Analyze Graphs of Relationships Between Two Variables
I can use a table and a graph to determine and analyze the relationship between two variable quantities.
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🎯 Content Objective / Objetivo de contenido
I can use a table and a graph to determine and analyze the relationship between two variable quantities.
Today's Flow
Total pacing: ~45 min · Progress bar at top tracks your place
LAUNCH
⏱ ~10 min
⏱️ 3 MIN · THINK-PAIR-SHARE
Football tickets cost $45 each: 1 ticket is $45, 2 tickets are $90, 3 tickets are $135. When you write the ordered pair for 3 tickets, which number goes first — 3 or 135 — and why?
Check for Understanding #1
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Football Game Tickets
The ticket shows the cost to attend a college football game. How does the total cost of tickets relate to the number of tickets purchased?
Concept Launch
💡 Two pictures of the same relationship
A table shows a relationship with numbers. A graph shows the same relationship with points. Reading either one tells you how the dependent variable responds to the independent variable.
Representing Two-Variable Relationships with Tables & Graphs. Formula: Table (x, y) → Graph Point (x, y). 1. Construct table: Put independent variable in column 1 (x) and dependent in column 2 (y). 2. Identify the rate of change or rule connecting x to y. 3. Plot the ordered pairs on the coordinate plane with x on horizontal axis and y on vertical
Check for Understanding #2
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Now you try
VOCABULARY
⏱ ~8 min
| Term / Término | Meaning / Significado | Example / Ejemplo | Visual |
|---|---|---|---|
| graph gráfica |
A picture on a coordinate plane that shows how two quantities are related. Un dibujo en un plano de coordenadas que muestra cómo se relacionan dos cantidades. |
The football ticket graph shows cost rising as tickets increase. | Points (1, 45), (2, 90), (3, 135) climbing left to right — cost by tickets. |
| table of values tabla de valores |
A list of matched pairs of values that shows a relationship using numbers. Una lista de pares de valores que muestra una relación usando números. |
The ticket table pairs 3 tickets with $135. | months 1, 2, 4 beside cost 24.95, 49.90, 99.80 — one row per pair. |
| ordered pair par ordenado |
Two numbers written as (x, y) that name one point on a graph. The independent variable comes first. Dos números escritos como (x, y) que nombran un punto en la gráfica. La variable independiente va primero. |
(2, 90) means 2 tickets cost $90. | (3, 135): 3 tickets cost $135 — the independent quantity comes first. |
| axis eje |
One of the two number lines that frame a graph. Each axis is labeled with a quantity from the problem. Una de las dos rectas numéricas que enmarcan una gráfica. Cada eje se rotula con una cantidad del problema. |
One axis shows the number of tickets; the other shows the total cost. | Horizontal axis: months. Vertical axis: total cost. |
| coordinates coordenadas |
The numbers in an ordered pair that tell how far to move along each axis to reach a point. Los números de un par ordenado que indican cuánto avanzar por cada eje para llegar a un punto. |
The coordinates of the tram point (5, 6000) are 5 minutes and 6,000 feet. | In (4, 105), 4 says how far across and 105 how far up. |
Which Word Fits?
A picture on the coordinate plane showing how two quantities relate is a ___.
Use It In a Sentence
Check for Understanding #3
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Turn & Talk — Launch
Football tickets cost $45 each: 1 ticket is $45, 2 tickets are $90, 3 tickets are $135. When you write the ordered pair for 3 tickets, which number goes first — 3 or 135 — and why?
👂 Listen For
A strong answer names tickets as independent and explains that independent always goes first, then checks the arithmetic (3 × 45 = 135). A weak answer just says 'because that's the order' without connecting it to independent/dependent.
Extend: The Sandia Peak tram travels 1,200 feet each minute. Justify why the point for 5 minutes is (5, 6000) and not (6000, 5), using the same rule you used for the tickets.
EXPLORE & PRACTICE
⏱ ~18 min
Visual Modeling Workspace
Use the drawing tray below to annotate the visual model. Teacher: say "Click to reveal" on key steps.
Explore Activity
Sort all eight cards. For each ordered pair, multiply the first number by 45 and check whether it equals the second number.
✍️ Explore Discourse
Find a card you called a match and your partner did not. Pick one ordered pair from it and test it in cost = 45 × tickets — what does the point on the graph tell you that the table does not?
Whiteboard Moment
Show your work clearly. Be ready to explain your thinking to a partner.
Turn & Talk — Explore
You sorted cards like '(4, 190) is a point on the graph' into matches and non-matches for cost = 45 × tickets. How did you check each one?
👂 Listen For
Listen for students actually multiplying to verify rather than guessing by how 'close' the numbers looked. A strong answer catches that 4 × 45 = 180, not 190, so the point is off by exactly $10.
Extend: Card '(1, 55) is a point on the graph' was also NOT a match. What single mistake would produce 55 instead of the correct value — and how is that different from the mistake in (4, 190)?
Practice Check A
The Sandia Peak tram travels 1,200 feet each minute and the ascent takes 15 minutes. How long is the whole ascent in feet?
✍️ Show Your Work
Explain why your answer is correct using today's vocabulary.
Practice Check B
On the tram graph with minutes on the horizontal axis and feet on the vertical axis, which ordered pair lies on the graph?
✍️ Show Your Work
Explain why your answer is correct using today's vocabulary.
Ratio Table Builder
Fill the ratio table. Each row must be equivalent.
| Factor | A | B |
|---|---|---|
| ×1 | ||
| ×2 | ||
| ×3 |
✍️ Justify Your Thinking
Football tickets cost $45 each, so the graph follows c = 45t. Sort each ordered pair: is it on the ticket graph or not?
A classmate turned in the work below. One step has a mistake. Read every step, find it, name it, and fix it.
No worked steps provided for this lesson.
Choose ONE option to show what you know — then do it in the workspace below.
Use evidence from today's lesson to complete each frame.
Today's key idea is: "Representing Two-Variable Relationships with Tables & Graphs. Formula: Table (x, y) → Graph Point (x, y). 1. Construct table: Put independent variable in column 1 (x) and dependent in column 2 (y). 2. Identify the rate of change or rule connecting x to y. 3. Plot the ordered pairs on the coordinate plane with x on horizontal axis and y on vertical" — and it works because ___.
Because graph means ___, but a tricky part is ___, so I have to ___.
I can justify my method by showing ___, which proves my answer makes sense.
I can prove my answer is correct by ___, using table of values to check my work.
✍️ TWR · WRITE 3 SENTENCES · 7 MIN
Representing Two-Variable Relationships with Tables & Graphs. Formula: Table (x, y) → Graph Point (x, y). 1. Construct table: Put independent variable in column 1 (x) and dependent in column 2 (y). 2. Identify the rate of change or rule connecting x to y. 3. Plot the ordered pairs on the coordinate plane with x on horizontal axis and y on vertical because ___
Representing Two-Variable Relationships with Tables & Graphs. Formula: Table (x, y) → Graph Point (x, y). 1. Construct table: Put independent variable in column 1 (x) and dependent in column 2 (y). 2. Identify the rate of change or rule connecting x to y. 3. Plot the ordered pairs on the coordinate plane with x on horizontal axis and y on vertical but ___
Representing Two-Variable Relationships with Tables & Graphs. Formula: Table (x, y) → Graph Point (x, y). 1. Construct table: Put independent variable in column 1 (x) and dependent in column 2 (y). 2. Identify the rate of change or rule connecting x to y. 3. Plot the ordered pairs on the coordinate plane with x on horizontal axis and y on vertical so ___
🌱 TWR · GROW THE KERNEL · 6 MIN
Answer these to add detail
Sentence starters (tap to use)
Student Workspace
Sort all eight cards. For each ordered pair, multiply the first number by 45 and check whether it equals the second number.
| Column A | Column B |
|---|---|
✏️ Sketch Your Strategy
Differentiation Paths
Step-by-step with a worked model and sentence frames.
Football tickets cost $45 each. What is the total cost of 3 tickets?
The Sandia Peak tram travels 1,200 feet each minute and the ascent takes 15 minutes. How long is the whole ascent in feet?
Nakai's savings account has a principal of $750 and earns 3.5% simple interest each year. How much interest does Nakai earn in one year?
Partner Activity
Work with your partner on the practice problems at your differentiation path level. Explain each step using math vocabulary.
Check for Understanding #4
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Real-World Connection
🌍 Math in the Wild
Nakai opened a savings account with an initial deposit, or principal, of $750. The account earns 3.5% simple interest each year — interest is the amount earned for the use of the principal. A graph shows the interest Nakai has earned over a period of years, with points rising steadily.
✍️ Connection Reasoning
What observations can you make about the data using the graph?
___ is the independent variable, while ___ is the dependent variable. As the years go by, the interest earned ___, because each year adds $___ to the total.
Turn & Talk — Connect
Nakai's account starts with $750 and earns the same $26.25 every year. Why does the interest graph rise in a straight line instead of curving upward more steeply each year?
👂 Listen For
A strong answer connects the constant $26.25/year to the straight-line shape and can compute it (3.5% of $750). A weak answer says 'it just goes up' without explaining WHY the increase stays constant.
Extend: A point on Nakai's graph is (4, x). Find x, and explain how you know your answer is NOT the account's total balance after 4 years.
Summarize: Analyze Graphs of Relationships Between Two Variables
Representing Two-Variable Relationships with Tables & Graphs. Formula: Table (x, y) → Graph Point (x, y). 1. Construct table: Put independent variable in column 1 (x) and dependent in column 2 (y). 2. Identify the rate of change or rule connecting x to y. 3. Plot the ordered pairs on the coordinate plane with x on horizontal axis and y on vertical
In your own words:
CLOSURE & REFLECT
⏱ ~8 min
Today I learned that ___ because ___.
One thing I am still not sure about is ___.
On the football ticket graph, what does the point (3, 135) represent?
Bonus Exit Check
A table pairs hours with kilometers: (1, 48), (2, 96), (5, 240). How many kilometers are traveled each hour?
✍️ Show Your Work
Explain why your answer is correct using today's vocabulary.
Reflection & Self-Assessment
Continue Learning
Launch the Full Interactive Activity
Students continue practice in the HTML lesson engine with auto-check, hints, and differentiation.
Family Connection
Share tonight's family homework and discuss one vocabulary word at home.
Open Family Homework ↗Teacher Notes
⏱️ Pacing Guide
- Launch & vocab: 12 min
- I Do / We Do / You Do: 15 min
- Explore & practice: 15 min
- Connect & closure: 8 min
Total: ~45 min
🎯 Listen For · Common Errors
• A strong answer names tickets as independent and explains that independent always goes first, then checks the arithmetic (3 × 45 = 135). A weak answer just says 'because that's the order' without connecting it to independent/dependent.
• Listen for students actually multiplying to verify rather than guessing by how 'close' the numbers looked. A strong answer catches that 4 × 45 = 180, not 190, so the point is off by exactly $10.
• A strong answer connects the constant $26.25/year to the straight-line shape and can compute it (3.5% of $750). A weak answer says 'it just goes up' without explaining WHY the increase stays constant.
• Listen for students naming BOTH steps — computing the total AND placing the independent variable first — not just stating the final pair.
Common mistake: Reversing the coordinates of a point — writing (cost, tickets) instead of (tickets, cost). The independent variable always comes first in the ordered pair.
Answer Key (Teacher Appendix)
Hide this slide during presentation or move to the end of your copy.
✓ Practice 1: 18,000 feet — The total distance is 15 × 1,200 = 18,000 feet.
✓ Practice 2: (5, 6000) — After 5 minutes the tram has traveled 5 × 1,200 = 6,000 feet, so the point is (5, 6000).
✓ Practice 3: 48 — Each row is 48 times the number of hours: 2 × 48 = 96 and 5 × 48 = 240, so the rate is 48 kilometers per hour.
✓ Practice 4: You can only buy whole numbers of tickets, so the points between whole numbers do not make sense — You cannot buy 2.5 tickets, so only the whole-number points represent real purchases — the dashes show the trend without claiming the in-between points are possible.
✓ Exit ticket: 3 tickets cost a total of $135 — The independent variable — number of tickets — comes first in the ordered pair, and 3 × $45 = $135, so the point pairs 3 tickets with a total cost of $135.