Variable Connections Slides Algebra 1 Workshop CCSS 6.EE.C.9 Foundation
Unit 3: Linear Relationships
Variable Connections
How one quantity controls another in everyday life, and how we write equations to tell the whole story.
X
Independent Variable
The Cause / The Input you choose
Y
Dependent Variable
The Effect / The Output that changes
Big Idea #1 Cause & Effect in Math
Who is in control?
Independent (Input)
The Controller / Cause
This is what changes first. It is free to vary on its own.
Real Life: Hours you work at your job, gallons of gas pumped, minutes spent gaming.
Dependent (Output)
The Result / Effect
This depends on what happened first. Its value is decided by the independent variable.
Real Life: Total paycheck amount, total gas cost, XP points earned.
Quick Memory Anchor: The Independent variable comes IN first, the Dependent variable DEPENDS on it!
The Secret Test Sentence Frame Strategy
Use the "Depends On" Test
Whenever you are not sure, test this sentence out loud:
"The Dependent Variable depends on the Independent Variable."
✓ Makes Sense (Correct!)
"Your paycheck depends on the hours worked."
✕ Does NOT Make Sense (Backwards!)
"The hours worked depends on your paycheck."
If it sounds backward, flip the two variables around!
Model 1: Hourly Wages From Story to Equation
Scenario: Jordan works at a smoothie shop.
Jordan earns $15 per hour. Let h = hours worked, and let p = total pay.
Input / Output Table
Hours worked (h) Total Pay (p)
1 hour $15 × 1 = $15
2 hours $15 × 2 = $30
3 hours $15 × 3 = $45
Building the Equation:
Step 1: Start with the Dependent Variable p =
Step 2: Multiply Rate by Independent Variable p = 15 · h
Final Equation
p = 15h
Rule of thumb: Total (Output) goes by itself on one side of the equals sign!
Model 2: Pizza Toppings Rate of Change
Scenario: Pizza Palace calculates cost.
A plain cheese pizza has extra toppings for $2 per topping (t). What is the total extra cost (c)?
1. Independent (x)
Toppings (t)
You choose how many toppings to add.
2. The Rule (Rate)
$2 each
Every single topping adds $2.00 to the bill.
3. Dependent (y)
Extra Cost (c)
Cost depends directly on toppings chosen.
Equation Equation Form: Dependent = Rate × Independent
c = 2t
What if you pick 4 toppings? c = 2(4) = $8 extra!
We Do: Class Challenge Think - Pair - Share
Case File #1
A streaming app charges $8 per month.
Let m = number of months subscribed.
Question 1: Which variable is Independent?
Is it m (months) or C (cost)?
Question 2: What is the Equation?
Option A: m = 8C C = 8m
Sentence Check: "The Total Cost (C) depends on the Months Subscribed (m) ." → So C = 8m !
Wrap-Up Check Ready for Guided Practice
Today's Master Checklist
1
Independent (x)
The cause or choice. You decide it first.
2
Dependent (y)
The effect or outcome. It depends on x.
3
Equation
y stands alone:
Now take out your Variable Connections Worksheet and quick reference guide!
You've got this! Math is just describing patterns in the world.
Variable Anchor Reference Sheet Algebra 1 Quick Guide Standard CCSS 6.EE.C.9
Variables & Two-Variable Equations
Desk Anchor Sheet
Input x
Independent Variable
The CAUSE or quantity you control. It changes freely and happens first.
Ask yourself:
"What do I choose or count first?"
Real-life examples:
Hours worked at a job (h)
Number of pizzas ordered (p)
Gigabytes of phone data used (g)
Output y
Dependent Variable
The EFFECT or outcome. Its value depends directly on the other quantity.
Ask yourself:
"What is the final result or total?"
Real-life examples:
Total paycheck amount (P)
Total cost of dinner bill (c)
Phone battery remaining (b)
Strategy #1 • Golden Sentence Test Say it aloud
"The Dependent Variable depends on the Independent Variable."
✓ Makes Sense: "My paycheck depends on the hours I work."
✕ Backwards: "The hours I work depends on my paycheck."
Strategy #2 • 3-Step Equation Builder Blueprint
Formula: y = (Rate) × x
1
Name the Variables
Choose letters for your two quantities. Identify which is input (x) and output (y).
Output = t (total)
2
Find the Unit Rate
Look for clue words: "per", "each", "every", "a" . This is your multiplying number.
"$14 per hour"
3
Write the Equation
Put the Dependent Variable alone on the left. Multiply rate by Independent Variable.
t = 14h
Clue Words for the Rate (Multiplier):
• per (e.g. $12 per ticket)
• each (e.g. $3 each snack)
• every (e.g. 5 miles every hr)
• for each (e.g. 10 pts for each)
Table & Graph Clue:
• In a vertical table: Left column = Independent (x), Right = Dependent (y)
• On a graph: Horizontal x-axis is Independent, Vertical y-axis is Dependent.
Keep this reference sheet open during guided notes, group practice, and individual assignments.
Variable Connections Student Worksheet Algebra 1 • Guided Practice
Variable Connections Worksheet
Name:
Date:
Period:
★ Golden Sentence Frame:
"The [Dependent Variable] depends on the [Independent Variable]."
1 Part 1: The "Depends On" Test
Fill in the sentence frame, then label each variable
Scenario A: You work a summer job. Your total paycheck and the hours you worked.
The
depends on the
.
Independent (Input):
Dependent (Output):
Scenario B: Buying concert tickets online. The total cost and the number of tickets bought.
The
depends on the
.
Independent (Input):
Dependent (Output):
2 Part 2: Guided Table & Equation Construction
Follow the 3 steps
Real-World Story: Maya works at a local graphic design studio and earns $16 per hour . Let h = hours worked, and let p = total pay ($).
Hours (h)
Math Rule
Total Pay (p)
1 16 × 1 $16
2 16 × 2 $32
3
5
Build Maya's Equation:
1. Unit Rate (multiplier):
2. Dependent variable:
3. Write complete equation:
Unit 3: Equations & Expressions Page 1 of 2 • Continue on Page 2 →
Algebra 1 • Practice & Application
Variable Connections Worksheet (Cont.)
Page 2 of 2
Problem 1: Gaming Passes Table & Equation
An arcade game card costs $4 for every game pass (g) purchased. What is the total cost (c)?
Passes (g) Total Cost (c)
1 pass $4
2 passes
5 passes
Independent Variable (Input):
Dependent Variable (Output):
Write the Equation:
Problem 2: Mobile Data Table & Equation
A prepaid cell plan charges $10 for each gigabyte of high-speed data (d) . Let t = total bill amount ($).
Gigabytes (d) Total Bill (t)
1 GB $10
3 GB
6 GB
Independent Variable (Input):
Dependent Variable (Output):
Write the Equation:
Error Detective Spot & Fix the Mistake
Marcus is buying movie tickets for $12 each. Let t = number of tickets and c = total cost. Marcus wrote the equation: t = 12c.
What is wrong with Marcus's equation?
Correct equation:
Self-Assessment & Reflection Circle one level
🟢
I Got It!
I can find variables and write the equation.
🟡
Getting There
I can find variables with the sentence frame.
🔴
Need Help
I mix up independent and dependent.
Algebra 1 • Variable Connections Teacher Check / Signature: _______________________
Variable Connections Teacher Guide Educator Facilitation Plan
Variable Connections Teacher Guide
Grade 9 Algebra 1 (SPED / Co-Taught) Pacing: 50–60 Min Block
Target Standard: CCSS.Math.Content.6.EE.C.9 (Algebra Foundation) Bridges to HSA-CED.A.2 & HSF-LE.A.1
Instructional Objective: Given relatable high-school scenarios, students will identify the independent and dependent variables using the "depends on" sentence frame, complete an input-output table with 100% guided accuracy, and formulate a two-variable equation ($y = mx$) with 80% independent accuracy.
Suggested Lesson Breakdown 50-Minute Instructional Window
Hook (8m)
Slides 1–2. Cause/effect: "Who controls who?" in real life (paychecks, phone battery).
Direct (12m)
Slides 3–5. Introduce "Depends on" frame & anchor sheet. Model Jordan's smoothie job.
Guided (12m)
Slide 6 & Worksheet Page 1. Choral read frames, complete Maya graphic studio table.
Partner (13m)
Worksheet Page 2. Peer pairing on Problems 1 & 2. Error detective Marcus analysis.
Closure (5m)
Self-rating traffic light check and collection of exit reflection.
SPED Accommodations & Universal Design (UDL) Matrix
Visual & Representation
Color-coding: Blue = Independent ($x$), Green = Dependent ($y$), Amber = Rate.Anchor sheet: Kept on desk at all times during independent work.Reduced visual clutter: Uncluttered worksheet boxes with generous handwriting space.
Scaffolding & Expression
Sentence Stems: "The [output] depends on [input]" eliminates linguistic barrier.Step-by-step tables: Calculates rate before requiring abstract equation.Oral Rehearsal: Students read sentence frame aloud to partner before writing.
Age-Appropriate Engagement
High-school contexts: Hourly job wages, arcade passes, gigabytes of data.Error analysis: Critiquing peer work builds metacognition and confidence.Chunked tasks: Problems broken into 3 manageable micro-questions.
Common Misconceptions & Scripted Prompts
Misconception 1: Reversing Variables ($x = my$)
Students put the independent variable alone (e.g. $h = 16p$ for hourly pay).
Scripted Prompt: "Test Marcus's equation with numbers: If you work 1 hour, is your pay $16? If $h = 16p$, then 1 = 16(p), meaning you earned 6 cents! Does that make sense?"
Misconception 2: Confusing Rate with Addend
Students write $p = h + 16$ instead of multiplying $p = 16h$.
Scripted Prompt: "Look at your table. For 2 hours, did Maya earn 2 + 16 = $18, or did she earn 16 + 16 = $32? Repeated addition is multiplication!"
Measurable IEP Goal Alignment for Progress Monitoring:
Goal 1: "By [Date], given a real-world linear scenario and a graphic organizer, [Student] will correctly identify the independent and dependent variables with 80% accuracy across 4 consecutive trials." Goal 2: "By [Date], given a two-variable real-world relationship and an input-output table, [Student] will formulate a correct two-variable equation in the form $y = mx$ in 4 out of 5 opportunities."
Variable Connections • Facilitation Guide Page 1 of 2 • Solutions on Next Page →
Educator Resource • Answer Key
Master Worksheet Solutions & Support Protocol
Page 2 of 2
Student Worksheet Answer Key
Part 1: The "Depends On" Test Solutions
Scenario A (Paycheck & Hours):
Scenario B (Concert Tickets & Cost):
Part 2: Maya Graphic Studio ($16/hr)
Table Rows:
Equation Building:
Final Equation:
Page 2: Independent Practice & Error Analysis
Prob 1 (Arcade Passes):
Prob 2 (Mobile Data):
Marcus Error Analysis:
Co-Teaching Roles & Paraprofessional Prompts
General Ed & SPED Co-Teaching Model:
• Alternative Teaching / Station Teaching: Gen Ed teacher facilitates whole-group slides modeling; SPED specialist leads small-group guided table construction with tactile token manipulatives (e.g. counters for hourly wages).One Teach, One Assist: During independent practice, SPED specialist circulates with highlighter pens (blue for independent, green for dependent).
Paraprofessional & Aide Cheat Prompts:
• When student is stuck, point directly to Anchor Reference Sheet Section 2: "Read the sentence aloud. Which one sounds normal: 'Pay depends on hours' or 'Hours depend on pay'?" "per", "each" ) before writing numbers.
Multi-Tiered System of Support (MTSS) Quick Plan: Tiers 1, 2, & 3
Tier 1 (Core): Slides + Guided Notes + Desk Reference.
Tier 2 (Targeted): Provide pre-labeled tables; provide calculator for unit rates.
Tier 3 (Intensive): Fill in rate column; student matches variable cards to slots.
Algebra 1 SPED Lesson Set • CCSS.Math.Content.6.EE.C.9 End of Facilitation Materials
