3.1 Simplifying Algebraic Expressions

• newEditor
• Maths
• Secondary Education
• Age 12

Teach Like a GEM

Subject: Mathematics

Lesson: 3.1 Simplifying Algebraic Expressions

Date: _______________

Key vocabulary: expression | term | like terms | simplify | collect | coefficient

Do Now

As you come in: sit down, take out your exercise book, pencil case and ruler, and begin working silently on the questions below.

Write these additions as multiplications:

1) 5 + 5 + 5

2) 9 + 9 + 9 + 9 + 9

3) 10 + 10

4) 18 + 18 + 18 + 18 + 18

5) What does it mean to 'simplify' something in everyday life?

6) Write 7 + 7 + 7 + 7 as a multiplication.

7) What is 3y if y = 4?

8) True or False: 2x and 3y are 'like terms'?

Answers: 1) 3×5=15 2) 5×9=45 3) 2×10=20 4) 5×18=90 5) To make something easier/shorter 6) 4×7=28 7) 12 8) False (different letters)

We will be able to:

► Simplify expressions by collecting like terms

► Use index notation for repeated letter multiplication

► Simplify expressions involving multiplication and division

We can:

⭐ Spot terms with the SAME letter and power

⭐ Add or subtract the numbers in front of like terms

⭐ Write letters in alphabetical order, numbers before letters

Key Definitions

Algebraic expression: A combination of numbers and letters, e.g. 3x + 2y. There is no equals sign.

Term: Each part of an expression, separated by + or −, e.g. in 3x + 2y, the terms are 3x and 2y.

Like terms: Terms with EXACTLY the same letter(s) and power(s), e.g. 2x and 5x are like terms, but 2x and 2x² are NOT.

Simplify: To collect like terms together to write an expression more simply.

Coefficient: The number in front of a letter, e.g. in 5x, the coefficient is 5.

Memory tip: Write letters in ALPHABETICAL order, and numbers BEFORE letters, e.g. write 2a not a2.

I Do: Collecting Like Terms

Simplify x + x + x

Step 1: Think of x as a rod of unknown length. Three rods placed end to end have a total length of 3x.

x + x + x = 3x

Simplify 5a − 3a

Step 1: 5a and 3a are LIKE terms (same letter), so subtract the coefficients: 5 − 3 = 2.

5a − 3a = 2a

Try these:

n + n = 2n

y + y + y + y + y = 5y

2a + 3a = 5a

8b − 3b = 5b

I Do: Simplifying Expressions with Different Letters

Simplify 7y + 2y − 3y

Step 1: All three terms have the same letter (y), so they are all like terms.

Step 2: Work left to right: 7y + 2y = 9y, then 9y − 3y = 6y.

7y + 2y − 3y = 6y

Simplify 9a + 3 − 3a − 7a

Step 1: Group the like terms together: the ‘a’ terms are 9a, −3a, −7a. The number term is 3 (no like term to combine with).

Step 2: Combine: 9a − 3a − 7a = −1a = −a. The 3 stays on its own.

9a + 3 − 3a − 7a = −a + 3

Try this: Simplify 6y − 2y + 8 − 3b

y terms: 6y−2y=4y. No like term for 8 or −3b. 4y + 8 − 3b

I Do: Index Notation with Letters

Simplify b × b

Step 1: Just like 2×2×2 = 2³, repeated multiplication of a letter uses powers.

b × b = b²

Simplify t × t × t

t × t × t = t³

Try these:

m × m × m × m = m⁴

a × b = ab (write letters next to each other, in alphabetical order)

p × p × y × y = p²y²

m × 2 = 2m (numbers go before letters)

Key point: Write letters in alphabetical order, and put numbers before letters, e.g. a×2=2a, not a2.

I Do: Multiplying and Dividing Terms

Simplify 3b × 2b

Step 1: Write out the full multiplication: 3b×2b = 3×b×2×b.

Step 2: The order of multiplication doesn't matter, so rearrange: 3×2×b×b = 6×b².

3b × 2b = 6b²

Simplify 8b ÷ 4 (written as a fraction: 8b/4)

Step 1: 8b/4 means 8b ÷ 4. Work out 8 ÷ 4 first.

8b ÷ 4 = 2b

Try these:

2b × 5b = 10b²

9a × 3a = 27a²

12b ÷ 4 = 3b

9a ÷ 2 = 4.5a

We Do: Algebraic Addition Pyramids

Each brick is the SUM of the two bricks below it. Complete the pyramid for 2a, 3b, a:

Step 1: Add the bottom-left and bottom-middle bricks: 2a + 3b (these are NOT like terms, so they stay separate).

Step 2: Add the bottom-middle and bottom-right bricks: 3b + a = a + 3b.

Step 3: Add the two new bricks together for the top: (2a+3b) + (a+3b) = 3a + 6b.

Top brick = 3a + 6b

Discussion: Why can't we simplify 2a + 3b into a single term? (They are not like terms — different letters.)

We Do: Matching Equivalent Expressions

Which of these expressions are equivalent to 2x?

4x − 3x, x + x, 3x, x × x, 2x × 2x, 9x/3

4x − 3x = x (NOT equivalent to 2x)

x + x = 2x (equivalent!)

3x = 3x (NOT equivalent — different coefficient)

x × x = x² (NOT equivalent — this is squared)

2x × 2x = 4x² (NOT equivalent)

9x ÷ 3 = 3x (NOT equivalent to 2x)

Only x + x is equivalent to 2x in this list.

Discussion: Why are x×x and 2x completely different kinds of expressions?

All

1) Simplify: a) n+n b) y+y+y+y+y

2) Simplify: a) 2a+3a b) 5b+6b

3) Simplify: a) 5a−3a b) 8b−3b

4) Simplify: a) b×b b) t×t×t

5) Simplify: a) a×b b) m×2 c) x×5

Most

1) Simplify by collecting like terms: a) 2x+4x+2 b) 2b+6c−3c

2) Simplify by collecting like terms: a) 6y−2y+8−3b b) 4y−2+3y

3) Simplify: a) 2x²+3x² b) 4a+2b²+3b²

4) Simplify: a) t×t×b b) p×p×y×y c) q×7×p

5) Simplify: a) 2b×5b b) 9a×3a

Some

1) Simplify: a) 9x+3−3x−7x b) 9a−7b+2a+5

2) Simplify: a) 2b²+3b+b² b) 5x+2x²+7x c) 8x⁴−x⁴

3) Simplify: a) 3a×2a×3a b) 12b÷4 c) 9a÷2

4) Addition pyramid: bottom row 5x, 7, −2x. Find the top brick.

5) Write a simplified expression for the perimeter of a rectangle: length (2x+3), width x.

Challenge: Are 3x+2y and 2y+3x equivalent? Explain why or why not.

All

Q1. a) 2n b) 5y

Q2. a) 5a b) 11b

Q3. a) 2a b) 5b

Q4. a) b² b) t³

Q5. a) ab b) 2m c) 5x

Most

Q1. a) 6x+2 b) 2b+3c

Q2. a) 4y+8−3b b) 7y−2

Q3. a) 5x² b) 4a+5b²

Q4. a) t²b b) p²y² c) 7pq

Q5. a) 10b² b) 27a²

Some

Q1. a) −x+3 b) 11a−7b+5

Q2. a) 3b²+3b b) 7x+2x² c) 7x⁴

Q3. a) 18a³ b) 3b c) 4.5a

Q4. Top = 3x+14

Q5. P = 2x+2(2x+3) = 6x+6

Challenge. Yes — addition is commutative, so the order of terms doesn't change the value.

✋ Fist to Five Check

Show me your confidence with TODAY’S TOPIC:

0 fist = totally lost → 5 fingers = could teach it myself!

🎯 Cold Call

Pick 2–3 students at random to explain:

“Explain why 2x² and 3x are NOT like terms. Walk us through Some Q2b.”

Exit Ticket

Read each question. On my signal, hold up 1, 2, 3 or 4 fingers to show your answer.

1. Simplify 6a + 2a.

1) 8a 2) 12a 3) 8a² 4) 6a²

2. Simplify p × p × p.

1) 3p 2) p³ 3) p+3 4) 3p³

3. Simplify 5x − 2x + 3.

1) 3x + 3 2) 7x + 3 3) 3x − 3 4) 6x

Answers: 1) Option 1 (8a) 2) Option 2 (p³) 3) Option 1 (3x+3)