PAPER · REASONING · PRACTICE

Geometric objects, segments and midpoints — Teacher reference

Distinguish lines, rays and segments; count cube elements and use point order and midpoint conditions.

40 minutes · Year 7, upper volume · text pp. 79–90 · foundation lessons 40–45

Textbook reference: Year 7, upper volume · Chapter 4 · Introduction to geometry

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

Student classroom page 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

Place M between A and B, then move M outside AB. Compare AM + MB with AB. Ask which position condition made the first equality true.

Organise the key idea

Segment addition depends on where the points lie. A midpoint lies on the segment and divides it into two equal lengths; a ray has one endpoint and one extending direction.

Open the original chapter Open the interactive inquiry

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

Worked example1

AB = 12 cm and M is its midpoint. Find AM and state both parts of the midpoint condition.

Points are shown in the stated order; use the given lengths.

A midpoint requires position and equal lengths.
AM and MB together form AB.
Divide 12 by 2.

M lies on AB and AM = MB.
AM is half of AB: 12 ÷ 2 = 6 cm.

Check your work

Use the stated order or midpoint condition before adding or halving.
Give the correct length with its unit.

Guided attempt2

M lies between A and B. AB = 10 cm and AM = 3 cm. Find MB and fill the length relation first.

AM + ____ = AB.
3 + MB = ____.
MB = ____ cm.

Points are shown in the stated order; use the given lengths.

Use the word “between”.
The whole AB consists of AM and MB.
Subtract the known part 3 from 10.

Because M lies between the endpoints, AM + MB = AB.
MB = 10 − 3 = 7 cm.

Check your work

Use the stated order or midpoint condition before adding or halving.
Give the correct length with its unit.

Prerequisite3

A segment has two endpoints and does not extend indefinitely beyond them. Is this description correct?

Use the given definition rather than the appearance of the notation.
Check the signed value or the stated condition.
Support your conclusion with a calculation or a concrete example.

Identify the two endpoints.
There is no indefinitely extending direction in a segment.

Check your work

State the applicable definition or condition accurately.
Support the conclusion with the given numbers, a check or a concrete counterexample.

Prerequisite4

A learner says a ray has two endpoints and extends indefinitely in one direction. Correct the endpoint count.

Use the given definition rather than the appearance of the notation.
Check the signed value or the stated condition.
Support your conclusion with a calculation or a concrete example.

The starting point is the only endpoint.
A second marked point indicates direction; it is not an endpoint of the ray.

Check your work

State the applicable definition or condition accurately.
Support the conclusion with the given numbers, a check or a concrete counterexample.

Independent practice5

C lies between A and B. AB = 9 cm and AC = 4 cm. Find CB.

Points are shown in the stated order; use the given lengths.

Use the interior-point condition.
The two parts make the whole 9 cm.
Subtract 4 from 9.

AC + CB = AB.
CB = 9 − 4 = 5 cm.

Check your work

Use the stated order or midpoint condition before adding or halving.
Give the correct length with its unit.

Independent practice6

M is the midpoint of AB and AB = 18 cm. Find AM.

Points are shown in the stated order; use the given lengths.

The two parts are equal.
Each is half the whole.
Divide 18 by 2.

AM = MB and AM + MB = 18.
AM = 18 ÷ 2 = 9 cm.

Check your work

Use the stated order or midpoint condition before adding or halving.
Give the correct length with its unit.

Independent practice7

How many edges does a cube have? Count each distinct object once.

Geometry diagram; not drawn to scale except for construction tasks.

Do not count only the visible parts of the sketch.
Count four edges on each end and four joining them.
Check the result 12 against your groups.

Use a complete cube, including the parts behind the visible front faces.
There are 12 distinct edges.

Check your work

State the applicable definition or condition accurately.
Support the conclusion with the given numbers, a check or a concrete counterexample.

Independent practice8

A straight line has no endpoints and extends indefinitely in both directions. Is this description correct?

Use the given definition rather than the appearance of the notation.
Check the signed value or the stated condition.
Support your conclusion with a calculation or a concrete example.

Marked points on a line do not become its endpoints.
It continues beyond every such point in both directions.

Check your work

State the applicable definition or condition accurately.
Support the conclusion with the given numbers, a check or a concrete counterexample.

Exit question9

M is the midpoint of AB and AM = 4 cm. Find AB.

Points are shown in the stated order; use the given lengths.

Here the half is known, not the whole.
There are two equal 4 cm parts.
Double 4.

The midpoint gives MB = AM = 4 cm.
AB = 4 + 4 = 8 cm.

Check your work

Use the stated order or midpoint condition before adding or halving.
Give the correct length with its unit.

Exit question10

A, B, M have coordinates 0, 6, 8 on one line. A learner claims AM + MB = AB merely because they are collinear. Test the claim.

Use the given order and lengths; the sketch is not a measuring scale.

Collinear does not necessarily mean between.
AM = 8, MB = 2 and AB = 6.
Compare 10 with 6.

M lies outside AB, beyond B.
Compute all three distances before comparing.

Check your work

State the applicable definition or condition accurately.
Support the conclusion with the given numbers, a check or a concrete counterexample.

Further practice11

How many faces does a cube have? Count each distinct object once.

Geometry diagram; not drawn to scale except for construction tasks.

Do not count only the visible parts of the sketch.
Pair the front/back, left/right and top/bottom.
Check the result 6 against your groups.

Use a complete cube, including the parts behind the visible front faces.
There are 6 distinct faces.

Check your work

State the applicable definition or condition accurately.
Support the conclusion with the given numbers, a check or a concrete counterexample.

Further practice12

How many vertices does a cube have? Count each distinct object once.

Geometry diagram; not drawn to scale except for construction tasks.

Do not count only the visible parts of the sketch.
Count four vertices on each of two opposite faces.
Check the result 8 against your groups.

Use a complete cube, including the parts behind the visible front faces.
There are 8 distinct vertices.

Check your work

State the applicable definition or condition accurately.
Support the conclusion with the given numbers, a check or a concrete counterexample.

Further practice13

M lies between A and B, AM = 5 cm and AB = 13 cm. Find MB.

Points are shown in the stated order; use the given lengths.

Identify the whole and the known part.
The whole is 13 cm.
Subtract 5.

AM + MB = AB.
MB = 13 − 5 = 8 cm.

Check your work

Use the stated order or midpoint condition before adding or halving.
Give the correct length with its unit.

Further practice14

A, B, C are collinear in that order. AB = 3 cm and BC = 7 cm. Find AC.

Use the given order and lengths; the sketch is not a measuring scale.

Read the stated point order.
The two given parts meet at B.
Add 3 and 7.

B lies between A and C.
AC = AB + BC = 3 + 7 = 10 cm.

Check your work

Use the stated order or midpoint condition before adding or halving.
Give the correct length with its unit.

Extension15

AB = 12 cm, M is its midpoint, and P lies between A and M with AP = 2 cm. Find PM.

Points are shown in the stated order; use the given lengths.

Use the midpoint before using P.
AM is 6 cm.
Subtract AP from AM.

AM = 12 ÷ 2 = 6 cm.
Since P lies between A and M, PM = AM − AP = 6 − 2 = 4 cm.

Check your work

Use the stated order or midpoint condition before adding or halving.
Give the correct length with its unit.

Extension16

A cube has six faces with four edges on each face. Why is its edge count not 6 × 4 = 24? Find the distinct edge count.

Geometry diagram; not drawn to scale except for construction tasks.

Inspect a single edge where two faces meet.
That edge appears in each of the two face counts.
Correct the repeated counting by dividing 24 by 2.

Six faces give 24 incidences, not 24 distinct edges.
Divide by two because every edge is shared by two faces.

Check your work

State the applicable definition or condition accurately.
Support the conclusion with the given numbers, a check or a concrete counterexample.

Preparation and assessment notes

Prerequisites

Reading point labels, adding lengths and halving a positive length.

Look for in students’ work

Use the prerequisite checks to choose support, project the worked and guided items, then let learners work on paper. Select supplements or extensions as needed. No learner answers are saved by this presentation.

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

All reference answers

Worked example1

AB = 12 cm and M is its midpoint. Find AM and state both parts of the midpoint condition.

Points are shown in the stated order; use the given lengths.

Reference answer

6 cm.

M lies on AB and AM = MB.
AM is half of AB: 12 ÷ 2 = 6 cm.

Use the stated order or midpoint condition before adding or halving.
Give the correct length with its unit.

Guided attempt2

M lies between A and B. AB = 10 cm and AM = 3 cm. Find MB and fill the length relation first.

AM + ____ = AB.
3 + MB = ____.
MB = ____ cm.

Points are shown in the stated order; use the given lengths.

Reference answer

7 cm.

Because M lies between the endpoints, AM + MB = AB.
MB = 10 − 3 = 7 cm.

Use the stated order or midpoint condition before adding or halving.
Give the correct length with its unit.

Prerequisite3

A segment has two endpoints and does not extend indefinitely beyond them. Is this description correct?

Reference answer

Correct. Its two endpoints bound its finite length.

Identify the two endpoints.
There is no indefinitely extending direction in a segment.

State the applicable definition or condition accurately.
Support the conclusion with the given numbers, a check or a concrete counterexample.

Prerequisite4

A learner says a ray has two endpoints and extends indefinitely in one direction. Correct the endpoint count.

Reference answer

A ray has one endpoint and extends indefinitely in one direction.

The starting point is the only endpoint.
A second marked point indicates direction; it is not an endpoint of the ray.

State the applicable definition or condition accurately.
Support the conclusion with the given numbers, a check or a concrete counterexample.

Independent practice5

C lies between A and B. AB = 9 cm and AC = 4 cm. Find CB.

Points are shown in the stated order; use the given lengths.

Reference answer

5 cm.

AC + CB = AB.
CB = 9 − 4 = 5 cm.

Use the stated order or midpoint condition before adding or halving.
Give the correct length with its unit.

Independent practice6

M is the midpoint of AB and AB = 18 cm. Find AM.

Points are shown in the stated order; use the given lengths.

Reference answer

9 cm.

AM = MB and AM + MB = 18.
AM = 18 ÷ 2 = 9 cm.

Use the stated order or midpoint condition before adding or halving.
Give the correct length with its unit.

Independent practice7

How many edges does a cube have? Count each distinct object once.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

12 edges.

12

Use a complete cube, including the parts behind the visible front faces.
There are 12 distinct edges.

State the applicable definition or condition accurately.
Support the conclusion with the given numbers, a check or a concrete counterexample.

Independent practice8

A straight line has no endpoints and extends indefinitely in both directions. Is this description correct?

Reference answer

Correct: no endpoints and two extending directions.

Marked points on a line do not become its endpoints.
It continues beyond every such point in both directions.

State the applicable definition or condition accurately.
Support the conclusion with the given numbers, a check or a concrete counterexample.

Exit question9

M is the midpoint of AB and AM = 4 cm. Find AB.

Points are shown in the stated order; use the given lengths.

Reference answer

8 cm.

The midpoint gives MB = AM = 4 cm.
AB = 4 + 4 = 8 cm.

Use the stated order or midpoint condition before adding or halving.
Give the correct length with its unit.

Exit question10

A, B, M have coordinates 0, 6, 8 on one line. A learner claims AM + MB = AB merely because they are collinear. Test the claim.

Use the given order and lengths; the sketch is not a measuring scale.

Reference answer

False. AM + MB = 8 + 2 = 10, but AB = 6. M must lie between A and B for that addition relation.

M lies outside AB, beyond B.
Compute all three distances before comparing.

State the applicable definition or condition accurately.
Support the conclusion with the given numbers, a check or a concrete counterexample.

Further practice11

How many faces does a cube have? Count each distinct object once.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

6 faces.

6

Use a complete cube, including the parts behind the visible front faces.
There are 6 distinct faces.

State the applicable definition or condition accurately.
Support the conclusion with the given numbers, a check or a concrete counterexample.

Further practice12

How many vertices does a cube have? Count each distinct object once.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

8 vertices.

8

Use a complete cube, including the parts behind the visible front faces.
There are 8 distinct vertices.

State the applicable definition or condition accurately.
Support the conclusion with the given numbers, a check or a concrete counterexample.

Further practice13

M lies between A and B, AM = 5 cm and AB = 13 cm. Find MB.

Points are shown in the stated order; use the given lengths.

Reference answer

8 cm.

AM + MB = AB.
MB = 13 − 5 = 8 cm.

Use the stated order or midpoint condition before adding or halving.
Give the correct length with its unit.

Further practice14

A, B, C are collinear in that order. AB = 3 cm and BC = 7 cm. Find AC.

Use the given order and lengths; the sketch is not a measuring scale.

Reference answer

10 cm.

B lies between A and C.
AC = AB + BC = 3 + 7 = 10 cm.

Use the stated order or midpoint condition before adding or halving.
Give the correct length with its unit.

Extension15

AB = 12 cm, M is its midpoint, and P lies between A and M with AP = 2 cm. Find PM.

Points are shown in the stated order; use the given lengths.

Reference answer

4 cm.

AM = 12 ÷ 2 = 6 cm.
Since P lies between A and M, PM = AM − AP = 6 − 2 = 4 cm.

Use the stated order or midpoint condition before adding or halving.
Give the correct length with its unit.

Extension16

A cube has six faces with four edges on each face. Why is its edge count not 6 × 4 = 24? Find the distinct edge count.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

Each edge belongs to two faces, so 24 face-edge incidences count every edge twice. The distinct count is 24/2 = 12.

Six faces give 24 incidences, not 24 distinct edges.
Divide by two because every edge is shared by two faces.

State the applicable definition or condition accurately.
Support the conclusion with the given numbers, a check or a concrete counterexample.

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

A 40-minute selection of original core practice after studying the cited textbook lessons. It is not a replacement for every lesson in the chapter.

大湾区初中数学三维课堂（七年级上册）正文.pdf

Year 7, upper volume · text pp. 79–90 · foundation lessons 40–45

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

Geometric objects, segments and midpoints

Year 7, upper volume · text pp. 79–90 · foundation lessons 40–45 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

AB = 12 cm and M is its midpoint. Find AM and state both parts of the midpoint condition.

Points are shown in the stated order; use the given lengths.

Guided attempt2

M lies between A and B. AB = 10 cm and AM = 3 cm. Find MB and fill the length relation first.

AM + ____ = AB.
3 + MB = ____.
MB = ____ cm.

Points are shown in the stated order; use the given lengths.

Prerequisite3

A segment has two endpoints and does not extend indefinitely beyond them. Is this description correct?

Prerequisite4

A learner says a ray has two endpoints and extends indefinitely in one direction. Correct the endpoint count.

Geometric objects, segments and midpoints

Year 7, upper volume · text pp. 79–90 · foundation lessons 40–45 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

C lies between A and B. AB = 9 cm and AC = 4 cm. Find CB.

Points are shown in the stated order; use the given lengths.

Independent practice6

M is the midpoint of AB and AB = 18 cm. Find AM.

Points are shown in the stated order; use the given lengths.

Independent practice7

How many edges does a cube have? Count each distinct object once.

Geometry diagram; not drawn to scale except for construction tasks.

Independent practice8

A straight line has no endpoints and extends indefinitely in both directions. Is this description correct?

Exit question9

M is the midpoint of AB and AM = 4 cm. Find AB.

Points are shown in the stated order; use the given lengths.

Exit question10

A, B, M have coordinates 0, 6, 8 on one line. A learner claims AM + MB = AB merely because they are collinear. Test the claim.

Use the given order and lengths; the sketch is not a measuring scale.

Geometric objects, segments and midpoints — Further practice

Year 7, upper volume · text pp. 79–90 · foundation lessons 40–45

Further practice11

How many faces does a cube have? Count each distinct object once.

Geometry diagram; not drawn to scale except for construction tasks.

Further practice12

How many vertices does a cube have? Count each distinct object once.

Geometry diagram; not drawn to scale except for construction tasks.

Further practice13

M lies between A and B, AM = 5 cm and AB = 13 cm. Find MB.

Points are shown in the stated order; use the given lengths.

Further practice14

A, B, C are collinear in that order. AB = 3 cm and BC = 7 cm. Find AC.

Use the given order and lengths; the sketch is not a measuring scale.

Extension15

AB = 12 cm, M is its midpoint, and P lies between A and M with AP = 2 cm. Find PM.

Points are shown in the stated order; use the given lengths.

Extension16

A cube has six faces with four edges on each face. Why is its edge count not 6 × 4 = 24? Find the distinct edge count.

Geometry diagram; not drawn to scale except for construction tasks.

Geometric objects, segments and midpoints — Teacher answers

Year 7, upper volume · text pp. 79–90 · foundation lessons 40–45

Worked example1

AB = 12 cm and M is its midpoint. Find AM and state both parts of the midpoint condition.

Points are shown in the stated order; use the given lengths.

Reference answer

6 cm.

M lies on AB and AM = MB.
AM is half of AB: 12 ÷ 2 = 6 cm.

Use the stated order or midpoint condition before adding or halving.
Give the correct length with its unit.

Guided attempt2

M lies between A and B. AB = 10 cm and AM = 3 cm. Find MB and fill the length relation first.

AM + ____ = AB.
3 + MB = ____.
MB = ____ cm.

Points are shown in the stated order; use the given lengths.

Reference answer

7 cm.

Because M lies between the endpoints, AM + MB = AB.
MB = 10 − 3 = 7 cm.

Use the stated order or midpoint condition before adding or halving.
Give the correct length with its unit.

Prerequisite3

A segment has two endpoints and does not extend indefinitely beyond them. Is this description correct?

Reference answer

Correct. Its two endpoints bound its finite length.

Identify the two endpoints.
There is no indefinitely extending direction in a segment.

State the applicable definition or condition accurately.
Support the conclusion with the given numbers, a check or a concrete counterexample.

Prerequisite4

A learner says a ray has two endpoints and extends indefinitely in one direction. Correct the endpoint count.

Reference answer

A ray has one endpoint and extends indefinitely in one direction.

The starting point is the only endpoint.
A second marked point indicates direction; it is not an endpoint of the ray.

State the applicable definition or condition accurately.
Support the conclusion with the given numbers, a check or a concrete counterexample.

Independent practice5

C lies between A and B. AB = 9 cm and AC = 4 cm. Find CB.

Points are shown in the stated order; use the given lengths.

Reference answer

5 cm.

AC + CB = AB.
CB = 9 − 4 = 5 cm.

Use the stated order or midpoint condition before adding or halving.
Give the correct length with its unit.

Independent practice6

M is the midpoint of AB and AB = 18 cm. Find AM.

Points are shown in the stated order; use the given lengths.

Reference answer

9 cm.

AM = MB and AM + MB = 18.
AM = 18 ÷ 2 = 9 cm.

Use the stated order or midpoint condition before adding or halving.
Give the correct length with its unit.

Independent practice7

How many edges does a cube have? Count each distinct object once.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

12 edges.

12

Use a complete cube, including the parts behind the visible front faces.
There are 12 distinct edges.

State the applicable definition or condition accurately.
Support the conclusion with the given numbers, a check or a concrete counterexample.

Independent practice8

A straight line has no endpoints and extends indefinitely in both directions. Is this description correct?

Reference answer

Correct: no endpoints and two extending directions.

Marked points on a line do not become its endpoints.
It continues beyond every such point in both directions.

State the applicable definition or condition accurately.
Support the conclusion with the given numbers, a check or a concrete counterexample.

Exit question9

M is the midpoint of AB and AM = 4 cm. Find AB.

Points are shown in the stated order; use the given lengths.

Reference answer

8 cm.

The midpoint gives MB = AM = 4 cm.
AB = 4 + 4 = 8 cm.

Use the stated order or midpoint condition before adding or halving.
Give the correct length with its unit.

Exit question10

A, B, M have coordinates 0, 6, 8 on one line. A learner claims AM + MB = AB merely because they are collinear. Test the claim.

Use the given order and lengths; the sketch is not a measuring scale.

Reference answer

False. AM + MB = 8 + 2 = 10, but AB = 6. M must lie between A and B for that addition relation.

M lies outside AB, beyond B.
Compute all three distances before comparing.

State the applicable definition or condition accurately.
Support the conclusion with the given numbers, a check or a concrete counterexample.

Further practice11

How many faces does a cube have? Count each distinct object once.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

6 faces.

6

Use a complete cube, including the parts behind the visible front faces.
There are 6 distinct faces.

State the applicable definition or condition accurately.
Support the conclusion with the given numbers, a check or a concrete counterexample.

Further practice12

How many vertices does a cube have? Count each distinct object once.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

8 vertices.

8

Use a complete cube, including the parts behind the visible front faces.
There are 8 distinct vertices.

State the applicable definition or condition accurately.
Support the conclusion with the given numbers, a check or a concrete counterexample.

Further practice13

M lies between A and B, AM = 5 cm and AB = 13 cm. Find MB.

Points are shown in the stated order; use the given lengths.

Reference answer

8 cm.

AM + MB = AB.
MB = 13 − 5 = 8 cm.

Use the stated order or midpoint condition before adding or halving.
Give the correct length with its unit.

Further practice14

A, B, C are collinear in that order. AB = 3 cm and BC = 7 cm. Find AC.

Use the given order and lengths; the sketch is not a measuring scale.

Reference answer

10 cm.

B lies between A and C.
AC = AB + BC = 3 + 7 = 10 cm.

Use the stated order or midpoint condition before adding or halving.
Give the correct length with its unit.

Extension15

AB = 12 cm, M is its midpoint, and P lies between A and M with AP = 2 cm. Find PM.

Points are shown in the stated order; use the given lengths.

Reference answer

4 cm.

AM = 12 ÷ 2 = 6 cm.
Since P lies between A and M, PM = AM − AP = 6 − 2 = 4 cm.

Use the stated order or midpoint condition before adding or halving.
Give the correct length with its unit.

Extension16

A cube has six faces with four edges on each face. Why is its edge count not 6 × 4 = 24? Find the distinct edge count.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

Each edge belongs to two faces, so 24 face-edge incidences count every edge twice. The distinct count is 24/2 = 12.

Six faces give 24 incidences, not 24 distinct edges.
Divide by two because every edge is shared by two faces.

State the applicable definition or condition accurately.
Support the conclusion with the given numbers, a check or a concrete counterexample.