PAPER · REASONING · PRACTICE

Rational numbers: position, opposites and absolute value

Use signed quantities, number-line order, opposite numbers and distance from zero in a fixed core-practice lesson.

40 minutes · Year 7, upper volume · text pp. 1–14 · foundation lessons 1–7

Textbook reference: Year 7, upper volume · Chapter 1 · Rational numbers

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.

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

Mark −4 and 3 on a number line. Compare their distance from each other with their distances from zero. Move both points two units right and ask what stays the same.

Organise the key idea

A sign describes direction relative to zero; absolute value describes distance. Zero is rational and an integer, but neither positive nor negative.

Open the original chapter Open the interactive inquiry

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

Worked example1

A is at −4 and B at 3. Find their distance in units and explain why adding the signed coordinates is wrong.

3-(-4)

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

Split the path at zero.
Each part is a nonnegative distance.
Add 4 and 3, or calculate 3 − (−4).

The points are on opposite sides of zero.
Distance = right coordinate − left coordinate = 3 − (−4) = 7.
Their coordinate sum is −1; it is not a distance.

Check your work

Apply the relevant sign and order rules to every operation.
Show an intermediate calculation and obtain the exact stated value.

Guided attempt2

Find the absolute value of −6. Explain why a distance cannot be negative.

Write the relevant rule or relation: ____.
Show an intermediate line: ____.
State the answer and check it: ____.

Locate the number relative to 0.
Measure distance, not direction.
A positive input stays positive; a negative input gives its opposite.

Absolute value is distance from 0 on a number line.
That distance is 6, so the absolute value is 6.

Check your work

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

Prerequisite3

Is 0 a positive number? Explain your classification.

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.

Positive numbers are strictly greater than zero.
0 is not greater than itself.

Check your work

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

Prerequisite4

Taking a gain as positive, write the signed amount for a loss of 12 dollars.

Read the positive-direction convention first.
A loss is not a gain.
Use a negative sign with 12.

A loss is in the direction opposite to a positive gain.
Attach a negative sign to the magnitude 12.

Check your work

Apply the relevant sign and order rules to every operation.
Show an intermediate calculation and obtain the exact stated value.

Independent practice5

Write the opposite of −7 and explain its position relative to 0.

-(-7)

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

Use the definition of opposite numbers.
Keep the distance from 0 and reverse the direction.
Change the sign; 0 is its own opposite.

The two points have the same distance from 0.
The opposite of −7 is 7.

Check your work

Apply the relevant sign and order rules to every operation.
Show an intermediate calculation and obtain the exact stated value.

Independent practice6

Find the absolute value of −9. Explain why a distance cannot be negative.

Locate the number relative to 0.
Measure distance, not direction.
A positive input stays positive; a negative input gives its opposite.

Absolute value is distance from 0 on a number line.
That distance is 9, so the absolute value is 9.

Check your work

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

Independent practice7

Which is greater, −3 or −8? Calculate the greater number minus the smaller one.

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

Use position, not just the size of the digits.
The point farther right represents the greater number.
After deciding the order, calculate −3 − (−8).

−3 is to the right of −8 on the number line.
The greater-minus-smaller difference is −3 − (−8) = 5.

Check your work

Apply the relevant sign and order rules to every operation.
Show an intermediate calculation and obtain the exact stated value.

Independent practice8

A learner calls −3/4 an integer because it has a minus sign. Is the claim valid? Classify the number correctly.

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.

A sign alone does not distinguish an integer from a fraction.
−3/4 is not a whole-number multiple of 1.

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

Start with −5 and take the opposite twice. What number do you obtain? Explain the two reversals.

-(-(-5))

Do one reversal at a time.
First find the opposite of −5.
Then find the opposite of the result 5.

The first opposite is 5.
The opposite of 5 is −5, returning to the starting number.

Check your work

Apply the relevant sign and order rules to every operation.
Show an intermediate calculation and obtain the exact stated value.

Exit question10

Find the absolute value of 0. Explain why a distance cannot be negative.

Locate the number relative to 0.
Measure distance, not direction.
A positive input stays positive; a negative input gives its opposite.

Absolute value is distance from 0 on a number line.
That distance is 0, so the absolute value is 0.

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

Write the opposite of 4 and explain its position relative to 0.

-(4)

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

Use the definition of opposite numbers.
Keep the distance from 0 and reverse the direction.
Change the sign; 0 is its own opposite.

The two points have the same distance from 0.
The opposite of 4 is −4.

Check your work

Apply the relevant sign and order rules to every operation.
Show an intermediate calculation and obtain the exact stated value.

Further practice12

Find the absolute value of −2.5. Explain why a distance cannot be negative.

Locate the number relative to 0.
Measure distance, not direction.
A positive input stays positive; a negative input gives its opposite.

Absolute value is distance from 0 on a number line.
That distance is 2.5, so the absolute value is 2.5.

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

A is at −6 and B at −1. Find the distance between them.

-1-(-6)

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

The path does not need to pass through zero.
Subtract the left coordinate from the right coordinate.
Compute −1 + 6.

Both points are left of zero; the right point is −1.
Distance = −1 − (−6) = 5.

Check your work

Apply the relevant sign and order rules to every operation.
Show an intermediate calculation and obtain the exact stated value.

Further practice14

Which is greater, −1/2 or −3/4? Find the greater-minus-smaller difference.

Use a common denominator.
For negatives, the value closer to zero is greater.
Compare −2/4 with −3/4.

Write −1/2 as −2/4; −2/4 is to the right of −3/4.
The difference is −2/4 + 3/4 = 1/4.

Check your work

Apply the relevant sign and order rules to every operation.
Show an intermediate calculation and obtain the exact stated value.

Extension15

A learner writes |−4| = −4. Explain the error using distance from zero.

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 minus sign describes the point’s direction.
Absolute value removes direction and keeps distance 4.

Check your work

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

Extension16

P starts at −5, moves 2 units right, reflects in 0, then moves 1 unit left. Find its final coordinate.

-(-5+2)-1

Record a new coordinate after each action.
Reflection in zero means taking the opposite.
The three coordinates are −3, 3, then the final result.

After moving right: −5 + 2 = −3.
Reflecting in zero gives 3; moving left once gives 2.

Check your work

Apply the relevant sign and order rules to every operation.
Show an intermediate calculation and obtain the exact stated value.

Preparation and assessment notes

Prerequisites

Whole numbers, fractions, decimals and comparing positive numbers.

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.

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. 1–14 · foundation lessons 1–7

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

Rational numbers: position, opposites and absolute value

Year 7, upper volume · text pp. 1–14 · foundation lessons 1–7 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

A is at −4 and B at 3. Find their distance in units and explain why adding the signed coordinates is wrong.

3-(-4)

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

Guided attempt2

Find the absolute value of −6. Explain why a distance cannot be negative.

Write the relevant rule or relation: ____.
Show an intermediate line: ____.
State the answer and check it: ____.

Prerequisite3

Is 0 a positive number? Explain your classification.

Prerequisite4

Taking a gain as positive, write the signed amount for a loss of 12 dollars.

Rational numbers: position, opposites and absolute value

Year 7, upper volume · text pp. 1–14 · foundation lessons 1–7 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

Write the opposite of −7 and explain its position relative to 0.

-(-7)

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

Independent practice6

Find the absolute value of −9. Explain why a distance cannot be negative.

Independent practice7

Which is greater, −3 or −8? Calculate the greater number minus the smaller one.

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

Independent practice8

A learner calls −3/4 an integer because it has a minus sign. Is the claim valid? Classify the number correctly.

Exit question9

Start with −5 and take the opposite twice. What number do you obtain? Explain the two reversals.

-(-(-5))

Exit question10

Find the absolute value of 0. Explain why a distance cannot be negative.

Rational numbers: position, opposites and absolute value — Further practice

Year 7, upper volume · text pp. 1–14 · foundation lessons 1–7

Further practice11

Write the opposite of 4 and explain its position relative to 0.

-(4)

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

Further practice12

Find the absolute value of −2.5. Explain why a distance cannot be negative.

Further practice13

A is at −6 and B at −1. Find the distance between them.

-1-(-6)

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

Further practice14

Which is greater, −1/2 or −3/4? Find the greater-minus-smaller difference.

Extension15

A learner writes |−4| = −4. Explain the error using distance from zero.

Extension16

P starts at −5, moves 2 units right, reflects in 0, then moves 1 unit left. Find its final coordinate.

-(-5+2)-1

Rational numbers: position, opposites and absolute value — Teacher answers

Year 7, upper volume · text pp. 1–14 · foundation lessons 1–7

Worked example1

A is at −4 and B at 3. Find their distance in units and explain why adding the signed coordinates is wrong.

3-(-4)

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

Reference answer

7 units; from −4 to 0 is 4, then from 0 to 3 is 3.

7

The points are on opposite sides of zero.
Distance = right coordinate − left coordinate = 3 − (−4) = 7.
Their coordinate sum is −1; it is not a distance.

Apply the relevant sign and order rules to every operation.
Show an intermediate calculation and obtain the exact stated value.

Guided attempt2

Find the absolute value of −6. Explain why a distance cannot be negative.

Write the relevant rule or relation: ____.
Show an intermediate line: ____.
State the answer and check it: ____.

Reference answer

|−6| = 6.

Absolute value is distance from 0 on a number line.
That distance is 6, so the absolute value is 6.

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

Prerequisite3

Is 0 a positive number? Explain your classification.

Reference answer

No. Zero is neither positive nor negative; it is an integer and a rational number.

Positive numbers are strictly greater than zero.
0 is not greater than itself.

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

Prerequisite4

Taking a gain as positive, write the signed amount for a loss of 12 dollars.

Reference answer

−12 dollars.

-12

A loss is in the direction opposite to a positive gain.
Attach a negative sign to the magnitude 12.

Apply the relevant sign and order rules to every operation.
Show an intermediate calculation and obtain the exact stated value.

Independent practice5

Write the opposite of −7 and explain its position relative to 0.

-(-7)

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

Reference answer

The value is 7.

7

The two points have the same distance from 0.
The opposite of −7 is 7.

Apply the relevant sign and order rules to every operation.
Show an intermediate calculation and obtain the exact stated value.

Independent practice6

Find the absolute value of −9. Explain why a distance cannot be negative.

Reference answer

|−9| = 9.

Absolute value is distance from 0 on a number line.
That distance is 9, so the absolute value is 9.

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

Independent practice7

Which is greater, −3 or −8? Calculate the greater number minus the smaller one.

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

Reference answer

−3 is greater; the difference is 5.

-3-(-8)

−3 is to the right of −8 on the number line.
The greater-minus-smaller difference is −3 − (−8) = 5.

Apply the relevant sign and order rules to every operation.
Show an intermediate calculation and obtain the exact stated value.

Independent practice8

A learner calls −3/4 an integer because it has a minus sign. Is the claim valid? Classify the number correctly.

Reference answer

No. −3/4 is a negative rational number and a fraction, not an integer.

A sign alone does not distinguish an integer from a fraction.
−3/4 is not a whole-number multiple of 1.

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

Exit question9

Start with −5 and take the opposite twice. What number do you obtain? Explain the two reversals.

-(-(-5))

Reference answer

The value is −5.

-5

The first opposite is 5.
The opposite of 5 is −5, returning to the starting number.

Apply the relevant sign and order rules to every operation.
Show an intermediate calculation and obtain the exact stated value.

Exit question10

Find the absolute value of 0. Explain why a distance cannot be negative.

Reference answer

|0| = 0.

Absolute value is distance from 0 on a number line.
That distance is 0, so the absolute value is 0.

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

Further practice11

Write the opposite of 4 and explain its position relative to 0.

-(4)

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

Reference answer

The value is −4.

-4

The two points have the same distance from 0.
The opposite of 4 is −4.

Apply the relevant sign and order rules to every operation.
Show an intermediate calculation and obtain the exact stated value.

Further practice12

Find the absolute value of −2.5. Explain why a distance cannot be negative.

Reference answer

|−2.5| = 2.5.

Absolute value is distance from 0 on a number line.
That distance is 2.5, so the absolute value is 2.5.

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

Further practice13

A is at −6 and B at −1. Find the distance between them.

-1-(-6)

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

Reference answer

5 units.

5

Both points are left of zero; the right point is −1.
Distance = −1 − (−6) = 5.

Apply the relevant sign and order rules to every operation.
Show an intermediate calculation and obtain the exact stated value.

Further practice14

Which is greater, −1/2 or −3/4? Find the greater-minus-smaller difference.

Reference answer

−1/2 is greater; the difference is 1/4.

-\frac{1}{2}-(-\frac{3}{4})

Write −1/2 as −2/4; −2/4 is to the right of −3/4.
The difference is −2/4 + 3/4 = 1/4.

Apply the relevant sign and order rules to every operation.
Show an intermediate calculation and obtain the exact stated value.

Extension15

A learner writes |−4| = −4. Explain the error using distance from zero.

Reference answer

The distance from −4 to 0 is 4, so |−4| = 4. A distance cannot be −4.

The minus sign describes the point’s direction.
Absolute value removes direction and keeps distance 4.

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

Extension16

P starts at −5, moves 2 units right, reflects in 0, then moves 1 unit left. Find its final coordinate.

-(-5+2)-1

Reference answer

The value is 2.

2

After moving right: −5 + 2 = −3.
Reflecting in zero gives 3; moving left once gives 2.

Apply the relevant sign and order rules to every operation.
Show an intermediate calculation and obtain the exact stated value.