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4.8H — Eating Up the Lunchroom Budget
Human review completed .
Use matrix multiplication to turn quantities and unit prices into total costs. The packet labels this section 4.8H.
1. In class: A Solidify Understanding Task
From a row of quantities to a total cost
The packet continues the food-order example from Section 4.7H. The order matrix has rows for soccer and drama, and columns for chips, cookies, and drinks:
The price matrix uses those same food items as its rows, and Mainstreet and Grandpa’s as its columns. The prices are dollars per package of chips, per dozen cookies, and per gallon of drink:
Worked example 1 — Find one entry of the cost matrix
Original problem — Main 1: Find the four entries a, b, c, and d in the total-cost matrix. Start with a, the soccer order’s cost at Mainstreet.
Worked solution: Use the soccer row from Q and the Mainstreet column from P. Multiply each quantity by its matching unit price, then add:
| Food item | Quantity | Mainstreet unit price | Cost |
|---|---|---|---|
| Chips | 8 | $2.50 | $20.00 |
| Cookies | 7 | $3.00 | $21.00 |
| Drinks | 4 | $2.00 | $8.00 |
This is matrix multiplication: each entry of the product comes from one row of the first matrix and one column of the second. Multiply the matching entries and add those products.
Complete the product — Main 1–2
| Entry | Calculation | Total |
|---|---|---|
| a · Soccer / Mainstreet | \(8(2.50)+7(3.00)+4(2.00)\) | $49.00 |
| b · Soccer / Grandpa’s | \(8(2.00)+7(4.00)+4(1.50)\) | $50.00 |
| c · Drama / Mainstreet | \(10(2.50)+4(3.00)+4(2.00)\) | $45.00 |
| d · Drama / Grandpa’s | \(10(2.00)+4(4.00)+4(1.50)\) | $42.00 |
The result has event rows and store columns. For Main 2, explain that you choose an event’s row and a store’s column, multiply matching quantities and prices, and add. Repeat for each event/store combination.
When can two matrices be multiplied?
A row from the first matrix and a column from the second must have the same number of entries. Therefore, the number of columns in the first matrix must equal the number of rows in the second.
The product takes its row count from the first matrix and its column count from the second. Here two events multiplied by two stores produce a \(2\times2\) result. Keep Q first and P second: their categories are arranged to match quantities with prices.
Worked example 2 — Add more events
Original problem — Main 3: Can the five-event order matrix be multiplied by the original price matrix? If so, find the costs.
Worked solution: Yes. The order matrix is \(5\times3\), and the price matrix is \(3\times2\). The three food categories match, so the result is \(5\times2\).
For example, chess at Mainstreet costs \(3(2.50)+4(3.00)+2(2.00)=23.50\). Repeating the same procedure gives:
| Event | Mainstreet | Grandpa’s |
|---|---|---|
| Soccer | $49.00 | $50.00 |
| Drama | $45.00 | $42.00 |
| Chess | $23.50 | $25.00 |
| Cheerleaders | $18.00 | $19.00 |
| Football | $87.00 | $88.00 |
More food categories require more price rows — Main 4
No: the new order matrix is \(5\times5\), but the old price matrix is \(3\times2\). The inner dimensions do not match. The order includes rolls and cold cuts, whose prices are missing. Add a price row for each new item to make a \(5\times2\) price matrix; then the product would have five event rows and two store columns.
2. Ready — Equivalent equations and rearranging for y
This reviews applying the same operation to both sides and isolating a variable. For a refresher, see Section 4.1’s equation-solving examples. Try the problems, then check your operations and answers.
Check your work — Ready 1–10 answer key
1–3 — Operations that give the second equation
| Problem | Operations |
|---|---|
| 1 | Multiply both sides by 3. |
| 2 | Divide both sides by 4. |
| 3 | Subtract 6x from both sides, then divide both sides by 4. |
4–6 — Are the equations equivalent?
| Problem | Answer and reason |
|---|---|
| 4 | Not equivalent. Dividing the first equation by 6 gives \(9x-7y=15\), not \(9x+7y=15\). For example, \((4,3)\) satisfies the first equation but not the second. |
| 5 | Equivalent. Divide both sides by 3. |
| 6 | Not equivalent. Solving the first equation gives \(y=-\frac25x+2\). For example, \((0,2)\) satisfies the first equation but not the supplied second equation. |
7–10 — Slope-intercept form and justifications
| Problem | Operations | Answer |
|---|---|---|
| 7 | Add 5x to both sides. | \(y=5x+12\) |
| 8 | Subtract 6x from both sides; divide both sides by 2. | \(y=-3x+6\) |
| 9 | Add 12x to both sides; divide both sides by 4. | \(y=3x-4\) |
| 10 | Subtract 5x from both sides; divide both sides by −2. | \(y=\frac52x-5\) |
3. Set — Equipment-cost matrices
Apply the same row-by-column method to equipment quantities and prices. The quantity rows are baseball and football; columns are socks, pants, and helmets. The price columns are Big Sky Sportingoods and Play It Forever. Try both questions before opening the key.
Check your work — Set 11–12 answer key
11 — Total-cost matrix
Rows: baseball, football. Columns: Big Sky, Play It Forever. Every entry is a cost in dollars.
\(a=724.50\), \(b=957.50\), \(c=2099.00\), \(d=2782.00\)
12 — Detailed calculations for a and b
a: Use the baseball row and Big Sky column.
| Item cost calculation | Cost |
|---|---|
| \(13(3.50)\) | $45.50 |
| \(15(35.00)\) | $525.00 |
| \(7(22.00)\) | $154.00 |
b: Use the baseball row and Play It Forever column.
| Item cost calculation | Cost |
|---|---|
| \(13(3.00)\) | $39.00 |
| \(15(40.00)\) | $600.00 |
| \(7(45.50)\) | $318.50 |
Multiply each equipment count by that store’s matching unit price, then add the three costs. Each result goes in the row for the sport and the column for the store.
4. Go — Triangle patterns
Use the figure in your packet. Let n be the step number. Count carefully at steps 1, 2, and 3, then describe how each quantity changes before checking the key.
Check your work — Go 13–17 answer key
For 14, count both blue and white unit triangles. The formulas use positive integer step numbers n.
| Problem / quantity | Steps 1, 2, 3 | Rule |
|---|---|---|
| 13 — Width | 1, 2, 3 | \(n\) |
| 14 — Unit triangles | 1, 4, 9 | \(n^2\) |
| 15 — Perimeter | 3, 6, 9 | \(3n\) |
| 16 — 60° unit-triangle corners | 3, 12, 27 | \(3n^2\) |
| 17 — White triangles | 0, 1, 3 | \(\frac{n(n-1)}2\) |
Each side of the large triangle has length n, so its perimeter is 3n. Each unit triangle has three 60° corners.
For 17, each horizontal row has one more blue triangle than white triangle. There are n rows and \(n^2\) unit triangles in total. Remove the n extra blue triangles, then split the remaining count equally between blue and white: \(\frac{n^2-n}{2}=\frac{n(n-1)}2\).
Additional learning resources
| Resource | What it helps with |
|---|---|
| Khan Academy — Intro to matrix multiplication | The row-by-column procedure and why corresponding-entry multiplication is not the standard matrix product. |
| Khan Academy — Matrix multiplication dimensions | Checking whether a product is defined and finding the dimensions of its result. |
Teacher’s review of the source packet
Go 16, p. 40: “The number of 60-degree angles in the figure” needs a counting convention. Specify whether students should count unit-triangle corners, including white regions, or only the large triangle’s outside angles. The answer key uses unit-triangle corners. No substantial mathematical issues were identified in the main task, Ready, or Set.
Coverage
| Packet component | Where supported |
|---|---|
| Main 1–2 · pp. 36–37 | Row-by-column multiplication, all four food costs, and the meaning of the result. |
| Main 3–4 · p. 37 | Compatible dimensions, five-event costs, and the missing-price limitation. |
| Ready 1–10 · p. 38 | Review link; operations, equivalence decisions with reasons, and isolated-y answers. |
| Set 11–12 · p. 39 | Cost matrix and detailed calculations for a and b. |
| Go 13–17 · p. 40 | Tables and rules; explicit angle-count convention and a short white-triangle explanation. |
Data and prompts adapted from Mathematics Vision Project, Secondary Mathematics I, Module 4 student packet, Section 4.8H, pp. 36–40, under CC BY 4.0. Explanations and solutions added for this supplement.
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