PAPER · REASONING · PRACTICE

Expressions: terms, coefficients and values — Teacher reference

Read the structure of a monomial or polynomial, distinguish coefficient from degree, and substitute signed values.

40 minutes · Year 7, upper volume · text pp. 41–44 · foundation lessons 21–22

Textbook reference: Year 7, upper volume · Chapter 2 · Adding and subtracting polynomials

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 −3x²y and −3xy². Their coefficients and total degrees agree, but their variable powers differ. Test both at x = 2, y = 1.

Organise the key idea

A coefficient is a signed numerical factor. A monomial’s degree is the sum of its variable exponents; substitution replaces every occurrence of each variable.

Open the original chapter Open the interactive inquiry

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

Worked example1

For the monomial below, find its degree. Explain which part of the term is counted.

-3x^2y

Separate the numerical factor from the variable factors.
An unwritten variable exponent is 1.
Add all variable exponents.

Add the exponents of the variables; do not include the numerical coefficient.
The exponent sum is 2 + 1 = 3.

Check your work

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

Guided attempt2

Find the value of 2x + 3 when x = −2. Keep the negative value in brackets during substitution.

2x+3

x = ____, so 2x + 3 = 2 × (____) + 3.
The product is ____.
The final value is ____.

x is a value, not a sign attached to the 2.
The product is 2 × (−2).
Then add 3.

Replace x by −2: 2 × (−2) + 3.
Calculate −4 + 3 = −1.

Check your work

Keep each sign and collect only terms with the same variable powers.
Substitute the stated values into the complete expression and calculate.

Prerequisite3

For the monomial below, find its coefficient. Explain which part of the term is counted.

-x

Separate the numerical factor from the variable factors.
For −x, the coefficient is −1, not 1.
Keep a fractional or negative numerical factor intact.

The coefficient is the complete numerical factor, including its sign.
The numerical factor is −1.

Check your work

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

Prerequisite4

Evaluate a² when a = 3.

a^2

The exponent tells how many equal factors appear.
Square 3; do not multiply it by 2.
Calculate 3 × 3.

Substitute 3 for a.
3² = 3 × 3 = 9.

Check your work

Keep each sign and collect only terms with the same variable powers.
Substitute the stated values into the complete expression and calculate.

Independent practice5

For the monomial below, find its coefficient. Explain which part of the term is counted.

-5ab^2

Separate the numerical factor from the variable factors.
For −x, the coefficient is −1, not 1.
Keep a fractional or negative numerical factor intact.

The coefficient is the complete numerical factor, including its sign.
The numerical factor is −5.

Check your work

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

Independent practice6

For the monomial below, find its degree. Explain which part of the term is counted.

\frac{1}{2}x^2y^3

Separate the numerical factor from the variable factors.
An unwritten variable exponent is 1.
Add all variable exponents.

Add the exponents of the variables; do not include the numerical coefficient.
The exponent sum is 2 + 3 = 5.

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

Find 3a − 2b when a = −1 and b = 2.

3a-2b

Replace both variables, not just one.
Keep −1 inside brackets in the product.
Subtract 4 from −3.

Substitute: 3 × (−1) − 2 × 2.
−3 − 4 = −7.

Check your work

Keep each sign and collect only terms with the same variable powers.
Substitute the stated values into the complete expression and calculate.

Independent practice8

Each notebook costs 4 dollars and a single packing fee is 3 dollars. Write the total cost for n notebooks. Do not substitute a value for n.

Separate the repeated unit cost from the fixed fee.
Multiply 4 by n.
Add 3 once, not 3n.

The notebook cost is 4n.
Add the single packing fee once: 4n + 3.

Check your work

Keep each sign and collect only terms with the same variable powers.
Give an equivalent simplified expression.

Exit question9

For the monomial below, find its degree. Explain which part of the term is counted.

-2x^3y

Separate the numerical factor from the variable factors.
An unwritten variable exponent is 1.
Add all variable exponents.

Add the exponents of the variables; do not include the numerical coefficient.
The exponent sum is 3 + 1 = 4.

Check your work

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

Exit question10

Find −x² when x = −3. Explain why the answer is not 9.

-x^2

The outside minus is not part of the squared base x.
First square the substituted value −3.
Then apply the outside minus.

The expression is the negative of x².
At x = −3, x² = 9, so −x² = −9.

Check your work

Keep each sign and collect only terms with the same variable powers.
Substitute the stated values into the complete expression and calculate.

Further practice11

For the monomial below, find its coefficient. Explain which part of the term is counted.

6mn^2

Separate the numerical factor from the variable factors.
For −x, the coefficient is −1, not 1.
Keep a fractional or negative numerical factor intact.

The coefficient is the complete numerical factor, including its sign.
The numerical factor 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.

Further practice12

Find a/2 + b when a = −4 and b = 3.

\frac{a}{2}+b

The divisor is the constant 2.
Calculate −4 ÷ 2 first.
Then add 3.

Substitute: (−4)/2 + 3.
−2 + 3 = 1.

Check your work

Keep each sign and collect only terms with the same variable powers.
Substitute the stated values into the complete expression and calculate.

Further practice13

Find 3x² − 2x + 1 when x = 2.

3x^2-2x+1

Square x before multiplying by 3.
The two occurrences of x both mean 2.
Calculate 12 − 4 + 1.

x² = 4, so the expression becomes 3 × 4 − 2 × 2 + 1.
12 − 4 + 1 = 9.

Check your work

Keep each sign and collect only terms with the same variable powers.
Substitute the stated values into the complete expression and calculate.

Further practice14

For the monomial below, find its degree. Explain which part of the term is counted.

7

Separate the numerical factor from the variable factors.
This term contains no variable factors.
A nonzero constant has degree 0.

Add the exponents of the variables; do not include the numerical coefficient.
There are no variable factors. The nonzero constant 7 has degree 0.

Check your work

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

Extension15

Find a² − 2ab + b² when a = 3 and b = 1. Replace every occurrence of each letter.

a^2-2ab+b^2

The product ab uses both given values.
Calculate the two squares and the middle product separately.
Combine 9, −6 and 1.

Substitute: 3² − 2 × 3 × 1 + 1².
9 − 6 + 1 = 4.

Check your work

Keep each sign and collect only terms with the same variable powers.
Substitute the stated values into the complete expression and calculate.

Extension16

A rectangle has width x cm and length (x + 3) cm. Write and simplify its perimeter.

A rectangle has two equal lengths and two equal widths.
Add 2(x + 3) and 2x.
Keep the extra 3 cm on each of the two long sides.

Two lengths contribute 2(x + 3); two widths contribute 2x.
The perimeter is 2x + 6 + 2x = 4x + 6 cm.

Check your work

Keep each sign and collect only terms with the same variable powers.
Give an equivalent simplified expression.

Preparation and assessment notes

Prerequisites

Signed arithmetic, powers and using a letter for a quantity.

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

For the monomial below, find its degree. Explain which part of the term is counted.

-3x^2y

Reference answer

The degree is 3.

Add the exponents of the variables; do not include the numerical coefficient.
The exponent sum is 2 + 1 = 3.

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

Guided attempt2

Find the value of 2x + 3 when x = −2. Keep the negative value in brackets during substitution.

2x+3

x = ____, so 2x + 3 = 2 × (____) + 3.
The product is ____.
The final value is ____.

Reference answer

The value is −1.

-1

Replace x by −2: 2 × (−2) + 3.
Calculate −4 + 3 = −1.

Keep each sign and collect only terms with the same variable powers.
Substitute the stated values into the complete expression and calculate.

Prerequisite3

For the monomial below, find its coefficient. Explain which part of the term is counted.

-x

Reference answer

The coefficient is −1.

The coefficient is the complete numerical factor, including its sign.
The numerical factor is −1.

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

Prerequisite4

Evaluate a² when a = 3.

a^2

Reference answer

The value is 9.

9

Substitute 3 for a.
3² = 3 × 3 = 9.

Keep each sign and collect only terms with the same variable powers.
Substitute the stated values into the complete expression and calculate.

Independent practice5

For the monomial below, find its coefficient. Explain which part of the term is counted.

-5ab^2

Reference answer

The coefficient is −5.

The coefficient is the complete numerical factor, including its sign.
The numerical factor is −5.

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

Independent practice6

For the monomial below, find its degree. Explain which part of the term is counted.

\frac{1}{2}x^2y^3

Reference answer

The degree is 5.

Add the exponents of the variables; do not include the numerical coefficient.
The exponent sum is 2 + 3 = 5.

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

Independent practice7

Find 3a − 2b when a = −1 and b = 2.

3a-2b

Reference answer

The value is −7.

-7

Substitute: 3 × (−1) − 2 × 2.
−3 − 4 = −7.

Keep each sign and collect only terms with the same variable powers.
Substitute the stated values into the complete expression and calculate.

Independent practice8

Each notebook costs 4 dollars and a single packing fee is 3 dollars. Write the total cost for n notebooks. Do not substitute a value for n.

Reference answer

4n+3.

4n+3

The notebook cost is 4n.
Add the single packing fee once: 4n + 3.

Keep each sign and collect only terms with the same variable powers.
Give an equivalent simplified expression.

Exit question9

For the monomial below, find its degree. Explain which part of the term is counted.

-2x^3y

Reference answer

The degree is 4.

Add the exponents of the variables; do not include the numerical coefficient.
The exponent sum is 3 + 1 = 4.

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

Exit question10

Find −x² when x = −3. Explain why the answer is not 9.

-x^2

Reference answer

The value is −9.

-9

The expression is the negative of x².
At x = −3, x² = 9, so −x² = −9.

Keep each sign and collect only terms with the same variable powers.
Substitute the stated values into the complete expression and calculate.

Further practice11

For the monomial below, find its coefficient. Explain which part of the term is counted.

6mn^2

Reference answer

The coefficient is 6.

The coefficient is the complete numerical factor, including its sign.
The numerical factor is 6.

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

Further practice12

Find a/2 + b when a = −4 and b = 3.

\frac{a}{2}+b

Reference answer

The value is 1.

1

Substitute: (−4)/2 + 3.
−2 + 3 = 1.

Keep each sign and collect only terms with the same variable powers.
Substitute the stated values into the complete expression and calculate.

Further practice13

Find 3x² − 2x + 1 when x = 2.

3x^2-2x+1

Reference answer

The value is 9.

9

x² = 4, so the expression becomes 3 × 4 − 2 × 2 + 1.
12 − 4 + 1 = 9.

Keep each sign and collect only terms with the same variable powers.
Substitute the stated values into the complete expression and calculate.

Further practice14

For the monomial below, find its degree. Explain which part of the term is counted.

7

Reference answer

The degree is 0.

Add the exponents of the variables; do not include the numerical coefficient.
There are no variable factors. The nonzero constant 7 has degree 0.

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

Extension15

Find a² − 2ab + b² when a = 3 and b = 1. Replace every occurrence of each letter.

a^2-2ab+b^2

Reference answer

The value is 4.

4

Substitute: 3² − 2 × 3 × 1 + 1².
9 − 6 + 1 = 4.

Keep each sign and collect only terms with the same variable powers.
Substitute the stated values into the complete expression and calculate.

Extension16

A rectangle has width x cm and length (x + 3) cm. Write and simplify its perimeter.

Reference answer

(4x + 6) cm, for x > 0.

4x+6

Two lengths contribute 2(x + 3); two widths contribute 2x.
The perimeter is 2x + 6 + 2x = 4x + 6 cm.

Keep each sign and collect only terms with the same variable powers.
Give an equivalent simplified expression.

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. 41–44 · foundation lessons 21–22

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

Expressions: terms, coefficients and values

Year 7, upper volume · text pp. 41–44 · foundation lessons 21–22 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

For the monomial below, find its degree. Explain which part of the term is counted.

-3x^2y

Guided attempt2

Find the value of 2x + 3 when x = −2. Keep the negative value in brackets during substitution.

2x+3

x = ____, so 2x + 3 = 2 × (____) + 3.
The product is ____.
The final value is ____.

Prerequisite3

For the monomial below, find its coefficient. Explain which part of the term is counted.

-x

Prerequisite4

Evaluate a² when a = 3.

a^2

Expressions: terms, coefficients and values

Year 7, upper volume · text pp. 41–44 · foundation lessons 21–22 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

For the monomial below, find its coefficient. Explain which part of the term is counted.

-5ab^2

Independent practice6

For the monomial below, find its degree. Explain which part of the term is counted.

\frac{1}{2}x^2y^3

Independent practice7

Find 3a − 2b when a = −1 and b = 2.

3a-2b

Independent practice8

Each notebook costs 4 dollars and a single packing fee is 3 dollars. Write the total cost for n notebooks. Do not substitute a value for n.

Exit question9

For the monomial below, find its degree. Explain which part of the term is counted.

-2x^3y

Exit question10

Find −x² when x = −3. Explain why the answer is not 9.

-x^2

Expressions: terms, coefficients and values — Further practice

Year 7, upper volume · text pp. 41–44 · foundation lessons 21–22

Further practice11

For the monomial below, find its coefficient. Explain which part of the term is counted.

6mn^2

Further practice12

Find a/2 + b when a = −4 and b = 3.

\frac{a}{2}+b

Further practice13

Find 3x² − 2x + 1 when x = 2.

3x^2-2x+1

Further practice14

For the monomial below, find its degree. Explain which part of the term is counted.

7

Extension15

Find a² − 2ab + b² when a = 3 and b = 1. Replace every occurrence of each letter.

a^2-2ab+b^2

Extension16

A rectangle has width x cm and length (x + 3) cm. Write and simplify its perimeter.

Expressions: terms, coefficients and values — Teacher answers

Year 7, upper volume · text pp. 41–44 · foundation lessons 21–22

Worked example1

For the monomial below, find its degree. Explain which part of the term is counted.

-3x^2y

Reference answer

The degree is 3.

Add the exponents of the variables; do not include the numerical coefficient.
The exponent sum is 2 + 1 = 3.

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

Guided attempt2

Find the value of 2x + 3 when x = −2. Keep the negative value in brackets during substitution.

2x+3

x = ____, so 2x + 3 = 2 × (____) + 3.
The product is ____.
The final value is ____.

Reference answer

The value is −1.

-1

Replace x by −2: 2 × (−2) + 3.
Calculate −4 + 3 = −1.

Keep each sign and collect only terms with the same variable powers.
Substitute the stated values into the complete expression and calculate.

Prerequisite3

For the monomial below, find its coefficient. Explain which part of the term is counted.

-x

Reference answer

The coefficient is −1.

The coefficient is the complete numerical factor, including its sign.
The numerical factor is −1.

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

Prerequisite4

Evaluate a² when a = 3.

a^2

Reference answer

The value is 9.

9

Substitute 3 for a.
3² = 3 × 3 = 9.

Keep each sign and collect only terms with the same variable powers.
Substitute the stated values into the complete expression and calculate.

Independent practice5

For the monomial below, find its coefficient. Explain which part of the term is counted.

-5ab^2

Reference answer

The coefficient is −5.

The coefficient is the complete numerical factor, including its sign.
The numerical factor is −5.

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

Independent practice6

For the monomial below, find its degree. Explain which part of the term is counted.

\frac{1}{2}x^2y^3

Reference answer

The degree is 5.

Add the exponents of the variables; do not include the numerical coefficient.
The exponent sum is 2 + 3 = 5.

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

Independent practice7

Find 3a − 2b when a = −1 and b = 2.

3a-2b

Reference answer

The value is −7.

-7

Substitute: 3 × (−1) − 2 × 2.
−3 − 4 = −7.

Keep each sign and collect only terms with the same variable powers.
Substitute the stated values into the complete expression and calculate.

Independent practice8

Each notebook costs 4 dollars and a single packing fee is 3 dollars. Write the total cost for n notebooks. Do not substitute a value for n.

Reference answer

4n+3.

4n+3

The notebook cost is 4n.
Add the single packing fee once: 4n + 3.

Keep each sign and collect only terms with the same variable powers.
Give an equivalent simplified expression.

Exit question9

For the monomial below, find its degree. Explain which part of the term is counted.

-2x^3y

Reference answer

The degree is 4.

Add the exponents of the variables; do not include the numerical coefficient.
The exponent sum is 3 + 1 = 4.

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

Exit question10

Find −x² when x = −3. Explain why the answer is not 9.

-x^2

Reference answer

The value is −9.

-9

The expression is the negative of x².
At x = −3, x² = 9, so −x² = −9.

Keep each sign and collect only terms with the same variable powers.
Substitute the stated values into the complete expression and calculate.

Further practice11

For the monomial below, find its coefficient. Explain which part of the term is counted.

6mn^2

Reference answer

The coefficient is 6.

The coefficient is the complete numerical factor, including its sign.
The numerical factor is 6.

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

Further practice12

Find a/2 + b when a = −4 and b = 3.

\frac{a}{2}+b

Reference answer

The value is 1.

1

Substitute: (−4)/2 + 3.
−2 + 3 = 1.

Keep each sign and collect only terms with the same variable powers.
Substitute the stated values into the complete expression and calculate.

Further practice13

Find 3x² − 2x + 1 when x = 2.

3x^2-2x+1

Reference answer

The value is 9.

9

x² = 4, so the expression becomes 3 × 4 − 2 × 2 + 1.
12 − 4 + 1 = 9.

Keep each sign and collect only terms with the same variable powers.
Substitute the stated values into the complete expression and calculate.

Further practice14

For the monomial below, find its degree. Explain which part of the term is counted.

7

Reference answer

The degree is 0.

Add the exponents of the variables; do not include the numerical coefficient.
There are no variable factors. The nonzero constant 7 has degree 0.

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

Extension15

Find a² − 2ab + b² when a = 3 and b = 1. Replace every occurrence of each letter.

a^2-2ab+b^2

Reference answer

The value is 4.

4

Substitute: 3² − 2 × 3 × 1 + 1².
9 − 6 + 1 = 4.

Keep each sign and collect only terms with the same variable powers.
Substitute the stated values into the complete expression and calculate.

Extension16

A rectangle has width x cm and length (x + 3) cm. Write and simplify its perimeter.

Reference answer

(4x + 6) cm, for x > 0.

4x+6

Two lengths contribute 2(x + 3); two widths contribute 2x.
The perimeter is 2x + 6 + 2x = 4x + 6 cm.

Keep each sign and collect only terms with the same variable powers.
Give an equivalent simplified expression.