CCSS 6.NS.A.1 — Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, including mixed numbers.
You are the timber engineer on the Cedar Marsh Park crew. A raised wooden walkway must cross a protected wetland. Your job: cut planks, space support posts, and ration sealant so the boardwalk is safe, level, and built with no wasted lumber. Every decision is a fraction-division problem.
A wetland boardwalk is a raised wooden walkway. It is built from three parts you must engineer — and each one is a question of how many equal-size pieces fit into a length, or how big each share becomes when a total is split. Both are answered by dividing.
Dividing answers "how many equal groups fit?" To divide by a fraction, flip it and multiply (use the reciprocal). Example: 6 ÷ ½ asks "how many halves fit in 6?" — and 6 × 2 = 12.
Español: Dividir responde "¿cuántos grupos iguales caben?" Para dividir entre una fracción, invierte y multiplica (usa el recíproco). Ejemplo: 6 ÷ ½ = 6 × 2 = 12.The park issued the official build specs. Hit every number exactly — the wetland permit allows no extra cuts or wasted lumber. Notice every value is a length or amount being split into equal fraction-size pieces.
| Component | Constraint | Value |
|---|---|---|
| Trail span (length of walkway) | Total distance to bridge | 8¼ ft |
| Decking plank width | Each board across the walkway | ¾ ft |
| Support-post spacing | Largest gap allowed between posts | 2¾ ft |
| Railing baluster stock | One long board you cut up | 12 ft |
| Baluster length | Each upright railing piece | ⅔ ft |
| Sealant supply | Total weatherproofing on hand | 5½ gal |
| Sealant per section | Used to coat each trail section | ⅓ gal |
Two pictures explain every problem on this page. A measurement bar shows "how many fit," and a share bar shows "split into equal parts."
Drag the green handle on the plan to stretch the trail span, or use the sliders to set the span and plank width. The scale plan redraws the planks live and the console reports how many whole planks fit and how much board is left over. Use it to see what span ÷ width means before you compute.
Set the span to 8¼ and the plank width to ¾. Count the planks the console draws and check the leftover. Then do the same calculation by hand using the "flip and multiply" method for Q2.1 below.
Español: Pon el tramo en 8¼ y el ancho en ¾. Cuenta los tablones que aparecen, luego haz el cálculo a mano para Q2.1.Use the spec sheet values. Each question asks "how many equal-size pieces fit?" — set up the division, convert any mixed numbers to improper fractions, then multiply by the reciprocal.
Q2.1 — Decking planks across the 8¼ ft span (¾ ft each)
Q2.2 — Balusters cut from one 12 ft railing board (⅔ ft each)
Q2.3 — Support-post spacing. The 8¼ ft span must be divided into equal gaps of exactly 2¾ ft between posts. How many gaps does that make?
Q2.3 gives 3 gaps. Three equal gaps need how many posts? (Think fence-post counting: a gap has a post on each end.) Be ready to defend your post count in the build report.
Turn mixed numbers into improper fractions first: 8¼ = 33/4, 2¾ = 11/4. Then flip the second fraction and multiply.
Español: Convierte a fracción impropia (8¼ = 33/4), luego invierte la segunda fracción y multiplica.Now the trickier engineering judgment: dividing does not always give a whole answer, and sometimes you divide to share, not to count.
Q3.1 — You have 5½ gal of sealant. Each trail section needs ⅓ gal. Compute the exact quotient 5½ ÷ ⅓ (enter the mixed-number value as a decimal).
Q3.2 — You cannot coat half a section. How many sections can be fully coated? (Interpret the remainder.)
Q3.3 — Fair share: ¾ gal of touch-up stain is split equally among 6 crew members. How much does each get, in gallons? (Enter a decimal.)
"How many fit" → flip and multiply. "Split a total into equal shares" → still divide: a fraction ÷ a whole number means multiply by 1 over that number.
Español: Dividir una fracción entre un número entero = multiplicar por 1 sobre ese número. ¾ ÷ 6 = ¾ × ⅙.Q3.1 gives a decimal quotient; Q3.2 requires a whole-number answer. In your build report, explain: when does a real engineer keep the fractional part of a quotient, and when must they round down to the nearest whole unit? Give one example from this boardwalk where each rule applies.
New request from the park manager: minimize wasted lumber. The supplier is out of ¾ ft planks. Choose an available width for the 8¼ ft span; the panel shows the live quotient and leftover. Pick the width with the least leftover.
| Width | 8¼ ÷ width | Whole | Leftover |
|---|---|---|---|
| ½ ft | 16½ | 16 | ¼ ft |
| ⅝ ft | 13⅕ | 13 | ⅛ ft |
| 1 ft | 8¼ | 8 | ¼ ft |
Q4.1 — Which width leaves the least lumber wasted? Enter its leftover in feet as a decimal.
Confirm with the math: 8¼ ÷ ⅝ = 33/4 × 8/5 = 13.2 planks. The 0.2 of a plank is the part that does not fit — convert it back to feet (0.2 × ⅝). Why is the smallest leftover the best engineering choice here?
When all build checks are approved, hand your plan to the park. Enter your name in the field below, then press Submit build report & grade to save your score as a PDF or DOC.
| Level | Score | Descriptor |
|---|---|---|
| 4 — Exceeds | 7 / 7 | Divides whole and mixed numbers by fractions accurately using the reciprocal, distinguishes "how many fit" from "fair share," interprets the remainder correctly, and justifies the least-waste plank width. |
| 3 — Meets | 5–6 / 7 | Computes most quotients correctly with at most one slip. Reaches a valid build with correct plank, baluster, and section counts. |
| 2 — Approaching | 3–4 / 7 | Divides simple cases but forgets to use the reciprocal, misreads a mixed number, or does not interpret the remainder. |
| 1 — Beginning | 0–2 / 7 | Few correct quotients. Confuses multiplying and dividing by a fraction. Needs reteaching on the reciprocal method. |