038e. Expanding Brackets

Learning Intentions

• To understand what it means to expand brackets
• expand brackets using repeated terms or rectangle areas
• expand brackets using the distributive law

Pre-requisite Summary

• Understand that multiplication can be written in different ways, such as $3\times x$, $3x$, or $3(x)$
• Know that a bracket can represent a grouped quantity
• Understand addition of like terms such as $x+x+x=3x$
• Be able to simplify simple algebraic expressions
• Understand that the area of a rectangle is found by multiplying length by width
• Recognise that the distributive law connects multiplication with addition and subtraction

Worked Examples

Worked Example 1

a) Explain what it means to expand $3(x+2)$.
b) Write $3(x+2)$ as repeated terms.
c) Simplify the expanded expression.

Worked Example 2

Use repeated terms or a rectangle model to expand:
a) $2(a+4)$
b) $5(m+1)$

Worked Example 3

Use a rectangle area model to expand:
a) $x(x+3)$
b) $4(y+2)$

Worked Example 4

Use the distributive law to expand:
a) $3(x+5)$
b) $6(a+2)$
c) $4(p+q)$

Worked Example 5

Use the distributive law to expand:
a) $5(x-2)$
b) $7(a-3)$
c) $3(2m+4)$

Worked Example 6

Expand and simplify:
a) $2(x+3)+x$
b) $4(a+1)-2$
c) $3(2y-1)+y$

Problems

Problem 1

a) Explain what it means to expand $4(x+1)$.
b) Write $4(x+1)$ as repeated terms.
c) Simplify the expanded expression.

Problem 2

Use repeated terms or a rectangle model to expand:
a) $3(a+2)$
b) $4(m+5)$

Problem 3

Use a rectangle area model to expand:
a) $x(x+4)$
b) $5(y+3)$

Problem 4

Use the distributive law to expand:
a) $2(x+6)$
b) $5(a+3)$
c) $3(p+q)$

Problem 5

Use the distributive law to expand:
a) $4(x-1)$
b) $6(a-2)$
c) $2(3m+5)$

Problem 6

Expand and simplify:
a) $3(x+2)+x$
b) $5(a+2)-3$
c) $2(4y-1)+y$

Exercises

Understanding and Fluency

1. Expand using repeated terms:
   a) $2(x+3)$
   b) $3(a+1)$
   c) $4(m+2)$

2. Expand using repeated terms or a rectangle model:
   a) $5(x+1)$
   b) $2(y+6)$
   c) $3(p+4)$

3. Expand using the distributive law:
   a) $3(x+4)$
   b) $4(a+5)$
   c) $6(m+2)$

4. Expand using the distributive law:
   a) $5(x-3)$
   b) $2(a-7)$
   c) $4(y-1)$

5. Expand:
   a) $3(2x+1)$
   b) $2(4a+3)$
   c) $5(m+n)$

6. Expand:
   a) $4(3x-2)$
   b) $2(5a-4)$
   c) $3(p+2q)$

7. Expand and simplify:
   a) $2(x+4)+x$
   b) $3(a+2)+2a$
   c) $4(y+1)-3$

8. Expand and simplify:
   a) $5(x-2)+x$
   b) $2(3a+1)-4$
   c) $3(2m-1)+m$

Reasoning

9. Explain why $3(x+2)$ means $3x+6$.

10. A student says $4(x+5)=4x+5$. Explain the mistake.

11. Explain how a rectangle model can show that $x(x+3)=x^2+3x$.

12. A student expands $2(3x+4)$ as $6x+4$. Explain why this is incorrect.

Problem-solving

13. A rectangle has width $3$ and length $(x+4)$. Write and expand an expression for its area.

14. A rectangle has width $y$ and length $(y+2)$. Write and expand an expression for its area.

15. A student buys $4$ bags, and each bag contains $(x+3)$ marbles. Write and expand an expression for the total number of marbles.

16. A machine makes $5(a-1)$ parts each hour. Expand the expression for the number of parts made in one hour.

17. A garden has $3$ rows, each with $(p+2)$ plants. Write and expand an expression for the total number of plants.

18. A builder lays $2(4x+5)$ tiles in two equal sections. Expand the expression for the total number of tiles.

Potential Misunderstandings

• Students may think expanding brackets means removing the brackets without changing the expression correctly
• Students may multiply only the first term inside the bracket and forget the others
• Students may confuse addition inside brackets with multiplication outside brackets
• Students may not recognise that $3(x+2)$ means three groups of $(x+2)$
• Students may misuse the rectangle area model by not multiplying each part of the length by the width
• Students may apply the distributive law to addition incorrectly, for example writing $3(x+2)=3x+2$
• Students may make sign errors when expanding brackets with subtraction
• Students may expand correctly but not simplify the final expression

Next: 039. Algebraic Modelling with Expressions