C. Algebra · Algebra: Expressions, Equations and Coding

Are They Equivalent? Comparing Expressions Four Ways

C1.3 A1About 70 minutes

Learning goals

I can use algebra tiles to show that two expressions are equivalent.
I can use a table of values to find a counterexample, which proves two expressions are not equivalent.
I can compare the graphs of two expressions to see whether they are equivalent.
I can simplify expressions algebraically to prove they are equivalent, and justify my choice.

Words to know

equivalent expressions
Expressions that have the same value for every value of the variable, such as 2(x + 3) and 2x + 6.

counterexample
One example that shows a statement is false. One value of x that gives two different results proves the expressions are not equivalent.

substitute
Replace a variable with a number to find the value of an expression.

table of values
A table that lists values of x and the matching values of one or more expressions.

justify
Give clear reasons or evidence that support your answer.

Let's start with a disagreement

A family packs 2 winter car kits. Each kit has x hand warmers plus 3 granola bars. Three students write an expression for the total number of items:

Priya writes 2(x + 3).
Tomás writes 2x + 6.
Jesse writes 2x + 3.

Are all three right? They look different. In this lesson you'll learn four ways to decide whether expressions are equivalent, and how to back up your answer.

Way 1: Concrete (algebra tiles)

Build each expression with tiles. If both use exactly the same tiles (after removing any zero pairs), they are equivalent.

2(x + 3) means 2 groups of one x-tile and three 1-tiles.

Two groups of (x + 3) contain 2 x-tiles and 6 unit tiles altogether, so 2(x + 3) = 2x + 6. Jesse's 2x + 3 would need only 3 unit tiles, so it is not the same.

Way 2: Numerical (a table of values)

Substitute the same values of x into each expression.

x	2(x + 3)	2x + 6	2x + 3
−2	2	2	−1
0	6	6	3
1	8	8	5
5	16	16	13

The first two columns match for every value tested. The last column is different every time.

One value that gives different results is enough to prove two expressions are NOT equivalent. At x = 0, 2x + 3 gives 3 but 2(x + 3) gives 6. That is a counterexample.
Matching values make equivalence likely, but checking a few numbers is not a proof by itself. Choose a mix of values: negative, zero, and positive.

Way 3: Graphical

Graph y = each expression. Equivalent expressions give exactly the same graph. Different expressions give different graphs, even if they cross at a point.

y = 2x + 6 and y = 2(x + 3) are the same line: graph both and the second lands exactly on top of the first. y = 2x + 3 is a different line: parallel, 3 units lower, so it is not equivalent.

Way 4: Algebraic

Simplify each expression using properties of operations. If they simplify to the same expression, they are equivalent, for every value of x. This is the strongest evidence: it is a proof.

2(x + 3) = 2x + 2(3) = 2x + 6 (distributive property). ✓

2x + 3 cannot be simplified to 2x + 6, so it is not equivalent.

Worked example

Example 1: Prove with algebra, check with numbers

Are 4(x + 1) − x and 3x + 4 equivalent?

Numerical check: x = 0 gives 4(1) − 0 = 4 and 3(0) + 4 = 4. x = −2 gives 4(−1) − (−2) = −4 + 2 = −2 and 3(−2) + 4 = −2. They match so far.
Algebra: 4(x + 1) − x = 4x + 4 − x.
Collect the x-terms: 4x − x = 3x. So the expression is 3x + 4.

Yes, they are equivalent. The algebra proves it for every value of x; the numbers back it up.

Worked example

Example 2: One match is not enough

Is (x + 1)² equivalent to x² + 1?

Try x = 0: (0 + 1)² = 1 and 0² + 1 = 1. They match! But that is only one value.
Try x = 2: (2 + 1)² = 9, but 2² + 1 = 5. Different.
x = 2 is a counterexample.

Not equivalent. (x + 1)² means (x + 1)(x + 1), which is not the same as squaring each part. The graphs below show it.

The graphs of y = (x + 1)² and y = x² + 1 meet only at the point (0, 1). Everywhere else they are different, so the expressions are not equivalent.

Worked example

Example 3: Sort and justify

Which of these are equivalent to 5x − 10?

A: 5(x − 2) B: 5(x − 10) C: −5(2 − x) D: 10 − 5x

A: 5(x − 2) = 5x − 10. ✓
B: 5(x − 10) = 5x − 50. ✗ (Counterexample: x = 0 gives −50, not −10.)
C: −5(2 − x) = −10 + 5x = 5x − 10. ✓ (Multiplying −5 by −x gives +5x.)
D: 10 − 5x has the opposite sign on every term. ✗ (x = 0 gives 10, not −10.)

A and C are equivalent to 5x − 10.

Choosing the right evidence

To show expressions are not equivalent: give one counterexample.
To show they are equivalent: simplify them to the same expression (or show the same tiles, or the same graph).
Avoid testing only x = 0 or x = 1. These values hide many mistakes: at x = 1, x² and x and x³ are all equal to 1!

Hands-on activity

Expression sort card game

You need: 15 small pieces of paper or index cards, a pen, your algebra tiles if you made them.

Copy one expression on each card:

2(x + 3), 2x + 6, x + x + 6, 3x + 6 − x
3(x − 1), 3x − 3, 4x − 3 − x, −3(1 − x)
x² + 2x, x(x + 2), x² + x + x
decoys: 2x + 3, 3x − 1, x² + 2, (x + 3)²

Shuffle them. Sort them into groups of equivalent expressions.
For each group, prove it with algebra or tiles. For each decoy, find a counterexample against the group it looks like.
Challenge someone at home: give them two cards and ask “Same or different?” They must justify.

Model answer: the three groups are listed above. For example, 2x + 3 vs 2x + 6: at x = 0, 3 ≠ 6. 3x − 1 vs 3x − 3: at x = 0, −1 ≠ −3. x² + 2 vs x² + 2x: at x = 2, 6 ≠ 8. (x + 3)² vs x² + 2x: at x = 0, 9 ≠ 0.

Just for fun

2(x + 3) and 2x + 6 went to a costume party dressed completely differently.

The host recognized them anyway: whatever value you gave them, they always gave the same answer.

Your turn

Are 5(x + 2) and 5x + 2 equivalent? Justify.

Describe how algebra tiles show that 2(x + 4) = 2x + 8.

Which are equivalent to 6x + 12? 6(x + 2), 3(2x + 4), 6(x + 12), 2(3x + 6)

Evaluate x² + x and x(x + 1) for x = 3 and x = −2. Then prove they are equivalent.

Kai says x + x is the same as x². Give a counterexample.

If you graphed y = 3(x − 1) and y = 3x − 3, what would you see? Why?

If you graphed y = x + 4 and y = x + 2, what would you see? Are the expressions equivalent?

More practice: warm-up

Simplify 3(x + 1). Is it equivalent to 3x + 3?

Is x + x + x equivalent to 3x?

Is 4x − x equivalent to 4?

Evaluate 2(x − 5) and 2x − 10 at x = 7.

Is 2x + 1 equivalent to 1 + 2x? Which property tells you?

More practice: on your own

Are 2(3x − 1) + 4 and 6x + 2 equivalent?

Are 10 − 2(x + 3) and 4 − 2x equivalent? Check with x = 1 too.

Are (x + 3)² and x² + 9 equivalent?

A rectangle has length x + 5 and width x. Maya writes the perimeter as 2(x + 5) + 2x, Jordan writes 4x + 10, and Lee writes 2(2x + 5). Who is right?

Is −(x − 4) equivalent to 4 − x?

Challenge

The expression ax + b gives the same values as 3x + 7 when x = 0 and when x = 1. Find a and b. Why does this make the expressions equivalent?

For which values of x do x² and 2x give the same result? Explain why they are still not equivalent.

Write three different expressions equivalent to 12x − 8: one with brackets, one with three terms, and one with a negative number outside brackets. Prove each.

Check your understanding

Quiz

Which expression is equivalent to 3(x + 4)?

Two expressions give different values when x = 5. What can you conclude?

If two expressions are equivalent, their graphs:

Which value of x shows that (x + 1)² and x² + 1 are NOT equivalent?

Which expression is NOT equivalent to 4x − 8?

How do algebra tiles show 2(x + 3)?

For parents and teachers

Big idea: C1.3 asks students to compare algebraic expressions using concrete, numerical, graphical and algebraic methods to identify those that are equivalent, and to justify their choices. The key idea: one counterexample proves expressions are not equivalent, but a few matching values do not prove they are. Simplifying algebraically is a proof.

Common difficulties:

Distributing to only the first term: writing 2(x + 3) as 2x + 3.
Testing only x = 0 or 1, which can hide differences.
Thinking (x + 1)² = x² + 1. This is a very common error that continues into Grade 10 and beyond.

How to help: Play the card-sort game together. When your teen says two expressions match, ask “How do you know for sure?” and expect either algebra or tiles. When they say two don't match, ask for the counterexample.