101. Order of Operations and Substitution

Learning Intentions

• To understand the rules for order of operations
• evaluate numerical expressions using the order of operations
• substitute integers for pronumerals in order to evaluate expressions

Pre-requisite Summary

• Know the four basic operations: addition, subtraction, multiplication and division
• Be able to perform basic calculations with integers accurately
• Understand that brackets group parts of an expression together
• Know that a pronumeral stands for a number
• Be able to replace a pronumeral with a given value
• Understand the difference between an expression and an equation
• Be familiar with using integers, including negative integers, in calculations

Worked Examples

Worked Example 1

Evaluate using the order of operations:
a) $6+4\times 3$
b) $18÷3+5$

Worked Example 2

Evaluate using the order of operations:
a) $(7+5)\times 2$
b) $20-(8÷2)$

Worked Example 3

Evaluate using the order of operations:
a) $3+2\times 5-4$
b) $24÷6\times 3$

Worked Example 4

Substitute the given integer and evaluate:
a) Find $a+7$ when $a=4$
b) Find $3b$ when $b=-2$

Worked Example 5

Substitute the given integer and evaluate:
a) Find $2x+5$ when $x=3$
b) Find $y-6$ when $y=-1$

Worked Example 6

Substitute the given integer and evaluate using the order of operations:
a) Find $2m+3$ when $m=-4$
b) Find $p÷2+6$ when $p=8$
c) Find $3(q-1)$ when $q=5$

Problems

Problem 1

Evaluate using the order of operations:
a) $5+3\times 4$
b) $16÷4+7$

Problem 2

Evaluate using the order of operations:
a) $(9+1)\times 3$
b) $15-(12÷4)$

Problem 3

Evaluate using the order of operations:
a) $8-2\times 3+1$
b) $30÷5\times 2$

Problem 4

Substitute the given integer and evaluate:
a) Find $c+9$ when $c=5$
b) Find $4d$ when $d=-3$

Problem 5

Substitute the given integer and evaluate:
a) Find $3x+2$ when $x=4$
b) Find $z-8$ when $z=-2$

Problem 6

Substitute the given integer and evaluate using the order of operations:
a) Find $2n+1$ when $n=-5$
b) Find $k÷3+4$ when $k=12$
c) Find $2(r+3)$ when $r=-1$

Exercises

Understanding and Fluency

1. Evaluate each expression using the order of operations:
   a) $4+2\times 6$
   b) $21÷3+8$
   c) $9-3\times 2$

2. Evaluate each expression with brackets:
   a) $(6+2)\times 4$
   b) $18-(10÷2)$
   c) $(12-5)\times 3$

3. Evaluate each numerical expression:
   a) $5+3\times 2-1$
   b) $24÷6+7$
   c) $20-4\times 3$

4. Evaluate each numerical expression:
   a) $16÷4\times 5$
   b) $30-2\times 8$
   c) $(14+4)÷3$

5. Substitute and evaluate:
   a) Find $a+6$ when $a=7$
   b) Find $2b$ when $b=-4$
   c) Find $c-5$ when $c=3$

6. Substitute and evaluate:
   a) Find $3x+1$ when $x=2$
   b) Find $2y-4$ when $y=-3$
   c) Find $z÷2$ when $z=18$

7. Substitute and evaluate using the order of operations:
   a) Find $2m+5$ when $m=6$
   b) Find $3(n-2)$ when $n=4$
   c) Find $p÷3+7$ when $p=-9$

8. Substitute and evaluate using the order of operations:
   a) Find $4q-3$ when $q=-2$
   b) Find $2(r+1)$ when $r=5$
   c) Find $s÷4-2$ when $s=16$

Reasoning

9. Explain why $6+4\times 2$ is not equal to $(6+4)\times 2$.

10. A student says that in $18-6÷2$, you subtract first because subtraction comes before division when reading from left to right. Explain the mistake.

11. Noah substitutes $x=-3$ into $2x+5$ and writes $2+(-3)+5$. Explain why this is incorrect.

12. Explain why brackets are important when substituting a negative integer into an expression such as $3a$ or $a-4$.

13. A student evaluates $24÷6\times 2$ as $24÷12=2$. Describe the error.

Problem-solving

14. A game score is calculated using the expression $10+3b$, where $b$ is the number of bonus stars. Find the score when $b=4$.

15. The cost of hiring bikes is given by $5n+12$, where $n$ is the number of bikes hired. Find the cost when $n=3$.

16. A machine uses the rule $2x-7$. Find the output when the input is $9$.

17. A student evaluates the expression $(4+6)\times 3$ to find the total number of stickers in $3$ identical packs. How many stickers are there altogether?

18. The temperature at noon is modelled by $t÷2+8$. Find the value when $t=-6$.

19. A pattern is described by $3n+2$. Find the value of the pattern when $n=5$ and when $n=-1$.

Potential Misunderstandings

• Students may think operations should always be completed strictly from left to right
• Students may forget that multiplication and division are completed before addition and subtraction
• Students may ignore brackets or complete operations outside the brackets first
• Students may think multiplication must always be done before division, instead of working left to right for operations of equal priority
• Students may think addition must always be done before subtraction, instead of working left to right for operations of equal priority
• Students may confuse an expression with an equation and try to “solve” instead of evaluate
• Students may substitute the wrong value for a pronumeral
• Students may forget that $3x$ means $3\times x$
• Students may substitute a negative integer without brackets and lose the negative sign
• Students may perform substitution correctly but then apply the order of operations incorrectly
• Students may think brackets are only used for grouping operations, rather than also helping show substitution clearly