Catapult Trajectories: Don`t Let Parabolas Throw You

Catapult Trajectories: Don’t Let Parabolas Throw You
TEKS
Aa (2) Symbolic reasoning plays a critical role in algebra; symbols provide powerful ways to
represent mathematical situations and to express generalizations. Students use symbols in a
variety of ways to study relationships among quantities.
Aa (5) Students use a variety of representations (concrete, pictorial, numerical, symbolic,
graphical, and verbal), tools, and technology (including, but not limited to, calculators with
graphing capabilities, data collection devices, and computers) to model mathematical situations
to solve meaningful problems.
Aa (6) Underlying mathematical processes. Many processes underlie all content areas in
mathematics. As they do mathematics, students continually use problem-solving, language and
communication, and reasoning (justification and proof) to make connections within and outside
mathematics. Students also use multiple representations, technology, applications and modeling,
Ab4 (B) Write the equation of a parabola using given attributes, including vertex, focus,
directrix, axis of symmetry, and direction of opening;
Ab8 (C) Predict and make decisions and critical judgments from a given set of data using linear,
quadratic, and exponential models.
Duration: Approximately 3 days
Materials and Resources
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iPads/computers – one per group of students
3-D printed catapult
Two types of projectiles (1 set per group)
Graph paper (1 inch grid)
Colored markers
Tape
Worksheets (See attached)
1 yard or meter of craft paper
Engage
The Engage portion of the lesson is designed to create student interest in the concepts addressed
and make connections to past and present learning. See the first 2 minutes of the YouTube video
“Dan Meyer on Real World Math” at https://www.youtube.com/watch?v=jRMVjHjYB6w
Show about 2 minutes of the YouTube video to the class:
“Trebuchet Siege Artillery – Battle Castle with Dan Snow”
https://www.youtube.com/watch?v=pR26RMI9T8c
or ask questions like these:
1) What is a trebuchet? What is a catapult? Describe them.
2) Do you know any real life examples of catapults? What can they be used for?
1. (Catapults are any device that throws an object, although it commonly refers to the
medieval siege weapon. - Trebuchets are a TYPE of catapult, using gravity (with a
counterweight) or traction (men pulling down), to propel the arm and often employing a
sling at the end of the arm for greater distance)
2. Catapults are used for dropping bombs from boats on submarines, to propel aircrafts into
the air from aircraft carriers, and for fun. Kids like to hurl water balloons at each other
using catapults
Explore
The Explore portion of the lesson provides the student with an opportunity to be actively
involved in the exploration of the mathematical concepts addressed. This part of the lesson
is designed for groups of three or four students. For the data collection portion, if there
are 4 students in the group, they will identify 3 observers and one recorder. If there are 3
students, one will both observe and record.
Provide instruction, as students need guidance and assistance. Have them create a 1 yard or 1
meter number line on the floor. Have them use a sheet of large graph paper (1” grid). Tape it to
the edge of a table or two desks with the bottom edge as close to the floor as possible. Once in
this position move the table or desks to align the number line on the floor with the grid lines on
the paper taped to the table or desks. Align the catapult at the zero mark on the number line.
Practice launching the first projectile and having one spotter finding the point where it lands and
another spotter looking for the apex of flight on the craft paper. Once the teams become
successful, have students start collecting data using Worksheet #1. Repeat the process for the
second projectile. Be sure to mark the points on the number line and identify the projectile and
launch order. They should have 3 trials for each projectile so they need to be sure to label their
data accurately.
Worksheet #1 Catapult Lab Investigation
Name ____________________ Date _____________________
Purpose
1) To investigate characteristics of the function resulting from height vs horizontal distance of a
projectile, and
2) To find the equation of the curve of best fit formed by a projectile from an algebraic approach.
Materials (for each group)
1 3-D printed catapult
1 small projectile
1 large projectile
tape
colored markers
1 sheet of large graph paper (1 inch grid)
Pre Lab Questions
Draw a sketch of what you think will be the path of the projectile from the catapult to the ground.
Setup NOTE: Each group should set up the lab station in the same manner. Any deviation should
be discouraged if class averages will be calculated. All catapult should be taped the same
distance from the edge of the number line. The vertical scale is height in inches.
Figure 1
Catapult
Gradations – Try marking every two inches
In Figure 1 the dots represent the students’ observations as the projectile crosses the vertical
lines. There are three students who each observe the location of the projectile as it passes each
vertical lines on the graph paper. The dotted line represents the actual path of the projectile (only
three points of which are recorded).
NOTE: Each group should set up the lab station in the same manner so class averages can be
calculated. All catapults should be taped the same distance from the edge of the number line.
The vertical scale is height in inches.
Procedure
1.
2.
3.
4.
5.
6.
7.
8.
Orient the paper vertically.
Make the origin the lower left corner.
Draw three vertical lines on the grid at 3, 6, and 9 inches, each in a different color.
Hang and tape graph paper from the desk edge.
Tape the catapult so that the tip is 4 inches from the edge of the number line.
Put catapult in front of graph as shown.
Place the large projectile in the catapult.
Release the projectile from catapult and observe the projectile path. Each of the other
three students must mark the spot where the projectile crosses a given colored line.
9. Record the heights in data table.
10. Repeat steps 8 and 9 twice more for a total of three trials.
11. Repeat steps 7 to 10 using small projectile.
Projectile 1
Launch
Trial 1
Trial 2
Trial 3
Group Average
at x= 0
0
0
0
0
at x = d1
at x = d2
at x = d3
Maximum
d1=
d1=
d1=
Projectile 2
Launch
Trial 1
Trial 2
Trial 3
Group Average
at x= 0
0
0
0
0
at x = d1
at x = d2
at x = d3
Analysis:
In this situation identify the independent variable. _________________________
What is the dependent variable? ___________________________
1) Enter the four data points for the forward distance in one spreadsheet column (or list) in the
calculator and the corresponding four data points for the group average data for vertical distance
in a second column or list, using the small projectile data.
2) Enter the corresponding four data points for the class average data for vertical distance in a
third column or list.
3) Graph the vertical distance versus the forward distance as a scatter plot. Determine the scale,
label the axes, and sketch the graph. Draw a smooth curve that goes through the points as much
as possible.
4) List the types of functions you have studied previously (might include linear, exponential, or
quadratic). Write the characteristics of your graph that are the same as each function you listed.
Give the characteristics that are different.
5) Which functions that you have studied previously have a highest point to the graph and/or a
lowest point to the graph?
6) Does your graph have a highest or lowest point? If so, which does it have?
7) Trace the points on your scatter plot and estimate the coordinates of any high or low points.
8) Use a dotted line to draw any lines of symmetry you can find on your graph. Give the equation
of the line(s) of symmetry.
9) From the smooth curve you drew, estimate the x-intercepts of the graph.
10) Is there a possible relationship between the x-values of the intercepts and the high/low point
on your graph? Explain.
11) Find the quadratic equation of best fit using quadratic regression (QUADREG) on the
graphing calculator. Confirm your curve with the data points. Record your equation in the
appropriate cell below (round to the nearest tenth).
4) Repeat this procedure with L1 and L3.
5) Repeat this entire process for the large projectile and record the curve equation.
Projectile Size
Averaged Data
Small
Group
Large
Group
Fitted Curve Equation
Explain
The Explain portion of the lesson is directed by the teacher to allow the students to
formalize their understanding of the TEKS addressed in the lesson. Use the Facilitation
Questions to prompt student groups to share their responses.
The teacher will introduce the term parabola (a shape defined by a quadratic equation), describe
the characteristics of a parabola, and the general equation yielding a parabolic graph (y = ax2 + bx
– c). The teacher will define terms, such as maximum and minimum, which students will use
later. The teacher will also distribute the Introduction to Parabolas Worksheet (WS #2). This
worksheet will apply prior knowledge of graphing on the xy-plane, using ordered pair notation,
to parabolas. The teacher will introduce the vocabulary keywords and concepts by discussing
characteristics of quadratics key terms:
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Parabola: shape defined by a quadratic equation
Quadratic equation: equation that can be written in the standard form ax2 + bx + c = 0,
where a, b, and c are real numbers and a does not equal zero.
Leading coefficient: In a polynomial, the coefficient of the term with the highest degree
is called the leading coefficient.
Coordinates: pairs of numbers, which specify the position or location of a point or of an
object.
Vertex: the peak in the curve. The peak will be pointing either downwards or upwards
depending on the sign of the x2 term.
Maximum: The maximum value of a quadratic function f(x) = ax2 + bx + c where a < 0, is
the y- coordinate of the vertex.
Minimum - The minimum value of a quadratic function f(x) = ax2 + bx + c where a > 0, is
the y- coordinate of the vertex.
Roots: solutions of an equation (also x-values of the x-intercepts)
x-intercept: an x-intercept of a curve is a point at which the line crosses the x-axis.
The teacher will explain polynomial equation progression from linear to quadratic, comparing
and contrasting shape of the graph, degree of x, number of terms, slopes, etc. In addition, the
students will see that while two points can be used to find the equation of a line, three points are
needed for a parabola. The standard form will be used for the basic lessons. The vertex form
could be as an extension. The teacher will ask the students to give an example of an object that is
roughly parabolic and opens up. (Does it have a minimum or maximum value? Similarly, does a
parabolic-shaped object opening down have a minimum or maximum?)
Facilitation Questions – Explain Phase
The teacher will link the behavior of parabolas to a trajectory caused by catapulting an object.
The teacher should ask questions to tie concepts together:
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Why does a projected object take the path of a parabola?
Which upward or downward path might it have – under what conditions?
Student pairs will discuss how key concepts of vertex and x- intercepts might have practical
meaning in a trajectory situation.
Extension
In this portion of the lesson, students will learn more about quadratic functions and practice
skills related to the concepts they learned in the Explore and Explain sections.
Place students into groups of four. Pass out iPads, or computers to groups. Allow students to use
smart phones, or their own laptops to explore internet sites about real world parabolas. Some
suggested sites could be http://www.intmath.com/blog/mathematics/is-the-gateway-arch-aparabola-4306 , http://mysite.du.edu/~jcalvert/math/catenary.htm ,
http://mathworld.wolfram.com/Parabola.html. Have each student write down 2 things they
learned about parabolas and have each student draw/copy/print at least 1 image of what that they
have seen around them (no duplications from the shared video) of parabolas. Have students share
in groups and then with the whole class what they have found out about parabolas.
Facilitation Questions
 Do all parabolas look like McDonald arches? Why or why not?
 Draw in the air with your finger two types of parabolas (think about a rollercoaster)
Depending on what students say be aware that a flattened catenary is common in construction.
The reason is that for engineering purposes and load distribution, a flattened catenary provides
greater load distribution with vectors going more directly into the substrate avoiding deviation
from 0 degrees.
 From the examples found in your group, show 2 types of examples of parabolas.
 Is the vertex of a parabola always at the highest point? Why or why not?
 Can the vertex be at the lowest point? Show an example and explain.
 Will this parabola open upward or downward: y = 3x2 + 12x – 2? How can you tell?
 Will this parabola open upward or downward: y = -2x2 + 4x – 4? How are you sure?
To connect the learning to real life, the teacher will play the video for the class and or post it to
the class website:
YouTube video of “Algebra 2: Parabolas in the Real World” @
https://www.youtube.com/watch?v=fV9YuF__fM4
and/or
“Parabolas in the Real World” @ https://www.youtube.com/watch?v=lbMir1UAO4I
and/or “Parabola’s in the real world” @ https://www.youtube.com/watch?v=Djnwlj6OG9k
Students will recognize catapult paths as tracing downward-opening quadratic equation graphs.
They will be able to identify parameters of quadratic equations (i.e., a, b, and c) and special parts
of the graphs (e.g., x- and y-intercepts, maximum).
How could catapults apply in real life? Why would accuracy of the projected path be important?
The teacher will link the path of a catapult to parameter changes in quadratic functions.
Extension/ Homework. As a homework assignment, each student can describe a situation
demonstrating a parabolic motion, give specific dimensions for the situation, graph a parabola
that fits the situation, and determine the quadratic function for the situation.
Additional Extension. Use Worksheet #2 and have students use the calculator to explore changes
in parameters. Give students a set of quadratic functions that have different values for the
parameters as ones they could use. They can also use textbook problems in this portion. The
most important part to do is the exploration and connections.
Worksheet #2 Name ____________________ Date _____________________
Identify the marked coordinates of the following parabola in ordered pair notation.
1)
point
A
B
C
D
E
F
x
y
x
y
G
2.
point
A
B
C
D
3). If a parabola opens down (like a lampshade), does it have a maximum or minimum?
How do you know this?
4). If a parabola opens up (like a tulip), does it have a maximum or minimum? Write
how you can you prove this to another student.
Answers for Worksheet
#2
1.
point
A
B
C
D
E
F
G
x
-3
-2
-1
0
1
2
3
y
9
4
1
0
1
4
9
2.
point
A
B
C
D
x
-3
-1
0
2
y
-5
3
4
0
3). If a parabola opens down (like a lampshade), does it have a maximum or minimum?
4). If a parabola opens up (like a tulip), does it have a maximum or minimum?
Evaluation
Worksheet #3 Summative Assessment
Name ____________________ Date _____________________
1. A jump rope is held by Susan and Juanita. The lowest point closer to Juanita. How
could this be?
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
A. Make a sketch labeling the position of the girls, the shape of the jump rope and
indicate the lowest point.
Summative Assessment
B. Where would the lowest point be if the students were of equal heights? Why?
2. Which of the following is a possible equation
for the parabola to the right?
a) y = 2x
b) y = −x
c) y = 1+x
e) y = 2x2 – 3
f) y = -3x2 + 1
3. Evaluate f(x) = -x2 + 8x + 2 when x = 3