Courseversity

Equivalent expressions

Primary EducationGrade 6MathematicsCV-PRI-G06-MATH-ALG-U01-L0445–60 min

You will learn

  • Collect like terms.
  • Use the distributive property.
  • Test equivalence.
  • Simplify.

Version 0.2 · expanded edition · built from cited public sources Content is based on the official frameworks and sources listed under Sources and standards. No official endorsement is claimed. Written in the style of common primary school assessments. No exam result is promised.

Student lesson

The lesson in 60 seconds

  1. Equivalent expressions have the same value for every number.
  2. Collect like terms.
  3. Distribute to remove brackets.
  4. Test with numbers.

Words to know

equivalent
Same value for every number.
like terms
Same letter part.
distribute
Multiply each part inside.
simplify
Make simpler.

1. Discover: same value

Are 2(n + 3) and 2n + 6 always equal?

2. Understand: equivalent expressions

  • 2(n + 3) = 2 × n + 2 × 3 = 2n + 6.
  • Test n = 5: 2 × 8 = 16 and 10 + 6 = 16.
  • 3n + n = 4n and n + n + n = 3n.
  • 3n + 2 and 5n are not like terms.

Idea 1: Distribute

3(n + 2) = 3 × n + 3 × 2 = 3n + 6. 2(n − 1) = 2n − 2.

Idea 2: Like terms

5n + 2n = 7n. n + n + n + n = 4n.

Idea 3: Test

3n + 2 and 5n: if n = 2, 8 and 10, so not equivalent. 2(n + 3) = 2n + 6.

Why test? One number that fails shows they differ.

Remember: Only like terms can be added.

3. Worked examples

Example 1 (I show you): Expand

3(n + 2)

  1. 3 × n and 3 × 2.

Answer: 3n + 6

Example 2 (We do it together): Collect

5n + 2n

  1. Add the coefficients.

Answer: 7n

Example 3 (You try): Test

3n + 2 and 5n

Show the steps and answer
  1. Try n = 2.

Answer: not equivalent

Did you know?

Like terms have the same letter part.

Why this matters

Equivalent expressions make algebra simpler and let us check work.

4. Explore: expand

You need paper.

  1. Multiply the outside by each inside term.
  2. Collect like terms.
  3. Test with a number.
  4. Write it.
Think about it: what did you notice?

2n + 6.

5. Practise

Say it in a sentence

Use these starters when you talk about the lesson. Say them out loud.

  • I multiply the outside by each term: ___.
  • Like terms are ___.
  • Simplified: ___.
  • They are / are not equivalent because ___.

Guided practice

Question 1. 2(n + 3)?

2n + 6.

Question 2. 3n + n?

4n.

Question 3. n + n + n?

3n.

Independent practice

Level 1: Start

  1. Expand 3(n + 2)
  2. Simplify 5n + 2n
  3. Simplify n + n + n + n
  4. Expand 2(n − 1)
  5. Is 3n + 2 equal to 5n?

Level 2: Grow

  1. 4(n + 1)
  2. 6n + 3n
  3. 7n − 2n

Level 3: Stretch

  1. Simplify 2(n + 3) + 3n
  2. 5(n − 2)
  3. Ben says 3n + 2 and 5n are equivalent. What is wrong?

Answers are in the answer key below. Try first, then check.

6. Apply: everyday mathematics

A rectangle is n + 3 long and 2 wide. Write its area as 2n + 6. Why?

Show a sample answer

2(n + 3) = 2n + 6.

Another one. Show that 3n + 3 and 3(n + 1) are equivalent.

Show a sample answer

3(n + 1) = 3n + 3

7. Common mistakes

Mistake: Writing 2(n + 3) = 2n + 3.
Fix: Multiply every term inside.
Another mistake: Multiplying only the first term.
Fix: Multiply every term inside.
Another mistake: Adding unlike terms.
Fix: Only like terms combine.

8. Lesson summary and revision card

I can…

  • I can collect like terms.
  • I can use the distributive property.
  • I can test equivalence.
  • I can simplify.

9. Check and master

Answer all 14 questions. You will see feedback after each one.

1. 3(n + 2) =

2. 5n + 2n =

3. n + n + n + n =

4. 2(n − 1) =

5. 3n + 2 and 5n are:

6. 4(n + 1)

7. 6n + 3n

8. 5(n − 2)

9. n + n + n

10. 7n − 2n

11. 3n + 3 and 3(n + 1) are…

12. Which is equivalent to 3(n + 4)?

Challenge 1. 2(n + 3) is equivalent to:

Challenge 2. Simplify 2(n + 3) + 3n

Your result appears here after you answer every question.

10. Key takeaways

  • Distribute to every term.
  • Collect like terms only.
  • Test with numbers.

11. Next lesson

Next lesson: Simple equations. Keep going with the next idea in this course.

Exam practice

These questions are written in the style of common school and international primary assessments. Try them on your own, without looking back. Then check your answers in the marking guide. The marks are shown in brackets.

  1. Expand 3(n + 2). [2 marks]
  2. Simplify 5n + 2n. [1 mark]
  3. Simplify n + n + n + n. [1 mark]
  4. Expand 2(n − 1). [2 marks]
  5. Ali says 3n + 2 and 5n are equivalent. Is he right? Explain. [3 marks]

Total: 9 marks.

Marking guide and model answers
  1. Multiply 3 by n and by 2 [1]; 3n + 6 [1]
  2. 7n [1]
  3. 4n [1]
  4. 2 × n and 2 × 1 [1]; 2n − 2 [1]
  5. No [1]. With n = 2, 3n + 2 = 8 but 5n = 10 [1]. They are not equivalent [1].

Score guide: 8 to 9 = secure. 6 to 7 = nearly there, revise the parts you missed. Under 6 = work through the lesson again with a grown-up.

Printable worksheet

  1. Expand 3(n + 2)
  2. 5n + 2n
  3. n + n + n + n
  4. Expand 2(n − 1)
  5. Is 3n+2 = 5n? (yes/no)
  6. Expand 3(n + 2)
  7. Simplify 5n + 2n
  8. Simplify n + n + n + n
  9. Expand 2(n − 1)
  10. 3n + 2 and 5n are

Homework

Expand 4(n + 5) and simplify 6n − 2n.

Answer key and marking scheme

Question Expected response Marks
Expand 3(n + 2) 3n + 6 2
5n + 2n 7n 2
n + n + n + n 4n 2
Expand 2(n − 1) 2n − 2 2
Is 3n+2 = 5n? (yes/no) no 2
Total 10

Practice answers: Worksheet: 1 3n + 6; 2 7n; 3 4n; 4 2n − 2; 5 no.

Quiz answers: 1 A, 2 B, 3 C, 4 A, 5 B, 6 A, 7 B, 8 C, 9 A, 10 B, 11 C, 12 A, Challenge 1 C, Challenge 2 C.

Extra worksheet answers (items 6 onward): 6 3n + 6; 7 7n; 8 4n; 9 2n − 2; 10 not equivalent.

Independent practice answers. Level 1: 1 3n + 6; 2 7n; 3 4n; 4 2n − 2; 5 no, not equivalent. Level 2: 1 4n + 4; 2 9n; 3 5n. Level 3: 1 5n + 6; 2 5n − 10; 3 With n = 2: 8 and 10. Not equal..

Mastery and remediation

Objective Bronze (Developing) Silver (Proficient) Gold (Advanced)
O1 Collect like terms Shows this with help from an adult or a picture Does this independently and explains how they know Explains it clearly and uses it in a new situation
O2 Use the distributive property Shows this with help from an adult or a picture Does this independently and explains how they know Explains it clearly and uses it in a new situation
O3 Test equivalence Shows this with help from an adult or a picture Does this independently and explains how they know Explains it clearly and uses it in a new situation
O4 Simplify Shows this with help from an adult or a picture Does this independently and explains how they know Explains it clearly and uses it in a new situation

Courseversity badge thresholds (overall quiz score): Gold = 90–100%; Silver = 80–89.9%; Bronze = 70–79.9%. A badge is never a full record of progress.

If a learner struggles

  1. Find the misconception: Writing 2(n + 3) = 2n + 3. Multiply every term inside.
  2. Explain another way: use real objects first, then pictures.
  3. Show a simpler example: Area of rectangles.
  4. Guided practice: work through one example together and let the learner try the next one.
  5. Reassess: use the quiz again or the worksheet.
  6. Move on when the learner shows Silver-level evidence independently.

For parents and guardians

In this lesson your child simplifies and tests equivalent expressions. You do not need to be good at mathematics to help.

  • Try numbers to test two expressions.
  • Ask: what are the like terms?
  • Use area models for brackets.
  • Ask “how do you know?” after each answer, and listen.
  • Praise effort, not only right answers. Mistakes are how we learn.
  • Ten to fifteen minutes a day is better than one long session.

Teacher package

1–5. Objectives, prerequisites, materials, preparation, duration

Lesson objectives. Learners will: collect like terms; use the distributive property; test equivalence; simplify.

Prerequisite knowledge. Lesson 3.

Materials. Counters (stones, seeds or bottle tops), paper, pencils, board or chalkboard. Low-resource options: use stones, seeds, sticks, bottle tops or drawings in the sand or on the board instead of printed materials.

Preparation. Gather the counters and draw the example pictures on the board before the lesson.

Suggested duration. 45–60 minutes. It can be split into two shorter sessions.

6–9. Teaching procedure, explanation notes, guided and independent activities

Introduction (5 min). Start with the Discover situation and let learners suggest ideas before you teach.

Explanation (10 min). Teach the Understand section with real objects first, then pictures, then symbols and words.

Guided activity (10 min). You need paper. Learners follow the Explore steps in pairs.

Independent activity (10 min). Learners complete the independent practice and the worksheet.

Assessment and reflection (5 min). Ask each learner to say one thing they learned and how they know it.

Key questions. Are the terms alike? Did I multiply every term?

Teacher explanation notes. Use the concrete, pictorial, abstract order: real objects first, then pictures, then symbols and words. Return to real objects whenever a learner needs support.

10–12. Differentiation, formative assessment, common misconceptions

Support. Use real objects and work with a partner; say each step aloud together. Extension. Ask learners to make up their own example and explain it to a partner.

Formative assessment. Listen to learner explanations, ask “how do you know?”, and check the independent work.

Misconceptions. Writing 2(n + 3) = 2n + 3. Multiply every term inside.

13–15. Assessment questions, answer keys, marking scheme

Use the quiz and worksheet on this page. The 9-mark exam practice scheme, the 10-mark scheme and all answers are in the answer sections above. Re-check every answer and the marking scheme before publication.

16–19. Remediation, enrichment, homework, teacher reflection

Remediation. Area of rectangles. Enrichment. Invite learners to find or create three more examples at home or in the classroom and explain them. Homework. See above.

Teacher reflection. After teaching, record: which concepts learners understood; which misconceptions occurred; which activities worked; which learners need more practice; and what to improve next time.

Sources and standards

This lesson is based on the sources below, and no official endorsement is claimed. Reference starting points named in the Courseversity standard: Common Core State Standards for Mathematics, Cambridge Primary and the IB Primary Years Programme. Where a standard code is named, it points to the official framework.

Official standards for this topic area. Common Core State Standards for Mathematics (official site): Grade 6 » Expressions & Equations. These pages list the standards that cover this topic area; read them alongside the lesson.

Research used to shape this lesson. Common Core 6.EE.A.3 and 6.EE.A.4; Cambridge Primary Mathematics Stage 6 (equivalent expressions). These sources informed the lesson.

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