Lesson title: Expanding expressions
Do Now
Copy then simplify:
\(12x \div 3\)
\(\dfrac{20a}{4}\)
\(15m \div 5\)
\(\dfrac{18p}{6}\)
\(24n \div 8\)
\(\dfrac{14x}{2}\)
Quick check - What’s the pattern?
Study these examples and spot the pattern:
\[3 \times 7 = 21\]
\[\begin{align*}
3 \times (4 + 3) &= 3 \times 4 + 3\times 3 \\
&= 12 + 9 \\
&= 21
\end{align*}\]
\[3 \times 4x = 12x\]
\[\begin{align*}
3 \times (x + 2) &= 3 \times x + 3 \times 2\\
&= 3x + 6
\end{align*}\]
Answer: What do you notice about what the 3 does in each example?
Today’s learning
By the end of this lesson, you will be able to:
- Expand simple brackets like \(3(x + 2)\)
- Write your working clearly in steps
- Check your answers by substituting
Key vocabulary
Copy these definitions:
Expand: To multiply out brackets
Brackets: The symbols ( ) that group terms together
Distribute: To multiply each term inside the bracket by the number outside
The expansion rule
Copy this rule:
When expanding brackets, multiply the number outside by EVERY term inside.
\[\begin{align*}
3(x + 2) &= 3 \times x + 3 \times 2 \\
&= 3x + 6
\end{align*}\]
The key word is EVERY - don’t miss any terms!
Step-by-step method
Listen:
Step 1: Write the bracket as separate multiplications
Step 2: Work out each multiplication
Step 3: Write the final answer
Example: \(4(a + 3)\)
\[\begin{align*}
4(a + 3) &= 4 \times a + 4 \times 3 &&\text{Step 1} \\
&= 4a + 12 &&\text{Steps 2 & 3}
\end{align*}\]
Your turn - Follow the steps
Copy and complete using the 3-step method:
Expand: \(5(x + 2)\)
\[\begin{align*}
5(x + 2) &= \underline{\hspace{3cm}} + \underline{\hspace{3cm}} \\
&= \underline{\hspace{3cm}}
\end{align*}\]
Expand: \(3(m + 4)\)
\[\begin{align*}
3(m + 4) &= \underline{\hspace{3cm}} + \underline{\hspace{3cm}} \\
&= \underline{\hspace{3cm}}
\end{align*}\]
Expand: \(2(n + 7)\)
\[\begin{align*}
2(n + 7) &= \underline{\hspace{3cm}} + \underline{\hspace{3cm}} \\
&= \underline{\hspace{3cm}}
\end{align*}\]
Expanding with subtraction
Examine the following:
\[\begin{align*}
4(x - 3) &= 4 \times x + 4 \times (-3) \\
&= 4x + (-12) \\
&= 4x - 12
\end{align*}\]
Key point: The minus sign stays with the number that follows it.
Practice with subtraction
Complete with full working:
Check your working
Learn this checking method:
To check \(3(x + 2) = 3x + 6\), substitute \(x = 1\):
Original: \(3(1 + 2) = 3(3) = 9\)
Expanded: \(3(1) + 6 = 3 + 6 = 9\) ✓
Both give 9, so this gives us confidence that it’s correct
You try checking
Check if this expansion is correct:
\[4(x + 3) = 4x + 12\]
Method: Substitute \(x = 2\) into both sides.
Original side: \(4(2 + 3) =\)
Expanded side: \(4(2) + 12 =\)
Do they match?
Spot the mistake
Find the error and write the correct solution:
Jamie’s working:
\[\begin{align*}
5(x + 4) &= 5x + 4
\end{align*}\]
What went wrong? Show the correct working.
Mixed practice
Complete these expansions with full working:
\(2(x + 5)\)
\(-3(a + 2)\)
\(4(m - 1 + n)\)
\(-2(p - 3)\)
\(3(x + y + 4)\)
\(-5(k + 1)\)
Expanding with negative numbers
Copy this example carefully:
\[\begin{align*}
-3(x + 4) &= -3 \times x + (-3) \times 4 \\
&= -3x + (-12) \\
&= -3x - 12
\end{align*}\]
Key point: The negative sign multiplies EVERYTHING inside.
Practice with negatives
Complete with full working:
More than two terms
Copy this example:
\[\begin{align*}
2(x + 3 + y) &= 2 \times x + 2 \times 3 + 2 \times y \\
&= 2x + 6 + 2y
\end{align*}\]
Remember: Multiply by EVERY term inside - however many there are!
Practice with multiple terms
Complete these:
Challenge questions
For those ready for more:
Summary
Today we learned:
- Expanding means multiplying out brackets
- Multiply the outside number by EVERY term inside
- Show your working in clear steps
- Check answers by substituting values
Key difference from before:
- Before: \(3 \times 2x = 6x\) (one term)
- Today: \(3(x + 2) = 3x + 6\) (multiple terms)
Exit task
On scrap paper, write your name, then complete:
Expand: \(4(x + 3)\)
Expand: \(2(m - 5)\)
Check question 1 by substituting \(x = 1\)