Algebra 1 Concept Lesson – Unit 2
Making the Team – Getting Started
LESSON OVERVIEW
Overview: This is a two-part lesson, with informal exploration in Part 1, followed by more formal mathematical work in Part 2. In Part 1, students
first explore the situation by creating a table that shows at least ten possible pairs of Long and Freestyle Dance scores, then each student in the
group plots her/his ten points (S,L) on a group graph. Students then use the given formula to calculate Total scores for each of their combinations
of scores on the Short and Long Programs and add the total scores to their tables. They identify the points on the graph that represent
combinations that qualify for the team, and those combinations that produce a total score of exactly 60 points. They then look for patterns in their
group graphs and generate initial conclusions about the situation. This is followed by a class discussion in which selected groups present their
results. All of the transparencies with group graphs are superimposed on each other on the overhead to form a single class graph. Groups then
look for patterns in the class graph and use it to evaluate their group’s thinking and make additional conjectures.
Part 2 begins with each small group writing the inequality that describes conditions for “making the team”. Students then decide how to shade
their graph to indicate the region that contains points that “make the team”. They then use their equation or graph to answer a series of questions
about the situation and interpret specific points on the graph in terms of the problem situation. Next, they describe what each of the regions of the
graph represent—the region containing points that are circled, and the region containing points that aren’t circled. They also identify the meaning
of the points that are starred. Finally, they use their graph to answer the “Key Questions”.
TX Standards Addressed: §111.32. Mathematics, Algebra I
(A.7) Linear functions. The student formulates equations and inequalities based on linear functions, uses a variety of methods to
solve them, and analyzes the solutions in terms of the situation.
The student is expected to:
(A) analyze situations involving linear functions and formulate linear equations or inequalities to solve problems;
(B) investigate methods for solving linear equations and inequalities using concrete models, graphs, and the properties of
equality, select a method, and solve the equations and inequalities; and
(C) interpret and determine the reasonableness of solutions to linear equations and inequalities
Mathematical Goals of the Lesson:
 Begin to develop an understanding of inequalities in the coordinate plane and their use in representing mathematical situations and
solving problems.
 Begin to develop an understanding of the relationship between inequalities and their graphs.
 Write inequalities to describe specific conditions in a situation.
 Evaluate algebraic expressions in two variables and interpret the results in terms of a mathematical situation.
 Begin to develop the ability to solve linear inequalities graphically.
 Interpret regions of the graph of linear inequalities in terms of the mathematical situation and identify the solution region for a
particular problem.
 Reason and solve problems using graphs and equations.
 Reason mathematically and use and make connections among a variety of mathematical representations.
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Academic Language Goals of the Lesson:
 Develop academic vocabulary to be used in the descriptions.
 Describe the conditions in a situation algebraically, orally or in writing.
 Explain the process used in solving the task, orally or in writing.
Assumption of Prior Knowledge:
 Solving linear equations in one variable
 Experience graphing inequalities in one
variable
 Experience graphing linear equations
and use them to represent situations
 Identify and interpret key characteristics
of graphs of linear equations, i.e., x- and
y- intercepts
Academic Language:
 Maximum and Minimum
 Inequality
 Intercepts
 Regions, solution regions, and boundary
lines
Materials:
 Task
 Calculators
 One transparency graph for each group,
with axis and scale labeled (blackline
master provided)
 A black overhead marker for each group
 Scores table (blackline master provided)
Connections to the LAUSD Algebra 1, Unit 2, Instructional Guide
Understand and graph linear
equations in one and two variables
using a variety of techniques

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Sketch the region defined by a
linear inequality
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Key:
Suggested teacher questions are shown in bold print.
Questions and strategies that support ELLs are underlined and identified by an asterisk.
Possible student responses are shown in italics
Phase
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SET-UP PHASE: Setting Up the Mathematical Task
INTRODUCING THE TASK
 Ask students whether they have either watched or been involved in competitions that involved multiple performances,
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such as skating, diving, cheerleading tryouts, play tryouts, etc. Ask them to describe how they think the different
performances are used to determine a single final score. Listen for different types of combinations: those involving taking
a simple average, i.e., weighing each component equally, and those involving weighted averages, i.e., some components
are more important than others.
Ask a student to read Part 1 of the task out loud as others follow along.
Ask several students to explain what they are being asked to do in their own words*.
Specifically ask, “What does this formula mean?” (The Freestyle Dance counts less than the Prepared Dance when you
figure the total score; The Freestyle Dance is worth 40% of the total score; the Prepared Dance is worth 60% of the total
score.)
Clarify any confusions students may have, but do not suggest a method for solving the problems. Ask for any words or
terms that the students do not understand. Clarify the meanings so that the task will be accessible to all students*.
To assist ELLs’ participation in the class discussion*:
 Allow time for students to talk first in small groups (pairs) and then have the groups report to the whole class.
 Reinforce appropriate language as students communicate their ideas (e.g., revoice a student’s contribution in
complete, grammatically correct language). Ask students if you have captured what they said.
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EXPLORE PHASE: Supporting Students’ Exploration of the Task
STRUCTURE
Phase
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PRIVATE THINK TIME
 Ask students to work individually for about 10 minutes so that they can make sense of the problem for themselves.
 Each student should create an individual table of values with at least 10 entries. Provide them with a blank table to use.
 Circulate around the classroom and clarify confusions. Be careful NOT to give away too much information or suggest a way to solve
the problem.
SMALL-GROUP WORK
 After about 10 minutes, ask students to work with their partner or in their small groups to construct their group graph on the
transparent graph you provided. Each student should plot the points from his or her table on the graph.
 Students should then calculate the total score for each set of points, and enter this total into their table. They then should each
circle or star the points that show combinations that give qualifying Total scores (60 or greater).
 As students are working, circulate around the room.
o Be persistent in asking questions related to the mathematical ideas, problem-solving strategies, and connections between
representations.
o Be persistent in asking students to explain their thinking and reasoning.
o Be persistent in asking students to explain, in their own words, what other students have said.
o
Be persistent in asking students to use appropriate mathematical language.
What do I do if students have difficulty getting started?
 Ask: “What is a possible Freestyle Dance score?” “Prepared Dance score?” “How can you figure out the Total score that you get
from those two scores?” “Does this Total score qualify for the team?”
What do I do if students finish early?
 Look at students’ graphs and be sure they have plotted a sufficient number of points to (1) represent the full range of outcomes for
the situation and (2) make the boundary line 60=.4X + .6Y visible on the graph. If not, prompt students to find combinations that
minimally qualify for the team (i.e., where T=60).
 Ask a “quiet” group member to explain the work of the group. If s/he can’t explain, challenge the group to make sure that that group
member understands, leave, and return later to ask again.
MONITORING STUDENTS’ RESPONSES
 As you circulate, attend to students’ mathematical thinking and to the strategies and representations used, in order to identify those
responses that will be shared during the Part 1 Share, Discuss, and Analyze Phase. During this phase, groups will (1) present their
group graphs and (2) describe how they generated the table values that appear on the graph. For this task, you will need to:
o Identify a graph to be used to begin the Share Phase. Look for a graph based on tables that were primarily generated
unsystematically, i.e., simply selecting various combinations of Short and Prepared Dance scores.
o Identify graphs based on tables with more systematic generation of combinations, e.g., containing multiple combinations for a
single Freestyle Dance score, using extreme values (e.g., 0 or 100) as scores, etc.
o Identify graphs based on tables in which students solved the equation 60= .4F + .6P to determine Prepared Dance scores that
would produce a qualifying Total score for particular Freestyle Dance scores.
o Identify graphs in which the boundary lines 60= .4F + .6P and/or S=60 are visible.
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Phase
Task Question:
1. Make a table that shows possible performance on the Short and Long Programs. Include at least 10 entries in your table.
2. Use a Black Marker to plot your possible performances (S, L) on your group graph. The x-axis represents the Freestyle
Dance score (F) and the y-axis represents the Prepared Dance score (L).
Possible Solutions
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Look for indictors of students’ understanding:
 that the Freestyle and Prepared Dance
scores are independent of each other, i.e.,
a given score on one can be associated
with multiple scores on the other.
 of a systematic organized way of charting
information.
 of a procedure for plotting points on a
graph
Possible Questions
Ask questions such as:
 How did you generate the
different combinations?

What do the 0 and the 100
mean? (or 100 and 0?)

Explain in your own words
what (___) said (another
student).*
Misconceptions/
Errors
 Not realizing that
 Confusing the xand y-ax
Making a Table:
F
P
100
0
60
60
70
60
60
90
60
70
90
80
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Could someone else
have the same
Freestyle Dance score
but a different
Prepared Dance score
(or vice versa)?

Indicate one of the
points. Ask, “How can
we find the Freestyle
Dance score that
corresponds with this
point? What is the
Prepared Dance score
that corresponds with
the point?”
the Short and
Prepared Dance
scores are
independent of
each other.
Many combinations could be generated.
0
100
60
50
40
80
100
40
50
50
10
40
0
Questions to Address
Misconceptions/Errors
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Phase
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Task Question:
3. A skater’s Freestyle Dance score (F) and Prepared Dance score (P) are combined to give the Total Score (T) using this formula:
.4F + .6P = T
Add a Total Score column to your table. Determine the total score for each of your entries and add these scores to your table.
Possible Solutions
Possible Questions
Look for indictors of students’ understanding:
 of the Total score being a combination of
the Freestyle and Prepared Dance scores
using the indicated formula.
Ask questions such as:
 How did you calculate a
Total Score?
Many combinations can be generated. The
table below shows some key combinations,
including those on the boundary line 60=.4F +
.6P
Making a Table:
F
P
0
100
60
50
40
80
100
40
50
50
10
40
0
100
0
60
60
70
60
60
90
60
70
90
80

Explain in your own
words what ( __) said
(another student).*
Misconceptions/Errors
 Not realizing that the
formula means that the
Total adds together 40%
of the Freestyle Dance
score and 60% of the
Prepared Dance score.
Questions to Address
Misconceptions/Errors
 Can a skater receive
200 points if s/he
dances two perfect
programs?
 What does .4F mean?
 What does .6P mean?
 A skater has a Freestyle
T
 Having difficulty using
60
40
60
56
58
68
76
70
56
62
58
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Dance score of 50 and a
Prepared Dance score
of 70. Explain how to
find the Total score.
the calculator to
compute the total (make
sure you try the
students’ calculators so
that you know how to
perform this calculation
on the model they will
be using).
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 What operation is
indicated here? (.4F)
 Will you multiply first or

add first?
Have you asked your
group mates to explain
how to evaluate this
expression?
6
Phase
Task question:
4. To make the team, the skater’s Total score must be at least 60 points.
a. CIRCLE the points that show combinations of scores that give Total scores that qualify for the team.
b. Put a STAR on those points where the Total score is exactly 60.
5. Examine your graphs. What patterns do you notice? Discuss your answers with your team members. Explain how you
arrived at your responses. What conclusions can you draw?
Possible Solutions
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Look for students who:
 have graphs that clearly show the
boundary and/or regions, i.e., have
sufficient numbers of points in each
region so the pattern in the location of
circled and non-circled points are visible.

clearly articulate the boundary
conditions, i.e., .4F +.6P =60, S=60.

have interesting patterns/observations
for the class to discuss.
Possible Questions
You might ask:
 How did you decide to
circle these points?
 Why aren’t these points
circled?
 Were there any points
for which you had a hard
time deciding whether to
star them or not?
Misconceptions/Errors
 If groups have not
plotted enough
points in the 2
regions or on the
boundary line, they
may not be able to
readily see the
mathematical
connections in the
graph.
Questions to Address
Misconceptions/Errors
Ask groups to find
and plot points:
 where .4F + .6P = 60
 where .4F + .6P  60
 where .4F + .6P  60
 What patterns do you
notice?
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Sharing, Discussing, and Analyzing Part 1 of the Task
Orchestrating the mathematical discussion: a possible Sequence for sharing student work, Key Questions to achieve the goals of
the lesson, and possible Student Responses that demonstrate understanding.
The purpose of this sharing/discussion is to share approaches to the task, but most importantly, to combine the individual group graphs into a
class graph that will contain more points, and thus make patterns in the graph more visible.
Phase
Sequencing of Student Work
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Each group describes their
strategy for completing the table
and then displays their graph.
Collect the graph after it is
displayed to include in the class
graph.
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Order of group presentations:
Graph based on tables:
1. Primarily generated
unsystematically, i.e., simply
selecting various combinations
of Short and Prepared Dance
scores.
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2. With more systematic
generation of combinations,
e.g., containing multiple
combinations for a single
Freestyle Dance score, using
extreme values (e.g., 0 or 100)
as scores, etc.
3. Graphs based on tables in
which students solved the
equation 60= .4F + .6P to
determine Prepared Dance
scores that would produce a
qualifying Total score for
particular Freestyle Dance
scores.
Rationale and
Mathematical Ideas
This discussion will lay the
foundation for the key ideas of
graphing systems of
inequalities:
 Boundary lines
 Regions
 Solution region
The following key mathematical
ideas may also surface that you
can expand upon later, if you
chose:
 Weighted average.
 Independent vs. dependent
variables (this situation
involves two independent
variables, unlike most of the
situations students have
encountered so far.
 Discrete versus continuous
representations.
 Graphs that represent just
the problem situation (0<
P<100) and (0<F<100) vs.
the complete graphs
F, P  IR without limits.
Possible Questions and Student Responses
How did you select the values in your table? How did you
generate the total scores?
 I just kept selecting pairs of numbers. I wanted to have a
pattern in my table so I went up by 10’s. I first tried
minimum and maximum values, e.g., 0 and 100, for the
Short and Prepared Dance scores.
 After I calculated the Total scores for the points I had, I saw
that none of my points would make the team, so I tried
higher Short and Prepared Dance scores to see if the Total
scores would make the team.
 I picked a Freestyle Dance score, and then used the
formula to figure out what Prepared Dance score the person
would have to get to qualify for the team. I set T=60, put in
the value for S, then solved the equation for L.
How did you decide whether to circle a point? How did you
decide if it should be starred?
 I checked to see if the Total was greater than or equal to
60.
What patterns or conclusions did you reach by looking at your
graph?
 I noticed that all the points with stars are on a line.
(Probe—to remind students that the points with stars
indicate where T=60.)
 I noticed that the stars formed a slanty line that went
downhill (Probe – does this line have a positive slope or a
negative slope?).
 I noticed that all the points that are circled are above a
slanty line. (Probe—to get to “above the line where T=60.)
4. Graphs in which the boundary
line 60= .4F + .6P is visible.
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Sequencing of Student
Work
Rationale and
Mathematical Ideas
CLASS GRAPH
This is a crucial phase of the lesson. By
superimposing 4 or 5 of the group graphs,
After the group
the two regions and boundary line should
presentations, collect
be much more prominent, the stray points
the transparencies with
can be discussed, the patterns can be
the group graphs from 4 discussed, and student understanding of
or 5 groups and create a the mathematics in the context of the
“superimposed Class
situation reinforced.
Graph” to display on the
overhead.
Possible Questions and Student Responses
Look at the class graph. How does it compare to
your group graph? What patterns do you see in
the class graph? Do they support or contradict
what you noticed in your group graph? Why?

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Because there are more points, the
boundary lines are more visible.
Now we can see two 2 different regions in
the graph, one where the points are circled
and the other where the points are not
circled.
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Part 2: Be sure to return each group’s graph. They will add to it in this part of the lesson.
Phase
Task question:
What condition has to be true for a skater to make the team? Write the inequality that represents this condition. Explain how
this inequality is related to your graph.
6. On your group graph, shade the region of the graph that represents Total scores that qualify for the team.
Possible Solutions
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Look for students who:
 clearly articulate the condition orally.
(Total score > 60)

can also mathematically represent this
condition as an inequality.
.4F + .6P > 60

are able to use the circled points on their
graphs to help them graph the inequality.

recognize that to graph an inequality in
the coordinate plane requires the
shading of a region.

are wondering what to do when T=60;
i.e., it is an important point; it is in the
solution region.

extend their shading to include the entire
first quadrant or the entire coordinate
plane, vs. those who limited their
shading to the square bounded by the
lines: F=0, F=100, P=0, P=100.
LEARNING RESEARCH AND DEVELOPMENT CENTER
Possible Questions
You might ask:
 What does it mean to
make the team? How
could you write that
using mathematics?

What is the
relationship between
your circled points
and the shaded
region?

What do the points
with the stars
indicate? What do
they tell you?

How did you decide
how far to shade?

Were there any parts
of the graph for which
you had a hard time
deciding whether to
shade them or not?
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Misconceptions/Errors
Questions to Address
Misconceptions/Errors

Not considering
points on the line
T=60 as possible
solutions.

What is the lowest
or minimum Total
score that makes
the team?

Shading beyond
the maximum and
minimum “relevant”
values.

What is the
maximum
possible score on
the x (F) and on
the y (P) axis?

What is the
minimum possible
score on the x (F)
and y (P) axis?

What would it
mean if a point
was out here
(outside of the
relevant domain
and range of the
context)?
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10
Phase
Task question:
7. Use the graph or equation to answer these questions:
a. Suppose a skater scores 50 on the Short Program. What score does s/he need on the Prepared Dance to make the team?
How
did you find your answer?
b. Pick a point in each of the regions and describe what that point represents.
c. What are the x- and y-intercepts? What do they represent?
8. Describe what the shaded and non-shaded regions on your graph represent.
7c
Possible Solutions
Look for students who:
 Use the graph to answer the questions.
 Use the equation to answer the questions.
 Go back to their table to answer the questions.
7a. Minimum score L=66.7 gives T=60.
7b. Shaded region: Total score ≥ 60 – they made
the team.
Non-shaded region: Total score < 60 – they
didn’t make the team.
Possible Questions
You might ask:
(#7a)
 Which points on the graph
represent scores of 50 on the
Short Program?
 If a skater scores 50 on the
7c. x-intercept of the line 60=.4S+.6P is (150,0)
which means a score of 150 on the Freestyle
Dance and 0 on the Prepared Dance - which is
impossible. y-intercept of the line 60=.4F +.6P is
(0,100) which means a Freestyle Dance score of 0
and a Prepared Dance score of 100 to make the
team.
Freestyle Dance, how many
scores can he or she obtain on
the Prepared Dance to make the
team? What is the minimum
score? How do you know?
(#7b)
 What do you know about the
Total score for each point? What
does it tell you about the
dancer’s performance?
(#7c)
8. See 7b above.
 Where are the x and y intercepts
Misconceptions/
Errors
 Not having
two clearly
delineated
regions on the
graph.
 Students may
not
understand
that a region
with no
shading is
also a region.
Questions to
Address
Misconceptions/
Errors
Ask student
(group) to graph
the line .4F + .6P =
60
Ask students to
show you on their
graph where all the
unacceptable Total
scores lie – where
the acceptable
Total scores lie.
Point to a point in
the non-shaded
region and ask
students to explain
why it is not
circled.
on the graph? What Freestyle
and Prepared Dance scores
would each of them represent?
(#8)
See questions for task question 6
and 7b.
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Sharing, Discussing, and Analyzing Part 2 of the Task
Orchestrating the mathematical discussion: a possible Sequence for sharing student work, Key Questions to achieve the goals of
the lesson, and possible Student Responses that demonstrate understanding.
Revisiting the Mathematical Goals of the Lesson:
 Begin to develop an understanding of inequalities in the coordinate plane and their use in representing mathematical situations and solving
problems.
 Begin to develop an understanding of the relationship between inequalities and their graphs.
 Write inequalities to describe specific conditions in a situation.
 Evaluate algebraic expressions in two variables and interpret the results in terms of a mathematical situation.
 Begin to develop the ability to solve linear inequalities graphically.
 Interpret regions of the graph of linear inequalities in terms of the mathematical situation, and identify the solution region for a particular
problem.
 Reason and solve problems using graphs and equations.
 Reason mathematically and use and make connections among a variety of mathematical representations.
LEARNING RESEARCH AND DEVELOPMENT CENTER
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Phase
Sequencing of
Student Work
Rationale and
Mathematical Ideas
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This is a teacher-led
discussion of the
whole task. Students
should be asked to
reprise their work on
the task and to
summarize their
thinking.
Students should explain their
thinking to each question using
their graph, table or inequality to
support their answers.
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Graphs from various
groups should be
shown to the class for
group discussion and
critique of the qualities
of the final graph.
Important aspects of
the graphs:
boundary lines
x-axis
y-axis
60= .4F + .6P
(starred)
2 regions
non-shaded –
.4F + .6P < 60
shaded – .4F + .6P 
60
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A discussion should follow
about whether the graph,
inequality or equation, or
their table was most helpful in
answering each of the
questions.
There should also be an encore
of the discussion of the 2
regions of the graph in the
context of the problem. This
discussion is to ensure that all
students understand the
significance of the regions and
where the solution region lies.
A teacher-led discussion should
address solid boundary lines,
i.e., are the points on the line
.4x + .6y = 60 included in the
solution. The use of solid lines
can then be introduced as a
convention for representing
points on the boundary that are
part of the solution, if this has
not been introduced previously.
You can relate it to solid-circle
(vs. open-circle) endpoints on
graphs of inequalities on the
number line
© 2007 University of Pittsburgh
Possible Questions and Student Responses
This phase is the debriefing of the entire task. The “Key
Questions” are the framing questions for the class discussion.
Q1: What is the minimum Freestyle Dance score a dancer can
receive and make the team? What score is required on the
Prepared Dance to make the team with the minimum Freestyle
Dance score? Explain how you know this.
A skater can score a 0 on the Freestyle Dance but must receive a
100 on the Prepared Dance to make the team.
 I can see this on the graph – there’s a starred point at (0,100).
 I can see this in the equation since a Prepared Dance of 100
will give a total score of 60 even if someone gets 0 on the
Freestyle Dance since 0 + .6(100) = 60.
Ask students where to look on their graph to answer this
question.
Ask students to explain if a dancer can score a zero on the
Prepared Dance and still make the team. Explain.
No, a zero on the Prepared Dance will disqualify a skater.
 There are no starred points on the graph where L = 0.
 You can’t get 60 points if you get 0 on the Prepared Dance
even if you score 100 on the Freestyle Dance since .4(100) =
40.
Ask students where to look on their graph to answer this
question.
Is this question easier to answer using their graph or the
equation?
Relate this question to the student question, “Right now I
have an average grade of 52 in your class. What do I have to
make on my final in order to pass this course?”
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13
Phase
Sequencing of Student
Work
Rationale and
Mathematical Ideas
S
H
A
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Possible Questions and Student Responses
Q2: A skater has a particular Freestyle Dance score.
How can s/he figure out the minimum Prepared Dance
score needed to make the team?
The graph can give you an approximate answer, but solving
the equation .4F +.6P = 60 will give an exact answer.
D
I
S
C
U
S
S
How can you graph the equation .4F + .6P = 60?
I could find two points that work and connect them with a
line.
I could find the x and y intercepts by putting in zero for F and
solving for P (giving the y-intercept), then a zero for P and
solving for F (giving the x-intercept).
A
N
D
I could change this into the y = mx + b form and then I get
P = 100 -
A
N
A
L
Y
Z
E
2
F
3
How can you use the graph to answer the question?
I find my Freestyle Dance score on the x-axis and then go
straight up until I hit the line. I then go over to the y-axis to
find the Prepared Dance score that goes with this passing
point.
How could you use the equation to answer the
question?
I substitute my Freestyle Dance score for S in the equation,
then I solve the equation for L. This will give me the exact
Prepared Dance score that will give me a Total score of 60.
LEARNING RESEARCH AND DEVELOPMENT CENTER
© 2007 University of Pittsburgh
Funded by the James Irvine Foundation
14
CLOSURE
Have students reflect on the mathematics of the lesson; find links to math that they have explored before; think of tasks that might be related to
the big ideas of the lesson.
It is important for students to step back and reflect on the ideas that surfaced and to situate their learning within past experiences and to think
forward to ways that they might build on these ideas in future tasks. This helps them to focus on the interconnectedness of mathematical ideas.
CLOSURE
Pose the following question and allow time for students to discuss it in small groups:
 What would happen to the graph if you changed the Qualifying Total score to 70?
HOMEWORK
1. What is the minimum Prepared Dance score a dancer can receive and still make the team? What score must they receive in
the Freestyle Dance? Explain.
2. Are the minimum scores for making the team the same for both the Freestyle and Prepared Dances? Explain why or why
not.
LEARNING RESEARCH AND DEVELOPMENT CENTER
© 2007 University of Pittsburgh
Funded by the James Irvine Foundation
15
LEARNING RESEARCH AND DEVELOPMENT CENTER
© 2007 University of Pittsburgh
Funded by the James Irvine Foundation
16
“High School Dance Challenge”
SCORES
Freestyle Prepared
LEARNING RESEARCH AND DEVELOPMENT CENTER
© 2007 University of Pittsburgh
“High School Dance Challenge”
SCORES
Freestyle Prepared
Funded by the James Irvine Foundation
17