the X's, not draw the axis.
- Use physical counters to build a bar graph from a real small data set before reading a printed one.
- Compare two scaled graphs of the same data drawn with different scales and discuss why they look different but show the same information.
- Create a line plot from a class's own measurements (such as hand span or pencil length) and write two questions a partner must answer from it.
- Introduce two-step comparison questions, such as combining two categories before comparing to a third.
Assessment: exit ticket
A short exit ticket sampling picture graphs, bar graphs, and line plots.
1. A pictograph key says 1 picture = 5 books. A row has 6 pictures. How many books is that?
Answer: 30 books, since 6 x 5 = 30.
2. A bar graph scaled in 2s shows Bar A at 14 and Bar B at 10. How many fewer is Bar B than Bar A?
Answer: 4 fewer, since 14 - 10 = 4.
3. A line plot shows 2 X's at 3 inches and 5 X's at 3 1/2 inches. How many measurements are shown in total?
Answer: 7 measurements, since 2 + 5 = 7.
Teacher notes and timings
- Rough timing across three lessons: Lesson 1 scaled picture graphs (section 1), Lesson 2 scaled bar graphs (section 2), Lesson 3 line plots plus the exit ticket (section 3 and assessment).
- Language to keep saying: check the key first, count every gridline, count every X. These three phrases target the unit's main misconceptions directly.
- The line plot figure in this unit is drawn on the number-line engine, since ChalkBee's diagram engine does not yet have a dedicated stacked-dot line-plot component; the label above each mark stands in for a stack of X's and is called out as such in the caption.
- Curriculum note: ACARA's AC9M3ST01 covers many-to-one pictographs and column graphs at the same Year 3 level as the US standard; the fraction-measurement line plot is a US-specific emphasis without a direct ACARA equivalent at this year, so lean on the US worked examples for that half of the unit.
- Present mode and print both work: use Present to build a graph live from class data, then print the worksheets for independent reading practice.