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5.1 — Pet Sitters
Human review completed .
Represent limits on space and money, check possible pet counts, and justify a boarding plan.
1. In class: A Develop Understanding Task
Represent the constraints
A constraint is a limit that a proposed choice must satisfy. Here the twins have limits on space and on the money available to buy pens and runs. Let x be the number of cats and y the number of dogs. The ordered pair \((x,y)\) records one possible boarding plan.
| Quantity | Cats | Dogs | Available limit |
|---|---|---|---|
| Space per animal | 6 ft² | 24 ft² | 360 ft² |
| Pen/run purchase cost | $32 | $80 | $1,280 |
| Daily boarding income | $8 | $20 | To be compared across plans |
The two resource limits become:
“Up to” includes the limit, so use \(\le\). The counts must also be nonnegative whole numbers. A system of inequalities is a collection of inequalities considered together; a solution must make all of them true.
Worked example 1 — Check the father’s suggestion
Original proposal: Board 12 cats and 12 dogs.
Worked solution: Substitute \(x=12\), \(y=12\) into both constraints:
| Check | Calculation | Does it fit? |
|---|---|---|
| Space | \(6(12)+24(12)=360\) | Yes — exactly the available space. |
| Purchase cost | \(32(12)+80(12)=1344\) | No — $64 over the budget. |
Meeting the space constraint alone is not enough. This plan cannot be funded with the $1,280 budget.
Worked example 2 — Compare plans that do fit
Sample alternatives: Compare an all-dog plan, an all-cat plan, and a mixed plan. These are examples of the kinds of choices the task asks you to investigate.
| Plan (cats, dogs) | Space | Purchase cost | Daily income |
|---|---|---|---|
| \((0,15)\) | 360 ft² | $1,200 | $300 |
| \((40,0)\) | 240 ft² | $1,280 | $320 |
| \((10,12)\) | 348 ft² | $1,280 | $320 |
All three meet both resource limits. As in the packet, these income calculations assume all boarding spaces are filled. More dogs do not automatically mean more income: dogs also need more space and cost more to accommodate.
The boarding prices are exactly one quarter of the corresponding purchase costs. Therefore:
No affordable plan can bring in more than $320 per day. The mixed plan \((10,12)\) reaches that limit, so it is a valid best recommendation. It is not the only one: for example, \((20,8)\) also meets both constraints and earns $320 per day.
When giving your recommendation, state the counts, check both constraints, calculate the daily income, and explain why your choice is reasonable.
2. Ready — Graph linear systems
For a refresher, see Section 3.2’s graphing examples. A solution is a point on both lines. Try the graphs before opening the key.
Check your work — Ready 1–4 answer key
Plots are not shown. These are the points of intersection.
(c) \((1,1)\)
(a) \((2,7)\)
\((0,3)\)
\((2,-2)\)
3. Set — Combinations and packing
Try finding your own combinations first. Check every condition, not just one. The key gives sample answers; many other answers are possible.
Check your work — Set 5–7 answer key
5 — Theater tickets
Let c be child tickets and a adult tickets. The conditions are \(5c+10a\ge2000\) and \(c+a\le350\), with whole-number counts.
| Child tickets | Adult tickets | Total seats |
|---|---|---|
| 50 | 175 | 225 |
| 100 | 150 | 250 |
| 150 | 125 | 275 |
| 200 | 100 | 300 |
| 300 | 50 | 350 |
Each sample brings in exactly $2,000 and uses no more than 350 seats.
6 — Basketball points
Each row totals 102 points: three times the first count, twice the second, plus the free throws.
| 3-point shots | 2-point shots | Free throws |
|---|---|---|
| 10 | 20 | 32 |
| 12 | 18 | 30 |
| 14 | 16 | 28 |
| 16 | 14 | 26 |
| 18 | 12 | 24 |
7 — Fill the 12 × 12 grid
There are many possible solutions. The packing below is one example; other arrangements that fill the grid without gaps or overlaps are also valid.
One complete packing: 36e, with zero of a, b, c, and d. Shape e is a 2 × 2 square. Place six squares across each row of blocks and six rows of blocks down the grid.
This covers all \(12\times12=144\) unit squares, so no packing can cover more. Every supplied piece has area 4; \(36\times4=144\). The layout is not drawn here.
4. Go — Line graphs and arithmetic sequences
This reviews line graphing and arithmetic sequences. For help with sequence rules, see Section 3.2. Try the exercises before checking the key.
Check your work — Go 8–16 answer key
8–11 — Test the point and find two other points
Plots are not shown. The sample points below are different from (3,5).
| Problem | Is (3,5) a solution? | Two other points |
|---|---|---|
| 8 | Yes | \((0,-1)\) and \((4,7)\) |
| 9 | No | \((-9,-1)\) and \((9,5)\) |
| 10 | No | \((-1,8)\) and \((4,-7)\) |
| 11 | No | \((-5,7)\) and \((10,-2)\) |
12
Values in order: 17, 9, 1, −7
Explicit rule: \(y=25-8x\)
13
Values in order: 17, 15, 13, 11, 9, 7, 5, 3, 1, −1, −3, −5, −7
Explicit rule: \(y=19-2x\)
14
Values in order: 17, 13, 9, 5, 1, −3, −7
Explicit rule: \(y=21-4x\)
15
No. The changes per term are −8, −2, and −4, so the points lie on lines with different slopes. The last value −7 occurs at different inputs: 4, 13, and 7.
16
\((1,17)\)
All three sequences begin at that same point.
Additional learning resources
| Resource | What it supports |
|---|---|
| Khan Academy — Writing two-variable inequalities word problem | Uses a chess-score example to translate a situation into an inequality involving two quantities. |
Teacher’s review of the source packet
No substantial mathematical or instructional issues were identified. The main task allows several equally good boarding plans; the supplement gives examples rather than a unique required recommendation.
Coverage
| Packet component | Where supported |
|---|---|
| Main task · p. 1 | Constraint definitions, both resource models, checked proposals, and a maximum-income argument. |
| Ready 1–4 · p. 2 | Linked graphing review and intersection answers; plots omitted. |
| Set 5–7 · p. 3 | Five ticket combinations, five scoring combinations, and one complete packing. |
| Go 8–16 · pp. 4–5 | Point tests, sample graphing points, sequence values/rules, and explained comparisons. |
Data and prompts adapted from Mathematics Vision Project, Secondary Mathematics I, Module 5 student packet, Section 5.1, printed pp. 1–5, under CC BY 4.0. Explanations, solutions, and any diagrams were prepared for this supplement.
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