Integrated Algebra I
Function Families 1 & 2
Teacher Edition
Notes on Exploring Functions with Fiona
INTRODUCTION: The Georgia Performance Standards for Grade 8 Mathematics
include:
 recognize a relation as a correspondence between varying quantities,
 recognize a function as a correspondence between inputs and outputs where the
output for each input must be unique,
 distinguish between relations that are functions and those that are not
functions,
 recognize functions in a variety of representations and contexts,
 identify relations and functions as linear or non-linear, and
 translate among verbal, tabular, graphic, and algebraic representations of
functions.
(Standard M8A3, parts a, b, c, d, h, i)
This learning task launches the unit by giving students an opportunity to work with a
variety of relations, most of them functions. Each of these relations is presented in a
real-world context since most of the students’ prior work with relations and functions
has been in such contexts. Function notation is introduced as a new topic. Through
work with the variety of examples given, students should develop understanding and
skill in using standard algebraic function notation while revisiting the concepts and
skills inherent in the Grade 8 standards listed above.
Items 1 and 2 below explore some relations involving heights of various fictional
human beings. The two different contexts allow review of the important distinction
between continuous and discrete input values, and exploration of possible alternative
perspectives in item 2 lead to review of the distinction between relations that are and
are not functions. Function notation is introduced at the end of Item 2; here common
English usage for describing the context of the function closely parallels standard
terminology for the notation. The remaining items allow students extensive practice
with function notation in meaningful contexts. Throughout the task, there is a
deliberate emphasis on correct terminology in service of conceptual understanding.
Supplies Needed:
graph paper
Exploring Functions with Fiona
1. One vacation when visiting his grandmother, Todd found markings on the inside of a closet
door showing the heights of his mother, Julia, and her brothers and sisters on their birthdays
growing up. From the markings in the closet, Todd wrote down his mother’s height form ages 2
to 16. His grandmother found the measurements at birth and the one year by looking in his
mother’s baby book. The data is provided in the table below, with heights rounded to inches.
Integrated Algebra I
Age (yrs.)
Height (in.)
x
y
0
21
1
30
2
35
Function Families 1 & 2
3
39
4
43
5
46
6
48
7
51
8
53
9
55
10
59
11
62
12
64
13
65
14
65
a. Which variable is the independent variable, and which is the dependent variable?
Explain your choice.
Comments:
It is important to note that the information given states that the heights are
measured at the same point each year, on Julia’s birthday. This property is
important for this question in clarifying that each x-value value corresponds to
an exact age in years; that the input values are unambiguous. In question c
below, it allows us to conclude that connecting the dots corresponds to
representing points in time between Julia’s birthdays.
Solution:
The age in years is the independent variable, and the height in inches is the
dependent variable. Since height was measured once each year on Julia’s
birthday, there is a unique height as output for each age in years used as input.
The height cannot be used as the independent variable, because, for some
heights, there is more than one associated age.
b. Make a graph of the data.
Julia's height
Height in inches
70
60
50
40
30
20
10
0
0
2
4
6
8
10
Age in years
12
14
16
c. Should you connect the dots on your graph? Explain.
Comments:
This is the first of many questions in this unit asking students whether the data
points graphed should be connected. These questions are designed to focus
15
66
16
66
Integrated Algebra I
Function Families 1 & 2
students on the differences between situations where the domain is continuous
and where the domain consists of discrete numbers. The important issues are
(i) whether the points created by connecting the dots are both meaningful and
(ii) whether they provide reasonably accurate representations of the functional
relationship. Note that, given the nature of this data, allowing students to use
technology to create the graph would be quite appropriate. Excel scatter plot
was used to create the graph; however, any similar statistical graphing tool
would also be a good choice.
Solution:
Yes, the dots for the (age, height) pairs should be connected. Julia had a height
at every moment from the time she was 0 years old until 16 years old, and her
height at any time between her birthdays would be approximated by the values
shown on the line joining the measured heights.
d. Describe how Julia’s height changed as she grew up.
Comments: Students will likely apply varying amounts of prior knowledge
about growth to answering this question. Those students whose families have
carefully tracked the students’ growth patterns will be aware that periods of no
growth are often followed by growth spurts and may give a more detailed
description of the changes in growth rate. Thus, teachers should anticipate and
accept varying levels of specificity for this answer. The correct term to describe
Julia’s height function is that it is a nondecreasing function since, on any
subinterval in the interval [0, 16], the function is either increasing or constant.
For most groups of students, this context allows an informal introduction to this
important term used in describing characteristics of functions. While student
answers may not include all the details given above, students should (i)
describe the nondecreasing nature of the growth (in their own words, use of the
term “nondecreasing” is not expected) and (ii) note that the rate of growth in
height is not constant, with the most rapid growth rate during the first year.
Solution:
Julia’s height increased continuously as she grew up until age 13, then did not
change by age 16 except for an increase of one inch during her 15th year. Her
height increased most rapidly during the first year. From age 1 to 6, the
number of inches she grew each year decreased, going from an increase of 5
inches in her second year to an increase of only 2 inches in her sixth year.
From ages 6 to 13, she grew 2, 3, or 4 inches per year with the numbers going
up and down again twice. She did not increase in height during her 14th year,
grew an inch in her 15th year, and did not increase in height during her 16th
year.
e. How tall was Julia on her 11th birthday? Explain how you can see this in both the
graph and the table.
Solution:
Integrated Algebra I
Function Families 1 & 2
Julia was 62 inches tall at age 11, as shown by the y-value corresponding to x =
11 in the table and by the point (11, 62) of the graph.
f. What do you think happened to Julia’s height after age 16? Explain. How could
you show this on your graph?
Comments: The graph shown includes values for Julia’s age through 35 years.
It is reasonable to assume that the mother of a high school student is about this
age or older. Some students may extend the graph further and predict a slight
decrease in height in Julia’s senior years since many women lose some height
in their later years due to osteoporosis. The important point is understanding
how to represent on the graph that the height values remain constant after age
16, but other discussions can be beneficial in providing additional depth of
understanding for interpreting graphs.
Solution:
Most females reach their adult heights by age 16, so it is likely that Julia’s
height remained the same after age 16. The graph could be extended as a
horizontal line at y = 66, as shown in the graph below.
Julia's height
Height in inches
70
60
50
40
30
20
10
0
0
5
10
15
20
Age in years
25
30
35
Integrated Algebra I
Function Families 1 & 2
Integrated Algebra I
Function Families 1 & 2