Algebra Maths Differentiated Worksheets - Free Printable

Let's go through the image and solve each section step by step. The worksheet is divided into Memory, Simplify expressions, Skill 1, 2, 3, Literacy, and Stretch sections.

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🔹 Memory Section: Substitution

> "To substitute, replace the variable with its corresponding value."

Example given:
- Let $ a = 5 $
- $ 4a = 4 \times a = 4 \times 5 = 20 $

✔ This shows that substitution means replacing variables with their given values.

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🔹 Simplify the following expressions (ROK)

We are asked to simplify:

1) $ 3 \times a $
→ $ 3a $ (This is already simplified)

2) $ b + b + b + b $
→ $ 4b $ (since $ b + b + b + b = 4b $)

3) $ a + b $
→ Already simplified → $ a + b $

4) $ a \times a $
→ $ a^2 $

5) What is the formula for the perimeter of this regular shape?

The shape is a regular pentagon (5 equal sides), labeled with side length $ h $.
Perimeter = sum of all sides = $ 5h $

✔ So, the formula is: $ P = 5h $

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🔹 Skill 1: Let $ x = 4 $, $ y = -3 $

Substitute and evaluate:

1) $ 3x + 6 = 3(4) + 6 = 12 + 6 = 18 $
2) $ 3x - 15 = 3(4) - 15 = 12 - 15 = -3 $
3) $ 2y - 2 = 2(-3) - 2 = -6 - 2 = -8 $
4) $ 4x + 9 = 4(4) + 9 = 16 + 9 = 25 $
5) $ 3 - x = 3 - 4 = -1 $
6) $ 3 - y = 3 - (-3) = 3 + 3 = 6 $
7) $ 5 - 2x = 5 - 2(4) = 5 - 8 = -3 $
8) $ 7 + y = 7 + (-3) = 4 $

✔ Skill 1 Answers:
1) 18
2) -3
3) -8
4) 25
5) -1
6) 6
7) -3
8) 4

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🔹 Skill 2: Let $ x = 2 $, $ z = 5 $

Evaluate:

1) $ x^2 + 5 = 2^2 + 5 = 4 + 5 = 9 $
2) $ x^2 - 3 = 4 - 3 = 1 $
3) $ 2x^2 + 3 = 2(4) + 3 = 8 + 3 = 11 $
4) $ 6x^2 - 7 = 6(4) - 7 = 24 - 7 = 17 $
5) $ x^3 - 7 = 8 - 7 = 1 $
6) $ z^2 + 3 = 25 + 3 = 28 $
7) $ 3z^2 - 80 = 3(25) - 80 = 75 - 80 = -5 $

✔ Skill 2 Answers:
1) 9
2) 1
3) 11
4) 17
5) 1
6) 28
7) -5

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🔹 Skill 3: Let $ s = -2 $, $ t = -4 $

Evaluate:

1) $ s^2 + 4 = (-2)^2 + 4 = 4 + 4 = 8 $
2) $ s^2 - 7 = 4 - 7 = -3 $
3) $ 2s^2 + 5 = 2(4) + 5 = 8 + 5 = 13 $
4) $ 6s^2 - 2 = 6(4) - 2 = 24 - 2 = 22 $
5) $ t^2 + 7 = (-4)^2 + 7 = 16 + 7 = 23 $
6) $ t^2 - 20 = 16 - 20 = -4 $
7) $ t^3 + 1 = (-4)^3 + 1 = -64 + 1 = -63 $

✔ Skill 3 Answers:
1) 8
2) -3
3) 13
4) 22
5) 23
6) -4
7) -63

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🔹 Literacy: Explain the links between substitution and football

Answer:

Substitution in algebra is like assigning specific roles or positions to players in football.

For example:
- Imagine a football team where each player has a role: striker, midfielder, defender.
- If we assign variables like $ S $ for striker, $ M $ for midfielder, and $ D $ for defender, then an expression like $ 2S + 3M + 2D $ could represent a team lineup.
- When we substitute actual numbers (e.g., $ S = 1 $, $ M = 2 $, $ D = 2 $), we get a real team setup: $ 2(1) + 3(2) + 2(2) = 2 + 6 + 4 = 12 $ players — which makes sense.

Another example:
- A player’s performance score might be modeled as $ 3g + 2a $, where $ g $ = goals and $ a $ = assists.
- If a player scores 4 goals and 3 assists: $ 3(4) + 2(3) = 12 + 6 = 18 $ — a total performance score.

✔ So, substitution is like plugging in real-world data (like player stats or team composition) into a mathematical model (like a formula), just as you’d plug in real players into a football formation.

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🔹 Stretch Questions

#### 1) Expressions:
Given:
$ 10 - 2x $, $ 3x + 10 $, $ x^2 $, $ -3x $, $ x^{-1} $, $ x^0 $

Let $ x = 5 $. Evaluate each:

- $ 10 - 2x = 10 - 10 = 0 $
- $ 3x + 10 = 15 + 10 = 25 $
- $ x^2 = 25 $
- $ -3x = -15 $
- $ x^{-1} = \frac{1}{5} = 0.2 $
- $ x^0 = 1 $ (any number to power 0 is 1)

So values:
0, 25, 25, -15, 0.2, 1

Now arrange in ascending order:

→ $ -15, 0, 0.2, 1, 25, 25 $

So ascending order:
1) $ -3x $
2) $ 10 - 2x $
3) $ x^{-1} $
4) $ x^0 $
5) $ 3x + 10 $ and $ x^2 $ (both 25)

a) Ascending order:
$ -3x < 10 - 2x < x^{-1} < x^0 < 3x + 10 = x^2 $

b) Can you reverse the order?
Yes, but only if we change the value of $ x $. For example, if $ x = 1 $, then:
- $ -3x = -3 $
- $ 10 - 2x = 8 $
- $ x^2 = 1 $
- $ x^{-1} = 1 $
- $ x^0 = 1 $
- $ 3x + 10 = 13 $

Then ordering would be different.

But with $ x = 5 $, no — because some values are fixed (like $ x^0 = 1 $ always). So we cannot reverse the order unless we change $ x $.

✔ Answer: No, you cannot reverse the order with $ x = 5 $, because the values are fixed.

c) Could you make every expression have the same value?

Try to find $ x $ such that all expressions are equal.

Let’s suppose:
- $ x^0 = 1 $ always
- So all others must also be 1

Set $ 10 - 2x = 1 $ → $ 2x = 9 $ → $ x = 4.5 $

Check $ 3x + 10 = 3(4.5) + 10 = 13.5 + 10 = 23.5 $ ≠ 1 → not equal

Try $ x = 1 $:
- $ x^0 = 1 $
- $ -3x = -3 $
- $ 10 - 2x = 8 $
- Not equal

Try $ x = 0 $:
- $ x^{-1} $ undefined → invalid

So no single value of $ x $ can make all expressions equal.

✔ Answer: No, it's impossible to make all expressions equal because $ x^{-1} $ is undefined at $ x = 0 $, and $ x^0 = 1 $, so others must be 1, but they aren't.

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#### 2) Find values of $ a $ and $ b $ when $ p = 10 $

Given:
- $ a = \dfrac{3p^3}{2} $
- $ b = \dfrac{2p^2(p - 3)}{7p} $

First, compute $ a $:

$ a = \dfrac{3(10)^3}{2} = \dfrac{3(1000)}{2} = \dfrac{3000}{2} = 1500 $

Now $ b $:

$ b = \dfrac{2(10)^2(10 - 3)}{7(10)} = \dfrac{2(100)(7)}{70} = \dfrac{1400}{70} = 20 $

✔ So:
- $ a = 1500 $
- $ b = 20 $

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✔ Final Summary of Answers:

#### Simplify ROK:
1) $ 3a $
2) $ 4b $
3) $ a + b $
4) $ a^2 $
5) $ P = 5h $

#### Skill 1 ($ x=4, y=-3 $):
1) 18
2) -3
3) -8
4) 25
5) -1
6) 6
7) -3
8) 4

#### Skill 2 ($ x=2, z=5 $):
1) 9
2) 1
3) 11
4) 17
5) 1
6) 28
7) -5

#### Skill 3 ($ s=-2, t=-4 $):
1) 8
2) -3
3) 13
4) 22
5) 23
6) -4
7) -63

#### Literacy:
Substitution is like assigning real players to positions in football — e.g., using variables to represent roles and substituting real numbers for actual players.

#### Stretch:
1a) Ascending: $ -3x, 10-2x, x^{-1}, x^0, 3x+10, x^2 $
1b) Cannot reverse with $ x=5 $ — values are fixed
1c) No, expressions can’t all be equal due to $ x^{-1} $ and $ x^0 $ constraints
2) $ a = 1500 $, $ b = 20 $

Let me know if you'd like this formatted as a printable answer sheet!