133r. Review Evaluating and Comparing Algebraic Expressions

Learning Intentions

• substitute values to evaluate algebraic expressions
• To understand what it means for two expressions to be equivalent
• To understand how the commutative and associative laws for arithmetic can be used to determine equivalence
• show that two expressions are not equivalent using substitution

Pre-requisite Summary

• Know that a pronumeral or variable stands for a number
• Be able to replace a pronumeral with a given value
• Understand that multiplication may be written without the $\times$ sign, for example $3x$
• Know the difference between an expression and an equation
• Recall the commutative law for addition and multiplication
• Recall the associative law for addition and multiplication
• Be able to evaluate numerical expressions accurately
• Understand that equivalent expressions always have the same value for the same substitution

Worked Examples

Worked Example 1

Substitute the given value and evaluate:
a) Find $3x+4$ when $x=2$
b) Find $2a-5$ when $a=7$
c) Find $mn$ when $m=3$ and $n=4$

Worked Example 2

Substitute the given value and evaluate:
a) Find $4p+1$ when $p=-2$
b) Find $q-6$ when $q=9$
c) Find $2r+3$ when $r=0$

Worked Example 3

Decide whether each pair of expressions is equivalent:
a) $x+5$ and $5+x$
b) $(a+2)+3$ and $a+(2+3)$
c) $2m+3$ and $3m+2$

Worked Example 4

Use the commutative or associative law to show that the expressions are equivalent:
a) $4+y$ and $y+4$
b) $(b+6)+1$ and $b+(6+1)$
c) $2\times (3\times t)$ and $(2\times 3)\times t$

Worked Example 5

Use substitution to show that two expressions are not equivalent:
a) $x+2$ and $2x$
b) $3n+1$ and $n+3$
c) $ab+5$ and $a+b+5$

Worked Example 6

For each pair, decide whether the expressions are equivalent. Use a law if possible, or substitution if needed:
a) $m+7$ and $7+m$
b) $2(p+4)$ and $2p+4$
c) $(x+1)+2$ and $x+(1+2)$

Problems

Problem 1

Substitute the given value and evaluate:
a) Find $5x+2$ when $x=3$
b) Find $3a-4$ when $a=6$
c) Find $cd$ when $c=2$ and $d=9$

Problem 2

Substitute the given value and evaluate:
a) Find $2p+7$ when $p=-3$
b) Find $k-8$ when $k=12$
c) Find $4r-1$ when $r=0$

Problem 3

Decide whether each pair of expressions is equivalent:
a) $y+6$ and $6+y$
b) $(c+4)+2$ and $c+(4+2)$
c) $3m+2$ and $2m+3$

Problem 4

Use the commutative or associative law to show that the expressions are equivalent:
a) $8+t$ and $t+8$
b) $(p+5)+4$ and $p+(5+4)$
c) $5\times (2\times q)$ and $(5\times 2)\times q$

Problem 5

Use substitution to show that two expressions are not equivalent:
a) $x+4$ and $4x$
b) $2n+5$ and $n+2$
c) $ab+3$ and $a+b+3$

Problem 6

For each pair, decide whether the expressions are equivalent. Use a law if possible, or substitution if needed:
a) $w+9$ and $9+w$
b) $3(r+2)$ and $3r+2$
c) $(y+2)+4$ and $y+(2+4)$

Problems

Understanding and Fluency

1. Substitute the given value and evaluate:
   a) $2x+3$ when $x=4$
   b) $5a-1$ when $a=3$
   c) $mn$ when $m=6$ and $n=2$

2. Substitute the given value and evaluate:
   a) $3p+4$ when $p=-1$
   b) $q-7$ when $q=10$
   c) $2r+5$ when $r=0$

3. Decide whether each pair of expressions is equivalent:
   a) $x+8$ and $8+x$
   b) $(a+3)+4$ and $a+(3+4)$
   c) $2m+1$ and $m+2$

4. Decide whether each pair of expressions is equivalent:
   a) $y+9$ and $9+y$
   b) $(b+5)+2$ and $b+(5+2)$
   c) $4n+3$ and $3n+4$

5. Use the commutative or associative law to justify equivalence:
   a) $6+t$ and $t+6$
   b) $(k+1)+7$ and $k+(1+7)$
   c) $3\times (4\times x)$ and $(3\times 4)\times x$

6. Use substitution to show that the expressions are not equivalent:
   a) $x+3$ and $3x$
   b) $2p+4$ and $p+2$
   c) $ab+1$ and $a+b+1$

7. For each pair, decide whether the expressions are equivalent. Use a law or substitution:
   a) $m+10$ and $10+m$
   b) $(q+2)+5$ and $q+(2+5)$
   c) $2(x+1)$ and $2x+1$

8. Evaluate both expressions for the same substitution:
   a) $x+4$ and $4+x$ when $x=7$
   b) $2n+3$ and $n+2$ when $n=5$
   c) $(a+6)+2$ and $a+(6+2)$ when $a=1$

Reasoning

9. Explain what it means for two expressions to be equivalent.

10. A student says that $x+5$ and $5x$ are equivalent because they both use $x$ and $5$. Explain the mistake.

11. Noah says that if two expressions give the same value for one substitution, then they must be equivalent. Is he correct? Explain.

12. Explain how the commutative law can be used to show that $p+7$ and $7+p$ are equivalent.

13. A student says that $(x+2)+3$ and $x+2+3$ are not equivalent because the brackets are different. Describe the error.

Problem-solving

14. A taxi fare can be written as $4+2k$, where $k$ is the number of kilometres travelled. Find the fare when $k=5$.

15. Two students write the total number of apples as $a+3$ and $3+a$. Explain whether these expressions are equivalent.

16. A pattern rule is written as $2n+1$. Another student writes $n+2$. Use substitution to decide whether the two rules are equivalent.

17. A builder writes the total length as $(l+2)+5$. Another builder writes $l+(2+5)$. Explain whether the expressions are equivalent.

18. A shop writes one total cost as $3x+2$ and another as $2x+3$. Use substitution to show whether the two expressions are equivalent.

19. A student claims that $2(a+4)$ and $2a+4$ are equivalent. Test the claim using substitution.

Potential Misunderstandings

• Students may think two expressions are equivalent if they look similar
• Students may think two expressions are equivalent if they use the same variables and numbers
• Students may confuse evaluating an expression with solving an equation
• Students may substitute a value into one expression correctly but not the other
• Students may forget that multiplication is implied in expressions such as $3x$
• Students may think the commutative law applies to subtraction in the same way as addition
• Students may think the associative law changes the order of terms, rather than the grouping
• Students may believe that matching one substitution is enough to prove equivalence
• Students may not recognise that one counterexample is enough to show two expressions are not equivalent