Proportions Mastery Practice Packet 8th Grade Independent Math
Proportions Mastery
Name:
Date: Period:
Core Rulebook: Solving Proportions
1. The Constant of Proportionality (\(k\))
For any proportional relationship: \(k = \frac{y}{x}\) or \(y = kx\)
Always divide the dependent value (\(y\)) by the independent value (\(x\)).
2. Cross-Multiplication Property
If \(\frac{a}{b} = \frac{c}{d}\), then cross products are equal: \(a \cdot d = b \cdot c\)
Multiply diagonals, write the equation, and isolate the variable.
A Identify the Constant of Proportionality (\(k\)) & Equation
Find \(k = \frac{y}{x}\) and write \(y = kx\)
1. Streaming Video Download Time
Minutes (\(x\)) 2 5 8 Gigabytes (\(y\)) 7 17.5 28
Calculate \(k = \frac{y}{x}\):
Equation (\(y = kx\)):
2. School Store Mechanical Pencils
Pencils (\(x\)) 3 6 10 Cost (\(y\)) $4.50 $9.00 $15.00
Calculate \(k = \frac{y}{x}\):
Equation (\(y = kx\)):
B Solve Proportions Using Cross-Multiplication
Cross multiply, show one step, and state the solution
\(\frac{x}{8} = \frac{9}{12}\) Solve for \(x\)
\(x =\)
\(\frac{5}{14} = \frac{15}{m}\) Solve for \(m\)
\(m =\)
\(\frac{4.2}{6} = \frac{k}{10}\) Solve for \(k\)
\(k =\)
\(\frac{18}{w} = \frac{24}{36}\) Solve for \(w\)
\(w =\)
Module: Proportions & Equations • 8th Grade Math Page 1 of 2
Part 2 • Applications
Real-World Proportion Challenges
Independent Problem Solving
Set up labels \(\rightarrow\) Cross-multiply \(\rightarrow\) Solve
Independent Strategy Tip: Always match corresponding units! For example: \(\frac{\text{miles}}{\text{hours}} = \frac{\text{miles}}{\text{hours}}\). Units in the numerators must match, and units in denominators must match.
7. Culinary Recipe Scaling
Rate • Cooking
A bakery recipe requires 3 cups of flour to make 24 artisanal cookies . You need to bake 60 cookies for a community event. How many cups of flour will you need?
Set up proportion with units:
Solution:
cups of flour
8. Map Scale & Distance
Scale Drawing
On an architectural blueprint, a scale indicates that 0.5 inches represents 14 feet of actual building length. If a conference room measures 3.5 inches on the blueprint, what is the room's actual length in feet?
Set up proportion with units:
Solution:
feet
9. Indirect Measurement (Shadow Reckoning)
Similar Triangles
At 2:00 PM on a sunny day, a 6-foot-tall student casts an 8-foot shadow . At the exact same time, the school flagpole casts a 36-foot shadow . How tall is the flagpole?
Set up proportion with units:
Solution:
feet tall
Self-Check & Progress Reflection
Check the steps you completed:
Cross-multiplied diagonals correctly Included proper units on all answers
Confidence Rating:
1 - Need Review
4 - Mastered
Module: Proportions & Equations • 8th Grade Math Page 2 of 2
Linear Equations Practice Packet 8th Grade Independent Math
Linear Equations Mastery
Name:
Date: Period:
The 4-Step Equation Roadmap
Golden Rule: Whatever you do to one side, do to the other
Step 1: Simplify
Distribute parentheses and combine like terms on each side.
Step 2: Collect Variables
Use addition or subtraction to move variables to one side.
Step 3: Collect Constants
Use addition or subtraction to move numbers to the other side.
Step 4: Isolate
Multiply or divide to get \(x\) completely by itself.
Worked Example: Follow the Steps Reference Model
\(3(2x - 4) + 5 = 4x + 9\)
\(6x - 12 + 5 = 4x + 9\)
\(6x - 7 = 4x + 9\)
\(2x - 7 = 9\)
\(2x = 16 \implies \mathbf{x = 8}\)
• Distribute: \(3 \cdot 2x = 6x\) and \(3 \cdot (-4) = -12\)
• Combine: \(-12 + 5 = -7\)
• Subtract: \(4x\) from both sides
• Add: \(7\) to both sides
• Divide: by \(2\)
A Independent Practice: Multi-Step & Variables on Both Sides
Show complete algebraic steps below
\(5x - 8 = 3x + 14\)
\(x =\)
\(4(p - 3) = 24\)
\(p =\)
\(2(3y + 1) = 4y + 18\)
\(y =\)
\(7 - 2(n + 4) = 5n - 29\)
\(n =\)
Module: Proportions & Equations • 8th Grade Math Page 1 of 2
Part 2 • Synthesis
Connecting Proportions & Equations
Proportional Equations & Modeling
Apply cross-products to expressions
Algebraic Proportions Reminder
When a proportion contains a binomial expression, use parentheses before cross-multiplying: \(\frac{x - 3}{4} = \frac{5}{2} \implies 2(x - 3) = 4 \cdot 5 \implies 2x - 6 = 20 \implies 2x = 26 \implies x = 13\) Don't forget to distribute to both terms inside the parentheses!
B Solve Algebraic Proportions
\(\frac{w + 5}{3} = \frac{16}{6}\)
\(w =\)
\(\frac{2y - 4}{5} = \frac{y + 1}{3}\)
\(y =\)
C Real-World Linear Modeling
7. Comparing Fitness Membership Plans
Balance Point Solutions Guide Facilitator & Self-Check Key
Proportions Solutions & Diagnostics
Packet 1 Answer Key
A Constant of Proportionality Solutions
1. Video Download Time \(k = 3.5\) GB/min
Work: \(\frac{7}{2} = 3.5\), \(\frac{17.5}{5} = 3.5\), \(\frac{28}{8} = 3.5\)
Equation: \(\mathbf{y = 3.5x}\)
2. Mechanical Pencils \(k = 1.50\) $/pencil
Work: \(\frac{4.50}{3} = 1.50\), \(\frac{9.00}{6} = 1.50\), \(\frac{15.00}{10} = 1.50\)
Equation: \(\mathbf{y = 1.5x}\) or \(\mathbf{y = 1.50x}\)
B Cross-Multiplication Solutions
Problem 3
\(12x = 72\)
\(\mathbf{x = 6}\)
Problem 4
\(5m = 210\)
\(\mathbf{m = 42}\)
Problem 5
\(6k = 42\)
\(\mathbf{k = 7}\)
Problem 6
\(24w = 648\)
\(\mathbf{w = 27}\)
C Part 2: Word Problem Detailed Breakdown
7. Recipe Flour
Proportion: \(\frac{3\text{ cups}}{24\text{ cookies}} = \frac{x\text{ cups}}{60\text{ cookies}} \implies 24x = 180\)
\(\mathbf{x = 7.5}\) cups or \(7\frac{1}{2}\) cups
8. Blueprint Distance
Proportion: \(\frac{0.5\text{ in}}{14\text{ ft}} = \frac{3.5\text{ in}}{x\text{ ft}} \implies 0.5x = 49\)
\(\mathbf{x = 98}\) feet \(49 \div 0.5 = 98\)
9. Shadow Reckoning
Proportion: \(\frac{6\text{ ft height}}{8\text{ ft shadow}} = \frac{h\text{ ft height}}{36\text{ ft shadow}} \implies 8h = 216\)
\(\mathbf{h = 27}\) feet tall Flagpole height
Diagnostic Guide: Common Student Misconceptions
Flipping \(k = \frac{x}{y}\) instead of \(\frac{y}{x}\):
If student wrote \(k = 0.285\) in #1, they divided minutes by GB. Remind: \(k\) is always output (\(y\)) divided by input (\(x\)).
Mismatched Word Problem Ratios:
If student set up \(\frac{\text{cups}}{\text{cookies}} = \frac{\text{cookies}}{\text{cups}}\), highlight corresponding units with colored pencil.
Module: Proportions & Equations • Complete Key Page 1 of 2
Facilitator & Self-Check Key
Linear Equations Solutions & Diagnostics
Packet 2 Answer Key
A Multi-Step & Both-Sides Solutions (Problems 1–4)
1. \(5x - 8 = 3x + 14\) \(\mathbf{x = 11}\)
Subtract \(3x\): \(2x - 8 = 14\) • Add \(8\): \(2x = 22\) • Divide by \(2\): \(x = 11\)
2. \(4(p - 3) = 24\) \(\mathbf{p = 9}\)
Distribute: \(4p - 12 = 24\) • Add \(12\): \(4p = 36\) • Divide by \(4\): \(p = 9\)
