Slope Showdown Slides Grade 7 Math Review
Standards: 7.RP.A.1 • 7.RP.A.2 • Slope Intro
Slope Showdown
Mastering unit rates, proportional relationships, equation models, and finding slope with rise over run.
1. Unit Rates
Rate per 1 unit quantity
2. Proportions
Constant ratio \(k = \frac{y}{x}\)
3. Solving
Cross-products & scale
4. Slope
\(\text{Rise} / \text{Run}\) on grids
Concept 01
Unit Rates & Proportionality
\(y = kx\)
Unit Rate
A comparison where the second quantity is 1 unit.
\(\frac{\$18}{3\text{ shirts}} = \frac{\$6}{1\text{ shirt}}\)
Constant of Proportionality
The ratio \(\frac{y}{x}\) must stay the same constant value (\(k\)) for all pairs.
\(k = \frac{y}{x} \implies y = kx\)
Graph Rules
A relationship is proportional if and only if its graph is a straight line through the origin.
Must pass through \((0,0)\)
Key Fact: On a proportional graph, the point \((1, k)\) always reveals the unit rate!
Concept 02
Solving Proportions
\(\frac{a}{b} = \frac{c}{d}\)
The Cross-Products Property
In any true proportion, cross-multiplied products are equal:
\(\frac{a}{b} = \frac{c}{d} \iff a \cdot d = b \cdot c\)
Scale Factor Method
Look for friendly multiplier relationships across numerators or denominators!
Real-World Walkthrough Recipe Scaling
A punch recipe uses 4 cups of juice for 14 servings. How many cups for 35 servings?
1 Set up ratio: \(\frac{4\text{ cups}}{14\text{ serv}} = \frac{x\text{ cups}}{35\text{ serv}}\)
2 Cross multiply: \(14 \cdot x = 4 \cdot 35\)
3 Solve for \(x\): \(14x = 140 \implies x = 10\text{ cups}\)
Always label your units to prevent upside-down ratios!
Proportions compare two equal rates or ratios: \(\text{Rate}_1 = \text{Rate}_2\)
Concept 03
Slope = Rise Over Run
\(m = \frac{\text{rise}}{\text{run}}\)
What is Slope?
Slope measures the steepness and direction of a line on a coordinate plane.
Rise (Vertical): Change in \(y\) (\(\Delta y\))
Run (Horizontal): Change in \(x\) (\(\Delta x\))
The Proportional Connection
For any line through \((0,0)\), Slope (\(m\)) is identical to the Constant of Proportionality (\(k\))!
Counting on the Grid
1
Pick two exact points
Look for intersection crosshairs (e.g., \((0,0)\) and \((2,6)\)).
2
Count the Rise
Count up (+) or down (-) from the first point to the second.
3
Count the Run
Count right (+) across the grid columns.
Example: \(\text{Rise } = +6\), \(\text{Run } = +2 \implies m = \frac{6}{2} = 3\)
Uphill from left to right = Positive Slope (+) Downhill from left to right = Negative Slope (-)
Quick Check
Showdown Challenge
Warm-Up Test
Challenge A: The Table Rate & Equation
Does this table represent a proportional relationship? If so, find \(k\) and write the equation:
Hours (\(x\))
2
5
8
Miles (\(y\))
14
35
56
Target: Calculate \(\frac{y}{x}\) for every column.
Challenge B: Rise Over Run Slope Finding
A line passes through \((0,0)\) and \((4, 18)\). What is the rise, run, and slope?
Point 1: \((0, 0)\) Point 2: \((4, 18)\)
\(\text{Rise} = 18 - 0 = 18\)
\(\text{Run} = 4 - 0 = 4\)
\(m = \frac{18}{4} = \frac{9}{2} = 4.5\)
Ready for independent practice! Open your review packet to begin.
Pages 1 & 2
Slope Showdown Review Packet Grade 7 Mathematics • Unit Review
Slope Showdown Review Packet
Ratios, Proportional Relationships & Rise Over Run
Name:
Date:
Period:
Score: / 25
Formulas: Unit Rate = \(y \div x\) Constant of Proportionality = \(k = \frac{y}{x}\) Equation = \(y = kx\) Slope = \(m = \frac{\text{rise}}{\text{run}}\)
1
Part 1: Unit Rates & Rates with Fractions
6 Points
1. A craft store sells 6 skeins of yarn for $16.50. Calculate the unit price per skein.
Show your division or setup below:
Unit Rate: $________ per skein
2. Marcus hikes \(\frac{3}{4}\) mile in \(\frac{1}{6}\) hour. What is his speed in miles per hour?
Show your complex fraction work:
Speed: ________ mph
3. Best Buy Showdown: Brand Alpha sells 18 oz of cereal for $4.14. Brand Beta sells 24 oz for $5.04.
Brand Alpha (18 oz for $4.14):
Rate: $________ / oz
Brand Beta (24 oz for $5.04):
Rate: $________ / oz
Which brand is the better value per ounce? Better Buy: Brand ____________
2
Part 2: Proportional Tables & Identifying \(k\)
6 Points
4. Maya works at a local garden center. Her earnings based on hours worked are shown below:
Hours (\(x\))
Pay (\(y\))
\(\frac{y}{x}\)
3
$42.00
______
5
$70.00
______
8
$112.00
______
Is this relationship proportional? YES / NO
Constant of proportionality (\(k\)): \(k = \) ________
Equation modeling Maya's pay: \(y = \) ________
Earnings for 11 hours: $________
5. Conceptual Check: Two fitness clubs charge monthly gym fees:
Club Power: $25/month. Equation: \(y = 25x\)
Club Flex: $20/month + $15 fee. Equation: \(y = 20x + 15\)
Which club represents a proportional relationship? Explain why the other is NOT proportional (origin):
Slope Showdown • Grade 7 Math Review Packet Page 1 of 2 Turn over for Proportions & Slope
Slope Showdown Review Packet • Page 2
Solving Proportions & Finding Slope Using Rise Over Run
Student Name:
3
Part 3: Solving Proportions
6 Points
6. Solve for \(x\):
\(\frac{x}{12} = \frac{15}{20}\)
Show work below:
\(x = \) ________
7. Solve for \(w\):
\(\frac{5.5}{2} = \frac{w}{8}\)
Show work below:
\(w = \) ________
8. Map Scale:
2 in represents 75 miles. If cities are 5 in apart, find real distance \(d\).
\(d = \) ______ mi
4
Part 4: Slope Using Rise Over Run
8 Points
9. Find the slope of Line A on the coordinate grid:
x y 0 1 2 3 4 2 4 6 8 (3, 6) Run = ? Rise = ?
Rise (\(\Delta y\)): ______ units
Run (\(\Delta x\)): ______ units
Slope (\(m = \frac{\text{rise}}{\text{run}}\)): \(m = \) ______
10. A line passes through \(P(2, 5)\) and \(Q(6, 17)\).
Calculate rise (\(\Delta y\)), run (\(\Delta x\)), and slope:
\(\text{Rise} = 17 - 5 =\) ________
\(\text{Run} = 6 - 2 =\) ________
Slope: \(m = \) ________
11. Real-World Rate: A scuba diver swims downwards. At 2 min her depth is 14 m; at 5 min her depth is 35 m.
Calculate slope (\(m = \frac{\Delta \text{depth}}{\Delta \text{time}}\)):
Slope = ________ m/min
Explain meaning in real-world context:
Include units of measurement
5
Part 5: Connecting \(k\), Unit Rate, & Slope
5 Points
12. Synthesis: A remote car travels according to \(y = 4.5x\), where \(x\) is seconds and \(y\) is meters.
a) Constant of Proportionality: \(k = \) ________
b) Unit Rate with Units: ________ m/s
c) Slope of Graph: \(m = \) ________
d) Why are the constant of proportionality \(k\) and the slope \(m\) equal for this graph?
Slope Showdown • Grade 7 Math Review Packet Page 2 of 2 Completed Review Packet
Slope Showdown Answer Key Teacher Scoring Guide & Worked Solutions
Slope Showdown Answer Key
Page 1 Solutions: Unit Rates, Tables & Proportionality
Total Points 25 Pts Total Page 1: 12 Pts • Page 2: 13 Pts
Scoring Tip: Award partial credit (0.5 pt) for correct algebraic setup even if minor computation slips occur.
1
Part 1 Solutions: Unit Rates & Rates with Fractions (6 Pts)
2 Pts Each
Problem 1: Skeins of Yarn $2.75 / skein
Setup: \(\frac{\$16.50}{6\text{ skeins}} = \$16.50 \div 6\)
Division: \(16.50 \div 6 = 2.75\)
Answer: $2.75 per skein
Watch for: Students dividing \(6 \div 16.50\) getting 0.36 skeins per dollar instead of cost per skein.
Problem 2: Complex Fraction 4.5 mph (or \(4\frac{1}{2}\))
Setup: \(\frac{\frac{3}{4}\text{ mi}}{\frac{1}{6}\text{ hr}} = \frac{3}{4} \div \frac{1}{6}\)
Reciprocal: \(\frac{3}{4} \times \frac{6}{1} = \frac{18}{4} = \frac{9}{2} = 4.5\)
Answer: 4.5 miles per hour
Watch for: Forgetting to multiply by the reciprocal of the denominator fraction.
Problem 3: Best Buy Comparison Better Buy: Brand Beta
Brand Alpha: \(\frac{\$4.14}{18\text{ oz}} = \$0.23\text{ per ounce}\)
Brand Beta: \(\frac{\$5.04}{24\text{ oz}} = \$0.21\text{ per ounce}\)
Comparison: $0.21 < $0.23 Brand Beta is cheaper by $0.02 per ounce. (2 Pts)
2
Part 2 Solutions: Tables & Constant of Proportionality (6 Pts)
4 Pts + 2 Pts
Problem 4: Maya's Pay Table Analysis (4 Pts) All ratios equal 14
Ratios \(\frac{y}{x}\):
\(\frac{42}{3} = 14\)
\(\frac{70}{5} = 14\)
\(\frac{112}{8} = 14\)
Proportional? YES (all ratios \(\frac{y}{x} = 14\))
Constant (\(k\)): \(k = 14\)
Equation: \(y = 14x\)
Pay for 11 hours: \(14 \times 11 = \$154.00\)
Problem 5: Conceptual Check (2 Pts) Club Power is Proportional
Exemplar Response: "Club Power (\(y = 25x\)) is proportional because at 0 months the cost is $0, meaning its graph passes directly through the origin \((0,0)\). Club Flex (\(y = 20x + 15\)) is NOT proportional because at 0 months the startup fee is $15, so its graph starts at \((0,15)\) rather than the origin."
Criteria: 1 pt for identifying Club Power; 1 pt for identifying the origin / \((0,15)\) y-intercept.
Slope Showdown • Teacher Answer Key Page 1 of 2 Turn over for Proportions & Slope Solutions
Slope Showdown Answer Key • Page 2
Solving Proportions, Rise Over Run & Synthesis Guide
13 Points Total
3
Part 3 Solutions: Solving Proportions (6 Pts)
2 Pts Each
Prob 6: Solve for \(x\) \(x = 9\)
Cross-multiply:
\(20 \cdot x = 12 \cdot 15\)
\(20x = 180\)
\(x = 180 \div 20 = 9\)
Scale factor: \(\frac{15}{20} = \frac{3}{4} = \frac{9}{12}\)
Prob 7: Solve for \(w\) \(w = 22\)
Cross-multiply:
\(2 \cdot w = 5.5 \cdot 8\)
\(2w = 44\)
\(w = 44 \div 2 = 22\)
Scale factor: Denominator \(\times 4\), so \(5.5 \times 4 = 22\)
Prob 8: Map Scale 187.5 mi
\(\frac{2\text{ in}}{75\text{ mi}} = \frac{5\text{ in}}{d\text{ mi}}\)
\(2 \cdot d = 75 \cdot 5\)
\(2d = 375 \implies d = 187.5\)
Unit rate approach: \(1\text{ in} = 37.5\text{ mi}\), \(37.5 \times 5 = 187.5\)
4
Part 4 Solutions: Slope Using Rise Over Run (8 Pts)
3 + 2 + 3 Pts
Prob 9: Grid Slope (3 Pts) Slope \(m = 2\)
Vertical Change (\(\text{Rise}\)): +6 units
Horizontal Change (\(\text{Run}\)): +3 units
Slope (\(m = \frac{\text{rise}}{\text{run}}\)): \(\frac{6}{3} = 2\)
Check: Also passes through \((1, 2)\), verifying unit rate / slope of 2.
Prob 10: Two Coordinates (2 Pts) Slope \(m = 3\)
\(\text{Rise } \Delta y = 17 - 5:\) 12
\(\text{Run } \Delta x = 6 - 2:\) 4
\(m = \frac{\text{Rise}}{\text{Run}} = \frac{12}{4}:\) 3
Common error: Subtracting upside down as \(\frac{x}{y} = \frac{4}{12} = \frac{1}{3}\).
Prob 11: Real-World Diver Rate (3 Pts) 7 meters / minute
Calculation (1.5 Pts): \(m = \frac{35 - 14}{5 - 2} = \frac{21\text{ meters}}{3\text{ minutes}} = 7\text{ m/min}\)
Context Meaning (1.5 Pts): "The diver descends at a constant rate of 7 meters every minute."
5
Part 5 Solutions: Synthesis Challenge (5 Pts)
5 Points Total
Problem 12: Remote Car Model (\(y = 4.5x\)) Comprehensive Mastery Check
a) Constant: \(k = 4.5\) (1 Pt)
b) Unit Rate: 4.5 m/s (1 Pt)
c) Slope: \(m = 4.5\) (1 Pt)
d) Conceptual Connection (2 Pts):
Exemplar: "Because this is a proportional relationship that passes through the origin \((0,0)\), the rate of change is constant throughout the entire graph. The slope (\(\frac{\text{rise}}{\text{run}} = \frac{y - 0}{x - 0} = \frac{y}{x}\)) is mathematically identical to the constant of proportionality \(k = \frac{y}{x}\)."
Slope Showdown • Teacher Answer Key Page 2 of 2 Scoring Guide Complete
