Rate Lab Guided Worksheet
Grade 7 Math Ratios & Proportions
Rate Lab: Unit Rates & Proportions
Name:
Date:
Period:
Core Concept: What is a Unit Rate?
A unit rate compares a quantity to exactly 1 unit of another quantity (denominator = 1). Divide the numerator by the denominator: \( \text{Unit Rate} = \frac{\text{Quantity } A}{\text{Quantity } B} \)
1 Warm-Up: Unit Rates with Whole Numbers
Divide to find rate per 1 unit
1A. Guided Example Model
A car travels 240 miles in 4 hours. Find the speed in miles per hour.
Divide: \( \frac{240 \text{ mi}}{4 \text{ hr}} = 240 \div 4 = 60 \text{ mph} \)
Unit Rate: 60 miles per hour
1B. Try It
A supermarket charges $18 for a pack of 12 bottles of juice. Find the cost per bottle.
Unit Price: $ / bottle
Core Concept: Unit Rates with Fractions & Rational Numbers
Fractions within rates create a complex fraction. To divide by a fraction, multiply by its reciprocal:
\( \frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \)
2 Fractions in Action: Unit Rates of Rational Numbers
Multiply by the reciprocal
2A. Step-by-Step Model Guided Walkthrough
Elena walks \( \frac{3}{4} \) of a mile in \( \frac{1}{3} \) of an hour. How many miles does she walk per hour?
1. Set Up Ratio
\( \frac{\frac{3}{4} \text{ mi}}{\frac{1}{3} \text{ hr}} \)
2. Multiply Reciprocal
\( \frac{3}{4} \times \frac{3}{1} = \frac{9}{4} \)
3. Mixed Number
\( 2\frac{1}{4} \text{ miles/hr} \)
2B. Your Turn
A recipe uses \( \frac{2}{3} \) cup of honey for \( \frac{1}{2} \) batch of granola. How much honey is needed for 1 full batch?
Answer: cups/batch
2C. Your Turn
A faucet leaks \( 1\frac{1}{2} \) gallons of water in \( \frac{3}{4} \) of an hour. At what rate is the faucet leaking per hour?
Answer: gal/hr
Turn over to connect Unit Rates to the Constant of Proportionality (\(k\)) Page 1 of 2
Part II
The Constant of Proportionality (\(k\))
Name:
Core Concept: Connecting Unit Rate to \(k\)
In a proportional relationship, the constant unit rate is called the constant of proportionality (\(k\)).
Finding \(k\):
\( k = \frac{y}{x} \)
Equation Form:
\( y = kx \)
3 Hunting for \(k\): Finding Constants in Tables
Test \( \frac{y}{x} \) for each pair
Table A: Lemonade Mix to Water
| Water (\(x\)) |
Mix (\(y\)) |
Ratio \( \frac{y}{x} \) |
| 2 |
7 |
\( \frac{7}{2} = 3.5 \) |
| 4 |
14 |
\( \frac{14}{4} = \_\_\_\_ \) |
| 6 |
21 |
\( \frac{21}{6} = \_\_\_\_ \) |
Proportional? YES / NO
Constant (\(k\)): \(k = \)
Equation (\(y = kx\)):
Table B: Drone Flight Distance
| Minutes (\(x\)) |
Miles (\(y\)) |
Ratio \( \frac{y}{x} \) |
| \( \frac{1}{2} \) |
\( \frac{1}{4} \) |
\( \frac{1/4}{1/2} = \_\_\_\_ \) |
| 2 |
1 |
\( \frac{1}{2} \) |
| \( 3\frac{1}{2} \) |
\( 1\frac{3}{4} \) |
\( \frac{7/4}{7/2} = \_\_\_\_ \) |
Proportional? YES / NO
Constant (\(k\)): \(k = \)
Equation (\(y = kx\)):
4 Rate Master Challenge: Application & Reasoning
Show your mathematical steps
4A. The Painting Showdown Problem Solving
Mateo paints \( \frac{2}{5} \) of a wall in \( \frac{1}{4} \) hour. Aaliyah paints \( \frac{5}{8} \) of a wall in \( \frac{1}{2} \) hour. Who works faster?
Mateo's Unit Rate:
Rate: walls/hr
Aaliyah's Unit Rate:
Rate: walls/hr
Conclusion: Faster painter: by walls/hr
4B. Interpreting and Predicting with \(y = kx\)
A submarine descends at a constant rate represented by \( y = 14.5x \), where \(y\) is depth (meters) and \(x\) is time (seconds).
1. Value of \(k\)
\(k = \)
2. Meaning in Context
Descends m each sec.
3. Depth after 12 s
\(y = \) m
Self Check: I can find unit rates with fractions and identify \(k\):
Page 2 of 2
Rate Lab Answer Key
Teacher Guide Answer Key & Solutions
Rate Lab: Unit Rates & Proportions
Full Worked Solutions
CCSS: 7.RP.A.1, 7.RP.A.2
Scoring Note: Award 1 point for setting up the ratio correctly, 1 point for multiplying by the reciprocal / dividing, and 1 point for correct units.
Total: 25 Pts
1 Part 1: Whole Number Rates (Solutions)
[4 Points Total]
1A. Guided Example (Car Travel)
Work:
\( \frac{240 \text{ miles}}{4 \text{ hours}} = 60 \text{ mph} \)
Answer: 60 miles per hour
1B. Try It (Juice Bottles) 2 Pts
Student Work:
\( \frac{\$18.00}{12 \text{ bottles}} = 18 \div 12 = 1.50 \)
Answer: $1.50 per bottle
2 Part 2: Rational Numbers & Complex Fractions (Solutions)
[8 Points Total]
2A. Walking Rate (Elena)
Complex Ratio \( \frac{3/4}{1/3} \)
Multiply Reciprocal \( \frac{3}{4} \times \frac{3}{1} = \frac{9}{4} \)
Solution \( 2\frac{1}{4} \) or 2.25 mph
2B. Honey Granola Recipe 4 Pts
Step 1: Ratio = \( \frac{2/3 \text{ cup}}{1/2 \text{ batch}} \)
Step 2: \( \frac{2}{3} \div \frac{1}{2} = \frac{2}{3} \times \frac{2}{1} = \frac{4}{3} \)
Step 3: Convert to mixed number: \( 1\frac{1}{3} \)
Answer: \( 1\frac{1}{3} \) (or \( \frac{4}{3} \)) cups/batch
2C. Leaking Faucet 4 Pts
Step 1: Improper fraction: \( 1\frac{1}{2} = \frac{3}{2} \)
Step 2: Ratio = \( \frac{3/2}{3/4} = \frac{3}{2} \times \frac{4}{3} \)
Step 3: \( \frac{12}{6} = 2 \) gallons per hour
Answer: 2 gallons per hour
Teacher Alert: Frequent Student Errors in Part 2
- Flipping the wrong fraction: Students often take the reciprocal of the numerator instead of the denominator. Remind them: "Leave the first alone, flip the second one!"
- Forgetting to change mixed numbers: In 2C, emphasize that students MUST convert \( 1\frac{1}{2} \) to \( \frac{3}{2} \) before attempting to multiply.
Rate Lab Teacher Answer Key • Page 1 of 2 Solutions & Scoring
Part II Solutions
Constant of Proportionality (\(k\)) Key
[13 Points Total]
3 Part 3: Finding \(k\) in Tables (Solutions)
[6 Points: 3 pts each]
Table A: Lemonade Mix (Key)
| Water (\(x\)) |
Mix (\(y\)) |
Ratio \( \frac{y}{x} \) (Work) |
| 2 |
7 |
\( 7/2 = 3.5 \) |
| 4 |
14 |
\( 14/4 = 3.5 \) |
| 6 |
21 |
\( 21/6 = 3.5 \) |
Proportional? YES (ratios are equal)
Constant (\(k\)): \(k = 3.5\) or \( \frac{7}{2} \)
Equation: \(y = 3.5x\) or \(y = \frac{7}{2}x\)
Table B: Drone Flight (Key)
| Min (\(x\)) |
Miles (\(y\)) |
Ratio \( \frac{y}{x} \) (Work) |
| \( \frac{1}{2} \) |
\( \frac{1}{4} \) |
\( \frac{1}{4} \times \frac{2}{1} = \frac{1}{2} \) |
| 2 |
1 |
\( 1/2 = 0.5 \) |
| \( 3\frac{1}{2} \) |
\( 1\frac{3}{4} \) |
\( \frac{7}{4} \times \frac{2}{7} = \frac{1}{2} \) |
Proportional? YES (constant ratio)
Constant (\(k\)): \(k = \frac{1}{2}\) or \(0.5\)
Equation: \(y = \frac{1}{2}x\) or \(y = 0.5x\)
4 Part 4: Challenge & Application Solutions
[7 Points Total]
4A. The Painting Showdown 4 Pts
Mateo's Unit Rate:
\( \frac{2/5}{1/4} = \frac{2}{5} \times \frac{4}{1} = \frac{8}{5} = 1\frac{3}{5} \text{ (1.6)} \)
Rate = 1.6 walls/hour
Aaliyah's Unit Rate:
\( \frac{5/8}{1/2} = \frac{5}{8} \times \frac{2}{1} = \frac{10}{8} = 1\frac{1}{4} \text{ (1.25)} \)
Rate = 1.25 walls/hour
Conclusion & Difference: Faster: Mateo by 0.35 (or \( \frac{7}{20} \)) walls per hour
4B. Submarine Descent (\(y = 14.5x\)) 3 Pts (1 pt each)
1. Constant (\(k\))
\(k = 14.5\)
2. Context Meaning
Descends 14.5 meters per second
3. Depth at 12 sec
\(14.5 \times 12 = 174\text{ m}\)
Next Steps & Small Group Remediation
If students struggle with \(k\): Review that \(k\) is always \( \frac{y}{x} \) (dependent / independent), never \( \frac{x}{y} \).
If students struggle with fractions: Use visual tape diagrams or double number lines to ground fraction division before the standard algorithm.
Rate Lab Teacher Answer Key • Page 2 of 2 Instructional Guide
