Linear Legends Slides Linear Legends
Regents Blueprint Mastery: Unit 1
Rate of Change
Systems
Transformations
01
Bell Ringer: Slope Check
A cell phone company charges a flat monthly fee of \( \$30 \) plus \( \$0.05 \) per minute of use.
Tasks:
Write a linear function \( C(m) \).
Identify the slope and its meaning.
Identify the y-intercept and its meaning.
Ready for Discussion
The Linear Blueprint
Slope-Intercept Form
\( y = mx + b \)
Standard Form
\( Ax + By = C \)
"Remember: Slope is ALWAYS the coefficient of the independent variable."
Constant Rate
No Exponents
System Investigation
How many solutions does the following system have?
\( y = 2x + 5 \)
\( 4x - 2y = -10 \)
One Solution
Lines intersect once
Infinite Solutions
Lines are identical
No Solution
Lines are parallel
Activity Time
Open your Line Leader Worksheet. We will tackle the first three scenarios together using the Graphing Calculator strategies.
15
Minutes
03
Problems
Line Leader Worksheet Line Leader Worksheet
Linear Functions & Systems Mastery
Name:
Date:
Phase 1: Rate of Change & Equations
1. A water tank starts with 500 gallons and is being drained at a constant rate of 25 gallons per hour.
A) Function Notation
Write a linear function \( W(h) \) to represent the gallons of water after \( h \) hours.
B) Solution
Determine how many hours it will take for the tank to be empty.
C) Graphical Representation
Sketch the graph below. Label the axes and intercepts.
2. Which linear equation passes through the points \( (2, 5) \) and \( (6, 13) \)?
\( y = 2x + 1 \)
\( y = 2x + 3 \)
\( y = \frac{1}{2}x + 4 \)
\( y = \frac{1}{2}x + 10 \)
Phase 2: Systems of Equations
3. A theater sells adult tickets for \( \$12 \) and student tickets for \( \$8 \). On Friday, they sold 250 tickets total and made \( \$2,600 \).
Write the System
Show Your Work
Final Count:
Adult Tickets: ___________ Student Tickets: ___________
4. Graphical System Analysis
Given the inequalities: \( y \ge 2x - 3 \) and \( y < -x + 4 \).
y x
Step 1
Identify if each boundary line is solid or dashed.
Step 2
Describe the solution set of this system of inequalities.
Check Point!
Is the point \( (0, 0) \) a solution? Justify your answer algebraically.
Slope Secrets Teacher Guide Slope Secrets
Teacher Guide & Answer Key: Linear Legends
Unit 1: Linear
Instructional Strategy
Common Misconceptions
Students often swap x and y in the slope formula. Emphasize "rise over run".
Mixing up independent vs. dependent variables in word problems.
Forgetting to flip the inequality sign when dividing by a negative number.
Regents Strategy
Encourage students to use the STAT -> EDIT function on their calculators to find equations of lines from points quickly. This is a massive time-saver on the Regents.
Answer Key
1
Water Tank Draining
A) Function
W(h) = 500 - 25h
B) Solution
500 - 25h = 0 → 500 = 25h → h = 20 hours
2
Points (2, 5) and (6, 13)
Correct: y = 2x + 1
Calculation: m = (13 - 5) / (6 - 2) = 8 / 4 = 2. Plug in (2, 5): 5 = 2(2) + b → b = 1.
3
Theater Ticket System
Equations
a + s = 250
12a + 8s = 2600
Final Answers
Adult: 150
Student: 100
4
Inequalities System
Boundaries: \( y \ge 2x - 3 \) (Solid) and \( y < -x + 4 \) (Dashed).
Check (0,0): 0 ≥ -3 (True) and 0 < 4 (True). Yes, it is a solution.
Lenny Educational Resources © 2026 Regents Blueprint Mastery
Polynomial Power Slides Polynomial Power
Regents Blueprint Mastery: Unit 2
Operations
Factoring
Zeros
02
Bell Ringer: Identity Quest
Which expression is equivalent to \( (2x^2 - 5x + 3) - (x^2 + 2x - 1) \)?
A) \( x^2 - 3x + 2 \)
B) \( x^2 - 7x + 4 \)
WATCH OUT!
"The most common mistake on the Regents is forgetting to distribute the negative sign to EVERY term in the second parentheses."
The Factoring Blueprint
Step 1
GCF
Always check for a Greatest Common Factor first.
Step 2
DOTS
Difference of Two Squares: \( a^2 - b^2 \)
Step 3
Trinomials
Find factors of \( c \) that sum to \( b \).
Zeros vs. Factors
If the zeros of a function are \( x = -3 \) and \( x = 5 \), what is the function in factored form?
Zeros
\( -3, 5 \)
Factors
\( (x + 3)(x - 5) \)
"Flip the sign to find the factor!"
Factor Force
It's time to build your polynomial mastery. Open your Factor Force Worksheet. Focus on precision with your signs!
Today's Target
Factoring Completely (Double Factoring)
Factor Force Worksheet Factor Force Worksheet
Polynomial Operations & Factoring
Name:
Date:
Phase 1: Polynomial Operations
1. Simplify the following expression:
\( 3(x^2 - 4x + 1) - 2(x^2 - 5) \)
Final Result:
2. Multiply and simplify:
\( (2x - 3)(x^2 + 4x - 1) \)
Show Your Work
Pro Tip
Use the area model (box method) or distribute each term in the first binomial to each term in the trinomial.
Phase 2: Factoring Completely
Factor each expression completely. Remember: GCF first!
Problem 3
\( 2x^2 - 18 \)
Problem 4
\( x^3 - 5x^2 - 6x \)
Problem 5
\( 3x^2 + 15x + 18 \)
Problem 6
\( 4x^2 - 25 \)
Phase 3: Graphs & Zeros
7. A polynomial function has zeros at \( x = -2 \), \( x = 0 \), and \( x = 4 \).
A) Write the Function
Write an expression for \( p(x) \) in factored form.
B) Sketch
Sketch a possible graph of \( p(x) \) below.
Multiple Choice Check
Which graph represents a function with factors \( (x - 1) \) and \( (x + 3) \)?
Graph A
Graph B
Graph C
Graph D
Polynomial Pro Teacher Guide Polynomial Pro
Teacher Guide & Answer Key: Polynomial Power
Unit 2: Polynomials
Pedagogical Focus
The "Factor Completely" Trap
Students often stop after the first step (e.g., pulling out a GCF). Remind them that if the remaining expression has an \( x^2 \), they likely need to factor again. "Factor Completely" on the Regents almost always implies two steps.
Visualizing Zeros
Use graph sketches to help students see that factors determine x-intercepts. A zero at \( x=4 \) must come from factor \( (x-4) \) because plugging in 4 makes that factor zero.
Answer Key
1
Simplifying Expressions
x^2 - 12x + 13
Breakdown: \( (3x^2 - 12x + 3) - (2x^2 - 10) = 3x^2 - 2x^2 - 12x + 3 + 10 \)
3-6
Factoring Results
Problem 3
2(x - 3)(x + 3)
Problem 4
x(x - 6)(x + 1)
Problem 5
3(x + 2)(x + 3)
Problem 6
(2x - 5)(2x + 5)
7
Function from Zeros
Part A: \( p(x) = x(x + 2)(x - 4) \)
Note: Any leading coefficient or constant multiplier \( a \) is acceptable, but usually \( a=1 \) is expected unless a point is given.
MC Check: The graph must intersect the x-axis at \( x = 1 \) and \( x = -3 \).
Lenny Educational Resources © 2026 Regents Blueprint Mastery
Quadratic Quest Slides Quadratic Quest
Regents Blueprint Mastery: Unit 3
Vertex Form
Solutions
Modeling
03
Bell Ringer: The Rocket
A rocket is launched from a height of 10 feet. Its height \( h(t) \) after \( t \) seconds is given by:
\( h(t) = -16t^2 + 64t + 10 \)
What is the initial height?
What does the \(-16t^2\) represent?
Calculator Hack
"Go to \( Y= \), type the equation, then hit \( 2nd \rightarrow GRAPH \) (TABLE) to find the vertex and when the rocket hits the ground."
The Vertex Vault
Axis of Symmetry
\( x = \frac{-b}{2a} \)
Vertex Form
\( y = a(x-h)^2 + k \)
Regents Hint
If the question asks for the maximum or minimum , they are ALWAYS asking for the y-value of the vertex.
Solving Toolbelt
Factoring
Best for simple numbers.
Quadratic Formula
Works for EVERYTHING.
Square Roots
When \( b = 0 \).
Complete Square
To find Vertex Form.
Which method would you use for \( x^2 - 10x + 21 = 0 \)? Why?
Parabola Pilot
Grab your Parabola Pilot Worksheet. We're going to model trajectory and solve complex quadratics step-by-step.
Part 1
Standard to Vertex Form
Part 2
Real-World Application
Parabola Pilot Worksheet Parabola Pilot Worksheet
Quadratic Functions & Modeling
Name:
Date:
Phase 1: Forms & Transformations
1. Convert \( f(x) = x^2 - 8x + 10 \) into vertex form by completing the square.
Vertex Point:
Min or Max?
2. Transformation Analysis
Describe the transformations from the parent function \( y = x^2 \) to the function \( g(x) = -2(x + 3)^2 - 5 \).
Shifted Left 3 units
Shifted Right 3 units
Vertical Stretch by 2
Reflected over the x-axis
Shifted Down 5 units
Phase 2: Projectile Motion
3. A ball is thrown into the air. Its height in feet is modeled by \( H(t) = -16t^2 + 48t + 5 \), where \( t \) is time in seconds.
A) Time to Reach Maximum
Use the formula \( t = -b/2a \)
B) Maximum Height
Plug your answer from part A back into \( H(t) \)
Graph the Flight
Height (ft) Time (s)
4. Solving via Quadratic Formula
Solve for \( x \) to the nearest tenth : \( 2x^2 + 7x - 4 = 0 \).
\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
Root 1
Root 2
Quadratic Guru Teacher Guide Quadratic Guru
Teacher Guide & Answer Key: Quadratic Quest
Unit 3: Quadratics
Facilitation Notes
Contextual Meanings
Students often struggle with "What does this mean in the context of the problem?".Y-intercept: Starting point (t=0).Vertex: Turning point (max height or min cost).Roots: When the object hits the ground (H=0).
Rounding Awareness
Regents questions are specific about rounding. If it says "nearest tenth", don't provide the exact radical form. Remind students to do the final calculation in their calculator as the very last step to maintain precision.
Answer Key
1
Standard to Vertex
f(x) = (x - 4)^2 - 6
Vertex: (4, -6)
Type: Minimum (a > 0)
2
Transformations
Left 3
Reflection over x-axis
Vertical stretch by 2
Down 5
3
Projectile Motion
Part A: Time
1.5 seconds
t = -48 / (2 * -16) = -48 / -32
Part B: Height
41 feet
H(1.5) = -16(1.5)^2 + 48(1.5) + 5
4
Quadratic Formula
x = 0.5 and x = -4.0
a=2, b=7, c=-4. Discriminant = 49 - 4(2)(-4) = 49 + 32 = 81. Roots: (-7 ± 9) / 4.
Lenny Educational Resources © 2026 Regents Blueprint Mastery
Data Detectives Slides Data Detectives
Regents Blueprint Mastery: Unit 4
Distributions
Correlation
Linear Regression
04
Bell Ringer: Box Plot Mystery
5060708090100
Quick Check:
What is the Median of this set?
What is the Interquartile Range (IQR) ?
What percentage of data is above 80?
The Detective's Warning
"Correlation is not Causation!"
Just because ice cream sales and shark attacks both increase in summer doesn't mean ice cream causes shark attacks.
r = 0.95
Strong Positive
r = -0.21
Weak Negative
The r-value Rule
The closer to 1 or -1 , the stronger the relationship.
0 means no linear correlation.Sign (+/-) tells the direction of the trend.
Mastering the Regression
1
Input Data
Hit STAT → EDIT. Enter values into L1 and L2.
2
Calculate
Hit STAT → CALC → LinReg (ax+b).
3
Analyze
Find a, b, and r. Round as requested!
Pro Tip: If you don't see 'r', go to CATALOG → DiagnosticOn.
Stat Scout
It's time for some investigative work. Open your Stat Scout Worksheet and get those calculators ready!
Target 1
Line of Best Fit
Target 2
Outlier Identification
Stat Scout Worksheet Stat Scout Worksheet
Data Analysis & Linear Regression
Name:
Date:
Phase 1: Descriptive Statistics
1. The following data set represents the number of points scored by a basketball team in 8 games:
5862646470728892
Mean (\(\bar{x}\)):
Standard Deviation (\(\sigma_x\)):
Range:
Five-Number Summary
MIN
Q1
MED
Q3
MAX
Identify any potential outliers using the \( 1.5 \times IQR \) rule.
Phase 2: Bivariate Data
2. A study compared the number of hours students spent studying (\( x \)) and their score on the final exam (\( y \)).
Hours (x) Score (y) 2 65 5 80 1 60 8 88 4 75
A) Line of Best Fit
Write the equation, rounding to the nearest hundredth.
y = ________x + ________
B) Correlation Coefficient
Identify the \( r \) value to the nearest thousandth.
Interpretation
Based on the r-value, describe the strength and direction of the relationship.
Phase 3: Residuals & Causation
3. Which residual plot suggests that a linear model is appropriate for the data?
Choice A
(Curved Pattern)
Choice B
(Random Scatter)
Choice C
(V-Shape)
Choice D
(Expanding Spread)
4. Critical Thinking: Causation
"A study found a strong positive correlation between the number of firefighters at a fire and the amount of damage caused by the fire."
Does this mean that firefighters cause the damage? Explain your reasoning using statistical vocabulary.
Trend Tracker Teacher Guide Trend Tracker
Teacher Guide & Answer Key: Data Detectives
Unit 4: Statistics
Detective Insights
Calculator Setup
A common frustration is students' calculators being in the wrong mode. Ensure Diagnostics ON is active so students see the r value. Remind them that \( \sigma_x \) is the population standard deviation, which is what the Regents typically uses.
Residual Meaning
Explain that a linear model is good when the residual plot is random . If there's a pattern (like a curve), it means the original data was probably non-linear (likely quadratic or exponential).
Answer Key
1
Basketball Scores
Mean: 71.25
Standard Dev (\(\sigma_x\)): 11.66
Range: 34 (92-58)
5-Number: 58, 63, 67, 80, 92
Outliers: IQR = 17. Upper bound = 80 + 25.5 = 105.5. Lower bound = 63 - 25.5 = 37.5. No outliers.
2
Study Hours vs Score
Equation
y = 4.07x + 57.34
r-value
0.984
Interpretation: Strong positive correlation. As study hours increase, exam scores tend to increase significantly.
3-4
Conceptual Logic
Q3 Answer: Choice B (The random scatter indicates a linear model fits well).
Q4 Explanation: No, firefighters do not cause the damage. This is a lurking variable situation (the size of the fire). A larger fire causes more damage AND requires more firefighters. Correlation != Causation.
Lenny Educational Resources © 2026 Regents Blueprint Mastery
Regents Ready Slides Regents Ready
The Final Frontier: Strategy & Mock Exam
Exam Survival Guide
01
Pacing is Power
Spend no more than 3 minutes on any Multiple Choice question. Save time for Part II, III, and IV!
02
Annotate the prompt
Circle "nearest tenth", "factor completely", or "justify". These are instructions, not suggestions.
03
Check Your Work
Plug your solutions back into the original equation. If it doesn't balance, rethink the step.
Calculator Reminders
Check Mode: RADIAN vs DEGREE
Clear Lists (L1, L2) before stats
Zoom Standard (Zoom 6) for graphs
2nd + TRACE = CALCULATE Menu
The "Almost Got Me" List
Negative Signs
Subtracting a whole binomial? Distribute that negative to every term inside.
Rounding Early
Never round your intermediate steps. Only round the FINAL answer in the last line.
Domain/Range
Read the context! Time cannot be negative. Number of people must be a whole number.
"Don't leave a single question blank. Even a wrong attempt can earn partial credit if the process is shown!"
Final Frontier Mock
This is a simulation of the actual Algebra 1 Regents. You will have 40 minutes to complete 10 high-impact questions covering all four blueprints.
Linear
3 Qs
Polynomials
2 Qs
Quadratics
3 Qs
Stats
2 Qs
"Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding."
Good Luck, Legends.
Final Frontier Mock Exam Final Frontier Mock Exam
Algebra 1 Regents Simulation
Name:
Score:
Instructions:
Answer all questions. For multiple choice, select the best option. For constructed response, show all work. You may use a graphing calculator.
1. Which expression is equivalent to \( 16x^2 - 49 \)?
\( (4x - 7)(4x - 7) \)
\( (4x - 7)(4x + 7) \)
\( (8x - 7)(8x + 7) \)
\( (8x - 49)(2x + 1) \)
2. The solution to \( 3(x - 2) < 2x + 5 \) is:
\( x < 11 \)
\( x > 11 \)
\( x < 7 \)
\( x > 7 \)
3. A function has zeros at \( -4 \) and \( 2 \). Which could be its equation?
\( f(x) = (x - 4)(x + 2) \)
\( f(x) = (x + 4)(x - 2) \)
\( f(x) = x^2 - 2x - 8 \)
\( f(x) = x^2 + 2x + 8 \)
4. Solving Systems (2 Points)
Solve the following system of equations algebraically for \( x \) and \( y \):
2x + 3y = 12
5x - y = 13
Show Work Here
x =
y =
5. The Quadratic Trajectory (4 Points)
An object is launched with an initial height of 6 feet and follows the path \( h(t) = -16t^2 + 32t + 6 \). Determine the maximum height of the object and justify your answer.
Show Work and Justification
6. Statistical Analysis (2 Points)
The scores of five students on a quiz are 80, 85, 90, 75, and 100. Find the standard deviation of this data set to the nearest hundredth .
Standard Deviation:
7. Factoring Completely (4 Points)
Factor the expression \( 3x^3 - 33x^2 + 90x \) completely.
Final Strategy Check
"Did you circle your final answers? Did you show every step of your algebra? If you used a calculator to find the answer, did you describe what you typed?"
Score Shield Answer Key Score Shield
Answer Key & Grading Rubric: Final Frontier
Final Mastery
Part I: Multiple Choice
Q1: Factoring DOTS
(4x - 7)(4x + 7)
Q2: Inequality
x < 11
3x-6 < 2x+5 → x < 11
Q3: Zeros
f(x) = (x + 4)(x - 2)
Part II: Constructed Response
Question 4: System of Equations
2 POINTS
Answers: x = 3, y = 2
Method: Elimination or Substitution.
Example: y = 5x - 13. Substitute into first: 2x + 3(5x - 13) = 12 → 2x + 15x - 39 = 12 → 17x = 51 → x=3.
Partial Credit (1pt):
Correct setup but math error.
Correct value for one variable only.
Question 5: Max Height
4 POINTS
Step 1: Time to Max. \( t = -32 / (2 * -16) = 1 \text{ second} \).
Step 2: Max Height. \( h(1) = -16(1)^2 + 32(1) + 6 = -16 + 32 + 6 = \mathbf{22 \text{ feet}} \).
The justification must state that the vertex represents the turning point/maximum of the parabola.
Question 7: Factor Completely
4 POINTS
Final Answer
3x(x - 5)(x - 6)
1. GCF: \( 3x(x^2 - 11x + 30) \)
2. Trinomial: \( 3x(x - 5)(x - 6) \)
Lenny Educational Resources © 2026 Regents Blueprint Mastery
