6.EE.C.9Common CoreMathExpressions and EquationsGrade 6
6.EE.C.9: Dependent and Independent Variables in Tables, Graphs and Equations
In plain English: 6.EE.C.9 is the Common Core grade 6 math standard that asks students to use two variables for two quantities that change together in a real situation, write an equation that gives the dependent variable in terms of the independent variable, and connect it to a table and a graph, as in d = 65t for a car at a constant speed. It prepares for proportional relationships in grade 7 and functions in grade 8.
Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.
Common Core State Standards for Mathematics · Domain: Expressions and Equations (EE) · Cluster: Represent and analyze quantitative relationships between dependent and independent variables. Also written as 6.EE.9 · Official standard
Students study two quantities that change together, such as the number of bags of popcorn you buy and the total cost. A variable is a letter that stands for a quantity that can change. The independent variable is the quantity you choose or control, and the dependent variable is the quantity that changes because of it: the cost depends on how many bags you buy. Students write an equation (a math sentence with an equal sign) that gives the dependent variable in terms of the independent variable, such as c = 2b.
Next, students show the same relationship in a table of values and on a graph in the first quadrant of the coordinate plane (the part of the grid where both numbers are 0 or more), where each row of the table becomes an ordered pair (x, y), a point written with the independent value first. They work through the official example of a car at a constant speed (the same speed the whole time), list and graph pairs of times and distances, and write d = 65t. Students then relate the three forms: the number that multiplies the independent variable is the amount the dependent variable grows in each new row, and an equation like y = x + 2 gives a graph that starts at 2 instead of 0. All equations have one of two grade 6 forms, y = kx or y = x + c, with whole numbers or simple decimals.
Learning Objectives
By the end of this lesson, students will be able to:
Choose variables for two quantities in a real situation and decide which one is the independent variable and which one is the dependent variable
Write an equation that gives the dependent variable in terms of the independent variable
Make a table and a graph of the relationship and use them to answer questions
Explain how the equation, the table and the graph show the same relationship
List and graph ordered pairs of times and distances for motion at a constant speed and write an equation such as d = 65t
Prior Knowledge Required
Students should already be comfortable with:
Making two number patterns from two rules and forming ordered pairs from them 5.OA.B.3
Graphing points in the first quadrant of the coordinate plane 5.G.A.2
Writing expressions with letters that stand for numbers, such as 3n 6.EE.A.2
Using a variable to stand for an unknown number in a real problem 6.EE.B.6
Finding a unit rate, the amount for 1 unit, such as miles per hour 6.RP.A.2
Show a price sign from the school fair and ask students to answer without a formal method:
Warm-Up Prompt
"Popcorn at the school fair costs $2 a bag. Make a table for 1, 2, 3, 4 and 5 bags. Which amount do you choose, and which amount changes because of your choice?"
Collect tables on the board: 1 bag costs $2, 2 bags cost $4, 3 bags cost $6, 4 bags cost $8 and 5 bags cost $10. You choose the number of bags, and the cost changes because of it. Name the parts: a variable is a letter for a quantity that can change, so let b be the number of bags and c the total cost in dollars. The number of bags is the independent variable (the one you choose), and the cost is the dependent variable (it depends on b). Every row follows one rule, cost = 2 × bags, so the equation is c = 2b. Save this table: Diagram 2 graphs it later.
Direct Instruction20-25 minutes
Part 1: Two quantities, two variables. Ask for each situation: which quantity do you pick, and which one follows? Put the independent variable first in the table and on the horizontal axis (the x-axis), and the dependent variable second and on the vertical axis (the y-axis). Then write the equation with the dependent variable alone on one side: "dependent = rule using the independent variable".
Official example: constant speed
A car travels at a constant speed of 65 miles per hour. List ordered pairs of times and distances, graph them, and write an equation for distance d in terms of time t.
Equation: (0, 0), (1, 65), (2, 130), (3, 195), (4, 260); each distance is 65 times the time, so d = 65t (Diagram 1)
A relationship that adds
Jon is always 2 years older than his sister Kim. Let k be Kim's age and j be Jon's age. Write an equation and make a table for k = 5, 6, 7 and 8.
Equation: j = k + 2; the table gives 7, 8, 9 and 10, and each row adds 2, not multiplies by 2
From a table to an equation
Mia babysits. Hours worked h: 2, 3, 5, 8. Money earned e (dollars): 24, 36, 60, 96. Write an equation for e in terms of h.
A printer prints 20 pages each minute. Which variable is independent? Write an equation, then find the pages printed in 6 minutes and the time to print 180 pages.
Equation: Minutes m is independent and pages p depends on it: p = 20m, so 6 minutes gives 120 pages and 180 pages takes 9 minutes
Checking every row of a table
Which equation fits this table? x: 2, 4, 6. y: 6, 8, 10.
Equation: y = 3x fits only the first row (3 × 4 = 12, not 8); y = x + 4 fits every row, so y = x + 4
Part 2: From the table to the graph. Each row of a table is an ordered pair: the independent value first, the dependent value second. In the official example, the row t = 3, d = 195 becomes the point (3, 195). Plot the pairs as in Diagram 1. Because the car moves the whole time, the points can be joined with a line: after half an hour it has gone 32.5 miles. For things you count, such as bags of popcorn, plot only the whole-number points.
Read the equation in the table: for d = 65t, each time t goes up by 1 hour, d goes up by 65 miles. The number that multiplies the independent variable is the amount the dependent variable grows in each new row.
Read the equation in the graph: for d = 65t, the points start at (0, 0), because 0 hours means 0 miles, and they climb 65 miles for each hour. A bigger multiplier makes a steeper line.
Multiply or add? Diagram 2 puts the popcorn equation c = 2b (written y = 2x) and Jon's age equation j = k + 2 (written y = x + 2) on one graph. The first starts at (0, 0) and goes up 2 for each step. The second starts at (0, 2) and goes up 1 for each step.
Part 3: Analyze the relationship. Students use the table or the graph to answer questions: how far does the car go in 4 hours (260 miles)? How long does it take to go 130 miles (2 hours)? Point out that the equation answers the same questions without the table: d = 65 × 4 = 260.
Guided Practice15 minutes
Pairs work through three problems, one at a time. For each one they name the independent and dependent variables, write the equation, and show a table or graph. After each problem, one pair explains how its table matches its equation.
Problem 1: Omar rides his bike at a constant speed of 10 miles per hour. Fill in the table, write an equation for distance d in terms of time t, and plot the pairs on grid paper.
Problem 1: Omar rides at a constant speed of 10 miles per hour.
Time t (hours)
1
2
3
?
5
Distance d (miles)
10
?
30
40
?
Answers: 20 miles after 2 hours, 4 hours for 40 miles and 50 miles after 5 hours; the equation is d = 10t. Problem 2: each van on a field trip seats 7 students. Write an equation for the number of seats s in v vans. How many seats are in 6 vans? (s = 7v, so 42 seats.) Problem 3: a concert ticket bought online costs $3 more than the same ticket bought at the door. Write an equation for the online price o in terms of the door price p. What is the online price when the door price is $9? (o = p + 3, so $12.) Listen for pairs who write 3p in Problem 3: ask them to test their equation with p = 9.
Independent Practice10-15 minutes
Students solve four problems on their own. (1) Tickets to a school play cost $6 each. Write an equation for the total cost c of n tickets and make a table for n = 1 to 4. (c = 6n; 6, 12, 18, 24.) (2) A table shows weeks x: 1, 2, 3, 4 and dollars saved y: 11, 22, 33, 44. Write the equation. (y = 11x.) (3) A graph shows the points (0, 7), (1, 8) and (2, 9). Write the equation. (y = x + 7.) (4) Cara earns $10 for each lawn she mows. Name the independent and dependent variables and write the equation. (The number of lawns n is independent and the money m is dependent: m = 10n.)
Closure5 minutes
Exit ticket: (1) A muffin recipe uses 3 eggs per batch, so e = 3b. Which variable is independent? How many eggs do 4 batches need? (b, the number of batches; 12 eggs.) (2) Write an equation for this table: x: 1, 2, 3 and y: 9, 10, 11. (y = x + 8.) (3) In one sentence, say how the table in (2) shows the number 8.
Differentiation Strategies
For Struggling Students
Give a sentence frame: "The ____ depends on the ____." Students fill it in before they choose letters for the variables
Give tables with an extra column labeled "Rule" so students write the rule (× 12 or + 2) next to every row and test it on every row
Provide grid paper with the axes already labeled and numbered so students can focus on plotting each ordered pair
For Advanced Students
Ask students to write two stories, one for y = 4x and one for y = x + 4, and explain why their tables share only one row
Give a graph with two lines for two walkers and ask which walker is faster and how the equations show it
Ask students to switch the roles in the babysitting example: if Mia wants to earn a set amount, how many hours must she work, and which variable is now independent?
Assessment Guidance
What to Look For
Check that students can say which quantity depends on the other and put the dependent variable alone on one side of the equation. In tables, look for students who test their rule on every row, not just the first one. On graphs, check that the independent variable is on the horizontal axis and that points are plotted as (independent, dependent). Ask students to point to the number from the equation in the table (the amount added each row) and in the graph (how steep the line is, or where it starts).
02
Classroom Activities
3 Activities
1
Four-Way Match: Story, Equation, Table and Graph
15 minPairs
Each pair gets 16 cards: 4 stories, 4 equations, 4 tables and 4 graphs. Pairs match each story with its equation, table and graph, then explain one match to another pair.
Card Sets (16 cards)
Set A. Story: a car wash charges $9 per car. Equation: m = 9c. Table: c = 1, 2, 3 and m = 9, 18, 27. Graph: points (1, 9), (2, 18), (3, 27)
Set B. Story: a hiker walks at a constant 3 miles per hour. Equation: d = 3h. Table: h = 1, 2, 3 and d = 3, 6, 9. Graph: points (1, 3), (2, 6), (3, 9)
Set C. Story: Rosa is 4 years older than her cousin Ben. Equation: r = b + 4. Table: b = 6, 7, 8 and r = 10, 11, 12. Graph: points (6, 10), (7, 11), (8, 12)
Set D. Story: every player in a game gets a 3-point bonus added to the points earned. Equation: f = p + 3. Table: p = 3, 6, 9 and f = 6, 9, 12. Graph: points (3, 6), (6, 9), (9, 12)
Procedure
Shuffle the cards and spread them face up
Start with the stories: for each one, say which quantity is the independent variable
Find the equation that fits the story, then test it on every row of each table card before you match a table
Match each graph card by plotting its points on grid paper or by reading them from the card
Discussion Questions
Sets B and D both use the numbers 3, 6 and 9. How can you tell their tables apart? (In Set B, 3, 6 and 9 are the distances; in Set D, they are the points earned.)
If you extend the graphs, which two pass through (0, 0)? (Sets A and B, the two that multiply)
Which graph is steeper, Set A or Set B? How does the equation tell you? (Set A: it multiplies by 9, not 3)
Modification for Distance Learning
Put the 16 cards on a shared slide as movable images. Pairs drag the matching cards into four rows and record a short voice note that explains one match.
2
Hallway Walk: Time and Distance at a Constant Speed
20 minGroups of 3
Groups make their own constant speed data, like the official example. One student walks at a steady pace along a taped line, one times the walk, and one records. Groups build a table, graph the ordered pairs and write an equation for distance in terms of time.
Setup
Before class, put masking tape marks every 3 meters along a hallway, from 0 to 12 meters, measured with a measuring tape
Each group needs a stopwatch or phone timer, grid paper and a recording table with the columns "Time t (seconds)" and "Distance d (meters)"
Procedure
The walker starts at 0 and walks at a steady, normal pace; the timer calls out the time as the walker passes each mark (3, 6, 9 and 12 meters)
The recorder writes each pair in the table, with time as the independent variable
Repeat with a slow, steady walk
Plot both walks on one graph and draw a line through each set of points
Find the meters per second for each walk and write an equation d = (meters per second) × t
Sample Data (invented)
A normal walk: 0 meters at 0 seconds, 3 meters at 2 seconds, 6 meters at 4 seconds, 9 meters at 6 seconds and 12 meters at 8 seconds, which gives d = 1.5t (1.5 meters per second is a typical walking speed). A slow walk: 3 meters at 3 seconds, 6 at 6, 9 at 9 and 12 at 12, which gives d = t. Real times will not be this even, so groups round the meters per second to one decimal place.
Discussion Questions
In the sample data, which walk has the steeper line? (The normal walk: 1.5 meters each second instead of 1)
Why is time the independent variable here?
Use your equation: how far would the walker go in 20 seconds at the same speed? (30 meters for the sample normal walk)
Challenge Variation
Groups predict the time for a 21-meter walk from their equation, then test the prediction in the hallway and explain any difference.
3
Relationship Posters and Gallery Walk
20 minGroups of 3-4
Each group gets one situation card and makes a poster that shows the relationship four ways: variables with units, an equation, a table with 5 rows and a graph. Groups then visit the other posters and answer the question each poster asks.
Situation Cards (4 cards)
Card 1: smoothies cost $5 each at a snack bar (c = 5s)
Card 2: each floor of an office building is 3 meters tall (h = 3f)
Card 3: Grandpa Lou is 50 years older than his granddaughter Mia (g = m + 50)
Card 4: a swimming pool is 25 meters long, and a swimmer counts the lengths she swims (d = 25L)
Poster Checklist
Name both variables with units, and circle the independent variable
Write the equation with the dependent variable alone on one side
Make a table with 5 rows and a graph with labeled axes
Write one question that a visitor can answer with the table, the graph or the equation
Discussion Questions
Which poster's graph does not pass through (0, 0), and why? (Card 3: when Mia is 0, Grandpa Lou is 50)
On Card 3, what happens to the difference in ages as Mia gets older? (It stays 50 years)
On Card 4, should the points be joined with a line? (No: she counts whole lengths, so plot only the points)
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: The Official Example d = 65t as a Table and a Graph
A car travels at a constant 65 miles per hour. The table lists ordered pairs of times and distances, from (0, 0) to (4, 260), and the graph plots them to scale. Each hour adds 65 miles, so the points lie on a straight line through (0, 0), and the equation d = 65t gives every row.
Diagram 2: A Rule That Multiplies and a Rule That Adds
The popcorn cost c = 2b and Jon's age j = k + 2 from this lesson, written as y = 2x and y = x + 2. Both graphs are drawn to scale from the same table of x-values. The multiply rule starts at (0, 0) and goes up 2 for each step; the add rule starts at (0, 2) and goes up 1 for each step. The two lines meet only at (2, 4). The lines show each pattern; for popcorn, only the whole-number points are real purchases, since you cannot buy part of a bag.
04
Homework Assignment
~30 min
6.EE.C.9 Homework: Variables, Tables, Graphs and Equations
Directions: For every problem, name the independent and the dependent variable with units. Show your table or graph, and check your equation on every row of your table.
Part 1: Writing Equations (Problems 1-2)
A gardener plants 8 tomato plants in each row. Let r be the number of rows and p the number of plants. Which variable is independent? Write an equation for p in terms of r, make a table for r = 1 to 5, and find the number of plants in 11 rows.
Tia is 6 years older than her brother Deshawn. Let d be Deshawn's age and t be Tia's age. Write an equation for t in terms of d, and make a table for d = 7, 8, 9 and 10. How old is Tia when Deshawn is 12? How old is Deshawn when Tia is 20? Explain why t = 6d does not fit.
Part 2: Tables and Graphs (Problems 3-4)
A bathtub faucet fills the tub at a constant rate. Minutes x: 2, 4, 6, 8. Liters of water y: 26, 52, 78, 104. Write an equation for y in terms of x and graph the pairs. How much water is in the tub after 10 minutes? How many minutes does it take to reach 91 liters?
A graph shows the number of songs s on a playlist and the total playing time m in minutes, with the points (1, 4), (2, 8), (3, 12) and (5, 20). Every song is the same length. Write an equation for m in terms of s. Explain what the point (5, 20) means. How long do 12 songs play? Is the point (6, 22) on this graph? Explain.
Part 3: Putting It Together (Problems 5-6)
A train travels at a constant speed of 80 miles per hour. Write an equation for the distance d in terms of the time t. List the ordered pairs for t = 0, 1, 2, 3 and 4 and graph them. How far does the train go in 2.5 hours? How long does it take to go 360 miles?
Priya looks at this table and writes y = 6x. x: 1, 2, 3, 4. y: 6, 7, 8, 9. Is she right? Write the correct equation. Then explain how the table and a graph of the points each show that your equation is correct and hers is not.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Variables
Independent and dependent variables named with units in every problem
Variables named but one pair reversed or without units
Variables missing
Equations
Every equation correct, with the dependent variable alone on one side
Most equations correct, or one fits only the first row
Equations missing or incorrect
Tables and Graphs
Tables complete and points plotted as (independent, dependent) with labeled axes
One error in a table or graph
Tables or graphs missing
Explanations
Clear link between the equation, the table and the graph in Problems 4 and 6
An explanation that is partly correct
No explanation
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
Tanya buys roses for $3 each. She uses one variable for the number of roses and one for the total cost. Which quantity is the independent variable?
Answer: B
Tanya chooses how many roses to buy, and the total cost changes because of that choice, so the number of roses she buys is independent. Choice A is the dependent variable: the cost depends on the number of roses. Choice C is the fixed price of $3, which does not change, so it is not a variable here. Choice D is not one of the two quantities in the problem.
Question 2 of 20 · Multiple Choice
A cyclist rides at a constant speed of 14 miles per hour. Which equation gives the distance d in miles in terms of the time t in hours?
Answer: D
Each hour adds 14 miles, so the distance is 14 times the number of hours: d = 14t. Choice A adds 14 instead of multiplying, which gives only 15 miles after 1 hour and 16 after 2. Choice B swaps the roles of the variables. Choice C divides, which gives less than 1 mile in the first hour.
Question 3 of 20 · Multiple Choice
Which equation fits every row of this table? x: 3, 5, 8, 10. y: 21, 35, 56, 70.
Answer: A
In every row, y is 7 times x: 7 × 3 = 21, 7 × 5 = 35, 7 × 8 = 56 and 7 × 10 = 70, so y = 7x. Choice B fits only the first row (3 + 18 = 21), but 5 + 18 = 23, not 35. Choice C reverses the variables. Choice D adds 7, which gives 10 for x = 3.
Question 4 of 20 · Multiple Choice
Which equation fits every row of this table? x: 4, 5, 6, 7. y: 13, 14, 15, 16.
Answer: C
Each y is 9 more than its x: 4 + 9 = 13, 5 + 9 = 14, 6 + 9 = 15 and 7 + 9 = 16, so y = x + 9. Choice A multiplies by 9, which gives 36 for x = 4. Choice B uses the first y-value, 13, as the amount added and gives 17 for x = 4. Choice D reverses the roles: it makes x the larger number.
Question 5 of 20 · Multiple Choice
Which table matches the equation p = 4n?
Answer: A
In choice A, every p is 4 times its n: 4 × 1 = 4, 4 × 2 = 8 and 4 × 3 = 12. Choice B adds 4 to n, which is the table for p = n + 4. Choice C swaps the rows, so it fits n = 4p. Choice D starts at the right value but then adds 1 each time instead of 4.
Question 6 of 20 · Multiple Choice
Let x be the number of binders bought and y the total cost in dollars. The graph passes through (0, 0), (2, 10) and (4, 20). What does the point (4, 20) mean?
Answer: C
An ordered pair lists the independent value first: x = 4 binders and y = 20 dollars, so 4 binders cost $20 (each binder costs $5). Choice B reads the pair in the wrong order. Choices A and D each treat one number of the pair as the price of one binder.
Question 7 of 20 · Multiple Choice
Alex earns money at a constant rate, and m = 15h gives the money m in dollars for h hours of work. Which point is on the graph of this equation?
Answer: D
For h = 3, m = 15 × 3 = 45, and the independent value comes first, so (3, 45) is on the graph. Choice A lists the numbers in the wrong order. Choice B adds 15 to 3 instead of multiplying. Choice C divides 15 by 5 instead of multiplying 15 by the hours.
Question 8 of 20 · Multiple Choice
Nia's phone battery goes down while she watches videos. Which quantity is the dependent variable?
Answer: B
The battery percent changes because of how long she watches, so it depends on the minutes watched and is the dependent variable. Choice A is the independent variable: Nia chooses how long to watch. Choices C and D do not change during the situation, so they are not the two related quantities.
Question 9 of 20 · Multiple Choice
A bus travels at a constant 50 miles per hour, so d = 50t. How far does it go in 3.5 hours?
Answer: D
Put t = 3.5 into the equation: d = 50 × 3.5 = 175 miles. Choice A adds 50 and 3.5 instead of multiplying. Choice B uses only the 3 whole hours and leaves out the half hour (25 more miles). Choice C divides 50 by 3.5.
Question 10 of 20 · Multiple Choice
In the equation w = 4c, w is the number of wheels on c cars in a parking lot. Which statement is true?
Answer: A
The number of wheels depends on how many cars there are, and the equation gives w in terms of c, so w is dependent and c is independent. Choice B reverses the roles. Choice C calls the number 4 a variable, but 4 never changes. Choice D is wrong because one variable must depend on the other.
Question 11 of 20 · Multiple Choice
Which equation has a graph that does NOT pass through (0, 0)?
Answer: B
Put x = 0 into each equation. For y = x + 6, y = 0 + 6 = 6, so the graph starts at (0, 6), not (0, 0). For choices A, C and D, any number times 0 is 0, so each of those graphs passes through (0, 0).
Question 12 of 20 · Multiple Choice
In a table for y = 3x, what happens to y each time x goes up by 1?
Answer: C
The rows are 3 × 1 = 3, 3 × 2 = 6, 3 × 3 = 9, and so on, so y goes up by 3 for each step of 1 in x. Choice A describes an equation that adds, such as y = x + 3. Choice B is wrong because 6 doubled is 12, but the next row is 9. Choice D confuses the multiplier 3 with the value of y.
Question 13 of 20 · Multiple Choice
A hiker walks at a constant speed. The graph of distance d in miles against time t in hours passes through (2, 5) and (4, 10). Which equation fits?
Answer: A
Each point gives the same rate: 5 ÷ 2 = 2.5 and 10 ÷ 4 = 2.5 miles per hour, so d = 2.5t. Choice B fits (2, 5) but gives 7 for t = 4, not 10. Choice C divides time by distance, 2 ÷ 5 = 0.4, which is hours per mile. Choice D treats the 5-mile increase from (2, 5) to (4, 10) as the distance for 1 hour, but it took 2 hours.
Question 14 of 20 · Multiple Choice
Which situation can be modeled by y = x + 12, where x is the independent variable?
Answer: D
Mr. Diaz's age is always his son's age plus 12, so y = x + 12. Choices A and B multiply: the eggs are y = 12x and the miles are y = 12x. Choice C divides, y = x ÷ 12.
Question 15 of 20 · Short Answer
A car travels at a constant speed of 55 miles per hour. Name the independent variable, write an equation for the distance d in terms of the time t, and list the ordered pairs for t = 1, 2, 3 and 4.
Time t (hours) is the independent variable, and distance d (miles) depends on it. The equation is d = 55t. The ordered pairs are (1, 55), (2, 110), (3, 165) and (4, 220). On a graph they lie on a straight line through (0, 0).
Question 16 of 20 · Short Answer
The table shows the number of tickets n and the total cost c in dollars. n: 2, 4, 5. c: 16, 32, 40. Write an equation for c in terms of n. What do 9 tickets cost? Is the point (6, 46) on the graph?
Each ticket costs 16 ÷ 2 = 8 dollars, and 32 ÷ 4 = 8 and 40 ÷ 5 = 8 too, so c = 8n. Nine tickets cost 8 × 9 = $72. The point (6, 46) is not on the graph, because 6 tickets cost 8 × 6 = $48.
Question 17 of 20 · Short Answer
Leah is 11 years older than her cousin Omar. Write an equation for Leah's age l in terms of Omar's age o, and make a table for o = 3, 4, 5 and 6. Does the graph pass through (0, 0)? Explain.
l = o + 11. The table gives Leah's ages 14, 15, 16 and 17. The graph does not pass through (0, 0): when Omar is 0, Leah is 11, so it starts at (0, 11). Each year both ages go up by 1, so the difference stays 11.
Question 18 of 20 · Short Answer
The equation g = 6m gives the gallons of water g that flow from a garden hose in m minutes. What does the point (5, 30) on its graph mean? What is the unit rate, and where does it show up in the table?
The point (5, 30) means that in 5 minutes, 30 gallons of water flow from the hose. The unit rate is 6 gallons per minute, the number that multiplies m. In a table with m = 1, 2, 3, ..., the gallons go up by 6 in every row, and the row m = 1 shows 6 gallons.
Question 19 of 20 · Short Answer
Pump A fills a pool at 25 gallons per minute, so g = 25m. For Pump B, a table shows minutes m: 2, 4, 6 and gallons g: 30, 60, 90. Which pump is faster? Use equations or graphs to explain.
For Pump B, 30 ÷ 2 = 15, 60 ÷ 4 = 15 and 90 ÷ 6 = 15, so g = 15m. Pump A is faster: it pumps 25 gallons each minute and Pump B pumps 15. On one graph, Pump A's line is steeper. In 10 minutes, Pump A pumps 250 gallons and Pump B pumps 150.
Question 20 of 20 · Short Answer
A graph shows the points (1, 11), (2, 12), (3, 13) and (4, 14). Write an equation for y in terms of x, and find y when x = 20.
Each y is 10 more than its x (1 + 10 = 11, 4 + 10 = 14), so y = x + 10. When x = 20, y = 20 + 10 = 30. A common error is y = 11x, which fits only the first point.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 6.EE.C.9 mean?
6.EE.C.9 means students describe two quantities that change together with two variables and an equation. They decide which quantity is the independent variable and which is the dependent variable, write an equation such as d = 65t, and show the same relationship in a table and on a graph. Then they explain how the three forms match.
What is the difference between an independent and a dependent variable?
The independent variable is the quantity you choose or control, and the dependent variable changes because of it. When you buy apples, you choose how many pounds to buy, and the price you pay depends on that. A sentence frame helps: "The ____ depends on the ____." The quantity after "depends on" is the independent one.
Can the independent and dependent variables switch?
Yes, it depends on the question. If a runner runs for a set time, time is independent and distance depends on it. If the runner must cover a set distance, such as a 5 km race, you might treat distance as independent and ask how long it takes. In grade 6, problems usually make the choice clear, and students should say which one they chose.
Is 6.EE.C.9 about slope and linear functions?
Not yet, but it prepares for both. In grade 6, students notice that a bigger multiplier makes a steeper line and that y = x + c starts at c. The word slope and the idea of a function are taught later, in grade 7 (proportional relationships, 7.RP.A.2) and grade 8 (functions, 8.F.A.1).
How do tables, graphs and equations show the same relationship?
They are three views of one rule. Each row of a table is an ordered pair, each ordered pair is a point on the graph, and the equation gives the dependent value for any independent value. For an equation like y = kx, the number k is how much y grows in each row, and it decides how steep the line is.
Which axis does the independent variable go on?
The independent variable goes on the horizontal axis (the x-axis), and the dependent variable goes on the vertical axis (the y-axis). Ordered pairs follow the same order: independent value first. Swapping them puts points in the wrong places, so ask students to label both axes with the quantity and its unit.
Should students connect the points on the graph?
Only when the in-between values make sense. Time and distance can be any number, so a line through the points is fine: a car can drive for half an hour. Things you count, such as tickets or people, come in whole numbers, so students plot only the points. Discussing this is part of analyzing the relationship.
What are common mistakes with 6.EE.C.9?
A common mistake is finding a rule from the first row only: for the table x: 2, 4 and y: 6, 8, the rule "times 3" fits the first row but not the second. Other mistakes are reversing the variables, plotting pairs in the wrong order, and writing y = 3x when the story adds 3. Ask students to test every equation on every row.
Why does the standard use d = 65t as its example?
It is a clear case of two quantities that change together at a constant rate. The time is independent, the distance depends on it, and the rate of 65 miles per hour appears in the table (65 more miles each hour), in the graph (the steepness of the line) and in the equation. The official example asks students to list and graph pairs of times and distances before they write the equation.
How can parents help with 6.EE.C.9 at home?
Look for pairs of quantities that change together. On a car trip, ask how far you will go in 2, 3 and 4 hours at a steady speed. At the store, make a quick table of the cost of 1, 2 and 3 items. Ask which quantity depends on the other, and whether the rule multiplies or adds.
07
Related Standards
6 standards
These standards connect to 6.EE.C.9: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
5.OA.B.3Prerequisite
Generate two numerical patterns from two rules and graph the ordered pairs they form
Lesson coming soon
5.G.A.2Prerequisite
Graph points in the first quadrant to represent real-world and math problems
Lesson coming soon
Alongside
6.RP.A.3Parallel
Use ratio and rate reasoning, including tables, to solve real-world problems
