037. Multiplication and Division in Algebraic Expressions

Learning Intentions

• To know the different ways multiplication and division can be written
• simplify expressions with multiplication
• simplify expressions with division

Pre-requisite Summary

• Understand that letters can represent numbers in algebra
• Recognise terms, coefficients and variables in simple expressions
• Recall multiplication facts and division facts
• Know that multiplication in algebra is often written without a sign, for example $3x$
• Understand that division can be written using a division sign, a fraction bar or slash notation
• Be able to combine numbers using order of operations
• Understand that a product means multiply and a quotient means divide

Worked Examples

Worked Example 1

Identify different ways multiplication and division can be written:
a) $3\times x$
b) $3x$
c) ${\displaystyle \frac{12}{y}}$
d) $12÷y$

Worked Example 2

Simplify expressions with multiplication:
a) $4\times x$
b) $2\times 5a$
c) $3\times x\times y$

Worked Example 3

Simplify expressions with multiplication:
a) $6\times 2b$
b) $5\times 3x$
c) $2\times 4m\times n$

Worked Example 4

Simplify expressions with division:
a) $12÷x$
b) ${\displaystyle \frac{18a}{3}}$
c) ${\displaystyle \frac{20xy}{5}}$

Worked Example 5

Simplify expressions with division:
a) ${\displaystyle \frac{24x}{6}}$
b) ${\displaystyle \frac{35ab}{7}}$
c) ${\displaystyle \frac{16m}{2}}$

Worked Example 6

For each expression:
a) rewrite it in simplest algebraic form
b) state whether multiplication or division has been simplified
For $3\times 4x$, ${\displaystyle \frac{30y}{5}}$, and $2\times x\times 5$

Problems

Problem 1

Identify different ways multiplication and division can be written:
a) $5\times y$
b) $5y$
c) ${\displaystyle \frac{15}{k}}$
d) $15÷k$

Problem 2

Simplify expressions with multiplication:
a) $7\times x$
b) $3\times 4a$
c) $2\times p\times q$

Problem 3

Simplify expressions with multiplication:
a) $8\times 2b$
b) $4\times 6x$
c) $3\times 2m\times n$

Problem 4

Simplify expressions with division:
a) $14÷x$
b) ${\displaystyle \frac{24a}{4}}$
c) ${\displaystyle \frac{30xy}{6}}$

Problem 5

Simplify expressions with division:
a) ${\displaystyle \frac{28x}{7}}$
b) ${\displaystyle \frac{42ab}{6}}$
c) ${\displaystyle \frac{18m}{3}}$

Problem 6

For each expression:
a) rewrite it in simplest algebraic form
b) state whether multiplication or division has been simplified
For $2\times 5x$, ${\displaystyle \frac{36y}{9}}$, and $4\times x\times 3$

Exercises

Understanding and Fluency

1. Rewrite each multiplication expression in algebraic form:
   a) $4\times x$
   b) $7\times a$
   c) $3\times y$

2. Rewrite each division expression in another form:
   a) $18÷x$
   b) ${\displaystyle \frac{20}{m}}$
   c) $12÷p$

3. Simplify expressions with multiplication:
   a) $2\times x$
   b) $5\times 3a$
   c) $4\times x\times y$

4. Simplify expressions with multiplication:
   a) $6\times 2b$
   b) $3\times 7m$
   c) $2\times 5p\times q$

5. Simplify expressions with division:
   a) ${\displaystyle \frac{12x}{3}}$
   b) ${\displaystyle \frac{20a}{5}}$
   c) ${\displaystyle \frac{18mn}{6}}$

6. Simplify expressions with division:
   a) ${\displaystyle \frac{24y}{4}}$
   b) ${\displaystyle \frac{35ab}{7}}$
   c) ${\displaystyle \frac{42m}{6}}$

7. Simplify each expression:
   a) $3\times 2x$
   b) ${\displaystyle \frac{16x}{2}}$
   c) $4\times y\times 2$

8. Simplify each expression:
   a) $5\times 4a$
   b) ${\displaystyle \frac{27b}{9}}$
   c) $2\times 3m\times n$

Reasoning

9. Explain why $3\times x$ and $3x$ mean the same thing.

10. A student says $4\times a$ should be written as $4+a$. Explain the mistake.

11. Explain why ${\displaystyle \frac{18x}{3}}$ simplifies to $6x$.

12. A student simplifies ${\displaystyle \frac{20a}{5}}$ to ${\displaystyle \frac{4}{a}}$. Explain why this is incorrect.

Problem-solving

13. A shop sells pens for $3 each. Write and simplify an expression for the cost of buying $p$ pens.

14. A factory packs $4$ boxes of $6x$ items each. Write and simplify an expression for the total number of items.

15. A ribbon of length $24x$ cm is cut into $6$ equal pieces. Write and simplify an expression for the length of one piece.

16. A farmer has $35ab$ kg of feed shared equally into $7$ bags. Write and simplify an expression for the mass of each bag.

17. A machine makes $5$ groups of $2m$ parts. Write and simplify an expression for the total number of parts.

18. A quantity of $42x$ litres is poured equally into $6$ containers. Write and simplify an expression for the amount in each container.

Potential Misunderstandings

• Students may think multiplication in algebra must always use the $\times$ sign
• Students may confuse $3x$ with $3+x$ instead of $3\times x$
• Students may think division can only be written with the $÷$ sign and not as a fraction
• Students may add coefficients when they should multiply them
• Students may divide the variable by the number incorrectly when simplifying expressions such as ${\displaystyle \frac{18x}{3}}$
• Students may think ${\displaystyle \frac{20a}{5}}={\displaystyle \frac{20}{5a}}$ instead of dividing only the numerical coefficient
• Students may not keep the variable part attached when simplifying multiplication or division
• Students may change the meaning of an expression by rewriting it with the wrong operation

Next: 038e. Expanding Brackets