Lesson title: HCF of algebraic terms

Do Now - Checking after the sub

Copy then answer:

Expand these expressions:

Quick check - Study Ahmed’s working

Ahmed expanded \(4(2x + 3)\):

\[\begin{align*} 4(2x + 3) &= 4 \times 2x + 4 \times 3 \\ &= 8x + 12 \end{align*}\]

Answer these questions:

1. What did Ahmed multiply 4 by?
2. Now you expand: \(2(3y + 5)\)

Today’s learning

By the end of this lesson, you will be able to:

• Find the HCF (Highest Common Factor) of numbers
• Find the HCF of algebraic terms
  
• Explain what HCF means in your own words

Key vocabulary

Copy these definitions:

Factor: A number that divides exactly into another number

• Factors of 12: 1, 2, 3, 4, 6, 12

HCF (Highest Common Factor): The largest number that divides exactly into two or more numbers

• HCF of 12 and 18 is 6

Finding HCF of numbers

Copy this method:

Step 1: List all factors of each number

Step 2: Find the common factors

Step 3: Pick the highest one

Example: Find HCF of 8 and 12

\[\begin{align*} \text{Factors of 8:} &\quad 1, 2, 4, 8 \\ \text{Factors of 12:} &\quad 1, 2, 3, 4, 6, 12 \\ \text{Common factors:} &\quad 1, 2, 4 \\ \text{HCF:} &\quad 4 \end{align*}\]

Your turn with numbers

Complete with full working:

HCF with algebraic terms

Copy this new idea:

Algebraic terms can also have common factors

Example: \(6x\) and \(9x\)

Both terms have: - Number factors - The same variable \(x\)

So we can find their HCF too!

Finding HCF of algebraic terms

Copy this method:

Step 1: Find HCF of the numbers (coefficients)

Step 2: Find the common variables

Step 3: Combine them

Example: Find HCF of \(6x\) and \(9x\)

\[\begin{align*} \text{HCF of coefficients 6 and 9:} &\quad 3 \\ \text{Common variable:} &\quad x \\ \text{HCF of terms:} &\quad 3x \end{align*}\]

Study this solution

Bella found HCF of \(8a\) and \(12a\):

\[\begin{align*} \text{HCF of 8 and 12:} &\quad 4 \\ \text{Common variable:} &\quad a \\ \text{HCF of } 8a \text{ and } 12a: &\quad 4a \end{align*}\]

Answer these questions:

1. How did Bella find that 4 is the HCF of 8 and 12?
2. Why is \(a\) the common variable?
3. Now find HCF of \(10x\) and \(15x\)

Your turn - Simple cases

Complete with full working:

What if variables are different?

Copy this important rule:

Example: Find HCF of \(6x\) and \(9y\)

\[\begin{align*} \text{HCF of coefficients 6 and 9:} &\quad 3 \\ \text{Common variables:} &\quad \text{none} \\ \text{HCF of terms:} &\quad 3 \end{align*}\]

Rule: If variables are different, the HCF is just the HCF of the numbers

Mixed practice

Find the HCF of each pair:

Check your working

Copy this checking method:

To check if your HCF is correct:

Both original terms should divide exactly by your HCF

Example: HCF of \(8x\) and \(12x\) is \(4x\)

Check: \(8x ÷ 4x = 2\) ✓ and \(12x ÷ 4x = 3\) ✓

Both divide exactly, so \(4x\) is correct

Practice checking

Use the checking method:

Common mistakes

Which is correct? Circle the right answer:

Your turn - Challenge

Find the HCF and show your check:

Summary

Today we learned:

• HCF = Highest Common Factor
• For algebraic terms: find HCF of numbers, then include common variables
• Same variables: HCF includes the variable
• Different variables: HCF is just the number
• Always check: both terms should divide exactly by your HCF

Exit task

On scrap paper, write your name, then complete:

1. Find HCF of \(8x\) and \(12x\)
2. Find HCF of \(6a\) and \(9b\)
3. Check your first answer by dividing