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4.7H — Cafeteria Consumption and Costs
Human review completed .
Organize food-order data in matrices, average matching entries, scale an order, and compare its cost at two stores. The packet labels this lesson 4.7H.
1. In class: A Develop Understanding Task
What is a matrix?
A matrix is a rectangular arrangement of numbers in rows and columns, usually enclosed in brackets. Each number is an entry. Rows go across; columns go down. Its dimensions give the number of rows first, then the number of columns: a \(2\times3\) matrix has two rows and three columns.
In Main 1, you are asked to organize Elvira’s Post-It notes from three years. Choose a consistent order so the same position always means the same thing. Here we use soccer first and drama second, with chips, cookies, and drinks in that order.
| Event | Chips (packages) | Cookies (dozens) | Drinks (gallons) |
|---|---|---|---|
| Soccer | 10 | 6 | 4 |
| Drama | 8 | 3 | 5 |
The blue notes, from three years ago, can therefore be written as this matrix:
For example, the entry in row 2, column 3 is 5: the drama club’s order was 5 gallons of drinks. Keep the row order, column order, and units the same for every year. The brackets contain the numbers; the labels tell us what they mean.
Main 1 answers: With that same arrangement, the yellow notes (two years ago) and pink notes (last year) give:
Worked example 1 — Average the three years’ orders
Original problem — Main 2: Use the average amount ordered over the past three years to recommend this year’s order for each event, and explain how you calculated it.
Worked solution: Add the three values for each event and food item, then divide by 3. For soccer chips:
Matrix addition adds entries in the same position. The matrices must have the same dimensions; here their labels and units also match.
Write the three matrices side by side, keeping each event in the same row and each food item in the same column:
Add the entries that occupy the same position in all three matrices. For example, the top-left entries give soccer chips: \(10+6+8\). The bottom-right entries give drama drinks: \(5+3+4\). Do this for all six positions:
To average the orders, divide every entry of this sum by 3:
Multiplying a matrix by a single number means multiplying every entry by that number. This is called scalar multiplication; the scalar is the number. Here the scalar is \(\frac13\).
| Recommended order | Chips (packages) | Cookies (dozens) | Drinks (gallons) |
|---|---|---|---|
| Soccer | 8 | 7 | 4 |
| Drama | 10 | 4 | 4 |
For future years, keep the categories aligned, add the corresponding entries, and divide by the number of years included.
Worked example 2 — Include three guests per member
Original problem — Main 3: Each team or club member can bring three guests. Assuming each guest consumes about the same amount as the member, use your Main 2 order to find the new order.
Worked solution: Each member is now part of a group of four people: the member plus three guests. Multiply every entry of the average-order matrix by 4:
The soccer order becomes 32 packages of chips, 28 dozen cookies, and 16 gallons of drinks. The drama order becomes 40 packages, 16 dozen, and 16 gallons.
Worked example 3 — Compare the two stores
Original problem — Main 4: Find the total cost of the Main 2 order for each event at each store. Buy all items for a given event from one store.
| Item and unit | Mainstreet Market | Grandpa’s Grocery |
|---|---|---|
| Chips, per package | $2.50 | $2.00 |
| Cookies, per dozen | $3.00 | $4.00 |
| Drinks, per gallon | $2.00 | $1.50 |
Worked solution: Multiply each quantity by its matching price, then add the three costs. For the soccer order at Mainstreet:
| Event | Mainstreet calculation and total | Grandpa’s calculation and total |
|---|---|---|
| Soccer | \(8(2.50)+7(3.00)+4(2.00)\) $49.00 | \(8(2.00)+7(4.00)+4(1.50)\) $50.00 |
| Drama | \(10(2.50)+4(3.00)+4(2.00)\) $45.00 | \(10(2.00)+4(4.00)+4(1.50)\) $42.00 |
Mainstreet is cheaper for the soccer order; Grandpa’s is cheaper for the drama order. Match packages with price per package, dozens with price per dozen, and gallons with price per gallon.
2. Ready — Complete function tables
This reviews finding an output from an input and recovering an input from an output. For a recursive arithmetic rule, use the starting value and the constant change. Try the missing entries yourself, then check the key.
Check your work — Ready 1–6 answer key
Completed rows are listed in the packet’s order as \((input,output)\).
\((-3,13)\), \((0,7)\), \((-18,43)\), \((21,-35)\)
\((-8,-49)\), \((17,26)\), \((12,11)\), \((-4,-37)\)
\((3,-1)\), \((15,71)\), \((4,5)\), \((7,23)\)
\((2,32)\), \((19,-2)\), \((10,16)\), \((13,10)\)
\((-2,-49)\), \((0,-35)\), \((8,21)\), \((5,0)\)
\((10,14)\), \((24,0)\), \((44,-20)\), \((0,24)\)
For checking the recursive tables, equivalent explicit rules are \(h(x)=-13+6(x-1)\) and \(d(t)=34-2(t-1)\).
3. Set — Organize and average catering data
Use the same method as the in-class example: keep categories and units aligned, then average corresponding entries. Try your own organization before opening the key; other consistent arrangements are valid.
Check your work — Set 7–10 answer key
In every matrix below, rows are family gatherings, city events; columns are bags of chips, dozens of cookies, gallons of drink.
7 — Last year
8 — Two years ago
9 — Three years ago
10 — Average order
| Event | Chips (bags) | Cookies (dozens) | Drink (gallons) |
|---|---|---|---|
| Family gatherings | 5 | 6 | 6 |
| City events | 14 | 21 | \(\frac{58}{3}=19\frac13\) |
4. Go — Arithmetic and geometric sequence rules
For a refresher, see Section 3.2: explicit and recursive sequence rules. Check the starting index, try the problems, then consult the key.
Check your work — Go 11–16 answer key
All rules start at index 1. The index n is a positive integer; the recursive step applies for \(n\ge1\). For 15, n counts running sessions; for 16, the rule gives the number of recipients on day n.
11 — Arithmetic
Recursive: \(f(1)=1\)
\(f(n+1)=f(n)+2\)
Explicit: \(f(n)=1+2(n-1)\)
12 — Geometric
Recursive: \(f(1)=3\)
\(f(n+1)=2f(n)\)
Explicit: \(f(n)=3\cdot2^{n-1}\)
13 — Arithmetic
Recursive: \(f(1)=3\)
\(f(n+1)=f(n)+4\)
Explicit: \(f(n)=3+4(n-1)\)
14 — Geometric
Recursive: \(f(1)=5\)
\(f(n+1)=1.6f(n)\)
Explicit: \(f(n)=5(1.6)^{n-1}\)
15 — Arithmetic
Recursive: \(f(1)=1000\)
\(f(n+1)=f(n)+50\)
Explicit: \(f(n)=1000+50(n-1)\)
16 — Geometric
Recursive: \(f(1)=5\)
\(f(n+1)=7f(n)\)
Explicit: \(f(n)=5\cdot7^{n-1}\)
14 — Context note: The decimal values fit a geometric sequence. Since actual bacteria are counted in whole numbers, these values are better understood as population estimates.
Additional learning resources
| Resource | What it helps with |
|---|---|
| Khan Academy — Intro to matrices | Matrix entries, rows, columns, and dimensions. |
| Khan Academy — Matrix addition and subtraction | Matching entries when adding matrices. Also covers subtraction, which is not required here. |
Teacher’s review of the source packet
Missing definition — Set heading, p. 34: Apart from the table of contents, this is the first use of “matrix” or “matrices” in the Math I student packets. The main task asks students to organize information but does not name or define a matrix; Set then uses “Organizing information in matrices” without an explanation. Before students begin Set, explicitly define a matrix, its rows, columns, entries, and dimensions, and show how a labeled table becomes a matrix. The in-class explanation above supplies this missing introduction.
Ready’s heading says “Creating tables from graphs,” although the exercises supply rules and equations. Go 14’s fractional bacteria values are better labeled as population estimates. No substantial calculation errors were identified.
Coverage
| Packet component | Where it is supported |
|---|---|
| Main 1 · pp. 31–32 | Matrix definition, labels, units, and the three yearly order matrices. |
| Main 2–3 · p. 32 | Entry-by-entry addition, averaging, scalar multiplication, and the guest order. |
| Main 4 · p. 32 | Matching quantities and unit prices; all four event/store costs. |
| Ready 1–6 · p. 33 | Brief review and completed input/output rows in the answer key. |
| Set 7–10 · p. 34 | Three labeled yearly matrices and the average-order matrix. |
| Go 11–16 · p. 35 | Linked sequence review; classification, recursive rules, and explicit rules. |
Original data and prompts adapted from Mathematics Vision Project, Secondary Mathematics I, Module 4 student packet, Section 4.7H, pp. 31–35, licensed under CC BY 4.0. Explanations and solutions were prepared for this supplement.
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