Foundations Reference Sheet
9/10 Grade Foundations
Linear Equations Reference Guide
Name: ____________________
Date: ____________________
The Three Forms of Linear Equations
Master these structures to model any linear relationship with precision.
Slope-Intercept
\(y = mx + b\)
\(m\): The Slope (Rate of Change)
\(b\): The y-intercept \((0, b)\)
USE WHEN:
You are given the starting value and a constant rate of growth or decay.
Point-Slope
\(y - y_1 = m(x - x_1)\)
Warning: Watch your signs! \(x - (-2) = x + 2\)
\(m\): Slope \((x_1, y_1)\): Point
USE WHEN:
You are given a specific point and the slope, but not the starting value.
Standard Form
\(Ax + By = C\)
Rates: \(A\) and \(B\) are unit rates.
Total: \(C\) is the constant sum.
USE WHEN:
You are modeling a scenario with two categories reaching a fixed total (e.g., Budgets).
The Mystery Point Strategy
- Identify the given slope and point.
- Write the Point-Slope equation.
- Plug in the known coordinate of the mystery point.
- Solve for the missing variable (\(x\) or \(y\)).
Slope Calculation
Use the change in \(y\) over the change in \(x\):
\(m = \frac{y_2 - y_1}{x_2 - x_1}\)
Always double-check your sign flips in Point-Slope form. Negative coordinates result in addition within the parentheses!
Foundations Student Packet
9/10 Grade Foundations
Student Practice Packet
Name: ____________________
Date: ____________________
Do Now: Warm-Up
5 Minutes
1. Sign Sensitivity Challenge:
A line passes through \((-2, 7)\) and has a slope of 4. Which equation correctly models this line in point-slope form?
a) \(y - 7 = 4(x - 2)\)
b) \(y - 7 = 4(x + 2)\)
c) \(y + 7 = 4(x - 2)\)
d) \(y + 7 = 4(x + 2)\)
Show your work/reasoning below:
2. Convert to Slope-Intercept Form:
Isolate \(y\) in the equation below to find the slope-intercept form:
\(3x + 4y = 12\)
Final Form: \(y = \) ___________________
Guided Practice
Example 1: The Mystery Coordinate Instructional Model
"A linear relationship has a slope of 2 and contains the points (3, 4) and (a, 10). What is the value of a?"
Step 1: Write Point-Slope Equation
Step 2: Substitute \((a, 10)\) and solve
Final Answer: \(a = \) ________
Example 2: Budget Modeling Guided Practice
"Alex is buying pizza and wings for a party. Pizzas (\(x\)) cost $15 each. Wings (\(y\)) cost $10 per dozen. Alex has a total budget of $120. Write an equation."
\(15x + 10y = 120\)
\(10x + 15y = 120\)
Why did you choose your answer?
Independent Practice
1. Conversion Mastery
Task #1
Convert the following equation into slope-intercept form:
\(y - 5 = 3(x + 2)\)
Show distributive and additive steps clearly:
Resulting Equation: \(y = \) ___________________
2. Fundraiser Modeling
Task #2
"Tickets to a show cost $12 for adults (a) and $8 for kids (k). The goal is to raise exactly $1,200."
Which equation correctly models this situation?
A) \(12a + 8k = 1200\)
B) \(8a + 12k = 1200\)
C) \(y = 12x + 8\)
Explain your choice:
3. The Hidden Coordinate
Task #3
"A line has a slope of -3 and contains the points (2, 5) and (x, 14). Find the value of x."
Show all algebraic steps:
Value of \(x\): __________
Final Accuracy Checklist:
- Did I calculate the slope correctly using \(m = \frac{y_2 - y_1}{x_2 - x_1}\)?
- Did I flip the signs when plugging coordinates into point-slope form?
- If a total sum is provided, is it placed correctly as the constant?
Exit Ticket
Concept Assessment: Accuracy & Justification
Score: /10
1. Solving for a Variable:
A line has a slope of 1/2 and passes through (2, 6) and (a, 15). Solve for a.
2. Comparison & Logic:
"Madison is shopping with $300. Sweaters are $45 each and Pants are $25 each. Explain why Standard Form is the most efficient way to model this situation."
Foundations Teacher Key
9/10 Grade Foundations
Teacher Answer Key & Facilitation Guide
Official Exemplar
Instructional Focus Points
- 1. Sign Sensitivity: Emphasize that in point-slope form \(y - y_1 = m(x - x_1)\), subtracting a negative results in addition. This is the most common student error in the Do Now.
- 2. Variable Substitution: Guide students to set up the general equation with the known point and slope first, then plug in the coordinates of the mystery point to solve for the missing variable.
- 3. Modeling Scenarios: Standard Form is ideal for "sum-to-total" scenarios. Help students map coefficients to rates and the constant to the total sum.
This key aligns page-for-page with the Student Packet for easy reference. Answers are provided in bold red with clear calculation steps for classroom review.
Do Now: Solution Key
1. Sign Sensitivity Challenge:
a) y - 7 = 4(x - 2)
b) y - 7 = 4(x + 2) CORRECT
Teacher Note:
"The x-coordinate is -2. Plugging into point-slope gives \(x - (-2)\), which becomes \(x + 2\). Option B is correct."
2. Convert to Slope-Intercept Form:
1. Subtract 3x: \(4y = -3x + 12\)
2. Divide by 4: \(y = -\frac{3}{4}x + 3\)
\(y = -\frac{3}{4}x + 3\)
Guided Practice Key
Ex 1: Missing Coordinate Solve
STEP 1: EQUATION
\(y - 4 = 2(x - 3)\)
STEP 2: PLUG & SOLVE
\(10 - 4 = 2(a - 3)\)
\(6 = 2a - 6\)
\(12 = 2a\)
a = 6
Ex 2: Budget Modeling
\(15x + 10y = 120\)
Exemplar Justification:
"$15/pizza (x) and $10/wings (y) are coefficients representing unit costs. The sum equals the total budget of $120. Standard Form models this directly."
Independent Practice Key (1-2)
Problem #1: Conversion
\(y - 5 = 3x + 6\)
\(y = 3x + 11\)
Problem #2: Fundraiser
Choice A
\(12a + 8k = 1200\)
Independent Practice Key (3)
Problem #3: Mystery Point
\(14 - 5 = -3(x - 2) \rightarrow 9 = -3x + 6\)
\(3 = -3x\)
\(x = -1\)
Checklist section mirrors student packet for instructional alignment.
Exit Ticket Exemplar
1. Coordinate Quest Solution:
\(15 - 6 = 1/2(a - 2)\)
\(9 = 1/2a - 1 \rightarrow 10 = 1/2a\)
\(a = 20\)
2. Reasoning Exemplar:
