Learning Targets
- I can write and evaluate powers with whole-number exponents (like s²).
- I can write an algebraic expression from a real engineering rule and evaluate it for a given value.
- I can use the distributive property to write equivalent expressions.
- I can use properties of operations (commutative and associative) to regroup a sum the smart way.
Teacher Notes (not printed)
Pacing
- Launch (5 min): Read "The job on the roof." Confront the #1 misconception head-on with Step 1: an exponent is repeated multiplication, so s² means s × s, NOT s × 2.
- Design work (25 min): Steps 1 → 5. Step 2 lets students drag the panel size and watch the math update; Steps 3–4 build and evaluate expressions and use the distributive property; Step 5 regroups a sum using properties of operations.
- Debrief (5 min): Connect 6(s + 4) = 6s + 24 (Step 4) to "wiring two crews two ways," and the smart-regrouping in Step 5 to mental-math efficiency.
Answer Key
- Step 1 — Powers: 4² = 16; 6² = 36; 2³ = 8.
- Step 3 — Evaluate at s = 5: s² = 25; s² + 4 = 25 + 4 = 29; 3s² = 3 × 25 = 75; 0.4 × 29 = 11.6 (cost factor).
- Step 4 — Distribute: 6(s + 4) = 6s + 24 (24 + 6s is also accepted). At s = 3: 4(s² + 9) = 4(9 + 9) = 72, and 4s² + 36 = 36 + 36 = 72 — the two forms match, showing equivalence.
- Step 5 — Regroup the smart way: 17 + 25 + 3 = (17 + 3) + 25 = 20 + 25 = 45 (commutative + associative properties).
Common Misconceptions
- Reading s² as s × 2 (= 10 at s = 5) instead of s × s (= 25). The student self-check reminder reinforces "exponent first."
- Writing 3s² as (3s)² — square first, then multiply by 3, so 3 × 25 = 75, not 15² = 225.
Standard
CCSS 6.EE.A.1–3 — Write and evaluate numerical expressions involving whole-number exponents; write, read, and evaluate algebraic expressions; apply properties of operations to generate equivalent expressions.
Solar Array Engineer
Your school just hired you as a solar engineer. Your job: design the rooftop solar array that powers the building. You will size square panel blocks with powers and exponents, write and evaluate expressions for energy and cost, and use the distributive property to plan the wiring crews — exactly the way real renewable-energy engineers do.
The job on the roof
Solar panels are square. Engineers mount them in square blocks so the wiring stays even — a block that is s panels across is also s panels tall, so it holds s × s = s2 panels. That little raised 2 is an exponent: it tells you how many times to multiply the base by itself. One square block of side 5 is not 5 + 2; it is 52 = 25 panels.
- Power / Potencia: a base raised to an exponent, like 34.
- Base / Base: the number being multiplied (the big number).
- Exponent / Exponente: how many times to multiply the base by itself (the small raised number).
- Expression / Expresión: numbers and letters joined by operations, like s2 + 6.
- Evaluate / Evaluar: replace the letter with a number and find the value.
- Distributive property / Propiedad distributiva: a(b + c) = ab + ac.
Exponents are not multiplication signs
Before you wire anything, prove you can read a power. The diagram shows a square block of side 4. Count the panels — then write it as a power.
A power is repeated multiplication, not "times the little number." 42 means 4 × 4, so 16. 23 means 2 × 2 × 2, so 8.
Español: una potencia es multiplicación repetida. 23 = 2 × 2 × 2 = 8.Why do engineers like square blocks? Compare two layouts that each use 36 panels: a 62 square versus a 4-by-9 rectangle. Which one needs less edge wiring (smaller perimeter)? Justify with the math, not a guess.
Size the array & watch the math
Drag the slider to choose the side length s of your square solar block. The blueprint redraws live, and every expression below recomputes. The school also wants a trim row of 4 extra panels along one edge, so your total-panel expression is s2 + 4.
Total panels: s2 + 4 = 20
Energy: 0.4 × (total) = 8.0 kW
Install cost: 250 × (total) = $5,000
Turn the engineering rules into expressions
Real specs are written in plain words; your job is to translate each rule into an algebraic expression, then evaluate it for a chosen side length. Fill in the table for s = 5.
| Engineering rule (words) | Expression | Evaluate at s = 5 |
|---|---|---|
| Panels in a square block of side s | s2 | |
| Square block plus 4 trim panels | s2 + 4 | |
| Three identical square blocks side by side | 3s2 | |
| Output in kW: four-tenths of the total (s² + 4) | 0.4(s2 + 4) |
Order of operations reminder (open if you're stuck)
Work one step at a time. At s = 5: first 5 × 5 = 25. Then add or multiply. Write each tiny step so you don't skip the exponent.
Español: Primero el exponente: 5 × 5 = 25, luego suma o multiplica.The roof can hold at most 120 panels. Write an inequality using s2 + 4 and find the largest whole-number side length that still fits. Show your reasoning.
Plan the wiring crews two ways
You have 4 wiring crews. Each crew wires one square block of s2 panels plus the same 9 connector panels on the roof edge. So each crew handles (s2 + 9) panels, and all four crews together handle 4(s2 + 9).
The distributive property says you can wire it crew-by-crew, or total the blocks and total the connectors — same answer: 4(s2 + 9) = 4s2 + 36.
How to expand 6(s + 4)
Draw two arrows from the 6 to each term inside the parentheses. Multiply along each arrow. 6(s + 4) → 6s + 24.
Español: Reparte el 6 a cada término: 6 × s y 6 × 4.Go backwards (factor): the budget line reads 12s² + 30. Pull out the greatest common factor and write it as 6(2s² + 5). Prove the two forms are equal by evaluating both at s = 2.
Regroup the count the smart way
On install day the crews report panel counts out of order. Use the commutative and associative properties to regroup the numbers so they are easy to add in your head — the total never changes.
| Property | What it lets you do | Solar example |
|---|---|---|
| Commutative | Change the order | 17 + 25 + 3 = 17 + 3 + 25 |
| Associative | Change the grouping | (17 + 3) + 25 = 20 + 25 |
| Identity | Adding 0 / multiplying by 1 keeps it the same | s2 + 0 = s2 |
Look for pairs that make a 10 or a 20. Here, 17 + 3 = 20, then 20 + 25 = 45.
Español: Busca pares que sumen 10 o 20 primero. 17 + 3 = 20, luego + 25 = 45.Which property lets you rewrite 3 · (s · 4) as 12s? Name it and explain each move you make.
Generate your Solar Array Spec Sheet
Lock in a final side length s, then generate your spec sheet. A finished spec means a real, buildable plan an installer could read.
Student self-check — tick each one before you turn it in:
- I wrote at least one power (s², 2³, …) and evaluated it correctly (exponent first).
- I wrote an expression from words and evaluated it for my chosen side length.
- I used the distributive property to show 4(s² + 9) = 4s² + 36 and proved both forms give the same value.
- I used a property of operations (commutative / associative) to regroup a sum.
- My spec sheet numbers are consistent with the live blueprint.
Engineer's Sign-off
When your powers are evaluated, your expressions are correct, and you have applied the distributive property, submit your design report. Enter your name in the field below, then press Check & Submit to save your score as a PDF or DOC.
Performance Rubric — Solar Array Engineer (6.EE.A.1–3)
| Level | Score | Descriptor |
|---|---|---|
| 4 — Exceeds | 11 / 11 | Evaluates all powers correctly (exponent = repeated multiplication), writes and evaluates all four expressions accurately at s = 5, applies the distributive property in both directions, and uses properties of operations to regroup. Spec sheet is complete and internally consistent. |
| 3 — Meets | 9–10 / 11 | Evaluates most powers and expressions correctly with at most two slips. Applies the distributive property and reaches a correct regroup. Minor arithmetic error only. |
| 2 — Approaching | 6–8 / 11 | Can evaluate simple powers but confuses exponent with multiplication by the exponent (e.g., 4² = 8) in some cases. Partial success on expressions and/or distributive property. Needs more practice with order of operations. |
| 1 — Beginning | 0–5 / 11 | Consistently misreads powers or cannot write an expression from words. Little or no correct use of the distributive property. Needs reteaching on exponents and algebraic expressions. |