035. Equivalent Expressions and Algebraic Generalisation

Learning Intentions

• To know what it means for two expressions to be equivalent
• determine whether two expressions are equivalent using substitution
• generalise number facts using algebra

Pre-requisite Summary

• Understand that a variable can represent a number
• Be able to substitute values into algebraic expressions
• Be able to evaluate expressions using order of operations
• Understand that an expression does not have an equals sign
• Know that two numerical calculations can have the same value even if they look different
• Be able to recognise simple arithmetic patterns
• Understand that algebra can describe a rule that works for many numbers
• Be able to use letters to represent any whole number

Worked Examples

Worked Example 1

a) Explain what it means for the expressions $x+x$ and $2x$ to be equivalent.
b) Test the expressions when $x=3$.
c) Test the expressions when $x=7$.

Worked Example 2

Use substitution to decide whether the expressions are equivalent:
a) $3x+2$ and $2x+5$
b) $4n+4$ and $4(n+1)$

Worked Example 3

Use substitution to decide whether the expressions are equivalent:
a) $2(a+3)$ and $2a+6$
b) $5m-2$ and $2+5m$

Worked Example 4

Write an algebraic generalisation for each number fact:
a) an even number can be written as $2n$
b) the sum of two consecutive numbers
c) the product of $3$ and any whole number

Worked Example 5

Generalise the pattern:
a) $1+2=3$
b) $2+3=5$
c) $3+4=7$
Write a rule for the sum of two consecutive whole numbers.

Worked Example 6

Generalise a number fact using algebra:
a) the sum of two even numbers is even
b) the product of an odd number and $2$ is even
c) write each statement using algebra

Problems

Problem 1

a) Explain what it means for the expressions $y+y$ and $2y$ to be equivalent.
b) Test the expressions when $y=4$.
c) Test the expressions when $y=9$.

Problem 2

Use substitution to decide whether the expressions are equivalent:
a) $2x+6$ and $x+8$
b) $3(n+2)$ and $3n+6$

Problem 3

Use substitution to decide whether the expressions are equivalent:
a) $4(a+1)$ and $4a+4$
b) $6m-1$ and $1+6m$

Problem 4

Write an algebraic generalisation for each number fact:
a) an odd number can be written as $2n+1$
b) the sum of two consecutive numbers
c) the product of $4$ and any whole number

Problem 5

Generalise the pattern:
a) $2+3=5$
b) $3+4=7$
c) $4+5=9$
Write a rule for the sum of two consecutive whole numbers.

Problem 6

Generalise a number fact using algebra:
a) the sum of two odd numbers is even
b) the product of any whole number and $5$ is a multiple of $5$
c) write each statement using algebra

Exercises

Understanding and Fluency

1. Decide whether each pair of expressions is equivalent by substituting $x=2$ and $x=5$:
   a) $x+x$ and $2x$
   b) $3x+1$ and $x+3$
   c) $2(x+4)$ and $2x+8$

2. Decide whether each pair of expressions is equivalent by substituting suitable values:
   a) $4y+3$ and $3+4y$
   b) $5(y+1)$ and $5y+1$
   c) $3a+6$ and $3(a+2)$

3. Use substitution to test equivalence:
   a) $2m+2m$ and $4m$
   b) $6n-3$ and $3+6n$
   c) $7p+0$ and $7p$

4. Use substitution to test equivalence:
   a) $3(x+2)$ and $3x+6$
   b) $4(x+1)$ and $4x+1$
   c) $2q+5$ and $5+2q$

5. Write an algebraic expression for each statement:
   a) any even number
   b) any odd number
   c) three times any whole number

6. Write an algebraic generalisation for each fact:
   a) the next number after $n$
   b) two consecutive numbers
   c) three consecutive numbers

7. Generalise each number fact using algebra:
   a) an even number plus an even number
   b) an odd number plus an odd number
   c) an even number plus an odd number

8. Generalise each number fact using algebra:
   a) the sum of two consecutive numbers
   b) the product of $2$ and any whole number
   c) the sum of a number and $5$

Reasoning

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

10. A student says that $2(x+3)$ and $2x+3$ are equivalent because both contain $2$ and $3$. Explain the mistake.

11. Explain why substitution can be used to test whether two expressions are equivalent.

12. A student tests $3x+2$ and $2x+5$ with $x=3$ and gets the same value, then says the expressions must be equivalent. Explain why more care is needed.

Problem-solving

13. A student claims that $4(n+2)$ and $4n+8$ are equivalent. Use substitution with two values of $n$ to check the claim.

14. Write an algebraic rule for the perimeter of a square with side length $s$. Explain how this generalises repeated addition.

15. Two consecutive whole numbers are added. Write an algebraic expression for the sum and test it for $n=6$.

16. Write an algebraic expression for an odd number and the next odd number. Then write an expression for their sum.

17. A teacher says “the sum of any whole number and the next whole number is always odd”. Write this using algebra.

18. A pattern shows:
    $1+3=4$
    $2+4=6$
    $3+5=8$
    Write an algebraic generalisation for the sum.

Potential Misunderstandings

• Students may think two expressions are equivalent only if they look the same
• Students may think expressions with the same numbers are automatically equivalent
• Students may use substitution incorrectly by replacing only one occurrence of a variable
• Students may think one successful substitution proves equivalence in every case
• Students may confuse an expression with an equation
• Students may not recognise that algebraic generalisation describes a rule for all suitable numbers
• Students may write examples instead of a general algebraic rule
• Students may confuse consecutive numbers with multiples or with numbers that differ by $2$

Next: 036. Like Terms and Simplifying Expressions