Symbolic Equality and Inequality
Warm-up
True or false? Show 4 sentences rapidly: 40 + 32 = 42 + 30 (true); 56 - 20 = 56 - 22 (false); 35 + 40 = 30 + 45 (true); 67 + 0 = 67 (true). Students show thumbs up or down. Discuss one that surprised the class.
Explore
Relational thinking challenge cards: each card shows an equation with one value missing. 53 + 24 = 54 + ?. Students solve WITHOUT adding either side. Write the reasoning: 54 is 1 more than 53, so the second addend must be 1 less: 23. Verify by computing if uncertain.
Formalize
Explore the subtraction pattern: 50 - 20 = 49 - 19 (true). 50 - 20 = 48 - 18 (true). What is the pattern? Both numbers decrease by the same amount and the difference stays the same. This is equivalent to 50 - 20 = (50-2) - (20-2): subtracting the same amount from both. Record this insight as a class generalization.
Symbolic Equality and Inequality
Connect to the pan balance: any operation you do to one side must be done to the other to maintain balance. Adding 5 to the left side without adding 5 to the right destroys equality. This is the first algebraic principle: what you do to one side, you must do to both.
Practice
Students sort 10 number sentences into TRUE and FALSE, recording reasoning for each. Solve 4 relational thinking problems without computing both sides. Exit ticket: 44 + 19 = 45 + ?, find the missing number without adding.
Exit ticket
Students sort 10 number sentences into TRUE and FALSE, recording reasoning for each. Solve 4 relational thinking problems without computing both sides. Exit ticket: 44 + 19 = 45 + ?, find the missing number without adding.
Step 1: Resist computing! Ask first: "can we DECIDE by comparing the sides?" Left is 27 + 8; right is 30 + 5.
Step 2: Compare piecewise: 30 is 3 more than 27. For balance, the right side's other number must be 3 LESS than 8 — and 5 is exactly 3 less than 8.
Step 3: Verdict: TRUE, by pure structure. One side traded 3 from one number to the other; the total can't feel the trade.
Step 4: Only now verify by arithmetic for the skeptics: 27 + 8 = 35 and 30 + 5 = 35 ✓.
Name the move: this is the "give-and-take" (compensation) structure. Students who can SEE it are simultaneously practicing mental math and pre-algebra. Follow up: is 46 + 9 = 50 + 5 true? (46→50 took 4; 9→5 gave 4 back: TRUE.)
Step 1: Read the whole sentence before touching anything. The equals sign is a balance point: whatever the right side is worth, the left side must match it.
Step 2: Evaluate the side WITHOUT the mystery: 14 − 3 = 11. Now the sentence says 6 + ? = 11.
Step 3: Solve the friendly equation: 6 + 5 = 11, so ? = 5.
Step 4: Substitute back and read the balanced sentence: 6 + 5 = 14 − 3 → 11 = 11 ✓.
Watch for the classic error: answering 8 (from 6 + 8 = 14) because the student stopped reading at the equals sign — treating = as "answer time" instead of "balance." When it appears (it will), send them back to the pan balance: put 6-and-? in one pan, 14-take-3 in the other.
Step 1: Check the claim: 5 + 9 = 14 and 4 + 8 = 12. The pans don't balance: 14 ≠ 12. The sentence is FALSE.
Step 2: Instead of just declaring it wrong, repair it — and find ALL the one-number repairs: change the 4 to 6 (5 + 9 = 6 + 8 ✓), or the 8 to 10 (5 + 9 = 4 + 10 ✓), or the 5 to 3 (3 + 9 = 4 + 8 ✓), or the 9 to 7 (5 + 7 = 4 + 8 ✓).
Step 3: For each repair, articulate the reasoning: the right side was 2 LIGHT, so add 2 somewhere on the right — or make the left side 2 lighter.
Step 4: The takeaway: equality is something you can restore by balancing amounts, not just check. "How much is it off by, and where do I put the difference?" is the exact mindset later used for solving equations.