Success criteria
- I can show how grouping changes the order of work.
- I can keep an equation balanced and undo one operation.
- I can use one rule consistently across a table.
- I can describe what changes and what stays constant.
Explain
(18 + 7) × 4 and 18 + (7 × 4) use the same numbers but not the same instructions. Explain what the brackets tell the reader to do first.
A pan balance models an equation: both sides have the same value. For x + 18 = 47, remove 18 from both pans. The balance stays level and reveals x = 29. Check by replacing x in the original equation.
For x × 6 = 54, divide both sides by 6. The inverse operation isolates x without changing the equality.
| One-step form | Undo while preserving equality |
|---|---|
x + a = b | subtract a from both sides |
x − a = b | add a to both sides |
x × a = b | divide both sides by a |
x ÷ a = b | multiply both sides by a |
Independent practice
Evaluate (18 + 7) × 4.
Evaluate 18 + (7 × 4).
A machine multiplies by 4, then adds 7. What is the output for input 4?
The outputs are 7, 11, 15, 19. What is their constant difference?
Each input rises by 1, so multiplying by 4 makes each output rise by 4.
Start at 9, multiply by 5, then subtract 8.
Solve x + 18 = 47.
29 + 18 = 47.
Solve x × 6 = 54.
9 groups of 6 make 54.
Why must the same inverse operation be applied to both sides of an equation?
An equation is a balance; changing only one side breaks the equality.
Transfer
Invent a two-step function machine whose first four outputs form an increasing pattern. Give only the input-output table to someone else. They must infer and test your rule. Then hide one input, write a one-step equation for it, and show the inverse operation that recovers it.
Delayed check
After one week, explain the difference between 3 × (12 + 5) and (3 × 12) + 5, make a table for “multiply by 6, subtract 2,” and solve one additive and one multiplicative equation, including a subtraction or division form. Keep a drawn balance level and check each solution in the original equation.
Evidence and next step
Save the invented machine, another person’s guess, and your correction at jeremy/portfolio/math/unit-2/function-machine.md.
- Rule inferred, explained, and retained → Unit 2 review; unlock stretch only through the full gate.
- Table correct but rule description vague → revise using “for every input…” language.
- Bracket error → act out each instruction in order before retrying.