HCPSS Worthwhile Math Task
Flying T-shirts (Revisited)
Common Core Standard
Solve Systems of Equations.
A.REI.C.7 Solve a simple system consisting of a linear equation and a quadratic equation in two
variables algebraically and graphically.
Create equations that describe numbers or relationships.
A.CED.A.2 Create equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.
Interpret functions that arise in applications in terms of a context.
F.IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative
relationship it describes.
MP1:
MP2:
MP3:
MP4:
MP7:
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Look for and make use of structure.
Common Core Algebra II, Unit 3
The Task
At a Varsity Basketball game, free t-shirts are being shot into the audience from a floor launcher.
Chan wants to make sure his girlfriend catches the t-shirt, so they need to figure out which
bleacher row she should sit in. Chan launches from center court. The t-shirt’s travel path,
according to the manufacturer’s manual, is represented by the equation y(x)  0.05x 2  2.5x ,
where y represents the vertical height in feet in terms of the horizontal distance traveled, x, in
feet. The bleachers begin 32 feet from center court and each bleacher row has a height of 1.5
feet and a width of 2 feet as shown in the diagram*. Use a line to represent the bleachers which
 30 ft from center court as shown
touches the front edge of each row of seat and touches the floor
in the diagram*. The diagram is not necessarily drawn to scale.
Help Chan out. Using this information, determine which row Chan should tell his girlfriend
should sit in.
*See attached student handout containing diagram of the gym layout.
Facilitator Notes
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has
licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0
Unported License.
HCPSS Worthwhile Math Task
Note: This is a task that is included in the HCPSS Common Core Algebra I resources. The
intent of the lesson in Algebra I was to generate the linear equation and then intersect it with the
given quadratic using a graphing calculator. In Algebra II, the solution should be found
symbolically instead of/or in addition to graphically. All subsequent investigations in this task
should be solved symbolically and confirmed using a calculator.
1. Discuss the concept of projectile motion and its relationship to the launching of a tightly
wrapped t-shirt. Ask if anyone has ever seen or, even better, caught a launched t-shirt.
2. Distribute the student resource to students and allow students time to read over task.
3. Be sure to include any potential opportunities for class discussion, key ideas to highlight,
etc.
4. Provide students with a variety of materials, including but not necessarily limited to
graph paper and graphing calculators.
5. Group students into pairs or small groups to solve the problem. Have groups record their
strategy. Indicate that more than one method of solution must be used to verify their
solution. (Look for evidence of MP1 and MP4.)
6. As groups work, circulate to monitor what strategies are being used to solve the problem.
Steer groups toward a symbolic manipulation to solve the problem if they are not using
that on their own. Once groups have had an opportunity to solve the task, have groups
compare solutions and strategies (either through a jigsaw, a gallery walk, or group
presentations). (Look for evidence of MP3.)
7. Make sure to highlight key methods for solving systems of equations, particularly using
symbolic manipulation, in addition to graphing and using intersection and/or using tables
of values. (Look for evidence of MP4.)
Follow-Up Questions
1. What else should we consider when we judge whether or not Chan’s girlfriend will get
the t-shirt? (Look for evidence of MP2.)
2. If the support structure for the ceiling of the gym is 40 ft. above the floor, is there any
danger that the t-shirt will strike part of the support structure? (Look for evidence of
MP2.)
3. How high does the shirt fly? If it struck part of the ceiling support structure or a lower
hanging speaker at its highest point and fell straight down, would it land in the bleachers
or on the floor? Describe where it would land on the bleachers or floor. (Look for
evidence of MP2 and MP4.)
4. If the bleachers were folded up, how far from center court would the t-shirt hit the floor?
(Look for evidence of MP2.)
Extension
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has
licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0
Unported License.
HCPSS Worthwhile Math Task

1. An equivalent form of the t-shirt’s travel path y(x)  0.05x 2  2.5x is
y(x)  0.05(x  25) 2  31.25 . What form of the equation is the new form?
Confirm they are equivalent. (Look for evidence of MP4.)
2
2. What do you notice about the coefficient
 in the equation y(x)  0.05(x  25)  31.25
and key points about the t-shirts path? (Look for evidence of MP7.)
3. How do you think the equation y(x)  0.05(x  25) 2  31.25 would change if the t-shirt
reached a maximum height of 40 feet? Justify your reasoning. (Look for evidence of

MP7.)

Solutions
Task Solutions:
Impose a coordinate graph on the diagram with center court at the origin. The given equation is
the parabola in the diagram (drawn only partially).
1.5 3
 . Using the given
The equation of the line that defines the bleachers is a line with slope
2
4
point (30, 0) where the bleachers begin rising with that slope, the equation of the line is
y = ¾ x – 22.5.
The intersection of the linear equation and the given parabolic equation will give the both the
horizontal distance (x) and vertical height (y) where the T-Shirt arrives at the stands.
The point of intersection can be found symbolically using substitution and the quadratic formula
or factoring as follows:
.75x- 22.5 = -.05x2 + 2.5x
75x – 2250 = -5x2 + 250x
5x2 - 175x – 2250 = 0


Using quadratic formula
or
Using factoring
5(x2-35x-450) = 0
5 (x – 45) (x +10) = 0
x = 45, -10 (only x = 45 is reasonable)
The point of intersection is (45,11.25), which means that the t-shirt has travelled 45 ft.
horizontally when it strikes the line representing the bleachers at a height of 11.25 ft..
This places the shirt in the 8th row by this logic:
This point would be 45-30=15 ft into the bleachers horizontally which would place in the 8th
row. Each row is 2ft wide so 15 ft would be in the middle of the 8th row, which ends at 16 ft.
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has
licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0
Unported License.
HCPSS Worthwhile Math Task
Or the point would be 11.25 ft high which would place it vertically in the 8th row. Each row is
1.5 ft. high so 11.25 ft would be in the middle of the 8th riser which ends at 12 ft..
Confirming solutions:
 Graphing both equations on the calculator and using an appropriate window will allow
the use of the intersect function to locate the solution to the system
 Using the equations in solution #1, a table of values, generated either by hand or on a
graphing calculator, will yield that the x value 45 will give the same y value 11.25 for
both equations. This amounts to trial and error in finding the intersection.
Follow Up Questions Solutions:
1. A variety of answers are possible here. The question is designed to get discussion going
for the subsequent questions. One possible answer might be that depending on the height
of Eric’s girlfriend or the height of the person in front of her, she might need to sit a row
closer. Also a discussion of the lateral positioning of might surface. This will lead to a
discussion that these equations are have only two dimensions in their description of the
flight, horizontal and vertical, not the third lateral dimension.
2. No. Using the maximum finding capabilities of a graphing calculator or a table, the
maximum height indicated by the flight equation is 31.25 ft.
3. The t-shirt goes 31.25 ft. high as in #2. This is asked to ensure the answer is quantified.
This height is reached at x = 25 ft., so the shirt still be over the floor as it falls vertically
down. There are a variety of ways to describe its position including, 25 ft from center
court or 7 ft from the bleachers.
4. 50 ft. This is the x-intercept of the flight equation. It can be found by setting the equation
equal to zero and solving the quadratic equation by factoring or using the quadratic
formula. This will result not just in x = 50 at a zero, but also x = 0. This should lead to a
discussion of the launch height of zero. These answers can be confirmed by using the
zero finding capabilities of a graphing calculator.
Extension Solutions:
1. The new form is the completed square form. To demonstrate that the equations are
equivalent symbolically:
 The square can be completed on the first equation.
 The new equation can be expanded in to standard form.
To confirm these answers on a graphing calculator:
 The equations may be graphed simultaneously with the result that only one graph
appears.
 A comparison of the table of values will show identical values.
2. Observations may vary, but encourage discourse to highlight these three:
 The coefficient of the x2 is the same.
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has
licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0
Unported License.
HCPSS Worthwhile Math Task


The number subtracted from x in the parenthesis is the horizontal location of the
maximum.
The number added at the end is the vertical maximum height.
3. One possible answer: y = -.05(x – 25)2 +40
This makes the flight a little unrealistic, as the initial height would be 8.5 ft. This is
acceptable as we are reasoning abstractly at this stage.
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has
licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0
Unported License.