Texas Grab-and-Go Substitute Packet · Teacher Answer Key
Answer Key: Linear Relationships in Four Forms
Grade: 8
Subject: Mathematics
For teacher use only
Math
How to use this key
Answers are grouped by section and question number, with full worked steps. Accept any correct method that reaches the right value — alternate strategies (using different table rows, counting up on the graph, or reasoning from the story) are noted where they apply. Watch-for notes flag common misconceptions. A basic calculator is allowed, so grade the reasoning and steps, not just the arithmetic. Estimated grading time: ~5–7 min per packet.
Start Warm-Up: Retrieval ~1 min
1 Evaluate the rule y = 2x + 5 at x = 4.
2 × 4 = 8, then 8 + 5 = 13. So y = 13. Full credit for
showing the multiply-then-add order. Watch-for: a student who writes 2 × (4 + 5)
= 18 added before multiplying — remind them to multiply mx first.
2 Read the point at x = 2 from Figure 1.
At x = 2, the marked point is (2, 10). Accept the ordered pair
(2, 10). Alternate: a student may also get this from y = 3x + 4 → 3(2) + 4 = 10.
Build Study the Math ~1 min
3 Water depth at 6 minutes (y = 3x + 4).
y = 3 × 6 + 4 = 18 + 4 = 22 inches. Alternate: extend the
table by adding 3 each minute (16 → 19 → 22). Both are correct.
Apply Use What You Know ~3 min
4 Match the four forms (gym membership, Table B).
4a. Using rows (2, 30) and (4, 40): m = (40 − 30) ÷ (4 − 2) = 10 ÷ 2 =
5 dollars per visit. (Any two rows work: e.g., (0, 20) and (6, 50) → 30 ÷ 6 = 5.)
4b. y-intercept b = 20 (the cost at x = 0 visits, read straight from the table). Equation: y = 5x + 20.
4c. The slope means the cost goes up $5 per visit (each visit adds $5). The y-intercept means the $20 sign-up fee you pay before any visits (at 0 visits the cost is already $20).
4d. It is NOT proportional, because the y-intercept is 20, not 0 — the line does not pass through the origin (0, 0). (A proportional relationship must start at 0.)
4b. y-intercept b = 20 (the cost at x = 0 visits, read straight from the table). Equation: y = 5x + 20.
4c. The slope means the cost goes up $5 per visit (each visit adds $5). The y-intercept means the $20 sign-up fee you pay before any visits (at 0 visits the cost is already $20).
4d. It is NOT proportional, because the y-intercept is 20, not 0 — the line does not pass through the origin (0, 0). (A proportional relationship must start at 0.)
5 Error analysis: a mis-found slope for Table B.
The mistake: The student reversed rise and run — they divided
run ÷ rise instead of rise ÷ run. Slope is the change in y (rise)
over the change in x (run), so the differences are flipped.
Correct work: m = rise ÷ run = (40 − 30) ÷ (4 − 2) = 10 ÷ 2 = 5 dollars per visit (not 0.2).
Full credit requires (1) naming that they flipped rise and run (used run/rise), and (2) the correct slope of 5. Sense-check to offer: 0.2 dollars per visit would mean 20¢ a visit, which does not match the table jumping $10 for every 2 visits.
Correct work: m = rise ÷ run = (40 − 30) ÷ (4 − 2) = 10 ÷ 2 = 5 dollars per visit (not 0.2).
Full credit requires (1) naming that they flipped rise and run (used run/rise), and (2) the correct slope of 5. Sense-check to offer: 0.2 dollars per visit would mean 20¢ a visit, which does not match the table jumping $10 for every 2 visits.
6 Multi-step application (predict from y = 5x + 20).
6a. y = 5(12) + 20 = 60 + 20 = $80 for 12 visits.
6b. Solve 5x + 20 = 75 → subtract 20: 5x = 55 → divide by 5: x = 11 visits. Check: 5(11) + 20 = 55 + 20 = 75. ✔ Alternate: a student may count up from $20 by $5 steps until reaching $75 (11 steps) — accept it.
6b. Solve 5x + 20 = 75 → subtract 20: 5x = 55 → divide by 5: x = 11 visits. Check: 5(11) + 20 = 55 + 20 = 75. ✔ Alternate: a student may count up from $20 by $5 steps until reaching $75 (11 steps) — accept it.
Explain Justify Your Strategy (Q7) ~1–2 min
7 Bike rental for 3 hours: Plan A y = 4x + 10 vs. Plan B y = 6x.
Model answer. Claim: For a 3-hour rental, Plan B costs less; and
Plan B is the proportional one.
Evidence: Plan A: y = 4(3) + 10 = 12 + 10 = $22. Plan B: y = 6(3) =
$18. $18 < $22, so Plan B is cheaper for 3 hours.
Reasoning: Plan B has y-intercept b = 0, so it passes through the origin and is
proportional (y = 6x). Plan A has y-intercept b = 10 (a $10 sign-up), so it is
not proportional. Even though Plan A's slope ($4/hr) is smaller than Plan B's
($6/hr), Plan A's $10 head start makes it cost more at 3 hours.
Note: Full credit requires both computed costs ($22 and $18), naming Plan B as cheaper, and identifying Plan B as proportional (b = 0). Extension worth noting: the two plans are equal when 4x + 10 = 6x → x = 5 hours; beyond 5 hours Plan A becomes cheaper. Students are not required to find this, but it is a strong discussion point.
Note: Full credit requires both computed costs ($22 and $18), naming Plan B as cheaper, and identifying Plan B as proportional (b = 0). Extension worth noting: the two plans are equal when 4x + 10 = 6x → x = 5 hours; beyond 5 hours Plan A becomes cheaper. Students are not required to find this, but it is a strong discussion point.
Justification scoring rubric (3 points)
| Score | Claim | Evidence | Reasoning |
|---|---|---|---|
| 3 | Names Plan B as cheaper for 3 hours AND names Plan B as proportional. | Both costs computed correctly ($22 and $18) with the numbers shown. | Explains proportional means b = 0 and correctly links slope and y-intercept to the comparison. |
| 2 | Names Plan B as cheaper (proportional part missing or partly right). | One cost computed, or both with a minor arithmetic slip. | Reasoning present but incomplete or missing the y-intercept link. |
| 1 | A plan named with weak or no support. | Evidence largely missing or incorrect. | Little or flawed reasoning. |
| 0 | No/incorrect claim. | No evidence. | No reasoning. |
Close ACE ~1 min
ACE Articulate / Connect / Extend.
Articulate: Accept any correct plain-language method: "pick two rows, subtract
the y-values to get the rise, subtract the x-values in the same order to get the run,
then divide rise ÷ run."
Connect: Valid items where y = mx + b was used include Q1 (as y = 2x + 5), Q3, Q4b, Q6, or Q7. Any of these earns credit if correctly named.
Extend: Any genuine non-proportional linear situation with a starting amount and a steady rate — e.g., a taxi with a base fare plus a per-mile rate, a phone plan with a monthly fee plus a per-gigabyte charge, savings that start at some amount and grow by a fixed weekly deposit. Must have a nonzero starting amount to count as non-proportional.
Connect: Valid items where y = mx + b was used include Q1 (as y = 2x + 5), Q3, Q4b, Q6, or Q7. Any of these earns credit if correctly named.
Extend: Any genuine non-proportional linear situation with a starting amount and a steady rate — e.g., a taxi with a base fare plus a per-mile rate, a phone plan with a monthly fee plus a per-gigabyte charge, savings that start at some amount and grow by a fixed weekly deposit. Must have a nonzero starting amount to count as non-proportional.
Challenge Early Finisher (optional)
EF Write a context for y = 8x + 15 (must be non-proportional).
(1) A sample situation: "A plumber charges a $15 service fee to come out,
plus $8 per hour of work. Let x = hours worked and y = total charge in dollars."
Many correct stories exist (a $15 flat delivery fee plus $8 per item; a gift card
starting balance, etc.).
(2) Meaning: The slope 8 is the rate of $8 per hour (each hour adds $8). The y-intercept 15 is the $15 starting amount / fee charged before any hours (the cost at x = 0).
(3) Predict y at x = 5: y = 8(5) + 15 = 40 + 15 = $55.
Full credit requires a clear x and y, a story with a real starting amount (so it is non-proportional), a correct interpretation of 8 and 15, and the correct value 55.
(2) Meaning: The slope 8 is the rate of $8 per hour (each hour adds $8). The y-intercept 15 is the $15 starting amount / fee charged before any hours (the cost at x = 0).
(3) Predict y at x = 5: y = 8(5) + 15 = 40 + 15 = $55.
Full credit requires a clear x and y, a story with a real starting amount (so it is non-proportional), a correct interpretation of 8 and 15, and the correct value 55.
Watch Common misconceptions
- Slope vs. y-intercept. Students may swap the two — reporting the starting amount as the rate, or the rate as the starting amount. In y = 5x + 20, m = 5 is the per-visit rate and b = 20 is the sign-up fee. Anchor: the number multiplied by x is the slope; the number added on is the y-intercept (the value at x = 0).
- Reversing rise and run. The Q5 error: computing run ÷ rise (giving 0.2) instead of rise ÷ run (giving 5). Slope is change in y over change in x — always y-difference on top. Subtract the x-values and y-values in the same order.
- Assuming all lines are proportional. Students often treat any straight line as proportional. A line is proportional only if it passes through (0, 0), i.e., b = 0. Figure 1 and Table B both have b ≠ 0, so they are linear but not proportional (Q4d and Q7 check this).
- Adding before multiplying. In Q1/Q3/Q6, evaluate mx first, then add b — do not add x and b before multiplying.
- Stopping at the wrong step when solving. In Q6b, after 5x = 55 a student may forget to divide by 5. Encourage a check by substituting the answer back into the equation.
Teacher follow-up based on likely errors
If many students miss Q5, do a quick 5-minute practice labeling rise and run on two or three graphs, always keeping the y-difference on top. If Q4d or Q7 show the "all lines are proportional" idea, re-anchor that only a line through (0, 0) is proportional (b = 0), and contrast it with a line that has a sign-up fee. The Q4c interpretations reveal who can attach meaning to slope and y-intercept in context — a good warm-up discussion next class. The Q7 justifications show who can compare two models using both slope and starting amount.
G08_MATH_Linear_01 — Linear Relationships in Four Forms
Teacher Answer Key