12.1 Inputs and Outputs 1
INPUTS AND OUTPUTS 1
In this lesson, students determine the amount of time needed to save for the purchase of a
camera, using input-output equations, tables, graphs, and words. Students learn the slopeintercept form of a line in a meaningful context, and they are led to appreciate the use of the
formula.
This lesson occurs near the beginning of a cluster that focuses on linear functions. Within
this cluster, students represent algebraic ideas visually, numerically, algebraically, and
verbally (the fourfold way). In previous lessons, students used the fourfold way to represent
geometric patterns. In future lessons, students will develop an informal understanding of
equations of the form y = mx + b in various contexts by making tables, writing equations,
and sketching graphs that lead to linear relationships. Based on this work, concepts
associated with linear functions will eventually be formalized.
Math Goals
• Use tables, graphs, equations, and words to solve problems.
(Gr4 MG2.1; Gr5 AF1.5; Gr6 AF2.0; Gr7 AF1.5; Gr7 MR2.0)
(Standards for
posting in bold)
•
Informally introduce the slope-intercept form of a line.
(Gr4 AF1.5)
Future Week
Summative
Assessment • Week 21: Direct Variation
(Gr5 AF1.5)
• Week 32: Slope
(Gr4 AF1.5)
Unit 3: Expressions and Equations (Teacher Pages)
Week 12 – TP1
12.1 Inputs and Outputs 1
PLANNING INFORMATION
Student Pages
Estimated Time: 45 – 60 Minutes
Reproducibles
Materials
Colored pencils
* SP1: Ready, Set, Go
* SP2: Saving for a Camera:
Instructions
* SP3: Saving for a Camera:
Tables
* SP4: Saving for a Camera:
Graphs
SP5: Saving for a Camera:
Questions
SP6: Saving for a Printer:
Instructions and Tables
SP7: Saving for a Printer:
Graphs
SP8: Saving for a Printer:
Questions
SP9: Brian’s Problem:
Instructions and Tables
SP10: Brian’s Problem:
Graphs and Questions
Prepare Ahead
Homework
SP6: Saving for a Printer:
Instructions and Table
SP7: Saving for a Printer:
Graph
SP8: Saving for a Printer:
Questions
Assessment
Strategies for
English Learners
* SP25:
Students not yet speaking
English can indicate answers to
questions by pointing to
evidence on the graph.
R52-53:
A54:
Knowledge
Check 12
Knowledge
Challenge 12
Weekly Quiz 12
Management Reminders
Allow adequate time for
summarizing. Students may
need a lot of help making
important connections between
the different representations
(numbers in the table; symbols
in the equations; graphs; using
vocabulary and expressing
solutions properly).
Strategies for
Special Learners
Link the formula for a linear
function to a familiar context:
y = mx + b
y = (money invested monthly)x
+ (start amount in the bank)
* Recommended transparency: Blackline masters for overheads 126-129 and 137 can be found in
the Teacher Resource Binder.
Unit 3: Expressions and Equations (Teacher Pages)
Week 12 – TP2
12.1 Inputs and Outputs 1
THE WORD BANK
function
A function f on a domain D is a rule that assigns to each element x of D
a unique value y = f(x). (Read f(x) as “f of x” or “the value of f at x”, NOT
“f times x”.) Thus a function is an input-output rule that assigns to each
input x a unique output y = f(x).
Example:
graph of a
function
The function y = mx + b assigns to each real number x
the value y = f ( x ) = mx + b .
The graph of a function f on a domain D is the collection of points with
coordinates (x, y), where x is in D and y = f(x) is the value of the
function at x.
Example:
The graph of the function y = mx + b is the straight line
in the plane consisting of the pairs (x, mx + b).
The graph of f(x) = x , x ≥ 0, is depicted below.
y
2
y=
x
2
3
1
0
linear function
1
x
4
A linear function (in variables x and y) is a function that can be expressed
in the form y = mx + b . The graph of y = mx + b is a straight line with
slope m and y-intercept b.
3
2
Example: The graph of the linear function y = x − 3 is a straight
line with slope m =
3
2
and y-intercept b = -3.
y
4
2
0
2
2
4
y=
3
x
2
6
8
–3
x
4
Unit 3: Expressions and Equations (Teacher Pages)
Week 12 – TP3
12.1 Inputs and Outputs 1
MATH BACKGROUND
Functions and Relations
Math Background 1
Preview/Warmup
There are two routes for developing the function idea. The route we have followed is to
define a function as an input-output rule, and to define the graph of the function as ordered
pairs of input and output values. Another more sophisticated route, which is common in
school mathematics, is to define first a “relation” to be a set of ordered pairs, and then to
define a function to be a relation with the “vertical line property. “ A function, according to
this definition, is what we have defined as the graph of the function. The two routes lead to
essentially the same class of objects. However, we have chosen the input-output definition
because it is the definition used in college mathematics, particularly calculus. Further, the
input-output definition is better adapted to understanding algebraic operations on functions
and the operation of composition of functions.
Unit 3: Expressions and Equations (Teacher Pages)
Week 12 – TP4
12.1 Inputs and Outputs 1
PREVIEW / WARMUP
Whole Class
¾ SP1*
Ready, Set, Go
Math Background 1
•
Introduce the goals and standards for the lesson. Discuss important
vocabulary as relevant.
•
Students fill in the t-table, and discuss why the input/output equation
should be y = 3x + 5.
INTRODUCE
Whole Class
¾ SP2*
Saving for a
Camera:
Instructions
•
Preview the directions , and link the information to the equation.
How much money must Julie save to purchase a digital camera?
$240. Christina? $240
¾ SP3*
Saving for a
Camera: Table
If Julie deposited $100 in the bank, and 0 months have passed,
what is her balance? Be sure students understand the meaning of account
balance. After 0 months she will not have saved any extra, so she will still have $100.
Using the equation and substituting 0 for x, show students that 10(0) + 100 = 100.
•
Do the same for another month or two.
Do we have to compute the balance for months 1, 2, 3, 4, etc. to
find the balance after month 3? No. If desired, we can use the formula to
check any month. Students may find at this time that going one month at a time
sequentially will be inefficient. However, allowing them to be inefficient now will help
them recognize the usefulness of the formula.
How much will Julie have after five months? 10(5) + 100 = 150
For how many months must Julie save until she has the $240?
Verify using the equation. 14 months: 10(14) + 100 = 240
¾ SP4*
Saving for a
Camera: Graph
Colored pencils
•
Students make a graph (using a colored pencil) that shows how much
Julie is saving each month.
What is the coordinate on the graph that shows how much Julie
has at the start? Graph the coordinate (0, 100). This is a good time to discuss
appropriate labeling and scaling for the graph. Convention dictates that the number of
months, the independent variable, will go on the horizontal (x) axis. The total amount
saved in dollars, the dependent variable, will go on the vertical (y) axis.
Do the coordinates form any pattern? They appear to be in a line—and they
are. However, in this situation, the actual points are not connected because we are
recording monthly data and fractions of months do not apply in this context. However, it
is permissible to draw a trend line, solid or dashed, to show the linear relationship.
Unit 3: Expressions and Equations (Teacher Pages)
Week 12 – TP5
12.1 Inputs and Outputs 1
EXPLORE
Pairs
¾ SP3*
Saving for a
Camera: Table
•
Encourage students to use the formula rather than checking each month
sequentially. The formula is an efficient way to find the desired solution, rather
than a slower, iterative process.
¾ SP4*
Saving for a
Camera: Graph
¾ SP5
Saving for a
Camera: Questions
Students use the equation to make a table and graph (with a different
colored pencil) to show how much Christina is saving each month.
•
Students answer comprehension questions that link the context of the
problem to the tables and graphs.
SUMMARIZE
Whole Class
¾ SP2*
Saving for a
Camera:
Instructions
•
Discuss solutions.
How can an equation be helpful in these problems? It is more
efficient/quicker to find the solutions.
Who had the most money saved in the bank from the start? Julie.
Was she the first to save $240? No. Why not? Julie only saved $10 per
¾ SP3*
Saving for a
Camera: Table
month, so Christina eventually passed Julie.
¾ SP4*
Saving for a
Camera: Graph
Who had the least money saved in the bank from the start? Christina.
Did she get to $240 last? No, Christina was first. Why? Christina saved more
per month than Julie and eventually passed Julie.
¾ SP5
Saving for a
Camera: Questions
Compare the amounts saved by each girl per month. Now compare
the steepness of the graphed lines. Do you notice a pattern? Slower
rate of savings translates into a flatter line. Faster rate of savings translates into a
steeper line. The rate of change is called the slope of the line.
PRACTICE
Individuals/Pairs
¾ SP6
Saving for a
Printer: Instructions
and Table
•
Students represent another context (saving for a printer) using tables
and graphs, and they answer questions about the problem using their
mathematical representations. This is appropriate for homework.
¾ SP7
Saving for a
Printer: Graph
¾ SP8
Saving for a
Printer: Questions
Unit 3: Expressions and Equations (Teacher Pages)
Week 12 – TP6
12.1 Inputs and Outputs 1
EXTEND
Whole Class
¾ SP9
Brian’s Problem:
Instructions and
Table
•
¾ SP10
Challenge students to write an equation, make a table, and draw a
graph to determine how long it will take Brian to save for the camera, the
printer, and both if he has $100 in the bank and is going to save $20
each month.
Why is ten months incorrect? Students may forget that to save for both, you
will use the original $100 that was in the bank for the first item, and then for the second
item it will take another seven months (i.e., you cannot use the $100 twice).
Brian’s Problem:
Graph and
Questions
CLOSURE
Whole Class
¾ SP1*
Ready, Set
•
Review the goals and standards for the lesson.
SELECTED SOLUTIONS
SP1
Warmup
SP5
Saving for a Camera:
Questions
SP8
Saving for a Printer:
Questions
SP9
Brian’s Problem:
Instructions and
Tables
SP10
Brian’s Problem:
Questions
x
y
10
35
1
8
y = 3x + 5
0
5
9
32
11
38
20
65
1. Julie ($100 > $40)
2. Christina ($25/month > $10/month);
Christina’s graph is steeper than Julie’s;
when comparing amount saved values
in the tables, Christina’s savings rise at
a faster rate than Julie’s.
1. Cary ($25 > $10)
2. Theresa ($20/month > $15/month);
Theresa’s graph is steeper than Cary’s;
when comparing amount saved values
in tables, Theresa’s savings rise at a
faster rate than Cary’s.
3. At 4 months they both have $140. This
is visible in the table. If 4 is substituted
for x in each equation, they both have a
y value of 140. The graphs intersect at
the point (4, 140).
4. 14 months
5. 8 months
6. Christina
3. At 3 months they both have $70. This is
visible in the table. If 3 is substituted in
for x in each equation, they both have a
y value of 70. The graphs intersect at
(3, 70).
4. 7 months
5. 9 months (must round up)
6. Theresa
1. y = 20x + 100
1. Discuss graph.
2. 7 months
Unit 3: Expressions and Equations (Teacher Pages)
3. 3 months
4. 15 months
Week 12 – TP7
STUDENT PAGES
12.1 Inputs and Outputs 1
INPUTS AND OUTPUTS 1
Ready (Summary)
Set (Goals)
We will use input-output equations,
tables, and graphs to find out how
much time is needed to save for a
camera and for a printer.
• Use tables, graphs, equations, and
words to solve problems.
• Informally introduce the slopeintercept form of a line.
Go (Warmup)
Rule: Multiply each input number by 3 and then add 5 to get each output number.
Input Number (x)
Output Number (y)
10
(10)(3) + 5 = 35
1
0
9
11
20
What is an equation for this rule? Use x for input and y for output.
y = _____________________________
Unit 3: Expressions and Equations (Student Packet)
Week 12 – SP1
12.1 Inputs and Outputs 1
SAVING FOR A CAMERA: INSTRUCTIONS
A digital camera costs $240.
Julie wants to save for the camera. She has $100 in the bank to start, and she is
going to save $10 each month.
Christina also wants to save for the camera. She has $40 in the bank to start, and she
is going to save $25 each month.
How many months will it take Julie and Christina to each save up for the digital
camera?
•
Let m represent the amount of money that Julie and Christina are going to deposit in
their bank accounts each month.
•
Let b represent the amount that Julie and Christina each already have in the bank to
start.
•
In the tables on the next page, keep track of the amounts that each girl has when
they start to save, and how much they have each month until they each reach their
goal.
•
Let x represent the number of months that Julie and Christina have been saving and
y represent the total amount saved.
Unit 3: Expressions and Equations (Student Packet)
Week 12 – SP2
12.1 Inputs and Outputs 1
SAVING FOR A CAMERA: TABLES
To find the total amount saved, use the equation form y = mx + b.
JULIE
m = $10 per month
b = $100 in the bank
CHRISTINA
m = $25 per month
b = $40 in the bank
y = 10x + 100
y = 25x + 40
x
(# of months)
y
(total amount saved)
0
10(0) + 100 = 100
1
10(___) + 100 = ___
Unit 3: Expressions and Equations (Student Packet)
x
(# of months)
y
(total amount saved)
Week 12 – SP3
12.1 Inputs and Outputs 1
SAVING FOR A CAMERA: GRAPHS
Use the data from the previous page to make graphs representing the total amount of
money that Julie and Christina will save each month. Use one color for Julie’s graph
and another color for Christina’s graph.
$300
$280
$260
$240
Total amount saved (y)
$220
$200
$180
$160
$140
$120
$100
$80
$60
$40
$20
0
2
4
6
8
10
12
14
16
18
20
Number of months (x)
Unit 3: Expressions and Equations (Student Packet)
Week 12 – SP4
12.1 Inputs and Outputs 1
SAVING FOR A CAMERA: QUESTIONS
1. Who starts out with more money in the bank? How do you know?
2. Who is saving at a faster rate? How do you know?
3. When will both girls have saved the same amount of money? How do you know?
4. How long will it take Julie to save for the camera?
5. How long will it take Christina to save for the camera?
6. Who will be the first to save enough money for the camera?
Unit 3: Expressions and Equations (Student Packet)
Week 12 – SP5
12.1 Inputs and Outputs 1
SAVING FOR A PRINTER: INSTRUCTIONS AND TABLES
A photo printer costs for $150.
Theresa wants to save for a photo printer. She has $10 in the bank to start, and she is
going to save $20 each month.
Cary also wants to save for the photo printer. She has $25 in the bank to start, and
she is going to save $15 each month.
How many months will it take Theresa and Cary to each save up for the photo printer?
To find the total amount saved, use the equation form y = mx + b.
Theresa
m = $20 per month
b = $10 in the bank
Cary
m = $_____ per month
b = $_____ in the bank
y = _______________
y = _______________
x
(# of months)
y
(total amount saved)
Unit 3: Expressions and Equations (Student Packet)
x
(# of months)
y
(total amount saved)
Week 12 – SP6
12.1 Inputs and Outputs 1
SAVING FOR A PRINTER: GRAPHS
Use the data from the previous page to make graphs representing the total amount of
money that Theresa and Cary will save each month. Use one color for Theresa’s graph
and another color for Cary’s graph.
$300
$280
$260
$240
Total amount saved (y)
$220
$200
$180
$160
$140
$120
$100
$80
$60
$40
$20
0
2
4
6
8
10
12
14
16
18
20
Number of months (x)
Unit 3: Expressions and Equations (Student Packet)
Week 12 – SP7
12.1 Inputs and Outputs 1
SAVING FOR A PRINTER: QUESTIONS
1. Who starts out with more money in the bank? How do you know?
2. Who is saving at a faster rate? How do you know?
3. When will both girls have saved the same amount of money? How do you know?
4. How long will it take Theresa to save for the printer?
5. How long will it take Cary to save for the printer?
6. Who will be the first to save enough money for the printer?
Unit 3: Expressions and Equations (Student Packet)
Week 12 – SP8
12.1 Inputs and Outputs 1
BRIAN’S PROBLEM: INSTRUCTIONS AND TABLE
Brian wants to save for both a camera and a photo printer. A digital camera costs $240 and
a photo printer costs $150. He has $100 saved in the bank and is going to save $20 each
month.
Cost of camera: _________
x
(# of months)
y
(amount
saved)
Cost of printer: __________
Total amount Brian needs to save: __________
To find the total amount saved, use the equation form:
y = mx + b
m = the amount of money Brian is going to deposit in
his bank account each month
b = the amount of money Brian already has in the bank
x = the number of months he has been saving
y = total amount saved
Write an equation to show the total amount of money Brian
has saved at the end of each month.
___________________________________________
Unit 3: Expressions and Equations (Student Packet)
Week 12 – SP9
12.1 Inputs and Outputs 1
BRIAN’S PROBLEM: GRAPH AND QUESTIONS
1. Make a graph to show the total amount of money Brian has saved each month.
Total amount saved (y)
$400
$300
$200
$100
0
5
10
15
20
Number of months (x)
2. How long will it take Brian to save for the camera?
3. How long will it take him to save for the printer?
4. How long will it take him to save for both the printer and the camera?
Unit 3: Expressions and Equations (Student Packet)
Week 12 – SP10