PAPER · REASONING · PRACTICE

Grouped data and histograms: frequency and area

Count boundary values once, label the vertical measure, and use frequency per class width for unequal classes.

40 minutes · Year 7, lower volume · text pp. 95–98 · textbook lessons 48–49

Textbook reference: Year 7 lower · Chapter 10 Collecting, organising and displaying data · text lessons 48–49

Chapter core practice: use after the referenced textbook lessons, or for chapter review. See the lower-volume plan

One worked example, one guided attempt, eight core questions, four follow-ups and two extensions. Write on paper; the teacher controls the reveal.

Teacher reference Arrange in My Classroom

Lesson routePhase

4′Check prerequisites
5′Observe and predict
8′Worked example and guided attempt
15′Independent practice
5′Discuss and correct
3′Exit questions

Observe · explain

A question before practising

18 minutes, fixed data on paper: use 1,1,1,1,2,2,3,3,3,3,4,5. Predict whether regrouping changes the sample size (3 min). Tally widths 2 over [0,2),[2,4),[4,6], then widths 3 over [0,3),[3,6]; compare frequencies 4,6,2 versus 6,6 and density heights (7 min). For a second given table, compare widths 10 and 20 with frequencies 20 and 30: calculate heights 2 and 3/2 and recover counts through areas (5 min). Conclude which totals stay fixed and why a shorter density bar can contain more observations (3 min).

Organise the key idea

Count boundary values once, label the vertical measure, and use frequency per class width for unequal classes.

Open the original chapter

Use the lesson question here for paper discussion; the original activity opens separately.

Ordinary route: use the first fixed case for the five-minute demonstration. Advanced route: follow the full 18-minute investigation; record predictions before checking.

Worked example1

Group the data $1,1,1,1,2,2,3,3,3,3,4,5$ into $0\leq x<2$ , $2\leq x<4$ , $4\leq x\leq6$ . Give frequencies and density heights (frequency ÷ class width).

Interval	Frequency	Density
[0,2)	____	____
[2,4)	____	____
[4,6]	____	____

Use the stated conditions and the labelled scales.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

Place each boundary in the class indicated by its inequality.
Counts are 4,6,2 and sum to 12.
Divide each frequency by width 2; bar area then equals frequency.

Reference answer

Frequencies 4,6,2; width 2 each; heights 2,3,1.

(h_1,h_2,h_3)=(2,3,1)

Interval	Frequency	Density
[0,2)	4	2
[2,4)	6	3
[4,6]	2	1

Use the stated conditions and the labelled scales.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Guided attempt2

Draw a density histogram for $0\leq x<2$ : 4; $2\leq x<4$ : 6; $4\leq x\leq6$ : 2. Label frequency per unit on the vertical axis.

All class widths equal $2-0=\square$ .
Complete heights $h_1=\frac{4}{2}=\square,\ h_2=\frac{6}{2}=\square,\ h_3=\frac{2}{2}=\square$ .
Draw touching bars over the three intervals; label y as frequency per unit and check area total ____.

Use the stated conditions and the labelled scales.

Use class width 2, not the number of classes 3.
The vertical measure is density: divide frequencies 4,6,2 by their own widths.
The three bar areas must recover 4,6,2 and sum to 12 observations.

The class widths are all 2.
Heights are 4/2=2, 6/2=3, 2/2=1.
Use touching bars and retain the class boundaries.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Prerequisite3

For data 4,7,9,12, find the maximum minus minimum.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

The maximum is 12 and minimum 4.
The difference is 8.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Prerequisite4

A class contains 3 of 12 observations. Find its relative frequency.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

The given categories account for every observation exactly once.
Use $\frac{3}{12}$ as the relative frequency.
The requested result is $\frac{1}{4}$ .

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice5

Data are $0,1,2,2,3,4,5,6$ . Use classes $0\leq x<2$ , $2\leq x<4$ , $4\leq x\leq6$ . Give all three frequencies.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

The two 2s go in the second class; 4 goes in the third.
The final class includes 6. Counts 2+3+3=8 retain every observation exactly once.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice6

The given density histogram has class width 2 throughout and heights 1,2,1. What fraction of the observations have value at least 2?

Use the stated conditions and the labelled scales.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

Multiply heights by widths to recover frequencies 2,4,2.
The final two classes contain 6 of the total 8, so 6/8=3/4.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice7

Class A spans 0 to 10 with frequency 20. Class B spans 10 to 30 with frequency 30. Find density heights for a histogram. Which class has more observations?

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

Widths are 10 and 20.
Heights are 20/10=2 and 30/20=3/2; compare frequencies 20 and 30 for counts.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice8

Draw a density histogram for $0\leq x<2$ : 2, $2\leq x<4$ : 4, $4\leq x\leq6$ : 2. Label both axes.

Use the stated conditions and the labelled scales.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

All widths are 2, so divide frequencies by 2.
Draw heights 1,2,1 on their stated intervals; their areas sum to 8.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Exit question9

For classes $0\leq x<2$ and $2\leq x<4$ , where is x=2 counted?

Both classes.
Neither class.
Only the second class.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

The first class excludes its upper boundary 2.
The second includes its lower boundary, so count 2 once there.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Exit question10

A density histogram bar has width 4 and height 3 observations per unit. Find its frequency.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

Frequency equals width times density height.
4×3=12 observations.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice11

Three nonoverlapping classes have frequencies 4,7,5. Find the total sample size.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

The given categories account for every observation exactly once.
Add the mutually exclusive category counts: $4+7+5$ .
The requested result is $16$ .

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice12

6 of 20 observations fall in one class. Find its percentage.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

The given categories account for every observation exactly once.
Use $\frac{6}{20}$ ×100% to express a percentage.
The requested result is $30$ .

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice13

Classes [0,2), [2,4), [4,6] have frequencies 2,4,2. For the frequency-distribution line graph, give the three class-midpoint/frequency points to connect in order.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

Class midpoints are (0+2)/2=1, (2+4)/2=3, (4+6)/2=5.
Pair them with frequencies 2,4,2 and connect in class order.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice14

Choose the classes [0,5), [5,10), [10,15), [15,20] for values between 1 and 18. How many classes are there, and what is the common class width?

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

Read successive boundaries 0,5,10,15,20.
There are four intervals and every adjacent boundary difference is 5.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Extension15

One density bar spans 0–10 with height 1. A second spans 10–40 with height 1/2. Which contains more observations? Explain why comparing height alone fails.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

Multiply each height by its own width: 1×10=10 and (1/2)×30=15.
Density heights alone are not comparable as frequencies when widths differ.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Extension16

Use the same data $1,1,1,1,2,2,3,3,3,3,4,5$ . Regroup into $0\leq x<3$ and $3\leq x\leq6$ . Find frequencies and explain what total is preserved.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

The first class contains four 1s and two 2s.
The remaining six values form the second class; 6+6=12.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Preparation and assessment notes

Prerequisites

Read frequencies, compare interval boundaries and calculate widths, ratios and rectangle areas.

Look for in students’ work

Use the worked and guided tasks selectively. The eight core paper tasks are prerequisites, independent tasks and exits; further practice and extensions are optional.

Collect the paper exit questions and note completion and common errors. This page does not grade or store paper answers.

Paper for the class

Download all worksheets (ZIP · 318 PDFs · 47.2 MB)

All 53 Year 7–8 upper- and lower-volume packs in Traditional Chinese and English. Includes teacher answers; student core, further practice and teacher papers are in separate folders.

Core student worksheet: two A4 pages. Further practice is separate. Teacher copies include reference reasoning.

Print edition

Textbook reference and authorship

Original 40-minute core consolidation after the mapped textbook lessons; not a replacement for every first-teaching period. Only the supplied text has been mapped.

大湾区初中数学三维课堂（七年级下册）正文.pdf

Year 7, lower volume · text pp. 95–98 · textbook lessons 48–49

Original questions and explanations; the source textbook and answer PDFs are not reproduced here.

Grouped data and histograms: frequency and area

Year 7, lower volume · text pp. 95–98 · textbook lessons 48–49 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

Group the data $1,1,1,1,2,2,3,3,3,3,4,5$ into $0\leq x<2$ , $2\leq x<4$ , $4\leq x\leq6$ . Give frequencies and density heights (frequency ÷ class width).

Interval	Frequency	Density
[0,2)	____	____
[2,4)	____	____
[4,6]	____	____

Use the stated conditions and the labelled scales.

Guided attempt2

Draw a density histogram for $0\leq x<2$ : 4; $2\leq x<4$ : 6; $4\leq x\leq6$ : 2. Label frequency per unit on the vertical axis.

All class widths equal $2-0=\square$ .
Complete heights $h_1=\frac{4}{2}=\square,\ h_2=\frac{6}{2}=\square,\ h_3=\frac{2}{2}=\square$ .
Draw touching bars over the three intervals; label y as frequency per unit and check area total ____.

Use the stated conditions and the labelled scales.

Prerequisite3

For data 4,7,9,12, find the maximum minus minimum.

Prerequisite4

A class contains 3 of 12 observations. Find its relative frequency.

Grouped data and histograms: frequency and area

Year 7, lower volume · text pp. 95–98 · textbook lessons 48–49 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

Data are $0,1,2,2,3,4,5,6$ . Use classes $0\leq x<2$ , $2\leq x<4$ , $4\leq x\leq6$ . Give all three frequencies.

Independent practice6

The given density histogram has class width 2 throughout and heights 1,2,1. What fraction of the observations have value at least 2?

Use the stated conditions and the labelled scales.

Independent practice7

Class A spans 0 to 10 with frequency 20. Class B spans 10 to 30 with frequency 30. Find density heights for a histogram. Which class has more observations?

Independent practice8

Draw a density histogram for $0\leq x<2$ : 2, $2\leq x<4$ : 4, $4\leq x\leq6$ : 2. Label both axes.

Use the stated conditions and the labelled scales.

Exit question9

For classes $0\leq x<2$ and $2\leq x<4$ , where is x=2 counted?

Both classes.
Neither class.
Only the second class.

Exit question10

A density histogram bar has width 4 and height 3 observations per unit. Find its frequency.

Grouped data and histograms: frequency and area — Further practice

Year 7, lower volume · text pp. 95–98 · textbook lessons 48–49

Further practice11

Three nonoverlapping classes have frequencies 4,7,5. Find the total sample size.

Further practice12

6 of 20 observations fall in one class. Find its percentage.

Further practice13

Classes [0,2), [2,4), [4,6] have frequencies 2,4,2. For the frequency-distribution line graph, give the three class-midpoint/frequency points to connect in order.

Further practice14

Choose the classes [0,5), [5,10), [10,15), [15,20] for values between 1 and 18. How many classes are there, and what is the common class width?

Extension15

One density bar spans 0–10 with height 1. A second spans 10–40 with height 1/2. Which contains more observations? Explain why comparing height alone fails.

Extension16

Use the same data $1,1,1,1,2,2,3,3,3,3,4,5$ . Regroup into $0\leq x<3$ and $3\leq x\leq6$ . Find frequencies and explain what total is preserved.

Grouped data and histograms: frequency and area — Teacher answers

Year 7, lower volume · text pp. 95–98 · textbook lessons 48–49

Use the worked and guided tasks selectively. The eight core paper tasks are prerequisites, independent tasks and exits; further practice and extensions are optional.

Ordinary route: prerequisites 4, short inquiry 5, worked/guided 8, independent 15, correction 5, exit 3 minutes. Advanced route: inquiry 18, organise 4, selected practice 13, exit 5 minutes.

Prerequisites: Read frequencies, compare interval boundaries and calculate widths, ratios and rectangle areas.

Ordinary route: use the first fixed case for the five-minute demonstration. Advanced route: complete the full investigation.

Inquiry and discussion: 18 minutes, fixed data on paper: use 1,1,1,1,2,2,3,3,3,3,4,5. Predict whether regrouping changes the sample size (3 min). Tally widths 2 over [0,2),[2,4),[4,6], then widths 3 over [0,3),[3,6]; compare frequencies 4,6,2 versus 6,6 and density heights (7 min). For a second given table, compare widths 10 and 20 with frequencies 20 and 30: calculate heights 2 and 3/2 and recover counts through areas (5 min). Conclude which totals stay fixed and why a shorter density bar can contain more observations (3 min).

Worked example1

Group the data $1,1,1,1,2,2,3,3,3,3,4,5$ into $0\leq x<2$ , $2\leq x<4$ , $4\leq x\leq6$ . Give frequencies and density heights (frequency ÷ class width).

Interval	Frequency	Density
[0,2)	____	____
[2,4)	____	____
[4,6]	____	____

Use the stated conditions and the labelled scales.

Reference answer

Frequencies 4,6,2; width 2 each; heights 2,3,1.

(h_1,h_2,h_3)=(2,3,1)

Place each boundary in the class indicated by its inequality.
Counts are 4,6,2 and sum to 12.
Divide each frequency by width 2; bar area then equals frequency.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Interval	Frequency	Density
[0,2)	4	2
[2,4)	6	3
[4,6]	2	1

Use the stated conditions and the labelled scales.

Guided attempt2

Draw a density histogram for $0\leq x<2$ : 4; $2\leq x<4$ : 6; $4\leq x\leq6$ : 2. Label frequency per unit on the vertical axis.

All class widths equal $2-0=\square$ .
Complete heights $h_1=\frac{4}{2}=\square,\ h_2=\frac{6}{2}=\square,\ h_3=\frac{2}{2}=\square$ .
Draw touching bars over the three intervals; label y as frequency per unit and check area total ____.

Use the stated conditions and the labelled scales.

Reference answer

Adjacent bars on [0,2),[2,4),[4,6] have heights 2,3,1.

The class widths are all 2.
Heights are 4/2=2, 6/2=3, 2/2=1.
Use touching bars and retain the class boundaries.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Use the stated conditions and the labelled scales.

Prerequisite3

For data 4,7,9,12, find the maximum minus minimum.

Reference answer

$8$ .

12-4

The maximum is 12 and minimum 4.
The difference is 8.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Prerequisite4

A class contains 3 of 12 observations. Find its relative frequency.

Reference answer

$\frac{1}{4}$ .

(3/12)\cdot 1

The given categories account for every observation exactly once.
Use $\frac{3}{12}$ as the relative frequency.
The requested result is $\frac{1}{4}$ .

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice5

Data are $0,1,2,2,3,4,5,6$ . Use classes $0\leq x<2$ , $2\leq x<4$ , $4\leq x\leq6$ . Give all three frequencies.

Reference answer

2,3,3.

The two 2s go in the second class; 4 goes in the third.
The final class includes 6. Counts 2+3+3=8 retain every observation exactly once.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice6

The given density histogram has class width 2 throughout and heights 1,2,1. What fraction of the observations have value at least 2?

Use the stated conditions and the labelled scales.

Reference answer

$\frac{3}{4}$

\frac{3}{4}

Multiply heights by widths to recover frequencies 2,4,2.
The final two classes contain 6 of the total 8, so 6/8=3/4.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Use the stated conditions and the labelled scales.

Independent practice7

Class A spans 0 to 10 with frequency 20. Class B spans 10 to 30 with frequency 30. Find density heights for a histogram. Which class has more observations?

Reference answer

Heights 2 and 3/2; B has more observations although its bar is shorter.

h_A=2,\quad h_B=\frac{3}{2}

Widths are 10 and 20.
Heights are 20/10=2 and 30/20=3/2; compare frequencies 20 and 30 for counts.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice8

Draw a density histogram for $0\leq x<2$ : 2, $2\leq x<4$ : 4, $4\leq x\leq6$ : 2. Label both axes.

Use the stated conditions and the labelled scales.

Reference answer

Three touching bars have heights 1,2,1; horizontal axis is value, vertical axis frequency per unit.

All widths are 2, so divide frequencies by 2.
Draw heights 1,2,1 on their stated intervals; their areas sum to 8.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Use the stated conditions and the labelled scales.

Exit question9

For classes $0\leq x<2$ and $2\leq x<4$ , where is x=2 counted?

Both classes.
Neither class.
Only the second class.

Reference answer

Only the second class.

The first class excludes its upper boundary 2.
The second includes its lower boundary, so count 2 once there.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Exit question10

A density histogram bar has width 4 and height 3 observations per unit. Find its frequency.

Reference answer

$12$ .

4\cdot 3

Frequency equals width times density height.
4×3=12 observations.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice11

Three nonoverlapping classes have frequencies 4,7,5. Find the total sample size.

Reference answer

$16$ .

4+7+5

The given categories account for every observation exactly once.
Add the mutually exclusive category counts: $4+7+5$ .
The requested result is $16$ .

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice12

6 of 20 observations fall in one class. Find its percentage.

Reference answer

$30$ %.

(6/20)\cdot 100

The given categories account for every observation exactly once.
Use $\frac{6}{20}$ ×100% to express a percentage.
The requested result is $30$ .

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice13

Classes [0,2), [2,4), [4,6] have frequencies 2,4,2. For the frequency-distribution line graph, give the three class-midpoint/frequency points to connect in order.

Reference answer

$(1,2),(3,4),(5,2)$

Class midpoints are (0+2)/2=1, (2+4)/2=3, (4+6)/2=5.
Pair them with frequencies 2,4,2 and connect in class order.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice14

Choose the classes [0,5), [5,10), [10,15), [15,20] for values between 1 and 18. How many classes are there, and what is the common class width?

Reference answer

Four classes, each width 5.

Read successive boundaries 0,5,10,15,20.
There are four intervals and every adjacent boundary difference is 5.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Extension15

One density bar spans 0–10 with height 1. A second spans 10–40 with height 1/2. Which contains more observations? Explain why comparing height alone fails.

Reference answer

The second: areas/frequencies are 10 and 15.

Multiply each height by its own width: 1×10=10 and (1/2)×30=15.
Density heights alone are not comparable as frequencies when widths differ.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Extension16

Use the same data $1,1,1,1,2,2,3,3,3,3,4,5$ . Regroup into $0\leq x<3$ and $3\leq x\leq6$ . Find frequencies and explain what total is preserved.

Reference answer

6 and 6; total observations remain 12, while the distribution’s appearance changes.

The first class contains four 1s and two 2s.
The remaining six values form the second class; 6+6=12.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.