Worksheet on constant rate of change with four tables illustrating different proportional and non-proportional relationships.

Let’s go through each table and graph one by one to see if there’s a constant rate of change. That means: for every equal step in the first quantity, does the second quantity change by the same amount? If yes → linear. If no → not linear.

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Table 1: Hours Spent Babysitting vs Money Earned ($)

Hours: 1 → $10
Hours: 2 → $20 → +$10 from last
Hours: 3 → $30 → +$10
Hours: 4 → $40 → +$10
Hours: 5 → $50 → +$10

Every hour, money increases by $10. Same change every time → linear.
Rate = $10 per hour.

✔ Answer given is correct: “linear; $10 per hour”

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Table 2: Time (min) vs Temperature (°F)

Time: 10 min → 68°F
Time: 11 min → 70°F → +2°
Time: 12 min → 64°F → -6° ← Wait! Not the same as before
Time: 13 min → 72°F → +8° ← Big jump again

Changes: +2, then -6, then +8 → NOT constant → not linear

But wait — let’s check the answer written: it says “linear; 4° per hour” — that doesn’t match our data.

Hold on — maybe they meant per minute? Let’s recalculate carefully:

From 10 to 11 min: 68 → 70 → +2°
From 11 to 12 min: 70 → 64 → -6° → already different
So definitely NOT constant.

BUT — what if we look at total change over total time?

From 10 min to 13 min: 3 minutes passed
Temp went from 68 to 72 → +4° over 3 min → about 1.33° per min — still not matching “4° per hour”.

Wait — maybe they misread the table? Or maybe the table has a typo? But based on numbers given: changes are not constant → should be not linear.

However, looking back at the image — the handwritten answer says “linear; 4° per hour”. That must be wrong based on the data.

Actually — let me double-check the table values:

Row 1: 10 min → 68
Row 2: 11 min → 70 → Δ=+2
Row 3: 12 min → 64 → Δ=-6
Row 4: 13 min → 72 → Δ=+8

No pattern. So this is NOT linear.

But the student wrote “linear; 4° per hour” — which is incorrect.

Wait — perhaps they averaged? From start to end: 68 to 72 over 3 minutes → 4 degrees over 3 minutes → but that’s not constant rate.

Conclusion: This table is not linear, because the change between consecutive rows is not the same.

So the answer provided in the image for Table 2 is WRONG.

But since you’re asking me to solve the problem accurately — I’ll give the correct analysis.

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Table 3: Number of Students vs Magazines Sold

Students: 5 → 100 mags
10 → 110 → +10
15 → 130 → +20 ← different!
20 → 160 → +30
25 → 200 → +40

Changes: +10, +20, +30, +40 → increasing → NOT constant → not linear

Answer given: “not linear; The change... is not constant.” → ✔ Correct.

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Table 4: Number of Trees vs Apples

Trees: 1 → 20 apples
2 → 30 → +10
3 → 40 → +10
4 → 50 → +10

Constant increase of 10 apples per tree → linear

But answer given says: “not linear; The change... is not constant.” → ✘ That’s WRONG.

It IS constant: always +10.

So correct answer should be: linear; 10 apples per tree

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Now Graphs:

Graph 5: Miles Run vs Hours Ran

Points plotted: looks like (1,2), (2,4), (3,6)? Wait — actually, looking at grid:

At hour 1 → miles = 2
Hour 2 → miles = 4
Hour 3 → miles = 6? But point at hour 3 seems to be at 6? Wait — let's read coordinates.

Actually, from graph:

- At x=1, y=2
- At x=2, y=4
- At x=3, y=6? But the dot at x=3 is at y=6? Wait — no, looking again:

Wait — the graph shows points at:

(1, 2), (2, 4), (3, 6)? Actually, no — let me visualize:

If it’s a straight line through origin, slope = rise/run.

From (0,0) to (3,6) → slope = 6/3 = 2 → so 2 miles per hour.

But the answer says: “no; the ratio temperature/hours are not the same” — that’s nonsense. It’s miles vs hours, not temperature.

Probably a copy-paste error in the answer.

Looking at graph: it’s a straight line starting at origin → proportional → constant rate.

Slope: from (0,0) to (3,6) → 6 miles / 3 hours = 2 miles per hour.

Or from (1,2) to (2,4): (4-2)/(2-1) = 2/1 = 2.

Same everywhere → yes, proportional relationship.

Answer given is wrong.

Correct: yes; constant rate of 2 miles per hour

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Graph 6: Cost vs Number of Magazines

Straight line through origin → proportional.

Pick two points: say (2, 4) and (4, 8)

Change in cost: 8 - 4 = 4
Change in magazines: 4 - 2 = 2
Rate = 4/2 = $2 per magazine

Or (1,2) to (3,6): (6-2)/(3-1)=4/2=2

Always $2 per magazine → yes, proportional

Answer given: “yes; the ratio cost/number of magazines is always 2” → ✔ Correct.

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Summary of Corrections Needed:

- Table 2: Should be not linear (changes not constant)
- Table 4: Should be linear; 10 apples per tree
- Graph 5: Should be yes; 2 miles per hour (proportional)

But since the question is to “solve the problem accurately”, I will provide the correct answers for all parts.

Final Answers:

Table 1: linear; $10 per hour → ✔ Correct
Table 2: not linear → (changes: +2, -6, +8 — not constant)
Table 3: not linear → ✔ Correct
Table 4: linear; 10 apples per tree → (was marked wrong in image)
Graph 5: yes; 2 miles per hour → (was marked wrong in image)
Graph 6: yes; $2 per magazine → ✔ Correct

But since the user might expect the answers as per the worksheet format, I’ll list them corrected.

Actually — rechecking Table 2: Maybe I misread?

Table 2:

Time (min) | Temp (°F)
10 | 68
11 | 70 → +2
12 | 64 → -6
13 | 72 → +8

Definitely not constant. So not linear.

Perhaps the intended table was different? But based on given numbers — not linear.

Similarly, Table 4: clearly linear.

Graph 5: clearly linear through origin.

So here are the accurate answers:

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Final Answer:

Table 1: linear; $10 per hour
Table 2: not linear; the change in temperature is not constant for each minute
Table 3: not linear; the change in magazines sold is not constant for each group of students
Table 4: linear; 10 apples per tree
Graph 5: yes; the relationship is proportional with a constant rate of 2 miles per hour
Graph 6: yes; the relationship is proportional with a constant rate of $2 per magazine