• Substitute \(x = 14.5\): \(y = 25 \times 14.5\)

• Multiply: \(25 \times 14 = 350\); \(25 \times 0.5 = 12.5\); \(350 + 12.5 = 362.5\)

Answer: Distance \(y\) = 362.5 miles

(b) Flight hours needed to reach \(575\text{ miles}\):

• Equation: \(y = 25x\)

• Substitute \(y = 575\): \(575 = 25x\)

• Inverse operation: \(x = \frac{575}{25} = 23\text{ hours}\)

Answer: Flight Time \(x\) = 23 hours

Rate Navigator Answer Key • Problem 1 Page 1 of 4 Scoring: Full credit requires correct units and shown steps

Teacher Answer Key Grade 7 Math • RP.A.2

Problem 2: Berry Juice Express — Solution Guide

Total Points: 10 pts

Target Concept: Testing multiple points ensures constant of proportionality; multi-step packaging.

Unit Rate: \(k = 12\text{ fl oz/lb}\)  |  Model: \(y = 12x\)

Juice vs. Berries Slope = 12

0 2 4 6 8 10 Berries Used, x (lb) 0 24 48 72 96 120 Juice, y (fl oz) (1, 12) Unit Rate

Common Student Misconceptions:

• Multi-Step Rounding in 4(b): \(330 \div 16 = 20.625\). Students often round up to 21 bottles. Remind them that a bottle can only be sold if it is completely full; hence only 20 full bottles can be made with 10 oz left over.

• Verifying Proportionality: Notice how \(\frac{24}{2} = \frac{48}{4} = \frac{72}{6} = 12\). Stress that every point on a ray produces equivalent ratios.

1

Ordered Pairs & Point Meaning

[2 pts]

(a) Three ordered pairs from graph (accept any 3):

( 2 , 24 )

( 4 , 48 )

( 6 , 72 )

(b) Meaning of \((6, 72)\):

Pressing 6 pounds of berries produces exactly 72 fluid ounces of blackberry juice.

2

Constant of Proportionality (\(k\))

[2 pts]

Test Point 1: \(\frac{24}{2} = 12\) fl oz/lb

Test Point 2: \(\frac{48}{4} = 12\) fl oz/lb

Unit Rate (\(k\)): \(k\) = 12 fluid ounces per pound

3

Formulate Equation

[2 pts]

Form \(y = kx\): y = 12x (where \(y\) = fl oz, \(x\) = pounds)

4

Problem Solving Solutions

[4 pts total: 2 pts each]

(a) Pounds of berries for \(384\text{ fl oz}\):

• Equation: \(y = 12x\)

• Substitute \(y = 384\): \(384 = 12x\)

• Inverse operation: \(x = \frac{384}{12} = 32\text{ pounds}\)

Answer: Berries Needed \(x\) = 32 pounds

(b) Number of full \(16\text{-oz}\) bottles from \(27.5\text{ lb}\) of berries:

• Total juice produced: \(y = 12(27.5) = 330\text{ fl oz}\)

• Divide by bottle capacity: \(\frac{330}{16} = 20.625\) bottles

• \(20 \times 16 = 320\text{ fl oz}\); Leftover: \(330 - 320 = 10\text{ fl oz}\)

Answer: Full Bottles = 20 full bottles (with 10 oz remaining)

Rate Navigator Answer Key • Problem 2 Page 2 of 4 Scoring: Award partial credit (1 pt) for 330 oz even if bottle rounding is missed

Teacher Answer Key Grade 7 Math • RP.A.2

Problem 3: Eco-Glide Commute — Solution Guide

Total Points: 10 pts

Target Concept: Direct variation model \(y = kx\) for decimal inputs and inequality comparisons.

Unit Rate: \(k = 20\text{ km/kWh}\)  |  Model: \(y = 20x\)

Distance vs. Energy Slope = 20

0 1 2 3 4 5 6 7 8 Energy Expended, x (kWh) 0 20 40 60 80 100 120 140 160 Distance, y (km) (1, 20) Unit Rate

Common Student Misconceptions:

• Justification in 4(b): Check that students do not simply guess "No". Full credit requires computing the maximum potential range \(20 \times 5.75 = 115\text{ km}\) and showing that \(115\text{ km} < 120\text{ km}\) (short by 5 km).

• Alternative Valid Method: A student may find required energy: \(\frac{120}{20} = 6\text{ kWh}\), and state that \(6\text{ kWh} > 5.75\text{ kWh}\). Both approaches are fully valid!

1

Ordered Pairs & Unit Meaning

[2 pts]

(a) Three coordinates along the line (accept any 3):

( 1 , 20 )

( 2 , 40 )

( 4 , 80 )

(b) Meaning of \((1, 20)\):

For every 1 kilowatt-hour (kWh) of battery energy expended, the electric scooter travels a distance of 20 kilometers (the unit rate).

2

Constant of Proportionality (\(k\))

[2 pts]

Calculation: \(k = \frac{y}{x} = \frac{80}{4} = 20\)

Unit Rate with Units:

\(k\) = 20 km / kWh

Directly observable at point (1, 20) where \(x = 1\).

3

Direct Variation Model

[2 pts]

Form \(y = kx\): y = 20x (where \(y\) = km, \(x\) = kWh)

4

Problem Solving Solutions

[4 pts total: 2 pts each]

(a) Energy used for \(110\text{ km}\):

• Equation: \(y = 20x\)

• Substitute \(y = 110\): \(110 = 20x\)

• Divide: \(x = \frac{110}{20} = 5.5\text{ kWh}\)

Answer: Energy \(x\) = 5.5 kWh

(b) Can battery with \(5.75\text{ kWh}\) complete a \(120\text{-km}\) trip?

• Max potential distance: \(y = 20 \times 5.75 = 115\text{ km}\)

• Comparison: \(115\text{ km} < 120\text{ km}\) (short by \(5\text{ km}\))

• Alternatively: \(120 \div 20 = 6\text{ kWh}\) needed; \(6 > 5.75\text{ kWh}\)

Answer: [   ] Yes   [ X ] No    (Max range: 115 km)

Rate Navigator Answer Key • Problem 3 Page 3 of 4 Scoring: Full credit for 4(b) requires mathematical comparison

Teacher Answer Key Grade 7 Math • RP.A.2

Problem 4: 3D Lab Extrusion — Solution Guide

Total Points: 10 pts

Target Concept: Multi-step subtraction problem solving with constant deposition rate.

Unit Rate: \(k = 30\text{ g/hr}\)  |  Model: \(y = 30x\)

Filament vs. Time Slope = 30

0 2 4 6 8 10 Print Time, x (hr) 0 60 120 180 240 300 Filament, y (g) (1, 30) Unit Rate

Common Student Misconceptions:

• Stopping Prematurely in 4(b): Many students find \(30 \times 18.5 = 555\text{ grams}\) and stop there, forgetting the question asks for filament remaining on the spool (\(1{,}000 - 555 = 445\text{ g}\)).

• Decimal Time: \(16.5\text{ hours}\) is \(16\text{ hours and }30\text{ minutes}\), not \(16\text{ hours }5\text{ minutes}\). Accept either \(16.5\text{ hr}\) or \(16\text{ hr }30\text{ min}\).

1

Ordered Pairs & Point Meaning

[2 pts]

(a) Three coordinates from printer profile (accept any 3):

( 2 , 60 )

( 4 , 120 )

( 6 , 180 )

(b) Meaning of \((4, 120)\):

After 4 hours of continuous printing, the 3D printer has extruded 120 grams of filament.

2

Constant of Proportionality (\(k\))

[2 pts]

Calculation: \(k = \frac{y}{x} = \frac{240}{8} = 30\)

Unit Rate with Units:

\(k\) = 30 grams / hour

Unit rate coordinate: ( 1 , 30 ) — 30 grams extruded each hour.

3

Direct Variation Equation

[2 pts]

Form \(y = kx\): y = 30x (where \(y\) = grams, \(x\) = hours)

4

Problem Solving Solutions

[4 pts total: 2 pts each]

(a) Hours to print \(495\text{ grams}\) of filament:

• Equation: \(y = 30x\)

• Substitute \(y = 495\): \(495 = 30x\)

• Inverse operation: \(x = \frac{495}{30} = 16.5\text{ hours}\) (or 16 hr 30 min)

Answer: Printing Time \(x\) = 16.5 hours

(b) Filament remaining from \(1{,}000\text{-g}\) spool after \(18.5\text{ hours}\):

• Filament consumed: \(y = 30 \times 18.5 = 555\text{ grams}\)

• Subtract from total spool: \(1{,}000 - 555 = 445\text{ grams}\)

Answer: Filament Remaining = 445 grams

Rate Navigator Answer Key • Problem 4 Page 4 of 4 Scoring: Deduct 1 pt if student writes 555 g instead of remaining 445 g