Week 7 · Expressions in real life and equivalent expressions · 3 lessons
2
Are two expressions equivalent? Testing and balancing
Discover Mathematics P6: Term 1, Unit 3, Lessons 6–7
GoalDecide whether two expressions are equivalent by testing several values and by using a balance.
New words
equivalent expressions · two expressions that have the same value for every value of the lettertest · try the same number in both expressions and comparecounterexample · one value where two expressions differ, which proves they are not equivalent
Explanation
Two expressions are equivalent if they give the same value for every number we put in. To test, choose several values and work out both expressions. If they ever differ, one value is enough to prove they are NOT equivalent. If they always agree for the values you tried, they are probably equivalent, but one value that agrees proves nothing: x + x and x × x are both 4 when x = 2, yet when x = 3 they are 6 and 9. A pan balance helps: two pans with x + x and 2x stay level for any weight of x.
Worked examples
x
1
2
5
2(x + 3)
8
10
16
2x + 6
8
10
16
Is 2(x + 3) equivalent to 2x + 6? Test x = 1, 2 and 5. The values are 8, 10, 16 for both. They always agree, so they are equivalent.
x
1
2
2(x + 3)
8
10
2x + 3
5
7
Is 2(x + 3) equivalent to 2x + 3? When x = 1 the values are 8 and 5. They differ, so they are NOT equivalent.
x
2
3
5
x + x
4
6
10
x × x
4
9
25
One agreeing value is not proof. x + x and x × x are both 4 when x = 2. But for x = 3 they are 6 and 9, and for x = 5 they are 10 and 25. So they are NOT equivalent.
Class activity
Detective pairs. A grown-up writes two expressions such as 4(x + 1) and 4x + 4. Test them with three numbers each and decide whether they are equivalent. Then the grown-up writes a pair that agrees at one value only. Can you find the value where they differ?
Practice
1
Test 3(x + 1) and 3x + 3 with x = 2 and x = 5. Are they equivalent?
x = 2: 9 and 9. x = 5: 18 and 18. They agree, so they are equivalent.
2
Are x + 4 and 4 + x equivalent?
Yes. Addition can be done in any order, so they have the same value for every x.
3
Are 2x and x + 2 equivalent? Test x = 2 and x = 5.
x = 2: 4 and 4. x = 5: 10 and 7. They differ at x = 5, so they are not equivalent.
4
Mina tests x = 3 and finds 5x = 15 and x + 12 = 15. She says they are equivalent. What do you say?
One value is not enough. Try x = 1: 5x = 5 but x + 12 = 13. They are not equivalent.
5
A balance has x, x, x on the left and 3x on the right. Does it stay level whatever x weighs?
Yes. Three x’s are always the same as 3x, so x + x + x and 3x are equivalent.
