Proportion Power Slides
Indiana Standard 7.AF.5 • 7th Grade Math
Unit 3: Proportions
PROPORTION POWER
Master two powerful strategies to set up, compare, and solve for unknowns in proportional equations.
Scale Multipliers
Cross Products
Real-World Models
Focus: Writing & Solving Proportions Slide 1 of 7
Core Concept
Slide 2 of 7
What is a Proportion?
A proportion is an equation stating that two ratios are equal.
General Equation
\[ \frac{a}{b} = \frac{c}{d} \]
Read: "\(a\) is to \(b\) as \(c\) is to \(d\)"
Concrete Ratio Match
\[ \frac{3\text{ cups flour}}{4\text{ cups milk}} = \frac{6\text{ cups flour}}{8\text{ cups milk}} \]
Both simplify to the exact same value (\(0.75\))
Units in top match; units in bottom match!
Golden Rule: Keep corresponding units in corresponding positions!
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Strategy 1
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Equivalent Ratios (Scale Multiplier)
Best when numbers share an obvious mental math multiplier!
Horizontal Multiplier × 4
\[ \frac{5}{8} = \frac{x}{32} \]
-
Look at bottom: \(8 \times 4 = 32\)
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Multiply top by \(4\): \(5 \times 4 = 20\)
Solution: x =
Vertical Multiplier × 3
\[ \frac{7}{21} = \frac{4}{y} \]
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Compare top to bottom: \(7 \times 3 = 21\)
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Apply to second ratio: \(4 \times 3 = 12\)
Solution: y =
Pro Tip: If multipliers are not friendly whole numbers, use Cross Products! Slide 3 of 7
Strategy 2 • Universal Tool
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Cross Products Property
For any proportion \(\frac{a}{b} = \frac{c}{d}\), diagonal products are equal: \(a \cdot d = b \cdot c\)
1. Cross-Multiply
\[ \frac{6}{15} = \frac{10}{w} \]
\(6 \cdot w = 15 \cdot 10\)
2. Form Equation
\[ 6w = 150 \]
Multiply the known side
3. Divide to Solve
\[ w = \frac{150}{6} = 25 \]
Divide by coefficient \(6\)
\(6 \cdot 25 = 150\)
\(15 \cdot 10 = 150\)
Both sides equal 150!
Works reliably for any proportion, including decimals and fractions! Slide 4 of 7
Real-World Protocol
Slide 5 of 7
Word Problem 4-Step Blueprint
A car travels 180 miles on 6 gallons. How many gallons for 450 miles?
1. Label Units
Set up unit template:
\[ \frac{\text{miles}}{\text{gallons}} \]
2. Match Values
Insert given and \(g\):
\[ \frac{180}{6} = \frac{450}{g} \]
3. Solve Equation
Cross-multiply:
\[ 180g = 2700 \]
4. State Answer
Include exact units:
g = 15 gallons
Reasonableness Check: \(450 \text{ mi} \div 180 \text{ mi} = 2.5\), and \(6 \times 2.5 = 15\text{ gallons}\). Matches perfectly!
Check for Understanding
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Your Turn: Solve With a Partner
Choose the best strategy (Scale Factor or Cross Products):
Problem A
\[ \frac{4}{7} = \frac{16}{m} \]
Hint: Notice \(4 \times 4 = 16\)
m =
Problem B
\[ \frac{5}{9} = \frac{k}{45} \]
Hint: Notice \(9 \times 5 = 45\)
k =
Problem C
\[ \frac{8}{12} = \frac{14}{p} \]
Hint: Cross-multiply: \(8p = 168\)
p =
Key insight: Simplifying \(\frac{8}{12}\) to \(\frac{2}{3}\) first makes Problem C mental math! Slide 6 of 7
Mistake Buster
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Avoid These 3 Proportion Pitfalls
Flipping Units
Keep units in the exact same positions:
\[ \frac{\text{miles}}{\text{hours}} \neq \frac{\text{hours}}{\text{miles}} \]
Match top-to-top and bottom-to-bottom.
Adding Numbers
Ratios are multiplicative, not additive!
\[ \frac{3}{5} \neq \frac{3 + 5}{5 + 5} \]
Never just add a number to both terms.
Stopping Early
Don't forget the final division step:
\[ 4x = 72 \implies x = 18 \]
Divide by \(4\) to isolate the unknown \(x\).
Ready for Independent Practice!
Slide 7 of 7
Proportion Power Worksheet
PROPORTION POWER
INDIANA ACADEMIC STANDARDS • 7.AF.5 (SOLVE PROPORTIONS)
Name:
Date: Period:
STRATEGY 1: SCALE FACTOR MULTIPLIER
Look for a clean horizontal or vertical multiplier between corresponding terms: \( \frac{2}{5} \xrightarrow{\times 6} \frac{12}{30} \)
STRATEGY 2: CROSS PRODUCTS PROPERTY
Multiply opposite diagonals: If \( \frac{a}{b} = \frac{c}{d} \), then \( a \cdot d = b \cdot c \). Divide to isolate the variable.
PART 1: GUIDED PRACTICE • SHOW ALL STEPS
1. Solve using Scale Multiplier: \( \frac{3}{8} = \frac{x}{40} \)
Find multiplier from \(8 \to 40\): Multiplier =
Final Answer: x =
2. Solve using Cross Products: \( \frac{5}{12} = \frac{15}{m} \)
Write cross-multiplication equation:
Final Answer: m =
PART 2: SOLVE EACH PROPORTION FOR THE UNKNOWN VARIABLE Show cross-products or scale factors
- \( \frac{4}{9} = \frac{y}{36} \)
y =
- \( \frac{7}{6} = \frac{35}{w} \)
w =
- \( \frac{k}{14} = \frac{15}{21} \)
k =
- \( \frac{5}{8} = \frac{22}{p} \)
p =
- \( \frac{9}{n} = \frac{24}{40} \)
n =
- \( \frac{2.4}{5} = \frac{12}{z} \)
z =
Turn over for Part 3 (Real-World Applications) and Error Analysis Page 1 of 2
Proportion Power • Student Practice
Name:
PART 3: REAL-WORLD PROPORTIONAL MODELING Set up a labeled proportion with units, solve, and label answers
9. Recipe Scaling: A cafeteria chili recipe calls for 4 cans of beans to make 18 servings. How many cans of beans are needed to make 45 servings?
Proportion Setup:
\( \frac{\text{beans}}{\text{servings}} = \frac{\text{beans}}{\text{servings}} \)
Answer:
10. Map Scale: On an Indiana highway map, the scale is 0.5 inches = 16 miles. If the map distance between Indianapolis and Bloomington is 1.75 inches, what is the actual driving distance?
Proportion Setup:
\( \frac{\text{inches}}{\text{miles}} = \frac{\text{inches}}{\text{miles}} \)
Answer:
11. Unit Pricing: The school bookstore sells a pack of 6 mechanical pencils for $4.50. At this same proportional rate, how much would a pack of 14 mechanical pencils cost?
Proportion Setup:
\( \frac{\text{cost (\$)}}{\text{pencils}} = \frac{\text{cost (\$)}}{\text{pencils}} \)
Proportion Power Answer Key
PROPORTION POWER
TEACHER ANSWER KEY
INDIANA ACADEMIC STANDARDS • 7.AF.5 (STEP-BY-STEP SOLUTIONS)
Total Points: 25 pts
Grading Insight: Students may use either the Equivalent Ratio (scale factor) method or the Cross Products Property. Accept any mathematically valid pathway as long as the steps and final answer are correct.
PART 1: GUIDED PRACTICE SOLUTIONS (2 PTS EACH) Scale Multipliers & Cross Products
1. Scale Multiplier Method: \( \frac{3}{8} = \frac{x}{40} \)
1. Identify horizontal multiplier: \(8 \times 5 = 40\)
2. Multiply numerator by 5: \(3 \times 5 = 15\)
Check: \(3/8 = 0.375\) and \(15/40 = 0.375\)
Final Answer: x = 15
2. Cross Products Method: \( \frac{5}{12} = \frac{15}{m} \)
1. Multiply diagonals: \(5 \cdot m = 12 \cdot 15\)
2. Simplify equation: \(5m = 180\)
3. Divide by 5: \(m = 180 \div 5 = 36\)
Final Answer: m = 36
PART 2: ALGEBRAIC PROPORTIONS (2 PTS EACH) 6 Problems • 12 Pts Total
- \( \frac{4}{9} = \frac{y}{36} \)
\(9 \times 4 = 36\)
\(y = 4 \times 4 = 16\)
Or: \(9y = 144 \implies y = 16\)
y = 16
- \( \frac{7}{6} = \frac{35}{w} \)
\(7 \times 5 = 35\)
\(w = 6 \times 5 = 30\)
Or: \(7w = 210 \implies w = 30\)
w = 30
- \( \frac{k}{14} = \frac{15}{21} \)
Simplify: \(15/21 = 5/7\)
\( \frac{k}{14} = \frac{5}{7} \xrightarrow{\times 2} k = 10 \)
Or: \(21k = 210 \implies k = 10\)
k = 10
- \( \frac{5}{8} = \frac{22}{p} \)
\(5 \cdot p = 8 \cdot 22\)
\(5p = 176\)
\(p = 176 \div 5 = 35.2\)
p = 35.2
- \( \frac{9}{n} = \frac{24}{40} \)
Simplify: \(24/40 = 3/5\)
\( \frac{9}{n} = \frac{3}{5} \xrightarrow{\times 3} n = 15 \)
Or: \(24n = 360 \implies n = 15\)
n = 15
- \( \frac{2.4}{5} = \frac{12}{z} \)
\(2.4 \times 5 = 12\)
\(z = 5 \times 5 = 25\)
Or: \(2.4z = 60 \implies z = 25\)
z = 25
Proportion Power Answer Key • Part 1 & Part 2 Page 1 of 2
