Example 1
Expand 3(2x + 5).
- 3 × 2x = 6x.
- 3 × 5 = 15.
- 6x + 15.
Answer: 6x + 15
⚖️ Grade 7 · equations · session 2 of 12 · 60 minutes · free
Expanding and factoring with the distributive property
The distributive property says a(b + c) = ab + ac: multiply everything inside the bracket. Expanding removes brackets; factoring puts them back by taking out a common factor.
A complete lesson · 60 minutes
Teach it, see it, practise it, all on this page.
Open this page with your learner and work down it, or press Teach this lesson to go through it one step at a time on a big screen. Every section has something to do.
- Warm-up4 min
- Expanding9 min
- Negatives and two brackets9 min
- Factoring9 min
- Activity 17 min
- Activity 27 min
- Activity 35 min
- Quiz6 min
- Set the project4 min
By the end you can
- Expand a(bx + c), including negative a
- Expand and simplify expressions with two brackets
- Factor out the greatest common factor
- NJSLS-M 7.EE.A.1 (Common Core 7.EE.1)
Before you start
- Like terms: combining and simplifying expressions
Have ready
- Algebra tiles (on screen)
- Squared paper for area models
Grade 7 · Unit 2 · NJSLS-M and Common Core 7.EE
Warm-up · think first
A rectangle is 4 wide and (x + 3) long. What is its area?
Show the answer
Split it into a 4 by x part and a 4 by 3 part: 4x + 12. That is 4(x + 3) = 4x + 12.
Teacher tip: Draw the rectangle split into two parts.
Part 1 of 3 · learn, see, practise
Expanding: multiply everything inside
a(b + c) = ab + ac: the number outside multiplies every term inside.
An area model helps: a rectangle a by (b + c) splits into a by b and a by c.
3(2x + 5) = 6x + 15, not 6x + 5.
Practise it
Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.
Warm-up
Warm-up
Core
Core
Stretch 🔥
Answers (for the teacher)
- 4x + 8
- 5x - 15
- 12x + 6
- 2x - 3
- 3x + 4
Part 2 of 3 · learn, see, practise
Negatives and subtracting brackets
A negative outside changes every sign inside: −2(x − 4) = −2x + 8.
A minus sign before a bracket is −1 times the bracket: 5 − (x + 3) = 5 − x − 3 = −x + 2.
Expand first, then combine like terms.
Example 1
Expand and simplify 4(x + 1) − 2(x − 3).
- 4x + 4 and −2x + 6.
- Combine: 4x − 2x = 2x; 4 + 6 = 10.
- 2x + 10.
Answer: 2x + 10
Build it with tiles
2(x + 3) means two groups of x + 3. Build two groups, then Simplify.
- Build 2(x + 3): two groups of x + 3, then Simplify.
- Build 3(x − 1).
- Build −(x − 2), which is −x + 2.
Practise it
Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.
Warm-up
Core
Core
Core
Stretch 🔥
Answers (for the teacher)
- -3x - 6
- -2x + 8
- -x + 4
- 5x + 7
- 7x + 7
Part 3 of 3 · learn, see, practise
Factoring: take out the common factor
Factoring is expanding backwards: 6x + 15 = 3(2x + 5).
Find the greatest common factor (GCF) of the coefficients and constant, and write it outside.
Check by expanding your answer.
Example 1
Factor 12x + 18.
- The GCF of 12 and 18 is 6.
- 12x ÷ 6 = 2x and 18 ÷ 6 = 3.
- 6(2x + 3); check: 6 × 2x + 6 × 3 = 12x + 18 ✓
Answer: 6(2x + 3)
Practise it
Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.
Warm-up
Warm-up
Core
Core
Stretch 🔥
Answers (for the teacher)
- 4
- 5(x + 3)
- 3(3x - 2)
- 7(2x + 3)
- -4(x + 2)
Watch out
Common mistakes
Multiplying only the first term
3(2x + 5) is 6x + 15: the 3 multiplies the 5 too.
Sign slips with negatives
−2 × −4 = +8; write each product before combining.
Not taking the greatest factor
6x + 12 = 6(x + 2) is fully factored; 2(3x + 6) is not.
Class activity 1 of 3
Area model match
Use an area model to expand. Write the expanded form.
- Draw a rectangle split into parts.
- Find each part's area.
- Add the areas.
Answers (for the teacher)
- 7x + 14
- 12x - 20
- 5x + 10
- -3x + 12
Class activity 2 of 3
Expand and simplify relay
Expand each bracket, then combine like terms.
- Expand the first bracket.
- Expand the second.
- Combine like terms.
Answers (for the teacher)
- 5x + 12
- 2x - 9
- 2x + 10
- 2x + 9
Class activity 3 of 3
Factor detective
Factor each expression fully. Type it like 4(x + 3).
- Find the greatest common factor.
- Divide each term by it.
- Check by expanding.
Answers (for the teacher)
- 4(2x + 5)
- 5(3x - 5)
- 6(x + 1)
- 8(3x - 2)
After class · project
Garden plots
Design rectangular garden beds whose areas are expressions, and show expanded and factored forms.
What to make: Three garden beds with expanded and factored areas.
Stretch: Make one bed with a negative inside the bracket.
How it is marked
| Look for | Done well when |
|---|---|
| Expanding | Each area expanded correctly |
| Total | Combined correctly |
| Factoring | Greatest common factor used |
Project template
Copy it onto paper or print the practice sheet.
Bed 1: ___ by ___ area = ___(___) = ______ Bed 2: ___ by ___ area = ______ Bed 3: ___ by ___ area = ______ Total area simplified: ______ Factored total: ______
Homework
Practise at home
Try each on paper, then tap to see the answer and working. Or print the practice sheet.
Expand 6(x − 7).
6x - 42
Expand and simplify 3(2x + 1) − (x − 4).
5x + 7
Factor 21x + 28.
7(3x + 4)
Expand 1/4(8x + 12).
2x + 3
🔥 Challenge round · finished early?
The hardest questions of the lesson, no notes
Cover the notes and try these from memory. They are meant to make you think hard: draw, try small cases, and check your answer.
Stretch 🔥
Stretch 🔥
Stretch 🔥
Answers (for the teacher)
- 3x + 4
- 7x + 7
- -4(x + 2)
🧠 Think-hard question
Factor 6x + 9y − 12.
Talk it through or use the drawing board. There is more than one way to get there.
Quiz · 8 questions
Final quiz: check what you learned
Pick an answer to see if it is right and why. Your best score is saved on this device.
🖨 Print what we learned
Make a worksheet from this lesson
Pick what to practise, then print it or choose Save as PDF in the print window to download it.
For parents and teachers
The 60-minute lesson plan
Use it as it is, or stretch any part over two sessions. Each row links to the part of the page you need.
| Time | Part | The teacher | The learner |
|---|---|---|---|
| 0–4 min | Warm-up | Put the warm-up on screen; take two or three answers before revealing. | Think, then share. |
| 4–13 min | Expanding | Explain, show the example step by step, then the practice. | Practise up to the stretch question. |
| 13–22 min | Negatives and two brackets | Use the interactive tool together, then the practice. | Try the tool, then practise. |
| 22–31 min | Factoring | Explain and show the example, then the practice. | Practise; the stretch can be homework. |
| 31–38 min | Activity 1 | Pairs or solo; hint only after a wrong check. | Solve and check. |
| 38–45 min | Activity 2 | Do the first item together. | Finish the activity. |
| 45–50 min | Activity 3 | Quick challenge; finish for homework if needed. | Solve the puzzles. |
| 50–56 min | Quiz | Run it as an exit ticket. | Answer and read why. |
| 56–60 min | Set the project | Show the template and agree a due date. | Start the project. |
Questions to ask
- What does the number outside the bracket do?
- What happens with a minus sign before a bracket?
- How do you find what to factor out?
If it feels hard
Draw the area model every time before writing the expansion.
If it feels easy
Factor 6x + 9y − 12.
FAQ
Expanding and factoring with the distributive property: questions
What is the distributive property?
a(b + c) = ab + ac: the factor outside multiplies every term inside the bracket.
What is factoring in Grade 7?
Writing an expression as a product by taking out a common factor, such as 12x + 18 = 6(2x + 3).
Which standard is this?
New Jersey 7.EE.A.1 (Common Core 7.EE.1).
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