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Use Variables to Represent Two Quantities in a Real-World Problem

Show your work. For word problems, write the equation first before building a table or plotting points.

Grade 6·22 problems·~35 min·Common Core Math - Grade 6·standard·6-ee-c-9
Work through problems with immediate feedback
A

Recall / Warm-Up

1.

Which quantity is the input (independent variable) in the statement: 'The number of miles driven depends on how long you drive'?

2.

In the equation y=4xy = 4x, which variable is the independent variable?

3.

Complete the table for the equation y=5xy = 5x. When x=0x = 0, yy =   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   . When x=3x = 3, yy =   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   . When x=6x = 6, yy =   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .

y when x=0:
y when x=3:
y when x=6:
B

Fluency Practice

1.

A school printer prints 12 pages per minute. Which statement correctly identifies the independent and dependent variables?

2.

A taxi charges a flat fee of $3 plus $2 per mile.

The equation for the total cost cc in terms of miles mm is   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .

The cost for 7 miles is   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .

equation:
cost for 7 miles:
3.

A plant is 8 cm tall and grows 3 cm each week.

The equation for the plant height hh after ww weeks is   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .

The height after 5 weeks is   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .

equation:
height after 5 weeks:
Input-output table for d = 65t with four rows showing t values 0, 1, 2, 3 and empty d-value boxes to fill in.
4.

Complete the table for the equation d=65td = 65t (distance in miles, time in hours).
When t=0t = 0, dd =   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .
When t=1t = 1, dd =   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .
When t=2t = 2, dd =   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .
When t=3t = 3, dd =   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .

d when t=0:
d when t=1:
d when t=2:
d when t=3:
Input-output table for e = 12h with four rows showing h values 0, 2, 4, 5 and empty e-value boxes to fill in.
5.

Complete the table for the equation e=12he = 12h (earnings in dollars, hours worked).
When h=0h = 0, ee =   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .
When h=2h = 2, ee =   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .
When h=4h = 4, ee =   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .
When h=5h = 5, ee =   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .

e when h=0:
e when h=2:
e when h=4:
e when h=5:
C

Varied Practice

1.

A bakery starts each day with 120 muffins and sells 15 per hour. Which equation represents the number of muffins mm remaining after hh hours?

2.

The graph of a relationship passes through the points (0, 0), (1, 4), (2, 8), and (3, 12). Which equation best represents this relationship?

3.

A container already holds 5 gallons of water. A hose adds 2 gallons per minute.
The equation is w=2t+w = 2t +   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   , where ww is total gallons and tt is minutes.
The starting value (y-intercept) is   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   gallons.
The rate of change is   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   gallons per minute.

constant term:
y-intercept value:
rate of change:
4.

Use the equation s=25ms = 25m to complete the table.
When m=0m = 0, ss =   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .
When m=1m = 1, ss =   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .
When m=4m = 4, ss =   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .
When m=10m = 10, ss =   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .

s when m=0:
s when m=1:
s when m=4:
s when m=10:
Coordinate plane showing the line d equals 65 times t with four plotted points and a highlighted point connected to the axes by dashed guide lines.
5.

A graph shows the equation d=65td = 65t. The ordered pair (2, 130) appears on the graph. What does this point represent in context?

D

Word Problems

1.

Lena earns $9 per hour babysitting. Let hh = hours worked and ee = total earnings in dollars.

1.

Which variable is the independent variable?

2.

The equation for ee in terms of hh is   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .

Lena's earnings for 6 hours are   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .

equation:
earnings for 6 hours:
2.

A pool contains 200 gallons of water. Water drains at 25 gallons per hour. Let tt = time in hours and ww = gallons remaining.

The equation for ww in terms of tt is   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .

After 3 hours, the gallons remaining are   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .

equation:
gallons after 3 hours:
3.

Marcus collects cans for recycling. He starts the week with 10 cans and collects 8 more each day. Let dd = days and cc = total cans.

The equation for cc in terms of dd is   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .

After 7 days, Marcus has   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   cans.

equation:
cans after 7 days:
4.

The equation p=3w+10p = 3w + 10 represents the number of plants pp in a garden after ww weeks.

What does the y-intercept (the point where the line crosses the y-axis) represent in this context?

E

Error Analysis

1.

Priya is graphing the equation d=65td = 65t. She plots time tt on the y-axis and distance dd on the x-axis. Her first three points are: (0,0)(0, 0), (65,1)(65, 1), (130,2)(130, 2).

What error did Priya make, and what is the correct way to set up the graph?

Carlos's input-output table for c equals 3n showing rows for n equals 1 through 4 with values 3, 6, 9, 12. No row for n equals 0 is shown.
2.

Carlos builds a table for c=3nc = 3n but starts his table at n=1n = 1, as shown in the table.

What important value is missing from Carlos's table, and why does it matter?

F

Challenge / Extension

Input-output table showing four ordered pairs: x equals 0 gives y equals 7, x equals 1 gives y equals 10, x equals 2 gives y equals 13, x equals 3 gives y equals 16.
1.

The table shows values for a linear relationship.
Equation:   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲  
Value of yy when x=10x = 10:   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲  

equation:
value at x equals 10:
2.

Two equations are given: y=5xy = 5x and y=5x+3y = 5x + 3. Both have the same coefficient for xx.
Explain how their graphs are similar and how they differ. What does the difference tell you about the real-world situations they could represent?

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