Formulas, Tables,
and Graphs: Part 2
Objective To analyze a real-world situation by making and
using a data table and a graph.
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ePresentations
eToolkit
Algorithms
Practice
EM Facts
Workshop
Game™
Teaching the Lesson
Key Concepts and Skills
• Apply multiplication and division facts
to calculate rates. Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
1 2
4 3
[Operations and Computation Goal 2]
• Construct, extend, and interpret a
line graph. [Data and Chance Goals 1 and 2]
• Plot data values from a table. [Measurement and Reference Frames Goal 3]
• Express rates as formulas. [Patterns, Functions, and Algebra Goal 1]
Key Activities
Students compare two summer jobs by
analyzing their potential profits. They construct
a data table, develop earnings formulas,
graph the data, and interpret the graph.
Ongoing Assessment:
Informing Instruction
See pages 231 and 232.
Playing Getting to One
Student Reference Book, p. 321
Math Masters, p. 448
calculator overhead calculator
(optional)
Students review place-value concepts
of decimal numbers.
Math Boxes 3 10
Math Journal 1, p. 119
Students practice and maintain skills
through Math Box problems.
Ongoing Assessment:
Recognizing Student Achievement
Use Math Boxes, Problem 1. Curriculum
Focal Points
Differentiation Options
READINESS
Custom-Made Representations
Math Masters, p. 97
Students write a rule, make a table, or
construct a graph to represent a situation.
ENRICHMENT
The Shape of Change
Math Masters, pp. 98 and 99
Geometry Template or metric ruler
per partnership: 3 or 4 containers of various
shapes and sizes, measuring cups (mL),
and water
Students conduct an experiment, graph the
results, and analyze their findings.
[Patterns, Functions, and Algebra Goal 1]
Study Link 3 10
Math Masters, p. 96
Students practice and maintain skills
through Study Link activities.
EXTRA PRACTICE
5-Minute Math
5-Minute Math™, p. 203
Students sketch graphs to represent actions
over time.
Materials
Math Journal 1, pp. 120 and 121
Study Link 39
straightedge
Advance Preparation
For the optional Enrichment activity in Part 3, refer to Math Masters, page 99 for ideas about container shapes.
Teacher’s Reference Manual, Grades 4–6 pp. 161, 164, 278–289
230
Unit 3
Variables, Formulas, and Graphs
Interactive
Teacher’s
Lesson Guide
Mathematical Practices
SMP1, SMP2, SMP4, SMP5, SMP6, SMP7, SMP8
Getting Started
Content Standards
6.RP.3, 6.RP.3a, 6.NS.5, 6.NS.6, 6.NS.6c, 6.NS.8
Mental Math and Reflexes
Math Message
Students solve mental computation
problems with positive and negative numbers.
Suggestions:
Suppose you get a summer job that pays $5.20
1
per hour and you work 3_
2 hours each day, 5 days per week.
Estimate the number of weeks it would take you to earn a total
of $800.
100 + (-40) 60
-50 + (-25) -75
60 + (-200) -140
-49 + 74 25
62 + (-322) -260
-42.3 + 38.9 -3.4
Study Link 3 9 Follow-Up
Briefly go over the answers.
1 Teaching the Lesson
▶ Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
Have students share their estimation strategies. Estimates should
be between 8 and 9 weeks. Working at this rate, a person would
earn $819 at the end of 9 weeks.
Remind students that in Lesson 3-5, they learned how to
represent a rate with a data table, a formula, and a line graph.
In this lesson, students use these representations to analyze and
compare two rates.
▶ Comparing the Profits for
PARTNER
ACTIVITY
Summer Jobs
(Math Journal 1, p. 120)
Student Page
Date
Time
LESSON
3 10
䉬
Formulas, Tables, and Graphs
254
Haylee and Chloe want to earn money during summer vacation.
Consumer Education Link Partners complete journal page
120. Discuss students’ answers. Chloe will have made more
money by the end of 3 weeks, but Haylee will have made more
money by the end of the 10-week period.
1.
As part of a follow-up discussion, have students use the table to
develop rules for the money earned by Haylee and Chloe in terms
of the time it takes to earn the money.
2.
Haylee’s Summer Job
Chloe’s Summer Job
Haylee is going to mow lawns.
Her father will lend her $190 to
buy a lawn mower. She figures
that she can mow 10 lawns per
week and make $12 per lawn
after paying for oil and gasoline.
Chloe is going to work in an
ice cream shop. The owner will
provide a uniform free of charge
and pay her $5.20 per hour.
1
She will work 3 2 hours per day,
5 days per week.
Complete the table at the right to
show how much profit each girl will
have made after 2 weeks, 3 weeks,
and so on. (Assume they do not
have to pay taxes.)
Use the table to answer the
following questions.
a.
Who will have made more
money by the end of 3 weeks?
Chloe
Ongoing Assessment: Informing Instruction
Watch for students who are unable to develop these rules on their own. Write
the rules on the board and have students verify that the rules are correct for
several in (weeks) and out (dollars) values.
b.
How much money will that
girl have made?
$273
c.
Who will make more
money during the summer?
Haylee
Profit
(dollars)
Time
(weeks)
Haylee
Start
190
0
1
70
91
2
50
170
290
410
182
273
364
455
546
637
3
4
5
6
530
7
650
770
890
1,010
8
9
10
Chloe
728
819
910
Math Journal 1, p. 120
Lesson 3 10
231
Haylee
Chloe
in
out
(weeks) (dollars)
in
out
(weeks) (dollars)
It may help students to think in terms of “What’s My Rule?”
tables. In each table, the in values are amounts of time (weeks),
and the out values are profits earned (dollars). (See margin.)
Chloe’s rule is easy to determine: out = in ∗ 5.20 ∗ 3.5 ∗ 5
start
-190
start
0
1
-70
1
91
2
50
2
182
Haylee’s rule is less obvious, and students may not
discover it on their own: out = (in ∗ 10 ∗ 12) - 190
3
170
3
273
Chloe’s and Haylee’s rules are general patterns.
4
290
4
364
5
410
5
455
6
530
6
546
Have students examine the general pattern they may have used to
arrive at Haylee’s data. Ask them to write each computation in
the form of a special case:
7
650
7
637
0 weeks: (0 ∗ 10 ∗ 12) - 190 = -190 or -$190
8
770
8
728
1 week: (1 ∗ 10 ∗ 12) - 190 = -70 or -$70
9
890
9
819
4 weeks: (4 ∗ 10 ∗ 12) - 190 = 290 or $290
10
1,010
10
910
▶ Graphing Profit Data and
Interpreting the Graph
PARTNER
ACTIVITY
PROBLEM
PRO
P
RO
R
OBL
BLE
B
LE
L
LEM
EM
SO
S
SOLVING
OL
O
L
LV
VIN
V
IN
ING
(Math Journal 1, pp. 120 and 121)
Ongoing Assessment:
Informing Instruction
Watch for students who may not recognize
the intervals along the vertical axis of the
graph. Suggest that these students write the
in-between amounts between the labeled
increments before graphing the profit data.
Time
LESSON
Formulas, Tables, and Graphs
3 10
䉬
3.
continued
254
Use the grid below to graph the profits from Haylee’s and Chloe’s summer jobs.
Label the lines Haylee and Chloe.
lee
1,400
y
Ha
1,200
loe
1,000
Ch
Profit ($)
800
600
200
1
2
3
4
5
6
7
8
9
10
11
12
Time (Weeks)
One way to analyze data is to look at how quickly the graph of the data rises or falls.
a.
Which graph rises more quickly, Haylee’s graph or Chloe’s graph?
b.
What is one conclusion that you can draw about profit data that are represented
by a quickly rising graph?
Haylee’s
5.
Sample answer: There is a greater
increase in profit over a shorter period
of time.
1
About 62 weeks
Estimate the time at which the graphs intersect.
Math Journal 1, p. 121
232
How quickly a line graph rises or falls reveals a great deal
about the data it represents. In the case of Haylee’s and
Chloe’s profit data, the rising line graphs indicate that each
girl continues to make a profit as time passes.
13
–200
4.
The point at which the graphs intersect represents the time at
which Haylee’s profit equals Chloe’s profit.
The negative numbers on the y-axis indicate debt, rather than
profit. Haylee begins the summer with nearly $200 of debt,
while Chloe begins the summer with no debt or profit.
400
0
0
Comments for follow-up discussion:
Students need to plot only three time-and-profit results for each
girl. For example, plot (0,-190), (1,-70), and (2,50) for Haylee;
plot (0,0), (1,91), and (2,182) for Chloe. Students should then
use a straightedge to connect these points and extend the line
for each girl. They can use the graph to determine each girl’s
profit after 11, 12, and 13 weeks.
Student Page
Date
Have students graph the profit data from their tables onto the
grid on journal page 121. They should plot two graphs on the
same grid—one graph to represent Haylee’s profits and one graph
to represent Chloe’s profits.
Unit 3
Variables, Formulas, and Graphs
Haylee’s line graph crosses the horizontal axis just after
Week 1. This point of intersection with the x-axis indicates that
Haylee has earned enough money to pay her father back and
can begin to make a profit.
Student Page
Date
2 Ongoing Learning & Practice
▶ Playing Getting to One
Time
LESSON
Math Boxes
3 10
䉬
夹
1.
PARTNER
ACTIVITY
Darin charges $5 an
hour to baby-sit on
weekdays and $7 an
hour on weekends. The
spreadsheet is a record
of the baby-sitting Darin
did during one week.
Complete the spreadsheet.
1
(Student Reference Book, p. 321; Math Masters, p. 448)
Distribute a calculator to each pair of students, as well as a game
record sheet (Math Masters, p. 448). Students read the directions
on Student Reference Book, page 321. Have a volunteer
demonstrate the game on an overhead calculator, if available.
Encourage students to play a practice game.
B
C
Number of Hours
Earnings ($)
2
Monday
4
3
Wednesday
2
4
Saturday
5
Total
20
10
35
65
5
11
B4
a.
Which cell contains the number of hours Darin worked on Saturday?
b.
Circle the formula Darin should NOT use to calculate his total earnings.
C5 C2 C3 C4
C5 6 ⴱ B5
C5 (5 ⴱ B2) (5 ⴱ B3) (7 ⴱ B4)
Write a formula that Darin can use to calculate his earnings for Monday.
c.
2.
A
Day of the Week
C2 5 ⴱ B2
142–144
Divide.
3.
18冄4
苶3
苶5
苶.6
苶
Order from least to greatest.
a.
0.23, 0.32, 0.023, 0.323
b.
36.837, 36.783, 36.878, 36.8375
0.023, 0.23, 0.32, 0.323
36.783, 36.837,
36.8375, 36.878
435.6 18 ▶ Math Boxes 3 10
INDEPENDENT
ACTIVITY
(Math Journal 1, p. 119)
4.
24.2
30
42
List the first 6 multiples of each number.
Name the greatest common factor
(GCF) of each pair of numbers.
5.
5, 10, 15, 20, 25, 30
6, 12, 18, 24, 30, 36
5
6
Name the least common multiple
(LCM) of 5 and 6.
Mixed Practice Math Boxes in this lesson are paired with
Math Boxes in Lesson 3-8. The skills in Problems 4 and 5
preview Unit 4 content.
Ongoing Assessment:
Recognizing Student Achievement
Math Boxes
Problem 1
30
a.
9 and 36
b.
50 and 20
c.
18 and 7
9
10
1
78
80
Math Journal 1, p. 119
Use Math Boxes, Problem 1 to assess students’ abilities to name a spreadsheet
cell (Problem 1a) and to identify a spreadsheet formula for calculating a total
(Problem 1b). Students are making adequate progress if they can correctly
complete Problems 1a and 1b. Some students may be able to write a spreadsheet
formula for calculating the total earnings for Monday (Problem 1c).
[Patterns, Functions, and Algebra Goal 1]
INDEPENDENT
ACTIVITY
(Math Masters, p. 96)
Study Link Master
Name
Date
STUDY LINK
Comparing Pet-Sitting Profits
3 10
䉬
Home Connection Students analyze pet-sitting
profits by making and using a table of data and a related
graph.
Time
Jenna and Thomas like to pet-sit for their neighbors. Jenna charges $3 per hour.
Thomas charges $6.00 for the first hour and $2 for each additional hour.
1.
Time
(hours)
Jenna’s Profit ($)
Thomas’s Profit ($)
$3
$6
$9
$12
$15
$6
$8
$10
$12
$14
1
2
3
4
5
2.
Extend both line graphs to find the profit each sitter will make for 6 hours.
Jenna (6 hours)
3.
$18
Thomas (6 hours)
$16
Which sitter, Jenna or Thomas, earns more
18
Jenna
4. Which line graph rises more quickly? Jenna’s
money for jobs of 5 hours or more?
5.
254
Complete the table below. Use the table to graph the profit values for each sitter.
16
14
12
Complete each statement. For every hour that
$3
passes, Jenna’s profit increases by
Thomas’s profit increases by
$2
;
.
Profit ($)
▶ Study Link 3 10
Thomas
10
Jenna
8
6
6.
At what point do the line graphs intersect?
4
(4,12)
2
0
0
1
2
3
4
5
6
Time (Hours)
Practice
7.
Evaluate when m 3.
a.
m4
81
b.
20m
8,000
c.
4m 4m
76
d.
10m 5m
875
m3
e. 2
m
3
Math Masters, p. 96
Lesson 3 10
233
Teaching Master
Name
Date
LESSON
Time
Reviewing Rules, Tables, and Graphs
3 10
䉬
Rule:
in
out
3 Differentiation Options
PARTNER
ACTIVITY
READINESS
▶ Custom-Made
5–15 Min
Representations
(Label for y-axis)
(Math Masters, p. 97)
0
0
(Label for x-axis)
This activity provides students with opportunities to
review the particular representation(s) of function with
which they may be struggling. Present students with a
situation and ask them to write a rule, complete a table, or
make a graph representing that situation. Some students might
need help labeling axes and assigning value increments
to each axis.
Examples:
Math Masters, p. 97
Every year Bart earns $150 more than he earned the
previous year.
Rule:
Previous year’s earnings (p) + $150 = Present year’s earnings
NOTE You can quickly modify this activity
by providing students with situations that
involve more than one operation.
Example:
Dalia’s hourly wage at her new job (x) is $1
more than twice the wage of her old job (w);
(x = 2w + 1).
in
out
( p)
( p + $150)
$0
$150
$75
$225
$150
$300
$200
$350
$300
$450
Gloria earns $5.00 for every hour she baby-sits.
Rule:
Number of hours baby-sitting (n) ∗ $5 = Gloria’s total earnings
234
Unit 3
Variables, Formulas, and Graphs
in
out
( n)
( n ∗ $5)
0
$0
4
$20
9
$45
10
$50
12
$60
PARTNER
ACTIVITY
ENRICHMENT
▶ The Shape of Change
Graph 1
20
(Math Masters, pp. 98 and 99)
Height of Water (cm)
30+ Min
1
Science Link To further explore how graph shapes
correspond to situations, have students conduct an
experiment. Provide students with three or four bottles or glasses
of various shapes, metric measuring cups (for example, a medicine
cup or beaker), the Geometry Template or metric ruler, and water.
Students pour the same amount of water into each bottle. With a
metric ruler, they measure the height of the water level after each
addition of water until each bottle is full. Students record and
graph results on Math Masters, page 98. Graphs should show that
the height of the water in a particular bottle increases more slowly
as the bottle becomes wider; the height increases more quickly as
the bottle becomes narrower. For example, the height of the water
level in Bottle 1 increases more quickly as water fills the base and
neck of the bottle. Since Bottle 2 is a cylinder, Graph 2 reveals a
steady change in water height as the volume of water increases.
(See margin.)
15
10
5
0
0 120 240 360 480 600 720 840 960
Volume (mL)
Graph 2
35
30
Height of Water (cm)
25
2
After completing the experiment and analyzing their results,
students match mystery graphs with various bottle shapes (Math
Masters, p. 99).
20
15
10
5
SMALL-GROUP
ACTIVITY
EXTRA PRACTICE
▶ 5-Minute Math
0
0 120 240 360 480 600 720 840 960
Volume (mL)
5–15 Min
To offer more practice using graphs to represent actions over time,
see 5-Minute Math, page 203.
Teaching Master
Teaching Master
Name
LESSON
3 10
䉬
Date
Time
Date
LESSON
Rate of Change Experiment
䉬
Bottle 2
Time
The Shape of Change
3 10
Using a metric measuring cup, pour 100 mL of water into each of 4 bottles of different
shapes. Each time you add water to a bottle, measure the height of the water level in
centimeters. On a separate sheet of paper, make a table to record each change in volume
and water height. Use your table to make a graph for each bottle on the grids below.
Bottle 1
Name
Assume the bottles are filled with water at a constant rate. Match the graphs with
their bottles. Write the letter of the graph under the bottle it represents.
1
3
4
5
A
C
B
Height of Water (cm)
0
0
0
Volume (mL)
0
0
Volume (mL)
Graph D
0
Height of Water (cm)
Bottle 4
0
Height of Water (cm)
Bottle 3
Height of Water (cm)
Volume (mL)
Graph B
0
Volume (mL)
Height of Water (cm)
0
0
Volume (mL)
Graph A
Height of Water (cm)
0
0
E
Height of Water (cm)
D
Height of Water (cm)
Height of Water (cm)
2
Graph C
0
0
Volume (mL)
Graph E
0
0
Volume (mL)
0
0
Volume (mL)
Math Masters, p. 98
Volume (mL)
Math Masters, p. 99
Lesson 3 10
235