PAPER · REASONING · PRACTICE

Rational numbers: operations, powers and large numbers — Teacher reference

Select core signed arithmetic, fractions, powers, scientific notation and rounding after the relevant textbook periods.

40 minutes · Year 7, upper volume · text pp. 15–40 · foundation lessons 8–20

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.

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

Compare −3² and (−3)². Predict each sign, calculate, then explain which part the exponent acts on.

Organise the key idea

Signs and operation order both matter. Powers are evaluated before a leading minus; a divisor must be nonzero. Scientific notation separates a scaled coefficient from a power of ten.

Open the original chapter Open the interactive inquiry

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

Worked example1

Evaluate the expression and explain why subtracting a negative becomes addition.

(-4)+7-(-2)

First add −4 and 7.
Use a − b = a + (−b).
The final operation is 3 + 2.

−4 + 7 = 3.
Subtracting −2 means adding 2, giving 3 + 2 = 5.

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

Evaluate, completing the power and product before the subtraction.

(-3)^2-2\times4

(−3)² = ____.
2 × 4 = ____.
____ − ____ = ____.

The brackets belong to the base of the power.
Calculate 9 and 8 before subtracting.
The final line is 9 − 8.

The base is −3, so (−3)² = 9.
2 × 4 = 8; then 9 − 8 = 1.

Check your work

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

Prerequisite3

Add the two negative numbers.

-6+(-3)

The two addends have the same sign.
Add their distances from zero.
Keep the negative sign.

Add the absolute values: 6 + 3 = 9.
Keep the common negative sign: −9.

Check your work

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

Prerequisite4

Calculate the quotient and explain its sign.

18\div(-3)

Determine the sign before the magnitude.
A positive divided by a negative is negative.
Calculate 18 ÷ 3.

The operands have different signs, so the quotient is negative.
18 ÷ 3 = 6, hence the quotient is −6.

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

Evaluate and retain the sign of each term.

-12-(-5)

Find the opposite of the subtracted number.
The opposite of −5 is 5.
Calculate −12 + 5.

Change subtraction to addition: −12 + 5.
The negative magnitude is larger, giving −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 product. Explain how many negative factors determine its sign.

(-2)\times(-3)\times(-4)

Count the negative factors, not the written minus characters elsewhere.
An odd number of negative factors gives a negative product.
Multiply the magnitudes 2, 3 and 4.

Three negative factors give a negative product.
2 × 3 × 4 = 24, so the product is −24.

Check your work

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

Independent practice7

Give the exact fraction value.

\frac{3}{4}-\frac{5}{6}

The denominators must match before subtraction.
The least common multiple of 4 and 6 is 12.
Subtract 10 from 9 in the numerator.

Use denominator 12: 3/4 = 9/12 and 5/6 = 10/12.
9/12 − 10/12 = −1/12.

Check your work

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

Independent practice8

Write 450000 in scientific notation. State the allowed range of the coefficient.

450000

Put the decimal point after the first nonzero digit.
Count how many places the decimal point moves.
For these positive large numbers, use a positive whole-number exponent.

Move the decimal point 5 places left to obtain 4.5.
Compensate by multiplying by 10⁵; check the original value.

Check your work

Use a coefficient from 1 inclusive to 10 exclusive.
Give an exponent that restores the original number exactly.

Exit question9

Evaluate. Explain why the two squared terms have different signs before addition.

-3^2+(-3)^2

An exponent acts before a leading minus.
Compare the bases 3 and −3.
Add −9 and 9.

−3² = −(3²) = −9, whereas (−3)² = 9.
The sum is −9 + 9 = 0.

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

Evaluate in the correct order.

(8-14)\div3+2^3

Start with the bracket and the power.
Do division before addition.
The last line is −2 + 8.

First 8 − 14 = −6 and 2³ = 8.
Then −6 ÷ 3 = −2; finally −2 + 8 = 6.

Check your work

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

Further practice11

Evaluate without treating the whole expression as zero.

(-5)\times0+4

Only the product has a zero factor.
Calculate multiplication first.
Do not forget the final +4.

The product (−5) × 0 is 0.
The remaining addition is 0 + 4 = 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

Evaluate the quotient before the subtraction.

(-24)\div(-6)-5

The two division operands have the same sign.
Their quotient is 4.
Then subtract 5.

The quotient of two negatives is positive: 24 ÷ 6 = 4.
4 − 5 = −1.

Check your work

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

Further practice13

Write 7200000 in scientific notation. State the allowed range of the coefficient.

7200000

Put the decimal point after the first nonzero digit.
Count how many places the decimal point moves.
For these positive large numbers, use a positive whole-number exponent.

Move the decimal point 6 places left to obtain 7.2.
Compensate by multiplying by 10⁶; check the original value.

Check your work

Use a coefficient from 1 inclusive to 10 exclusive.
Give an exponent that restores the original number exactly.

Further practice14

Round 3.146 to two decimal places. Identify the digit that decides the rounding.

Two decimal places means the hundredths place.
Use the third decimal digit to decide.
That digit is 6.

Keep the hundredths place and inspect the next digit, 6.
Since 6 is at least 5, 3.14 rounds up to 3.15.

Check your work

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

Extension15

Evaluate exactly and explain the sign of the product being subtracted.

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

Multiply before subtracting.
The product is negative.
Subtracting that product adds 3/2.

The product is (3/4) × (−2) = −3/2.
1/2 − (−3/2) = 1/2 + 3/2 = 2.

Check your work

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

Extension16

A learner writes 8 ÷ 0 = 0. Is this defined? Explain using the multiplication that division would require.

Check whether the divisor is allowed.
Translate division into multiplication.
Can 0 × q ever equal 8?

A ÷ B = q requires B × q = A.
No number multiplied by 0 gives 8.

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

Opposite numbers, absolute value and the four operations on 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.

All reference answers

Worked example1

Evaluate the expression and explain why subtracting a negative becomes addition.

(-4)+7-(-2)

Reference answer

The value is 5.

5

−4 + 7 = 3.
Subtracting −2 means adding 2, giving 3 + 2 = 5.

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

Guided attempt2

Evaluate, completing the power and product before the subtraction.

(-3)^2-2\times4

(−3)² = ____.
2 × 4 = ____.
____ − ____ = ____.

Reference answer

The value is 1.

1

The base is −3, so (−3)² = 9.
2 × 4 = 8; then 9 − 8 = 1.

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

Prerequisite3

Add the two negative numbers.

-6+(-3)

Reference answer

The value is −9.

-9

Add the absolute values: 6 + 3 = 9.
Keep the common negative sign: −9.

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

Prerequisite4

Calculate the quotient and explain its sign.

18\div(-3)

Reference answer

The value is −6.

-6

The operands have different signs, so the quotient is negative.
18 ÷ 3 = 6, hence the quotient is −6.

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

Independent practice5

Evaluate and retain the sign of each term.

-12-(-5)

Reference answer

The value is −7.

-7

Change subtraction to addition: −12 + 5.
The negative magnitude is larger, giving −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 product. Explain how many negative factors determine its sign.

(-2)\times(-3)\times(-4)

Reference answer

The value is −24.

-24

Three negative factors give a negative product.
2 × 3 × 4 = 24, so the product is −24.

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

Independent practice7

Give the exact fraction value.

\frac{3}{4}-\frac{5}{6}

Reference answer

The value is −1/12.

\frac{-1}{12}

Use denominator 12: 3/4 = 9/12 and 5/6 = 10/12.
9/12 − 10/12 = −1/12.

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

Independent practice8

Write 450000 in scientific notation. State the allowed range of the coefficient.

450000

Reference answer

4.5 × 10⁵; the coefficient is at least 1 and less than 10.

Move the decimal point 5 places left to obtain 4.5.
Compensate by multiplying by 10⁵; check the original value.

Use a coefficient from 1 inclusive to 10 exclusive.
Give an exponent that restores the original number exactly.

Exit question9

Evaluate. Explain why the two squared terms have different signs before addition.

-3^2+(-3)^2

Reference answer

The value is 0.

0

−3² = −(3²) = −9, whereas (−3)² = 9.
The sum is −9 + 9 = 0.

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

Exit question10

Evaluate in the correct order.

(8-14)\div3+2^3

Reference answer

The value is 6.

6

First 8 − 14 = −6 and 2³ = 8.
Then −6 ÷ 3 = −2; finally −2 + 8 = 6.

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

Further practice11

Evaluate without treating the whole expression as zero.

(-5)\times0+4

Reference answer

The value is 4.

4

The product (−5) × 0 is 0.
The remaining addition is 0 + 4 = 4.

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

Further practice12

Evaluate the quotient before the subtraction.

(-24)\div(-6)-5

Reference answer

The value is −1.

-1

The quotient of two negatives is positive: 24 ÷ 6 = 4.
4 − 5 = −1.

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

Further practice13

Write 7200000 in scientific notation. State the allowed range of the coefficient.

7200000

Reference answer

7.2 × 10⁶; the coefficient is at least 1 and less than 10.

Move the decimal point 6 places left to obtain 7.2.
Compensate by multiplying by 10⁶; check the original value.

Use a coefficient from 1 inclusive to 10 exclusive.
Give an exponent that restores the original number exactly.

Further practice14

Round 3.146 to two decimal places. Identify the digit that decides the rounding.

Reference answer

3.15; the third decimal digit is 6, so the hundredths digit increases by 1.

Keep the hundredths place and inspect the next digit, 6.
Since 6 is at least 5, 3.14 rounds up to 3.15.

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

Extension15

Evaluate exactly and explain the sign of the product being subtracted.

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

Reference answer

The value is 2.

2

The product is (3/4) × (−2) = −3/2.
1/2 − (−3/2) = 1/2 + 3/2 = 2.

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

Extension16

A learner writes 8 ÷ 0 = 0. Is this defined? Explain using the multiplication that division would require.

Reference answer

It is undefined. A quotient q would require 0 × q = 8, but 0 × q = 0 for every number q.

A ÷ B = q requires B × q = A.
No number multiplied by 0 gives 8.

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. 15–40 · foundation lessons 8–20

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

Rational numbers: operations, powers and large numbers

Year 7, upper volume · text pp. 15–40 · foundation lessons 8–20 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

Evaluate the expression and explain why subtracting a negative becomes addition.

(-4)+7-(-2)

Guided attempt2

Evaluate, completing the power and product before the subtraction.

(-3)^2-2\times4

(−3)² = ____.
2 × 4 = ____.
____ − ____ = ____.

Prerequisite3

Add the two negative numbers.

-6+(-3)

Prerequisite4

Calculate the quotient and explain its sign.

18\div(-3)

Rational numbers: operations, powers and large numbers

Year 7, upper volume · text pp. 15–40 · foundation lessons 8–20 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

Evaluate and retain the sign of each term.

-12-(-5)

Independent practice6

Find the product. Explain how many negative factors determine its sign.

(-2)\times(-3)\times(-4)

Independent practice7

Give the exact fraction value.

\frac{3}{4}-\frac{5}{6}

Independent practice8

Write 450000 in scientific notation. State the allowed range of the coefficient.

450000

Exit question9

Evaluate. Explain why the two squared terms have different signs before addition.

-3^2+(-3)^2

Exit question10

Evaluate in the correct order.

(8-14)\div3+2^3

Rational numbers: operations, powers and large numbers — Further practice

Year 7, upper volume · text pp. 15–40 · foundation lessons 8–20

Further practice11

Evaluate without treating the whole expression as zero.

(-5)\times0+4

Further practice12

Evaluate the quotient before the subtraction.

(-24)\div(-6)-5

Further practice13

Write 7200000 in scientific notation. State the allowed range of the coefficient.

7200000

Further practice14

Round 3.146 to two decimal places. Identify the digit that decides the rounding.

Extension15

Evaluate exactly and explain the sign of the product being subtracted.

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

Extension16

A learner writes 8 ÷ 0 = 0. Is this defined? Explain using the multiplication that division would require.

Rational numbers: operations, powers and large numbers — Teacher answers

Year 7, upper volume · text pp. 15–40 · foundation lessons 8–20

Worked example1

Evaluate the expression and explain why subtracting a negative becomes addition.

(-4)+7-(-2)

Reference answer

The value is 5.

5

−4 + 7 = 3.
Subtracting −2 means adding 2, giving 3 + 2 = 5.

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

Guided attempt2

Evaluate, completing the power and product before the subtraction.

(-3)^2-2\times4

(−3)² = ____.
2 × 4 = ____.
____ − ____ = ____.

Reference answer

The value is 1.

1

The base is −3, so (−3)² = 9.
2 × 4 = 8; then 9 − 8 = 1.

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

Prerequisite3

Add the two negative numbers.

-6+(-3)

Reference answer

The value is −9.

-9

Add the absolute values: 6 + 3 = 9.
Keep the common negative sign: −9.

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

Prerequisite4

Calculate the quotient and explain its sign.

18\div(-3)

Reference answer

The value is −6.

-6

The operands have different signs, so the quotient is negative.
18 ÷ 3 = 6, hence the quotient is −6.

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

Independent practice5

Evaluate and retain the sign of each term.

-12-(-5)

Reference answer

The value is −7.

-7

Change subtraction to addition: −12 + 5.
The negative magnitude is larger, giving −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 product. Explain how many negative factors determine its sign.

(-2)\times(-3)\times(-4)

Reference answer

The value is −24.

-24

Three negative factors give a negative product.
2 × 3 × 4 = 24, so the product is −24.

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

Independent practice7

Give the exact fraction value.

\frac{3}{4}-\frac{5}{6}

Reference answer

The value is −1/12.

\frac{-1}{12}

Use denominator 12: 3/4 = 9/12 and 5/6 = 10/12.
9/12 − 10/12 = −1/12.

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

Independent practice8

Write 450000 in scientific notation. State the allowed range of the coefficient.

450000

Reference answer

4.5 × 10⁵; the coefficient is at least 1 and less than 10.

Move the decimal point 5 places left to obtain 4.5.
Compensate by multiplying by 10⁵; check the original value.

Use a coefficient from 1 inclusive to 10 exclusive.
Give an exponent that restores the original number exactly.

Exit question9

Evaluate. Explain why the two squared terms have different signs before addition.

-3^2+(-3)^2

Reference answer

The value is 0.

0

−3² = −(3²) = −9, whereas (−3)² = 9.
The sum is −9 + 9 = 0.

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

Exit question10

Evaluate in the correct order.

(8-14)\div3+2^3

Reference answer

The value is 6.

6

First 8 − 14 = −6 and 2³ = 8.
Then −6 ÷ 3 = −2; finally −2 + 8 = 6.

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

Further practice11

Evaluate without treating the whole expression as zero.

(-5)\times0+4

Reference answer

The value is 4.

4

The product (−5) × 0 is 0.
The remaining addition is 0 + 4 = 4.

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

Further practice12

Evaluate the quotient before the subtraction.

(-24)\div(-6)-5

Reference answer

The value is −1.

-1

The quotient of two negatives is positive: 24 ÷ 6 = 4.
4 − 5 = −1.

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

Further practice13

Write 7200000 in scientific notation. State the allowed range of the coefficient.

7200000

Reference answer

7.2 × 10⁶; the coefficient is at least 1 and less than 10.

Move the decimal point 6 places left to obtain 7.2.
Compensate by multiplying by 10⁶; check the original value.

Use a coefficient from 1 inclusive to 10 exclusive.
Give an exponent that restores the original number exactly.

Further practice14

Round 3.146 to two decimal places. Identify the digit that decides the rounding.

Reference answer

3.15; the third decimal digit is 6, so the hundredths digit increases by 1.

Keep the hundredths place and inspect the next digit, 6.
Since 6 is at least 5, 3.14 rounds up to 3.15.

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

Extension15

Evaluate exactly and explain the sign of the product being subtracted.

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

Reference answer

The value is 2.

2

The product is (3/4) × (−2) = −3/2.
1/2 − (−3/2) = 1/2 + 3/2 = 2.

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

Extension16

A learner writes 8 ÷ 0 = 0. Is this defined? Explain using the multiplication that division would require.

Reference answer

It is undefined. A quotient q would require 0 × q = 8, but 0 × q = 0 for every number q.

A ÷ B = q requires B × q = A.
No number multiplied by 0 gives 8.

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