Parallel Path Packet Student Parallel Path
Name:
Date:
Phase 01: Initial Survey (Do Now)
Identify the slope (\(m\)) for each given equation or pair of points.
1. \(y = 3x - 5\)
\(m = \)
2. \(y = -\frac{2}{3}x + 8\)
\(m = \)
3. \((2, 4)\) and \((4, 10)\)
\(m = \)
4. \(2x + 4y = 12\)
\(m = \)
Technical Specifications
Parallel Lines
Lines that never intersect. Parallel lines have the EXACT SAME SLOPE .
\(m_1 = m_2\)
Perpendicular Lines
Use Opposite Reciprocal slopes:
Flip the sign (\(+\) to \(-\))
Flip the fraction (\(2/3\) to \(3/2\))
\(m_1 = -\frac{1}{m_2}\)
Phase 02: Guided Drafting (Parallel)
Blueprint Task A Writing Parallel Equations
Write the equation of a line that passes through the point \((4, -1)\) and is parallel to the line \(y = 2x + 3\) .
Step 1: Identify the slope of the given line
Step 2: Determine your new slope (\(m\))
Step 3: Solve for \(b\) or use Point-Slope Form
Final Specification:
y = ____________________
Phase 03: Guided Drafting (Perpendicular)
Blueprint Task B Writing Perpendicular Equations
Write the equation of a line that passes through \((6, 2)\) and is perpendicular to \(y = -3x + 1\) .
Step 1: Identify the slope of the given line
Step 2: Determine your "Opposite Reciprocal" slope
Step 3: Construct the equation
Final Specification:
y = ____________________
Phase 04: Quality Assurance Spot Check
Inspection Question: What is the perpendicular slope to the line \(y = 5x - 2\)?
A) 5
B) -5
C) -1/5
Explain your reasoning:
Phase 05: Independent Drafting
1. Parallel to \(y = \frac{1}{2}x - 4\) through \((-2, 5)\).
Result: \(y = \) _______________________
2. Perpendicular to \(y = 4x + 10\) through \((8, 1)\).
Result: \(y = \) _______________________
3. Perpendicular to \(y = -\frac{2}{5}x + 2\) through \((0, -3)\).
Result: \(y = \) _______________________
Phase 06: Inspection Check (Exit Ticket)
Structural Analysis
An architect is designing two walkways. Walkway A follows the path of the line \(y = -x + 4\). Walkway B must be perpendicular to Walkway A and pass through the point \((3, 7)\) . Write the equation for Walkway B.
Final Approval Specification:
y = ____________________
Parallel Path Key Answer Key
Phase 01: Initial Survey Solutions
1. \(y = 3x - 5\)
\(m = 3\)
2. \(y = -\frac{2}{3}x + 8\)
\(m = -\frac{2}{3}\)
3. \((2, 4)\) and \((4, 10)\)
\(m = 3\)
4. \(2x + 4y = 12\)
\(m = -\frac{1}{2}\)
Phase 02: Guided Drafting Solutions (Parallel)
Parallel to \(y = 2x + 3\) through \((4, -1)\)
Slopes
Old \(m = 2\)
Equation Setup
\(y - (-1) = 2(x - 4)\)
Final: \(y = 2x - 9\)
Phase 03: Guided Drafting Solutions (Perpendicular)
Perpendicular to \(y = -3x + 1\) through \((6, 2)\)
Slopes
Old \(m = -3\)
Equation Setup
\(y - 2 = \frac{1}{3}(x - 6)\)
Final: \(y = \frac{1}{3}x\)
Phase 04: QA Spot Check Solution
Correct Choice: C) \(-\frac{1}{5}\)
Reasoning: The original slope is \(5\). For a perpendicular line, we must find the opposite reciprocal. Flipping the sign makes it negative, and the reciprocal of \(5\) (which is \(5/1\)) is \(1/5\).
Phase 05: Independent Drafting Solutions
1. Parallel to \(y = \frac{1}{2}x - 4\) through \((-2, 5)\)
\(y = \frac{1}{2}x + 6\)
2. Perpendicular to \(y = 4x + 10\) through \((8, 1)\)
\(y = -\frac{1}{4}x + 3\)
3. Perpendicular to \(y = -\frac{2}{5}x + 2\) through \((0, -3)\)
\(y = \frac{5}{2}x - 3\)
Phase 06: Inspection Check Solution (Exit Ticket)
Walkway A Slope: \(m_A = -1\)
Perpendicular Slope: \(m_B = 1\)
\(y - 7 = 1(x - 3) \rightarrow y - 7 = x - 3\)
Approved Specification
\(y = x + 4\)
Equation Blueprint Packet Trimmed Trimmed Equation Blueprint
Mastering fractions through organization.
Name:
Date:
Do Now: Vocab Match
Reciprocal
Numerator
Denominator
Coefficient
Variable
Constant
A. A letter used to represent an unknown number.
B. The number on top of a fraction.
C. A number by itself (no variable attached).
D. The "flipped" version of a fraction.
E. The number on the bottom of a fraction.
F. The number directly in front of a variable.
Switch: Work on your Resource Sheet now!
Guided Practice
Example 1: Variable in Numerator
Solve: \( \frac{x}{4} - 5 = 2 \)
Example 2: Fractional Coefficient
Solve: \( \frac{2}{3}x + 4 = 10 \)
Independent Practice
LEVEL 1
1. \( \frac{x}{2} = 8 \)
LEVEL 1
2. \( \frac{1}{4}x = 3 \)
LEVEL 2
3. \( \frac{x}{5} + 3 = 6 \)
LEVEL 2
4. \( \frac{3}{5}x = 12 \)
LEVEL 3
5. \( \frac{2}{5}x - 2 = 8 \)
LEVEL 3
6. \( \frac{3x}{4} + 1 = 10 \)
Exit Ticket
Solve the equation below. Show every step and keep your work organized inside the grid.
Final Challenge: \( \frac{2}{3}x + 5 = 11 \)
Equation Blueprint Resource Sheet Refined Resource Sheet
Fractional Equations Cheat Sheet
1. Parts of an Equation
3
4
Numerator:
Denominator:
Variable:
Constant:
Coefficient:
2. Solving with Fractions
Case A: Variable in the Numerator
\( \frac{x}{4} = 5 \)
To remove the 4, you must both sides.
Case B: Fractional Coefficient
\( \frac{2}{3}x = 8 \)
Multiply both sides by the .
Flip the fraction! The reciprocal of 2/3 is 3/2 .
2 3
The Grid Master Rule
In an equation, every step should be written on a new line. Keep equals signs (\(=\)) in a straight line. Always bring down everything you haven't used yet!
Equation Blueprint Answer Key Refined ANSWER KEY
Equation Blueprint Suite
Resource Sheet Solutions
1. Parts of Equation:
Numerator: Top Number
Denominator: Bottom Number
Variable: Letter (unknown)
Constant: Number alone
Coefficient: Number with variable
2. Solving Rules:
Case A: Multiply both sides
Case B: Reciprocal (flip it)
Reciprocal of 2/3: 3/2
Grid Master Rule: Straight vertical line
Do Now Match
D Reciprocal
B Numerator
E Denominator
F Coefficient
A Variable
C Constant
Problem Solutions
1. \( \frac{x}{2} = 8 \)
\( 2 \cdot (\frac{x}{2}) = 16 \)
\( x = 16 \)
2. \( \frac{1}{4}x = 3 \)
\( 4 \cdot (\frac{1}{4}x) = 12 \)
\( x = 12 \)
3. \( \frac{x}{5} + 3 = 6 \)
\( \frac{x}{5} = 3 \)
\( 5 \cdot 3 = 15 \)
\( x = 15 \)
4. \( \frac{3}{5}x = 12 \)
\( \frac{5}{3} \cdot \frac{3}{5}x = 12 \cdot \frac{5}{3} \)
\( x = 20 \)
5. \( \frac{2}{5}x - 2 = 8 \)
\( \frac{2}{5}x = 10 \)
\( \frac{5}{2} \cdot 10 = 25 \)
\( x = 25 \)
6. \( \frac{3x}{4} + 1 = 10 \)
\( \frac{3}{4}x = 9 \)
\( \frac{4}{3} \cdot 9 = 12 \)
\( x = 12 \)
Exit Ticket Challenge Key
Problem: \( \frac{2}{3}x + 5 = 11 \)
Step 1: Subtract 5
\( \frac{2}{3}x = 6 \)
Step 2: Reciprocal Multiply (3/2)
\( \frac{3}{2} \cdot \frac{2}{3}x = 6 \cdot \frac{3}{2} \)
Step 3: Solve
\( x = \frac{18}{2} \)
\( x = 9 \)
