PAPER · REASONING · PRACTICE

Polynomials: collect terms and remove brackets — Teacher reference

Keep signed coefficients, distinguish unlike powers, apply the distributive law and check by substitution.

40 minutes · Year 7, upper volume · text pp. 45–54 · foundation lessons 23–27

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 4x − (x + 6) with 4x − x + 6. Test x = 2 and explain which constant sign a minus bracket requires.

Organise the key idea

Like terms have exactly the same variable powers. A minus before brackets multiplies every enclosed term by −1; a numerical multiplier distributes to every term.

Open the original chapter Open the interactive inquiry

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

Worked example1

Simplify by collecting the x terms and the constant terms separately.

3x+2-x+5

Do not combine constants with x terms.
The coefficient of −x is −1.
Combine 3 − 1 and 2 + 5 separately.

3x − x = 2x; 2 + 5 = 7.
The simplified expression is 2x + 7.

Check your work

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

Guided attempt2

Remove the brackets and simplify. Show how each enclosed sign changes.

5a-(2a-3)

−(2a − 3) = ____ a + ____.
5a − 2a + 3 = ____ a + ____.

The outside minus acts on both enclosed terms.
−(−3) becomes +3.
Now combine 5a and −2a.

Distribute the minus: 5a − 2a + 3.
Collect: 3a + 3.

Check your work

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

Prerequisite3

Combine the like terms.

2x+5x

Check the variable and exponent.
Add the numerical coefficients.
Keep x unchanged.

Both terms have x to the first power.
Add coefficients: 2 + 5 = 7, giving 7x.

Check your work

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

Prerequisite4

Can y + y² be combined into one monomial? Write an equivalent expression and explain.

y+y^2

Having the same letter alone is not enough.
Compare the two exponents.
Keep both terms.

The exponents are 1 and 2.
They are unlike terms, so reordering does not combine them.

Check your work

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

Independent practice5

Simplify.

-4x+7x-2

Attach the negative sign to 4x.
Combine only the x terms.
The constant stays separate.

−4 + 7 = 3 for the x coefficient.
Keep the constant −2: 3x − 2.

Check your work

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

Independent practice6

Simplify, keeping the squared and linear terms in separate groups.

2a^2+3a-5a^2+a

Do not combine a² with a.
For a² use 2 − 5.
For a use 3 + 1.

2a² − 5a² = −3a².
3a + a = 4a, giving −3a² + 4a.

Check your work

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

Independent practice7

Remove brackets and simplify.

4x-(x+6)

Both enclosed signs change.
The constant becomes −6.
Combine 4x − x.

Subtract both enclosed terms: 4x − x − 6.
Collect to obtain 3x − 6.

Check your work

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

Independent practice8

Apply the distributive law, then simplify.

2(3y-1)-4y

Multiply 2 by each enclosed term.
The constant product is −2.
Then combine 6y and −4y.

2(3y − 1) = 6y − 2.
6y − 2 − 4y = 2y − 2.

Check your work

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

Exit question9

Simplify and explain the sign of the final constant.

-(2x-5)+3x

Treat the outside minus as multiplication by −1.
−1 multiplied by −5 is +5.
Combine −2x + 3x.

−(2x − 5) = −2x + 5.
Add 3x: x + 5.

Check your work

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

Exit question10

Simplify first, then evaluate at a = −2.

5a-2(2a-3)

The factor −2 acts on both terms in the brackets.
The constant contribution is +6.
Substitute into a + 6.

Distribute: 5a − 4a + 6 = a + 6.
At a = −2, the value is −2 + 6 = 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.

Further practice11

Simplify.

7x-3x+2-9

Make one group for x and one for constants.
Use signed coefficients.
Check the constant is negative.

7x − 3x = 4x.
2 − 9 = −7, so the result is 4x − 7.

Check your work

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

Further practice12

Simplify.

a-(2a-b)+3b

−b inside the negative bracket becomes +b.
Group a terms and b terms separately.
1 − 2 = −1; 1 + 3 = 4.

Remove brackets: a − 2a + b + 3b.
Collect separately: −a + 4b.

Check your work

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

Further practice13

Expand each product, then simplify.

3(2x+1)-2(x-4)

Distribute both 3 and −2.
−2 × (−4) is +8.
Add the constants 3 and 8.

The products give 6x + 3 − 2x + 8.
Collect to get 4x + 11.

Check your work

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

Further practice14

Simplify and explain why xy and yx can be grouped.

2xy-3yx+5xy

The order of factors does not change the product.
Every term has x¹y¹.
Add the three signed coefficients.

Multiplication is commutative, so yx = xy.
The coefficient is 2 − 3 + 5 = 4.

Check your work

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

Extension15

A = 2x − 1 and B = x + 3. Find A − 2B. Keep the whole expression B in brackets first.

(2x-1)-2(x+3)

2B means twice every term of B.
The outer subtraction makes both contributions negative.
The x terms cancel.

Substitute the whole expressions: (2x − 1) − 2(x + 3).
Expand: 2x − 1 − 2x − 6 = −7.

Check your work

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

Extension16

A learner writes 3(x − 2) = 3x − 2. Correct the expansion, then use x = 1 to show why the learner’s version fails.

3(x-2)

Inspect the constant multiplication.
3 × (−2) is −6.
Compare −3 and 1 at the supplied test value.

3 must multiply both x and −2, giving 3x − 6.
At x = 1, 3(1 − 2) = −3 while 3 × 1 − 2 = 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.

Preparation and assessment notes

Prerequisites

Terms, coefficients, exponents and signed arithmetic.

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

Simplify by collecting the x terms and the constant terms separately.

3x+2-x+5

Reference answer

2x+7.

2x+7

3x − x = 2x; 2 + 5 = 7.
The simplified expression is 2x + 7.

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

Guided attempt2

Remove the brackets and simplify. Show how each enclosed sign changes.

5a-(2a-3)

−(2a − 3) = ____ a + ____.
5a − 2a + 3 = ____ a + ____.

Reference answer

3a+3.

3a+3

Distribute the minus: 5a − 2a + 3.
Collect: 3a + 3.

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

Prerequisite3

Combine the like terms.

2x+5x

Reference answer

7x.

7x

Both terms have x to the first power.
Add coefficients: 2 + 5 = 7, giving 7x.

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

Prerequisite4

Can y + y² be combined into one monomial? Write an equivalent expression and explain.

y+y^2

Reference answer

No. Keep y² + y; their exponents differ.

y^2+y

The exponents are 1 and 2.
They are unlike terms, so reordering does not combine them.

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

Independent practice5

Simplify.

-4x+7x-2

Reference answer

3x−2.

3x-2

−4 + 7 = 3 for the x coefficient.
Keep the constant −2: 3x − 2.

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

Independent practice6

Simplify, keeping the squared and linear terms in separate groups.

2a^2+3a-5a^2+a

Reference answer

−3a²+4a.

-3a^2+4a

2a² − 5a² = −3a².
3a + a = 4a, giving −3a² + 4a.

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

Independent practice7

Remove brackets and simplify.

4x-(x+6)

Reference answer

3x−6.

3x-6

Subtract both enclosed terms: 4x − x − 6.
Collect to obtain 3x − 6.

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

Independent practice8

Apply the distributive law, then simplify.

2(3y-1)-4y

Reference answer

2y−2.

2y-2

2(3y − 1) = 6y − 2.
6y − 2 − 4y = 2y − 2.

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

Exit question9

Simplify and explain the sign of the final constant.

-(2x-5)+3x

Reference answer

x+5.

x+5

−(2x − 5) = −2x + 5.
Add 3x: x + 5.

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

Exit question10

Simplify first, then evaluate at a = −2.

5a-2(2a-3)

Reference answer

The value is 4.

4

Distribute: 5a − 4a + 6 = a + 6.
At a = −2, the value is −2 + 6 = 4.

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

Further practice11

Simplify.

7x-3x+2-9

Reference answer

4x−7.

4x-7

7x − 3x = 4x.
2 − 9 = −7, so the result is 4x − 7.

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

Further practice12

Simplify.

a-(2a-b)+3b

Reference answer

−a+4b.

-a+4b

Remove brackets: a − 2a + b + 3b.
Collect separately: −a + 4b.

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

Further practice13

Expand each product, then simplify.

3(2x+1)-2(x-4)

Reference answer

4x+11.

4x+11

The products give 6x + 3 − 2x + 8.
Collect to get 4x + 11.

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

Further practice14

Simplify and explain why xy and yx can be grouped.

2xy-3yx+5xy

Reference answer

4xy.

4xy

Multiplication is commutative, so yx = xy.
The coefficient is 2 − 3 + 5 = 4.

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

Extension15

A = 2x − 1 and B = x + 3. Find A − 2B. Keep the whole expression B in brackets first.

(2x-1)-2(x+3)

Reference answer

−7.

-7

Substitute the whole expressions: (2x − 1) − 2(x + 3).
Expand: 2x − 1 − 2x − 6 = −7.

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

Extension16

A learner writes 3(x − 2) = 3x − 2. Correct the expansion, then use x = 1 to show why the learner’s version fails.

3(x-2)

Reference answer

Correct: 3x − 6. At x = 1, the original is −3 but 3x − 2 is 1.

-3

3 must multiply both x and −2, giving 3x − 6.
At x = 1, 3(1 − 2) = −3 while 3 × 1 − 2 = 1.

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

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. 45–54 · foundation lessons 23–27

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

Polynomials: collect terms and remove brackets

Year 7, upper volume · text pp. 45–54 · foundation lessons 23–27 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

Simplify by collecting the x terms and the constant terms separately.

3x+2-x+5

Guided attempt2

Remove the brackets and simplify. Show how each enclosed sign changes.

5a-(2a-3)

−(2a − 3) = ____ a + ____.
5a − 2a + 3 = ____ a + ____.

Prerequisite3

Combine the like terms.

2x+5x

Prerequisite4

Can y + y² be combined into one monomial? Write an equivalent expression and explain.

y+y^2

Polynomials: collect terms and remove brackets

Year 7, upper volume · text pp. 45–54 · foundation lessons 23–27 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

Simplify.

-4x+7x-2

Independent practice6

Simplify, keeping the squared and linear terms in separate groups.

2a^2+3a-5a^2+a

Independent practice7

Remove brackets and simplify.

4x-(x+6)

Independent practice8

Apply the distributive law, then simplify.

2(3y-1)-4y

Exit question9

Simplify and explain the sign of the final constant.

-(2x-5)+3x

Exit question10

Simplify first, then evaluate at a = −2.

5a-2(2a-3)

Polynomials: collect terms and remove brackets — Further practice

Year 7, upper volume · text pp. 45–54 · foundation lessons 23–27

Further practice11

Simplify.

7x-3x+2-9

Further practice12

Simplify.

a-(2a-b)+3b

Further practice13

Expand each product, then simplify.

3(2x+1)-2(x-4)

Further practice14

Simplify and explain why xy and yx can be grouped.

2xy-3yx+5xy

Extension15

A = 2x − 1 and B = x + 3. Find A − 2B. Keep the whole expression B in brackets first.

(2x-1)-2(x+3)

Extension16

A learner writes 3(x − 2) = 3x − 2. Correct the expansion, then use x = 1 to show why the learner’s version fails.

3(x-2)

Polynomials: collect terms and remove brackets — Teacher answers

Year 7, upper volume · text pp. 45–54 · foundation lessons 23–27

Worked example1

Simplify by collecting the x terms and the constant terms separately.

3x+2-x+5

Reference answer

2x+7.

2x+7

3x − x = 2x; 2 + 5 = 7.
The simplified expression is 2x + 7.

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

Guided attempt2

Remove the brackets and simplify. Show how each enclosed sign changes.

5a-(2a-3)

−(2a − 3) = ____ a + ____.
5a − 2a + 3 = ____ a + ____.

Reference answer

3a+3.

3a+3

Distribute the minus: 5a − 2a + 3.
Collect: 3a + 3.

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

Prerequisite3

Combine the like terms.

2x+5x

Reference answer

7x.

7x

Both terms have x to the first power.
Add coefficients: 2 + 5 = 7, giving 7x.

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

Prerequisite4

Can y + y² be combined into one monomial? Write an equivalent expression and explain.

y+y^2

Reference answer

No. Keep y² + y; their exponents differ.

y^2+y

The exponents are 1 and 2.
They are unlike terms, so reordering does not combine them.

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

Independent practice5

Simplify.

-4x+7x-2

Reference answer

3x−2.

3x-2

−4 + 7 = 3 for the x coefficient.
Keep the constant −2: 3x − 2.

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

Independent practice6

Simplify, keeping the squared and linear terms in separate groups.

2a^2+3a-5a^2+a

Reference answer

−3a²+4a.

-3a^2+4a

2a² − 5a² = −3a².
3a + a = 4a, giving −3a² + 4a.

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

Independent practice7

Remove brackets and simplify.

4x-(x+6)

Reference answer

3x−6.

3x-6

Subtract both enclosed terms: 4x − x − 6.
Collect to obtain 3x − 6.

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

Independent practice8

Apply the distributive law, then simplify.

2(3y-1)-4y

Reference answer

2y−2.

2y-2

2(3y − 1) = 6y − 2.
6y − 2 − 4y = 2y − 2.

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

Exit question9

Simplify and explain the sign of the final constant.

-(2x-5)+3x

Reference answer

x+5.

x+5

−(2x − 5) = −2x + 5.
Add 3x: x + 5.

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

Exit question10

Simplify first, then evaluate at a = −2.

5a-2(2a-3)

Reference answer

The value is 4.

4

Distribute: 5a − 4a + 6 = a + 6.
At a = −2, the value is −2 + 6 = 4.

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

Further practice11

Simplify.

7x-3x+2-9

Reference answer

4x−7.

4x-7

7x − 3x = 4x.
2 − 9 = −7, so the result is 4x − 7.

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

Further practice12

Simplify.

a-(2a-b)+3b

Reference answer

−a+4b.

-a+4b

Remove brackets: a − 2a + b + 3b.
Collect separately: −a + 4b.

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

Further practice13

Expand each product, then simplify.

3(2x+1)-2(x-4)

Reference answer

4x+11.

4x+11

The products give 6x + 3 − 2x + 8.
Collect to get 4x + 11.

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

Further practice14

Simplify and explain why xy and yx can be grouped.

2xy-3yx+5xy

Reference answer

4xy.

4xy

Multiplication is commutative, so yx = xy.
The coefficient is 2 − 3 + 5 = 4.

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

Extension15

A = 2x − 1 and B = x + 3. Find A − 2B. Keep the whole expression B in brackets first.

(2x-1)-2(x+3)

Reference answer

−7.

-7

Substitute the whole expressions: (2x − 1) − 2(x + 3).
Expand: 2x − 1 − 2x − 6 = −7.

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

Extension16

A learner writes 3(x − 2) = 3x − 2. Correct the expansion, then use x = 1 to show why the learner’s version fails.

3(x-2)

Reference answer

Correct: 3x − 6. At x = 1, the original is −3 but 3x − 2 is 1.

-3

3 must multiply both x and −2, giving 3x − 6.
At x = 1, 3(1 − 2) = −3 while 3 × 1 − 2 = 1.

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