HSA.CED.A.2: Creating and Graphing Equations in Two or More Variables
In plain English: HSA.CED.A.2 is the Common Core algebra standard that asks students to write equations in two or more variables that represent relationships between quantities, and to graph them on coordinate axes with labels and scales. The relationships can be linear or nonlinear, such as quadratic or exponential, and each axis names a quantity and its unit. It is usually taught in Algebra I and used again in Algebra II.
Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
Common Core State Standards for Mathematics · Domain: Creating Equations (CED) · Cluster: Create equations that describe numbers or relationships Also written as HSA-CED.A.2 or A-CED.2 · Official standard
In this lesson, students write equations that relate two or more quantities in a real situation, then graph those equations on coordinate axes that carry clear labels, units and scales. The work has two halves that the standard names together: building the equation (choosing variables, deciding which quantity depends on which, and writing the relationship) and communicating it with a graph that someone else can read without asking questions.
Students meet linear relationships in both slope-intercept form (C = 12v + 40) and standard form (15s + 10h = 600), a quadratic area relationship, an exponential growth relationship, and a formula with three variables. Throughout, they decide what the axes should show, pick a scale that fits the values in the problem, and decide whether the graph should be a set of points or a connected curve.
Learning Objectives
By the end of this lesson, students will be able to:
Define variables, with units, for two or more related quantities in a context
Write an equation in two or more variables that represents a described relationship, including linear, quadratic and exponential relationships
Choose axes, labels, units and a consistent scale that fit the values in a problem
Graph the equation accurately from a table of values or from intercepts
Decide whether the graph should be discrete points or a continuous line or curve, and justify the choice from the context
Prior Knowledge Required
Students should already be comfortable with:
Writing and solving equations in one variable from a context HSA.CED.A.1
Constructing a linear function from a description, a table or two points 8.F.B.4
Graphing proportional relationships and interpreting the unit rate as the slope 8.EE.B.5
Plotting ordered pairs in all four quadrants of the coordinate plane
Show the prompt and give students 4 minutes to work alone before comparing with a partner.
Warm-Up Prompt
"A climbing gym charges a $40 membership fee plus $12 for each visit. Make a table of the total cost for 0, 1, 2, 3 and 4 visits. What stays the same from row to row, and what changes?"
Collect the table on the board (40, 52, 64, 76, 88). Ask: "If I tell you the number of visits, can you always find the cost? What would a rule look like?" Name the two quantities, v = number of visits and C = total cost in dollars, and write C = 12v + 40. Point out that this equation has two variables, so it describes a whole relationship instead of a single answer.
Direct Instruction20 minutes
Teach a four-step routine for any relationship between quantities:
Name the quantities: Write "Let v = number of visits" and "Let C = total cost in dollars." Decide which quantity depends on the other.
Write the relationship: Look for a starting amount, a constant rate, a constant percent change, or a geometric formula. Test the equation with one row of a table.
Set up the axes: The independent quantity goes on the horizontal axis. Label both axes with the quantity and its unit, then choose a scale with equal spacing that covers the largest value you need.
Graph and check the domain: Plot points from a table or from intercepts. Connect them only if every value in between makes sense in the context.
Model each example below. For each one, write the equation, build a short table, and sketch axes with labels and a scale before plotting.
Linear, slope-intercept form
"A climbing gym charges a $40 membership fee plus $12 per visit. Relate the total cost C to the number of visits v."
Equation: C = 12v + 40 (for 5 visits, C = 100)
Linear, standard form
"A club sells T-shirts for $15 and hats for $10 and wants to take in exactly $600. Relate the number of T-shirts s to the number of hats h."
Equation: 15s + 10h = 600 (intercepts: s = 40 when h = 0, h = 60 when s = 0)
Quadratic
"A rectangular garden is enclosed by 40 meters of fencing. Relate the area A to the width w."
Equation: A = w(20 - w) (largest area 100 m² when w = 10)
Exponential
"A town has 8,000 residents and its population grows by 3% each year. Relate the population P to the number of years t."
Equation: P = 8000(1.03)t (after 2 years, P = 8,487.2, about 8,487 people)
Three variables
"A theater sells adult tickets for $12 and student tickets for $8. Relate the revenue R to the numbers of adult tickets a and student tickets s."
Equation: R = 12a + 8s (for 50 adults and 75 students, R = 1,200)
Graph the gym example together using Diagram 1 as the model, then the garden example using Diagram 2. Contrast the two graphs: the gym graph is a set of separate points because a person cannot make 2.5 visits, while the garden graph is a continuous curve because any width between 0 and 20 meters is possible. For the three-variable example, explain that a graph on a flat coordinate plane shows two variables at a time, so students fix one value (for example R = 1,200) and graph the relationship between a and s.
Guided Practice15 minutes
Pairs complete three short tasks on graph paper. (1) A tank holds 500 gallons and drains at 25 gallons per minute: write W in terms of t and graph it. (2) A student has only $5 and $10 bills totaling $85: write an equation relating the number of each bill and list every pair of whole numbers that works. (3) A table shows a culture of 300 bacteria that doubles every day: write N in terms of d. Circulate and check three things on every graph: both axes have a label with a unit, the tick marks are evenly spaced, and the scale reaches the largest value in the table. Debrief the tank problem by asking why the graph stops at t = 20.
Independent Practice10 minutes
Students work alone on two problems: a rideshare that costs $2.50 plus $1.80 per mile (write and graph for 0 to 20 miles) and a rectangle with a perimeter of 30 feet (write the area A in terms of the width w and graph it from a table). Require a complete graph for each: title, labeled axes with units, a stated scale, and points plotted accurately. Students who finish early add a sentence explaining whether their graph should be connected.
Closure5-10 minutes
Exit ticket: "A phone repair shop charges $35 to diagnose a problem plus $60 per hour of labor. (a) Write an equation for the cost C of a repair that takes h hours. (b) Sketch axes for 0 to 4 hours with labels and a scale that fits." A correct ticket shows C = 60h + 35 and a vertical axis that reaches at least $275.
Differentiation Strategies
For Struggling Students
Give a partially completed table for each context so students can see the pattern before writing the equation
Provide pre-drawn axes with blank label lines and ask students to fill in the quantity, the unit and the scale
Start with linear situations in slope-intercept form, then add standard form, then quadratic and exponential
Use the sentence frame "Each time ___ goes up by 1, ___ goes up by ___" to connect the rate to the equation
For Advanced Students
Graph two gym plans on the same axes (for example C = 12v + 40 and C = 20v) and describe what the graphs show about which plan is cheaper
Write a formula with three variables, such as V = s²h for a box with a square base, then graph h against s for a fixed volume
Compare a linear and an exponential model for the same starting value and explain how the choice of scale affects what the graph shows
Assessment Guidance
What to Look For
Check both halves of the standard. Students should write an equation that matches the context (test it with one known pair of values) and produce a graph that another reader could interpret without help: both axes labeled with quantity and unit, an even scale, and a domain that fits the situation. A common error is an uneven scale such as 0, 5, 10, 20, 50 on one axis, which bends a straight-line relationship. Another is connecting points when only whole-number inputs make sense.
02
Classroom Activities
3 Activities
1
Story, Table, Equation, Graph
20 minGroups of 3-4
Each group gets 12 cards: 3 stories, 3 tables, 3 equations and 3 graphs. Groups sort the cards into 3 complete sets and then find the one graph card that has a labeling or scale error.
Card Sets
Story: "A kayak rental costs $15 plus $8 per hour." Equation: C = 8h + 15. Table: (0, 15), (1, 23), (2, 31), (3, 39).
Story: "A club sells T-shirts for $15 and hats for $10 and wants exactly $600." Equation: 15s + 10h = 600. Table: (40, 0), (20, 30), (0, 60).
Story: "A culture of 300 bacteria doubles every day." Equation: N = 300(2)d. Table: (0, 300), (1, 600), (2, 1200), (3, 2400).
Make one graph card with unevenly spaced ticks on the vertical axis, so groups must find the error.
Procedure
Groups have 10 minutes to build the three sets and write one sentence per set explaining how they know the cards match
Each group names the flawed graph card and redraws it correctly on graph paper
Debrief: Which card type was easiest to match first? What in the equation told you the shape of the graph?
Modification for Distance Learning
Put the cards on a shared slide deck and have groups drag them into columns in breakout rooms. Each group pastes a photo of its corrected graph.
2
Scale Detectives
20 minPairs
Pairs graph the same relationship, a water tank that holds 500 gallons and drains at 25 gallons per minute (W = 500 - 25t), on three different sets of axes and decide which scale communicates the relationship best.
Three Sets of Axes
Axes A: time 0 to 5 minutes, water 0 to 100 gallons (too small: in the first 5 minutes the tank still holds 375 to 500 gallons, so no point of the graph fits)
Axes B: time 0 to 25 minutes by 5, water 0 to 500 gallons by 100 (fits the whole relationship)
Axes C: time 0 to 100 minutes, water 0 to 5,000 gallons (so large that the graph is squeezed into a corner)
Discussion Questions
Which axes show both intercepts, (0, 500) and (20, 0)? What does each intercept mean in the context?
Why should the graph stop at t = 20 instead of continuing below the horizontal axis?
How do you pick a scale before you start plotting? (Find the largest value of each variable, then choose a step that gives about 5 to 10 ticks.)
3
Build Your Own Relationship
20 minIndividual then share
Each student chooses a situation from their own life (a savings plan, a phone plan, a garden, a fundraiser) and produces a one-page poster with a written description, defined variables, an equation, a table of at least four rows and a fully labeled graph.
Requirements
At least two variables, each defined with a unit
A table whose values all satisfy the equation
A graph with a title, labeled axes, units, an even scale and a domain that fits the context
One sentence stating whether the graph is discrete or continuous, and why
Gallery Walk Variation
Post the posters. Classmates leave a sticky note that checks one row of the table against the equation and names one strength of the graph. Posters with an incorrect row are revised before they are collected.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Graphing C = 12v + 40 With Labels and Scales
Total cost at a climbing gym with a $40 fee and $12 per visit, drawn to scale. Each horizontal tick is 1 visit and each vertical tick is $20. The points are not connected because the number of visits must be a whole number.
Diagram 2: Graphing the Quadratic A = w(20 - w)
Area of a rectangular garden enclosed by 40 meters of fencing, drawn to scale. The width can be any value between 0 and 20 meters, so the graph is a continuous curve. The largest area, 100 square meters, occurs when the garden is a 10 m by 10 m square.
04
Homework Assignment
~30 min
HSA.CED.A.2 Homework: Writing and Graphing Relationships
Directions: For each problem, (a) define each variable with its unit, (b) write an equation, (c) answer the question, and (d) when asked for a graph, draw it on graph paper with a title, labeled axes, units and an even scale. State whether your graph should be connected.
Part 1: Writing Equations (Problems 1-3)
A bike-share service charges $1.25 to unlock a bike plus $0.30 per minute of riding. Write an equation for the cost C of a ride that lasts t minutes. What does a 20-minute ride cost?
A bakery sells muffins for $3 each and cookies for $2 each. One day it takes in exactly $180 from these two items. Write an equation relating the number of muffins m and the number of cookies c. Find both intercepts and explain what each one means.
A car is bought for $24,000 and loses 15% of its value each year. Write an equation for its value V after t years. What is the value after 3 years?
Part 2: Graphing With Labels and Scales (Problems 4-6)
A tank holds 300 gallons of water and drains at 12 gallons per minute. Write an equation for the water W left after t minutes. Graph it from t = 0 until the tank is empty, and state the scale you used on each axis.
A ball is kicked straight up, and its height in feet after t seconds is h = -16t² + 48t. Make a table for t = 0, 0.5, 1, 1.5, 2, 2.5 and 3, then graph the relationship. What is the greatest height, and when does the ball land?
A student earns $15 per hour tutoring and $12 per hour working at a cafe. Write an equation for total weekly earnings E if the student tutors for t hours and works c hours at the cafe. Then, for a week in which the student earns exactly $180, write and graph the relationship between t and c.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Variables
Every variable defined with a unit
Variables named, units missing
Variables not defined
Equation
Correct and checked with one data pair
Correct structure, one error
Incorrect or missing
Graph
Labeled axes, units, even scale, accurate points
One missing label or one scale error
Missing or unreadable graph
Interpretation
Answers the question in a sentence and justifies connected or discrete
Answer given without explanation
No interpretation
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
A plumber charges $75 for a house call plus $50 per hour of labor. Which equation gives the total cost C for a job that takes h hours?
Answer: B
The $75 is charged once and the $50 is charged for every hour, so C = 50h + 75. Choice A swaps the one-time fee and the hourly rate. Choice C adds the two numbers as if both were charged every hour.
Question 2 of 20 · Multiple Choice
A school fundraiser sells pies for $12 and cakes for $20 and takes in exactly $1,200. If p is the number of pies and c is the number of cakes, which equation models the situation?
Answer: A
Each pie brings in $12 and each cake brings in $20, so the revenue is 12p + 20c = 1200. Choice B attaches each price to the wrong variable. Choice C assumes every item costs $32.
Question 3 of 20 · Multiple Choice
For the equation 12p + 20c = 1200 from the previous question, what is the p-intercept (the value of p when c = 0)?
Answer: C
Set c = 0: 12p = 1200, so p = 100. It means 100 pies and no cakes bring in $1,200. Choice A, 60, is the c-intercept (1200 / 20), found by dividing by the wrong price.
Question 4 of 20 · Multiple Choice
A car travels at a steady 55 miles per hour. A student graphs d = 55t for 0 ≤ t ≤ 6 hours. What is the smallest top value for the vertical (distance) axis that still shows the whole graph?
Answer: D
At t = 6, d = 55(6) = 330 miles, so the vertical axis must reach at least 330. Choice B, 300, cuts off the last part of the graph. Choice A adds 55 and 6 instead of multiplying.
Question 5 of 20 · Multiple Choice
A savings account starts with $2,000 and earns 4% interest compounded once a year. Which equation gives the balance B after t years?
Answer: A
Growing by 4% each year means multiplying by 1 + 0.04 = 1.04 each year, so B = 2000(1.04)t. Choice B multiplies by 0.04, which would shrink the balance. Choice D uses 1.4, which is 40% growth.
Question 6 of 20 · Multiple Choice
Which point lies on the graph of y = 3x - 4?
Answer: D
Substitute x = 2: y = 3(2) - 4 = 2, so (2, 2) is on the graph. For (0, 4), the y-value at x = 0 is -4, not 4, which is a sign error. For (1, 1), y would be -1. For (3, 4), y would be 5.
Question 7 of 20 · Multiple Choice
A student graphs E = 18h, a lifeguard's total earnings for h hours of work. Which is the best label for the vertical axis?
Answer: C
A good axis label names the quantity and its unit: "Total earnings (dollars)." Choice A gives only the letter, which a reader cannot interpret. Choice B names the quantity on the other axis. Choice D describes the rate, $18, not the quantity being graphed.
Question 8 of 20 · Multiple Choice
A rectangle has a perimeter of 36 feet. Which equation gives its area A in terms of its width w?
Answer: B
Length plus width is half the perimeter, 18 feet, so the length is 18 - w and A = w(18 - w). Choice A uses the full perimeter for the length plus width. Choice D is the perimeter formula, which always equals 36 and does not give the area.
Question 9 of 20 · Multiple Choice
A phone's battery charge is modeled by B = 80 - 5t, where B is the charge in percent and t is the number of hours since unplugging. What does -5 represent?
Answer: A
The coefficient of t is the change in B for each hour, and it is negative, so the battery loses 5 percentage points per hour. Choice B confuses the rate with the starting charge, which is 80 percent. Choice C is wrong because the battery reaches 0 at t = 16. Choice D ignores the negative sign.
Question 10 of 20 · Multiple Choice
Large pizzas cost $14 each, drinks cost $2 each, and there is a $5 delivery fee. Which equation gives the total cost T of p pizzas and d drinks?
Answer: D
Each pizza adds $14, each drink adds $2, and the $5 fee is added once: T = 14p + 2d + 5. This equation has three variables. Choice A charges $16 for every item. Choice B subtracts the fee instead of adding it.
Question 11 of 20 · Multiple Choice
For B = 80 - 5t, at what value of t does the graph cross the horizontal (time) axis?
Answer: C
The graph crosses the time axis when B = 0: 80 - 5t = 0, so t = 16 hours. This is when the battery is empty. Choice A uses the rate as the answer. Choice B is the B-intercept, not the t-intercept.
Question 12 of 20 · Multiple Choice
A bowling alley charges C = 5g + 4, where g is the number of games bowled and $4 is the shoe rental. Should the plotted points be connected with a line?
Answer: A
In this context g can only be 0, 1, 2, 3 and so on, so the graph is a set of separate points. Choice B ignores the context: the equation alone has a line as its graph, but only whole-number points make sense here. Choice D gives a reason that has nothing to do with whether the graph is connected.
Question 13 of 20 · Multiple Choice
Which equation matches the table? x: 0, 1, 2, 3 and y: 5, 15, 45, 135.
Answer: B
Each y-value is 3 times the previous one and the value at x = 0 is 5, so y = 5(3)x. Choices A and D both give 15 at x = 1 but give 20 or 25 at x = 2, not 45: they treat the change as constant instead of a constant factor. Choice C swaps the starting value and the growth factor.
Question 14 of 20 · Multiple Choice
A florist sells roses for $4 each and tulips for $3 each and takes in exactly $240, so 4r + 3t = 240. Which point, written (r, t), lies on the graph?
Answer: C
Check (30, 40): 4(30) + 3(40) = 120 + 120 = 240. Choice A combines the two intercepts into one point, which gives 480. Choice B gives 160 + 90 = 250. Choice D uses the r-intercept as if it were the t-intercept; the t-intercept is (0, 80).
Question 15 of 20 · Short Answer
A canoe rental costs $20 plus $12 per hour. Write an equation for the cost C of renting for h hours. If you graph it for 0 to 6 hours, what is a sensible scale for the vertical axis?
C = 12h + 20 At h = 6, C = 12(6) + 20 = 92, so the vertical axis should reach at least $100. A scale of $10 per tick from $0 to $100 fits the values and gives 10 even intervals. The horizontal axis can use 1 hour per tick from 0 to 6.
Question 16 of 20 · Short Answer
A plant is 4 cm tall when it is planted and grows 1.5 cm per week. Write an equation for its height H after w weeks, and find its height after 6 weeks.
H = 1.5w + 4 After 6 weeks, H = 1.5(6) + 4 = 9 + 4 = 13 cm. The starting height is the value of H when w = 0, and the growth rate is the coefficient of w.
Question 17 of 20 · Short Answer
A drama club sells programs for $2 each and posters for $5 each and collects $40 in all. Write an equation relating the number of programs p and the number of posters q. Give two pairs of values that make sense.
2p + 5q = 40 Possible pairs include (p, q) = (20, 0), (15, 2), (10, 4), (5, 6) and (0, 8). Check (10, 4): 20 + 20 = 40. Only whole numbers make sense, and q must be even, because 2p is always even and 40 is even, so 5q must be even too. A pair such as (2.5, 7) satisfies the equation but not the context.
Question 18 of 20 · Short Answer
A culture starts with 400 bacteria and triples every day. Write an equation for the number of bacteria N after d days, and find N after 4 days.
N = 400(3)d After 4 days, N = 400(3)4 = 400(81) = 32,400 bacteria. Tripling means multiplying by 3 each day, so the relationship is exponential, not linear.
Question 19 of 20 · Short Answer
The height of a ball in feet is h = -16t² + 64t, where t is the time in seconds. Make a table for t = 0, 1, 2, 3 and 4, and use it to describe the graph.
Table: (0, 0), (1, 48), (2, 64), (3, 48), (4, 0) The graph is a parabola that opens down. It starts on the ground, reaches a greatest height of 64 feet at t = 2 seconds, and lands at t = 4 seconds. A vertical scale of 10 or 16 feet per tick and a horizontal scale of 0.5 or 1 second per tick fit these values. The curve is connected because time is continuous.
Question 20 of 20 · Short Answer
A student's graph has these tick labels on the horizontal axis: 0, 5, 10, 20, 40. What is wrong, and how should the student fix it?
The scale is not even. The gaps between ticks are 5, 5, 10 and 20, but the ticks are spaced the same distance apart, so the graph is stretched and a straight-line relationship would look curved. The student should use one step size for the whole axis, for example 0, 10, 20, 30, 40, and plot every point using that scale.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What is the difference between HSA.CED.A.1 and HSA.CED.A.2?
HSA.CED.A.1 asks students to write an equation or inequality in one variable and solve it to answer a single question, such as "How many visits cost $100?" HSA.CED.A.2 asks for an equation in two or more variables, such as C = 12v + 40, that describes the whole relationship, and then asks students to graph it. The output of A.1 is usually a number; the output of A.2 is an equation and a graph.
Which variable goes on which axis?
Put the independent quantity, the one you choose or control, on the horizontal axis and the quantity that depends on it on the vertical axis. For the gym, the number of visits determines the cost, so visits go on the horizontal axis. For a standard-form equation like 15s + 10h = 600, neither variable depends on the other, so either choice works as long as the axes are labeled clearly.
What does "with labels and scales" require on a graph?
Each axis needs a label that names the quantity and its unit, such as "Time (minutes)" or "Area (square meters)." Each axis also needs tick marks with numbers that are evenly spaced, and the scale must cover the values in the problem. A title is good practice. The test is whether someone who has not read the problem can tell what the graph shows.
How do students choose a good scale?
Find the smallest and largest values each variable will take in the problem. Then pick a step size, such as 1, 2, 5, 10, 20, 50 or 100, that gives roughly 5 to 10 tick marks across that range. The two axes can have different scales; for the gym graph, the horizontal step is 1 visit and the vertical step is $20. What matters is that each axis uses one step size throughout.
When should the points on a graph be connected?
Connect them only when every value between the points makes sense in the context. Time, length and area can take any value, so their graphs are continuous. Numbers of people, tickets or visits must be whole numbers, so their graphs are separate points. Students can still draw a light dashed line to show the pattern, as long as they state that only the whole-number points are solutions in context.
Does this standard include nonlinear equations?
Yes. The standard does not name a type of equation, so it applies to any relationship students study. Algebra I courses usually work with linear, quadratic and exponential relationships, and Algebra II adds other function types. This lesson includes a quadratic area model, A = w(20 - w), and exponential growth models such as P = 8000(1.03)t.
How do you graph an equation with three variables, such as R = 12a + 8s?
A flat coordinate plane shows two variables at a time. Fix the value of one variable and graph the relationship between the other two. For example, setting R = 1,200 gives 12a + 8s = 1,200, which students can graph with a on one axis and s on the other. Choosing a few different values of R produces a family of parallel lines, which is a good extension discussion.
What mistakes do students often make on this standard?
Attaching a rate to the wrong variable, such as writing 40f + 25b = 5000 when floor tickets cost $25
Mixing up the starting amount and the rate, as in C = 40v + 12
Using an uneven scale or leaving out units
Writing a linear equation for a situation with a constant percent change, which is exponential
Connecting points when only whole-number values make sense
Is this skill tested on the SAT?
Yes. The Algebra domain of the digital SAT includes writing and interpreting linear equations in two variables from a context and connecting an equation to its graph, and the Advanced Math domain includes quadratic and exponential models. Students who can move between a context, an equation and a labeled graph are practicing exactly those skills.
What comes after HSA.CED.A.2?
The next standard in the cluster, HSA.CED.A.3, uses equations and inequalities in two variables as constraints and asks which solutions are viable. HSA.REI.D.10 explains why the graph of an equation is the set of all its solutions. In the functions strand, HSF.IF.C.7 develops graphing of specific function types and HSF.LE.A.2 asks students to build linear and exponential functions from a description, a graph or a table.
07
Related Standards
6 standards
These standards connect to HSA.CED.A.2: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
8.F.B.4Prerequisite
Construct a function to model a linear relationship between two quantities
