Instructional Week 6: February 8-12
ISTEP + 10 Mathematic
Focus Topic: Interpret Graphs and Correlation
Paced Standards:
AI.F.2: Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the
function is increasing or decreasing, linear or nonlinear, has a maximum or minimum value). Sketch a graph that exhibits
the qualitative features of a function that has been verbally described. Identify independent and dependent variables
and make predictions about the relationship.
8.DSP.3: Write and use equations that model linear relationships to make predictions, including interpolation and
extrapolation, in real-world situations involving bivariate measurement data; interpret the slope and yintercept.+
AI.DS.2: Graph bivariate data on a scatter plot and describe the relationship between the variables.
PS: 1, 2, 3, 4, 5, 6, 7, and 8 +
Key Vocabulary
Bivariate Data – data that has two variables
Scatter Plot – a plot that shows a relationship between two sets of data.
Teacher Background: Sample Problems for AI.F.2
1. The figure shows the graph of T , the temperature (in degrees Fahrenheit) over one particular 20-hour
period in Santa Elena as a function of time t. (Illustrative Mathematics)
a.
b.
c.
d.
e.
Estimate T(14).
If t=0 corresponds to midnight, interpret what we mean by T(14)in words.
Estimate the highest temperature during this period from the graph.
When was the temperature decreasing?
If Anya wants to go for a two-hour hike and return before the temperature gets over 80 degrees,
when should she leave?
2. An epidemic of influenza spreads through a city. The figure below is the graph of I=f(w), where I is the
number of individuals (in thousands) infected w weeks after the epidemic begins.
a.
b.
c.
d.
e.
a. Estimate f(2) and explain its meaning in terms of the epidemic.
b. Approximately how many people were infected at the height of the epidemic? When did that occur? Write
your answer in the form f(a)=b.
c. For approximately which w is f(w)=4.5; explain what the estimates mean in terms of the epidemic.
d. An equation for the function used to plot the image above is f(w)=6w(1.3)−w. Use the graph to estimate the
solution of the inequality 6w(1.3)−w≥6. Explain what the solution means in terms of the epidemic.
Teacher Background: Sample Problems for AI.F.2 Teacher Notes
1. a. T(14) is a little less than 90 degrees Fahrenheit; maybe 88 or 89 degrees.
b. The temperature was almost 90 degrees at 2:00 in the afternoon.
c. The highest temperature was about 90 degrees.
d. The temperature was decreasing between 4:00 p.m. and 8:00 p.m. It might have continued to decrease
after that, but there is no information about the temperature after 8:00 p.m.
e. The temperature reaches 80 degrees just before 10:00 a.m. If Anya wants to go for a two-hour hike and
return before the temperature gets over 80 degrees, then she should start her hike before 8:00 a.m.
2.
a. To evaluate f(2), we determine which value of I corresponds to w=2. Looking at the graph, we see
that I≈7 when w=2. This means that approximately 7000 people were infected two weeks after the epidemic
began.
b. The height of the epidemic occurred when the largest number of people were infected. To find this, we look
on the graph to find the largest value of I, which seems to be approximately 8.5, or 8500people. This seems
to have occurred when w=4, or four weeks after the epidemic began. We can say that the height of the
epidemic corresponds to the evaluation f(4)=8.5.
c. To find a solution to f(w)=4.5, we must find the value of w for which I=4.5, or 4500 people were infected.
We see from the graph that there are actually two values of w at which I=4.5, namely w≈1and w≈10. This
means that 4500 people were infected after the first week when the epidemic was on the rise, and that
after the tenth week, when the epidemic was slowing, 4500 people remained infected.
d. We are looking for all the values of w for which f(w)≥6. Looking at the graph, this seems to happen for all
values of w≥1.5 and w≤8. This means that more than 6000 people were infected starting in the middle of
the second week and lasting until the end of the eighth week, after which time the number of infected
people fell below 6000.
Resources for AI.F.2
 http://www.opusmath.com/common-core-standards/8.f.5-describe-qualitatively-the-functionalrelationship-between-two-quantities
(THIS SHOWS GOOD EXAMPLE PROBLEMS—YOU MUST WORK
OUT THE PROBLEMS FOR YOURSELF)

http://www.charleston.k12.il.us/cms/Teachers/math/PreAlgebra/paunit3/RV3Key.PDF
TEACHER RESOURCE FOR PLANNING LESSON)
(THIS IS A GOOD
Teacher Background: Sample Problems for 8.DSP.3
1. The scatter plot below shows the relationship between the number of airports in a state and the
population of that state according to the 2010 Census. Each dot represents a single state. The number
of airports in each state comes from data
onhttp://www.nationalatlas.gov/atlasftp.html?openChapters=chptrans#chptrans. The data for
population comes from the 2010 census: http://www.census.gov/2010census/data/
a. How would you characterize the relationship between the number of airports in a state and the state's
population? (Select one):
i.
ii.
iii.
iv.
v.
The variables are positively associated; states with higher populations tend to have fewer airports.
The variables are negatively associated; states with higher populations tend to have fewer airports.
The variables are positively associated; states with higher populations tend to have more airports.
The variables are negatively associated; states with higher populations tend to have more airports.
The variables are not associated.
LaToya uses the function y=(1.35×10−6)x+6.1 to model the relationship between the number of airports, y and the
population in a state, x.
b. How many airports does LaToya’s model predict for a state with a population of 30 million people?
c. What does the number 6.1 that appears in LaToya’s function mean in the context of airports vs.
populations? (Select one.)
i.
ii.
iii.
iv.
v.
vi.
vii.
viii.
The average number of airports in a state is 6.1.
The median number of airports in a state is 6.1.
The model predicts a population of 6.1 people in a state with no airports.
The model predicts 6.1 airports in a state with no people.
The model predicts that 6.1 states have no airports.
The model predicts 6.1 more airports, on average, for each additional person in a state.
The model predicts 6.1 fewer airports, on average, for each additional person in a state.
The number 6.1 cannot be interpreted in this context.
d. What does the number 1.35×10−6 that appears in LaToya’s function mean in the context of airports vs.
populations? (Select one.)
i. The average number of airports in a state is 1.35×10−6.
ii. The median number of airports in a state is 1.35×10−6.
iii. The model predicts 1.35×10−6 airports in a state with no people.
iv. The model predicts 1.35×10−6 people in a state with no airports.
v. The model predicts that 1.35×10−6 states have no airports.
vi. The model predicts 1.35×10−6 more airports, on average, for each additional person in a state.
vii. The model predicts 1.35×10−6 fewer airports, on average, for each additional person in a state.
viii.The number 1.35×10−6 cannot be interpreted in this context.
e. Fill in the following newspaper headline based on this relationship:
On average, a state in the contiguous 48 US states has 1 additional airport for every
__________________ additional people.
2. Jerry forgot to plug in his laptop before he went to bed. He wants to take the laptop to his friend's house
with a full battery. The pictures below show screenshots of the battery charge indicator after he plugs in the
computer.
a. When can Jerry expect that his laptop battery is fully charged?
b. At 9:27 AM Jerry makes a quick calculation:
The battery seems to be charging at a rate of 1 percentage point per minute. So the battery should be
fully charged at 10:11 AM.
Explain Jerry's calculation. Is his estimate most likely an under- or over-estimate? How does it compare to
your prediction?
c. Compare the average rate of change of the battery charging function on the first given time interval and on
the last given time interval. What does this tell you about how the battery is charging?
d. How long would it take for the battery to charge if it started out completely empty?
Teacher Background: Sample Problems for 8.DSP.3 Teacher Notes
1. a. (iii)
b. 46.6 airports
c. (iv)
d. (vi)
e. 700 thousand
2.
a. In this situation we are looking at two variables: time, t and battery charge, b. There are several ways we can
choose units. A reasonable choice is "time in minutes since the laptop was plugged in" and "battery charge in
percentage of full, %"
The laptop started charging at 9:11 a.m. and it was initially 41% charged. If we let t be time since the laptop
was plugged in, this information corresponds to the point with coordinates (0,41). Similarly, we can translate
the other screenshots into coordinate points:(16,56), (25,64), (37,74), (44,79), (57,86), (66,91).
We can now make a scatter plot of the data to get an idea how the two variables are related
It looks like there is a linear relationship between the variables.
Using technology such as a graphing calculator we find that b=43.7+0.757t.
We can extend the line of best fit and see at what time it will reach an output value of 100%. This
happens about 75 minutes after the computer was plugged in.
b. At 9:27 a.m. Jerry has two data points: (0,41) and (16,56). So the battery charged 15 percentage points in 16
minutes. That is almost 1 percentage point per minute. At 9:27 a.m. the battery still needs to charge 44 more
percentage points, which will take approximately 44 minutes. So 44 minutes after 9:27 a.m. puts us at 10:11
a.m.
Jerry's estimate will probably be an underestimate for the time needed to fully charge the battery. He is
overestimating the rate at which the battery is charging. Therefore, his calculation comes up with a shorter
amount of time than is actually needed.
We can visualize Jerry's method with a line of slope 1 through the point (16,56).
c. The average rate of change of the battery charging function during the first time interval
is 56−4116−0=1516=0.9375 percentage points per minute. During the last time interval the average rate
of change is 91−8666−57=59≈0.556 percentage points per minute. This means that as the battery gets
charged, the rate at which it charges is going down. So it will take longer to go from 90% to 100% than it
took to get from 40% to 50%.
d. We can use the line of best fit to find an estimate for the time it would take for the battery to charge
from 0 to 100%. If we extend the line to the left, we see that it crosses the horizontal axis at about −60.
So we can estimate that it would have taken 60 minutes for the battery to go from 0% to 41% and we
already estimated that it would take about 75 minutes to go from 41% to 100%. So it would take about
135 minutes to charge the battery all the way from 0 to 100%.
This is probably not a bad estimate. Our average rate of change calculations show that we are probably
underestimating the time it takes to charge the last 20 percentage points. But on the other hand, we
probably overestimated the time it takes for the battery to go from 0 to 40%, so we should be close overall.
Resources for 8.DSP.3
 This link provides several short videos explaining various parts of this standard (including interpolation and
extrapolation):
https://learnzillion.com/lessonsets/455-represent-data-on-a-scatter-plot-fit-functions-to-the-data-andassess-fit

This provides practice problems for various parts of this standard:
http://www.mathworksheetsland.com/8/30intslope/ip.pdf

This provides an opportunity for students to practice with concepts pertaining to this standard:
http://www.shmoop.com/common-core-standards/ccss-8-sp-3.html

This link provides several short videos that address concepts regarding scatter plots:
https://learnzillion.com/lessonsets/696-use-lines-of-best-fit-to- solve-problems

The following tasks provide application concepts pertaining to this standard:
http://illuminations.nctm.org/Activity.aspx?id=4186
https://www.illustrativemathematics.org/illustrations/41
https://www.illustrativemathematics.org/illustrations/1520

https://www.illustrativemathematics.org/illustrations/1558
Process Standards to Emphasize with all standards
PS.1: Make sense of problems and persevere in solving them.
PS.2: Reason abstractly and quantitatively.
PS.3: Construct a viable argument and critique the reasoning of others.
PS.4: Model with mathematics.
PS.5: Use appropriate tools strategically.
PS.6: Attend to precision.
PS.7: Look for and make use of structure.
PS.8: Look for and express regularity in repeated reasoning.
Teacher Background: Sample Problems for AI.DS.2
1.
Weight vs. Height
Twelve students in Mrs. McAllister’s 6th grade class were used as subjects in an experiment to see if there is
correlation between height and weight. The students were measured to see how tall they were in inches.
The height is the independent variable. Next they were weighed in pounds and both the height and weight
were recorded in the table. The weight is the dependent variable.
a. Construct a graph on graph paper and plot the points.
f.
Is there a relationship between height and weight of the students in Mrs. McAllister’s 6th grade class?
If so, what is the relationship?
Teacher Background: Sample Problems for AI.DS.2 Teacher Notes
1. a.
Weight in pounds
Height in inches
b. There is a positive linear correlation between height and weight based on the students in
Mrs. McAllister’s class.
Resources for AI.DS.2
c. Web based practice for students with activities, discussions, and worksheets:
http://www.shodor.org/interactivate/activities/ScatterPlot/
d. Web based practice with Real World Examples. Teachers are able to print everything off:
http://illuminations.nctm.org/Activity.aspx?id=4186
e. Scatter Plot Foldable with Lines of Best Fit and Real World Examples:
http://everybodyisageniusblog.blogspot.com/2012/08/scatter-plots-correlation.html
 This tutorial explains how to understand linear models of bivariate data:
https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-data/cc-8th-patterns-in-data/e/linearmodels-of-bivariate-data
 This provides a link to several short videos with tutorials explaining bivariate data concepts:
https://learnzillion.com/lessonsets/143-model-and-interpret-bivariate-measurement-data
 This interactive activity allows students to practice with bivariate data with scatter plots:
http://www.shmoop.com/probability-statistics/scatter-plots-exercises.html
Process Standards to Emphasize with all standards
PS.1: Make sense of problems and persevere in solving them.
PS.2: Reason abstractly and quantitatively.
PS.3: Construct a viable argument and critique the reasoning of others.
PS.4: Model with mathematics.
PS.5: Use appropriate tools strategically.
PS.6: Attend to precision.
PS.7: Look for and make use of structure.
PS.8: Look for and express regularity in repeated reasoning.
Week 6 Instructional Assessment
ISTEP+ Grade 10
Name______________________________
Erica has been keeping a record of what grades she earned on a test and the number of hours she studied
for that test. Below is the scatterplot that shows the relationship between the grade and the number of
hours she studied for the test. Each dot represents a single test grade. Erica is currently preparing for her
Algebra Final Exam.
Grade vs. Hours Studied
(AI.DS.2) 1. How would you describe the relationship between Erica’s test grades and the number of hours
she studied?
(AI.F.2) 2. Identify which variable is the independent variable and which is the dependent variable. Explain.
__________________________________________________________________________________
_________________________________________________________________________________
(8.DSP.3) 3. Which equation best represents a line that fits the data points?
a. y  17 x  90
b. y  17 x  52
1
x5
2
1
d. y   x  90
2
c. y 
(8.DSP.3) 4. If Erica wants to earn a 90% on the Algebra Final Exam how many hours approximately must she study?
a. 5 hours
b. 6 hours
c. 7 hours
d. 8 hours
(8.DSP.3) 5. Interpret the slope of the line in this situation for Erica and her grades.
_______________________________________________________________________________________
_____________________________________________________________________________________
Solutions to Week 6 Instructional Assessment
(AI.DS.2) 1. (2 points)
A strong positive linear association.
(AI.F.2) 2. (2 points)
The number of hours Erica studies is the independent variable and the grade is the dependent variable. The
grade is dependent on the number of hours Erica studies.
(8.DSP.3) 2. b
(8.DSP.3) 3. d
(8.DSP.3) 4. (2 points)
For every hour that Erica studies she will increase her grade by approximately 17%.