Lesson 2.1 - ACTIVITY: Bungee Jumping
Obj.: • collect experimental data. • make a scatter plot of experimental data.
Constructing a mathematical model to describe a relationship between
two variables often begins with collecting data and recording the
values in a table. The data can then be graphed with a scatter plot.
Go to pages 23 & 24 of the textbook.
A bungee cord is elastic, meaning that it can be stretched and then returned to its
original length. You can use a large rubber band to simulate a bungee cord.
Answer questions 1-9 in the space below.
1.______ _____________
2.___________________
3._____________________
7.
4.
5.
6.
8.
9.______________________________
________________________________
________________________________
________________________________
________________________________
10.____ ______________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
**Practice for Lesson 2.1 pages 24 & 25, 1-8 all**
Lesson 2.2 - INVESTIGATION: Proportional Relationships
Obj.: • identify proportional relationships. • identify independent and
dependent variables. • find a constant of proportionality. • write an equation
that expresses a proportional relationship.
Vocabulary
• Constant of proportionality - The ratio of the values of the dependent variable to
the corresponding values of the independent variable in a proportional relationship.
• Dependent variable - The variable in an equation whose value is determined by the
value of the independent variable.
• Independent variable - The input variable in a function.
• Proportional relationship - The relationship between two variables for which the
ratio of the dependent variable to the independent variable is constant.
An equation is one of the most useful kinds of mathematical models. It
can allow you to predict the value of one quantity when you know the
value of another. In this lesson, you will use ratios to find an equation
that models the relationship between the weight on a rubber band and
how much the band stretches. Go to pages 26 & 27 of the textbook.
When a clear pattern exists on a graph of two variables, it is often possible to find an
equation that relates the variables. Clues to an equation may sometimes be found by
examining the ratios of the values of the variables.
1. Return to your data table from the
Activity in Lesson 2.1. Label the fourth
column “Stretch-to-Weight Ratio.”
Calculate the ratio of the total stretch
to the total weight for each row of the table
and record it. Write your ratios as decimals.
2.__________________________
5.________________________________
6.__________________________
3.__________________________
4.__________________________
7.__________________________________________________________________
_____________________________________________________________________
**Practice for Lesson 2.2 pages 27-29, 1-16 all**
Lesson 2.3 - Direct Variation Functions
Obj.: • identify direct variation. • write direct variation equations.
Vocabulary
• Direct proportion - One quantity is directly proportional to another if the ratios of the
two quantities are constant. A graph of the quantities is a line that includes the point
(0, 0).
• Direct variation - The equation y = kx expresses direct variation. The equation can
be read “y is directly proportional to x” or “y varies directly with x.”
• Direct variation function - An equation or function expressing direct variation.
• Function - A function is a relationship between input and output in which each input
value has exactly one output value.
• Input value - One of the possible values of the independent variable of a function.
• Origin - All direct variation graphs are straight lines that pass through the origin
(0, 0).
• Output value - The value of the dependent variable that corresponds to a specific
input value in a function.
• Variation - The way in which two variables are related.
• Varies directly - A way to describe values that exhibit direct variation.
Example 1
A. Use the regulating line for the Notre Dame proportioning system in Figure 2 to
estimate the width of a similar rectangle with a height of 200 feet.
about 120 ft
B. Use a proportion to calculate the width.
125 ft
Example 2
The table shows the cost of having various amounts of clothes washed at a
drop-off laundry.
a. Verify that cost is directly proportional to weight.
The ratios of cost to weight all equal 1.30.
b. Write an equation that models cost C as a function of weight w.
C = 1.30w
c. Draw a graph of cost vs weight.
d. Use your equation to determine the cost for
a load of 8 pounds of clothes. $10.40
**Practice for Lesson 2.3 pages 35-37, 1-11 all**
Lesson 2.4 - Review and Practice
Obj.: • solve problems that require previously learned concepts and skills.
Example 1
Fill in the blank.
𝑦
a. For two variables x and y, if the ratio ⁄𝑥 is constant, then y is ___________________ to x.
b. An input/output relationship in which each input value has exactly one output value is called a(n)
___________________.
Example 2
Choose the correct answer.
Which of the following is not a correct way of writing the ratio “3 to 4”?
B. 3⁄4
A. 3 : 4
C. 3 X 4
D. 3 to 4
Example 3
Change the decimal to a fraction in lowest terms.
a. 0.4
b. 0.22
Example 4
Change the fraction to a decimal.
Example 5
Evaluate the expression. a. |‒8| + 2
7
25
b. 4 − 2|-10|
Example 6
The value of the square root is between which two integers?
√78
Example 7
Solve the proportion.
a.
5
6
=
𝑥
42
b.
7
19
=
**Do Lesson 2.4 R. A. P. pages 38 & 39, 2-24 even**
14
𝑎
Lesson 2.5 - Slope
Obj.: • find the slope of a line given rise and run. • find the slope of a line given
two points on the line. • interpret the slope of a line as a rate of change.
Vocabulary
• Rate of change - The ratio of the change in the value of one quantity to the change
in the value of another quantity.
• Slope - A measure of the steepness of a line: the ratio of the vertical change (Δy) to
the horizontal change (Δx) between two points on a line.
Slope =
𝒓𝒊𝒔𝒆
𝒓𝒖𝒏
or
𝒚𝟐 − 𝒚𝟏
𝒙𝟐 − 𝒙𝟏
or
𝒗𝒆𝒓𝒕𝒊𝒄𝒂𝒍 𝒄𝒉𝒂𝒏𝒈𝒆
𝒉𝒐𝒓𝒊𝒛𝒐𝒏𝒕𝒂𝒍 𝒄𝒉𝒂𝒏𝒈𝒆
or
∆𝒚
∆𝒙
Example 1
Find the slope of a staircase that rises 90 inches over a run of 120 inches.
3
4
or 0.75
Example 2
The graph below shows the cost of having various amounts of clothes cleaned at a drop-off laundry.
Find the slope of the line.
**Practice for Lesson 2.5 pages 43 - 47, 1-20 all**