Proportional Connections Poster Eureka Math² Grade 7 · Module 1
PROPORTIONAL RELATIONSHIPS
Two quantities are proportional if they have a constant ratio.
Visual Guide Topic A Overview
In Tables
All ratios of the dependent variable \(y\) to the independent variable \(x\) must be equivalent.
The Proportional Rule
\(\frac{y}{x} = k\)
(where \(k\) is constant and \(x \neq 0\))
Time, \(x\) (hrs) Pay, \(y\) ($) Ratio \(\frac{y}{x}\) 1 15 \(\frac{15}{1} = 15\) 2 30 \(\frac{30}{2} = 15\) 3 45 \(\frac{45}{3} = 15\)
Since every single row yields the exact same value of 15, this table represents a proportional relationship!
On Graphs
A graph shows a proportional relationship if and only if it meets two strict rules:
1 A Straight Line
2 Passes through the Origin \((0,0)\)
\(x\) \(y\)
Origin (0,0)
Unit Rate (1, r)
If a graph is straight but does not cross \((0,0)\), it is NOT proportional!
Reading Key Coordinates on Proportional Graphs
0,0
The Origin
Indicates that when \(x = 0\), \(y = 0\). For example: "0 hours of work equals $0 of earned pay."
1, r
The Unit Rate Point
Indicates the value of \(y\) when \(x = 1\). This directly gives you the Unit Rate \(r\)!
Grade 7 Module 1 · Anchor Poster
Classroom Wall Companion
Proportional Connections Notebook Insert Cut along dashed line to glue into notebook
Eureka Math² Grade 7 · Guided Notes Module 1, Topic A
Name: ________________________
Date: __________________
Period: ___________
PROPORTIONAL CONNECTIONS
Fill in the blanks and complete the diagrams to build your core anchor notes.
Defining Proportionality
Two quantities are proportional if there is a constant ratio. In other words, the ratio of the dependent variable (\(y\)) to the independent variable (\(x\)) must always be equivalent.
Formula: \(\frac{y}{x} = \) __________________ (this is our constant, \(k\))
Part 1: Looking at Tables
Calculate the ratios of \(y / x\) for each row. Are they equal?
Time \(x\) (hrs) Pay \(y\) ($) Ratio: \(\frac{y}{x}\) 2 30 \(\frac{30}{2} = 15\) 4 60 ______ = ______ 6 90 ______ = ______
Is this table proportional?
Yes, because __________________
No, because __________________
Part 2: Looking at Graphs
A proportional graph must have two rules:
It must be a __________________ line.
It must pass through the __________________ \((0, 0)\).
x y Plot the coordinates from Part 1 table! 0 2 4 30 60
Use a straightedge/ruler to connect your points.
Part 3: Interpreting the Key Coordinates
A proportional graph always tells a story at specific coordinate points. Complete the math definitions:
A Point \((0, 0)\):
This represents that when you have 0 of \(x\), you have 0 of \(y\).
B Point \((1, r)\):
This represents the unit rate. The \(y\)-coordinate \(r\) tells us the value of \(y\) when \(x = \) __________.
Quick Check: Non-Proportional Graph Analysis
Look at the graph on the left. Is it proportional? Why or why not?
Write your answer: __________________________________________________________________________
Grade 7 Module 1 · Lesson 1 Notebook Resource Keep this for Unit Review!
Constant Core Poster Eureka Math² Grade 7 · Module 1
THE CONSTANT CORE
Understanding the constant of proportionality \(k\) and the equation \(y = kx\).
Visual Guide Topic B & C Overview
What is \(k\)?
The symbol \(k\) represents the Constant of Proportionality. It tells you how many units of \(y\) you get for exactly 1 unit of \(x\).
Formula for \(k\)
\(k = \frac{y}{x}\)
\(y\) divided by \(x\) for any point!
Also known as the Unit Rate or Constant Rate.
Crucial Rule
To find \(k\) from a table or coordinate point, always divide the dependent value \(y\) by the independent value \(x\).
The Proportional Equation
Every proportional relationship can be modeled by a clean, simple linear equation:
y = k x
\(y\) Dependent
\(k\) Constant Rate
\(x\) Independent
Warning: Watch out for Additive Terms!
PROPORTIONAL: \(y = 3x\)
vs
NOT PROPORTIONAL: \(y = 3x + 4\)
Adding or subtracting any number other than zero breaks proportionality!
Applying the core: Step-by-Step Scenario
Real-World Case
A recipe states that 3 cups of flour (\(x\)) can bake 36 delicious cookies (\(y\)).
Step 1: Find \(k\)
\(k = \frac{36}{3} = 12\)
The unit rate is 12 cookies per cup of flour.
Step 2: Write Eq
\(y = 12x\)
This equation represents any quantity of cookies.
Step 3: Solve New
\(12 \times 5 = 60\)
With 5 cups, we can bake exactly 60 cookies!
Grade 7 Module 1 · Anchor Poster 2
Classroom Wall Companion
Constant Core Notebook Insert Cut along dashed line to glue into notebook
Eureka Math² Grade 7 · Guided Notes Module 1, Topics B & C
Name: ________________________
Date: __________________
Period: ___________
THE CONSTANT CORE: \(k\) & \(y = kx\)
Discover how the constant of proportionality unlocks linear equations. Fill in the blanks.
Part 1: Defining the Constant \(k\)
The constant of proportionality is written as the variable \(k\). It represents the unit rate (how much \(y\) changes for 1 unit of \(x\)). To calculate \(k\), we divide the dependent variable by the independent variable.
Formula for \(k\)
\(k = \) ____________________
\(x\) (hours) \(y\) (miles) \(k = y / x\) 3 60 60/3 = 20 5 100 ____/____ = ____
Part 2: The Equation Structure
Once you know your constant of proportionality (\(k\)), you can write the proportional linear equation.
y = k x
\(y\) variable represents: ______________________
\(k\) represents: ______________________
\(x\) variable represents: ______________________
Part 3: Guided Practice Case Study
The Travel Scenario: A drone flies at a constant speed, covering 80 meters in exactly 4 seconds.
Step 1: Identify variables
Independent (\(x\)): ___________________
Step 2: Find constant \(k\)
k = ______ m ______ s = ______ m/s
Step 3: Write the equation
Equation: y = __________________
Step 4: Predict a new value
How far in 10 seconds?
Part 4: Spotting Proportional Equations
Circle every equation below that represents a proportional relationship:
A) \(y = 8x\)
B) \(y = 8x - 2\)
C) \(y = 0.5x\)
D) \(y = \frac{4}{x}\)
Explain why your choices are proportional: ______________________________________________________________
Grade 7 Module 1 · Lesson 2 Notebook Resource Keep this for Unit Review!
