Equality and Inequality
Warm-up
I will show you a number sentence. Thumbs up if true, thumbs down if false. 3+4=7 (true). 5+3=9 (false). 8=8 (true: pause here for discussion). 4+6=3+7 (true: this one generates debate). The last example is the central learning moment.
Explore
Pairs work with relational thinking cards: find the missing number in 6+4=5+? without calculating both sides. Reason about the relationship: if one addend increases by 1, what must happen to the other to keep balance? This is genuine algebraic thinking.
Formalize
Introduce the not-equal symbol formally. When the two sides are not equal, we use this symbol. Write 5+3 not-equal-to 10 and explain. Then: is 6+2 not-equal-to 7+2 true or false? True: they are not equal (8 is not 9). This double-negative requires careful reasoning.
Equality and Inequality
Connect the symbols to the pan balance. Every time we write = we are saying the two sides would balance. Every time we write the not-equal symbol we are saying one side would tip. Return to the balance and test equations physically to anchor the abstract symbols.
Practice
Students sort 8 number sentences into TRUE and FALSE, recording reasoning. Exit ticket: write your own TRUE number sentence that has + on both sides of the = sign.
Exit ticket
Students sort 8 number sentences into TRUE and FALSE, recording reasoning. Exit ticket: write your own TRUE number sentence that has + on both sides of the = sign.
Step 1: Expect resistance: many students say FALSE "because there's no answer after the equals sign." This reveals the operator misconception — they read = as "and now write the answer."
Step 2: Move to a pan balance. Put 4 cubes and 6 cubes in the left pan, 3 cubes and 7 cubes in the right pan. The pans level out.
Step 3: Say the sentence in balance language: "4 and 6 balances 3 and 7, because both sides are worth 10." The equals sign MEANS "balances" / "is worth the same as" — not "the answer is coming."
Step 4: Practice with true/false cards: 5 + 5 = 6 + 4 (true), 8 = 8 (true — this one shocks them), 9 = 5 + 3 (false, and they must say WHY: 9 doesn't balance 8).
Students who fix this now avoid the single most damaging misconception in early algebra.
Step 1: Read it as a balance: the left side must end up worth the same as the right side.
Step 2: Compare the sides piece by piece instead of computing: the right side has 9, which is 1 MORE than the left side's 8.
Step 3: So the left side's mystery number must be 1 more than the right side's 4 — it must be 5 — to keep the balance level.
Step 4: Verify by computing (this time): 8 + 5 = 13 and 9 + 4 = 13. ✓
The relational shortcut ("one side has 1 more here, so it needs 1 less there") is real algebraic reasoning. Students who compute both sides get the same answer — but students who compare are thinking structurally, and it's worth naming and praising that move.
Step 1: Build both quantities first: 13 cubes in one row, 9 cubes in a row directly underneath, aligned one-to-one. The extra 4 cubes sticking out make "13 is greater" visible.
Step 2: Introduce the symbol as a record of what they SEE: 13 > 9, read aloud left to right as "13 is greater than 9." The wide end sits with the bigger number; the point aims at the smaller.
Step 3: Now reverse the sentence: 9 < 13, read "9 is less than 13." Same fact, both directions — insist students read the symbol, not just place it.
Watch for: alligator-only students who can aim the mouth but can't READ the sentence. The mnemonic is fine for choosing the symbol; the exit skill is saying "greater than" and "less than" correctly in both directions.