138. Expanding Brackets and Simplifying Expressions

Learning Intentions

• To understand that the distributive law can be used to expand brackets
• expand brackets using the distributive law
• use expansion together with combining like terms to simplify expressions

Pre-requisite Summary

• Know that multiplication can be written beside brackets, for example $3(x+2)$
• Understand that a term is a part of an expression separated by $+$ or $-$ signs
• Be able to identify like terms
• Be able to combine like terms such as $3x+5x=8x$
• Understand that simplifying means writing an expression in a shorter equivalent form
• Recall that the distributive law means multiplying the factor outside the bracket by each term inside the bracket

Worked Examples

Worked Example 1

Use the distributive law to expand:

a) $3(x+4)$
b) $5(a+2)$
c) $2(m+7)$

Worked Example 2

Use the distributive law to expand:

a) $4(x-3)$
b) $6(y-2)$
c) $7(p-5)$

Worked Example 3

Expand, then simplify by combining like terms:

a) $3(x+2)+4x$
b) $5(a-1)+2a$
c) $2(m+6)+3m$

Worked Example 4

Expand, then simplify by combining like terms:

a) $4(x+3)-2x$
b) $6(y-2)+y$
c) $3(p-4)+5p$

Worked Example 5

Expand and simplify:

a) $2(a+5)+3(a+1)$
b) $4(x-2)+2(x+3)$
c) $5(m+1)-2(m-4)$

Worked Example 6

Write an expression from the words, then expand and simplify:

a) Three times the sum of $x$ and $2$, then add $5x$
b) Four times the difference between $y$ and $1$, then subtract $2y$
c) Twice the sum of $a$ and $6$, then add three times $a$

Problems

Problem 1

Use the distributive law to expand:

a) $2(x+5)$
b) $4(a+3)$
c) $6(m+1)$

Problem 2

Use the distributive law to expand:

a) $3(x-4)$
b) $5(y-3)$
c) $8(p-2)$

Problem 3

Expand, then simplify by combining like terms:

a) $2(x+3)+5x$
b) $4(a-2)+3a$
c) $3(m+4)+2m$

Problem 4

Expand, then simplify by combining like terms:

a) $5(x+1)-3x$
b) $4(y-2)+2y$
c) $2(p-6)+4p$

Problem 5

Expand and simplify:

a) $3(a+4)+2(a+2)$
b) $5(x-1)+3(x+2)$
c) $4(m+2)-3(m-1)$

Problem 6

Write an expression from the words, then expand and simplify:

a) Two times the sum of $x$ and $4$, then add $3x$
b) Five times the difference between $y$ and $2$, then subtract $y$
c) Three times the sum of $a$ and $5$, then add two times $a$

Exercises

Understanding and Fluency

1. Expand each expression:
   a) $2(x+3)$
   b) $5(a+1)$
   c) $4(m+6)$

2. Expand each expression:
   a) $3(x-2)$
   b) $6(y-4)$
   c) $7(p-1)$

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

4. Expand and simplify:
   a) $5(x-2)+x$
   b) $2(y+7)-3y$
   c) $6(p-1)+2p$

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

6. Expand and simplify:
   a) $6(y-2)+3(y+4)$
   b) $5(p+3)-2(p-1)$
   c) $3(t-5)+4(t+2)$

7. Write an expression from the words, then simplify:
   a) Four times the sum of $x$ and $2$
   b) Three times the difference between $a$ and $5$
   c) Two times the sum of $m$ and $4$, then add $m$

8. Write an expression from the words, then simplify:
   a) Five times the sum of $y$ and $1$, then subtract $2y$
   b) Three times the difference between $p$ and $2$, then add $4p$
   c) Two times the sum of $n$ and $6$, then add three times the difference between $n$ and $1$

Reasoning

9. Explain why $3(x+4)=3x+12$.

10. A student says that $4(x+2)=4x+2$. Explain the mistake.

11. Noah says that $5(a-3)=5a-3$ because only the first term is multiplied by $5$. Is he correct? Explain.

12. Explain why expanding brackets often creates like terms that can then be combined.

13. A student expands $2(x-5)$ as $2x+10$. Describe the error.

Problem-solving

14. A rectangle has width $x+3$ cm and length $4$ cm. Write and simplify an expression for the area.

15. A student buys $3$ packs of pencils. Each pack contains $p+2$ pencils. Write and simplify an expression for the total number of pencils.

16. A pattern has $4(n+1)$ squares in one part and $2n$ more squares in another part. Write and simplify an expression for the total number of squares.

17. A taxi fare is calculated as $5(d+2)$ dollars, where $d$ is the number of kilometres, and then a discount of $3d$ dollars is subtracted. Write and simplify the expression.

18. A garden has two sections. One section has area $3(x+4)$ square metres and the other has area $2x$ square metres. Write and simplify the total area.

19. A builder uses $2(m+5)$ bricks in one row and $3(m-1)$ bricks in another row. Write and simplify the total number of bricks used.

Potential Misunderstandings

• Students may multiply the number outside the bracket by only the first term inside the bracket
• Students may forget to multiply both terms inside the bracket
• Students may change the sign incorrectly when expanding brackets with subtraction
• Students may think $3(x+2)$ means $3x+2$ instead of $3x+6$
• Students may combine unlike terms after expanding
• Students may not recognise like terms created by expansion
• Students may confuse expanding with factorising
• Students may make arithmetic errors when multiplying the coefficient by the constant term