Project Title: Quadratic Catapult

Time Frame: 3-5 days

FULL STANDARDS AUDIT

TEKS Covered:

111.40.c.1.b,c,d,g: The student is expected to: use a problem-solving model that incorporates analyzing given information,

formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and

the reasonableness of the solution; select tools, including real objects, manipulatives, paper and pencil, and technology as

appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems;

communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols,

diagrams, graphs, and language as appropriate; display, explain, or justify mathematical ideas and arguments using precise

mathematical language in written or oral communication.

111.40.c.4.b,d,f: The student is expected to: write the equation of a parabola using given attributes, including vertex, focus,

directrix, axis of symmetry, and direction of opening; transform a quadratic function f(x) = ax2 + bx + c to the form f(x) = a(x -

h)2 + k to identify the different attributes of f(x); solve quadratic and square root equations.

111.40.c.8.b,c: The student is expected to: use regression methods available through technology to write a linear function, a

quadratic function, and an exponential function from a given set of data; and predict and make decisions and critical

judgments from a given set of data using linear, quadratic, and exponential models.

111.41.c.7.a: The student uses the process skills in applying similarity to solve problems. The student is expected to: apply the definition of similarity in terms of a dilation to identify similar figures and their proportional sides and the congruent corresponding angles.

111.43.c.5.a,c: Mathematical modeling in science and engineering. The student applies mathematical processes with algebraic techniques to study patterns and analyze data as it applies to science. The student is expected to: use proportionality and inverse variation to describe physical laws such as Hook's Law; use quadratic functions to model motion.

Common Core:

CCSS.Math.Content.HSA.REI.B.4.a

Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)2 =

q that has the same solutions. Derive the quadratic formula from this form.

CCSS.Math.Content.HSG.SRT.A.2

Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar;

explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs

of angles and the proportionality of all corresponding pairs of sides.

CCSS.Math.Content.HSG.MG.A.3

Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints

or minimize cost; working with typographic grid systems based on ratios).

CCSS.Math.Content.HSA.CED.A.2

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate

axes with labels and scales.

CCSS.Math.Content.HSA.CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. CCSS.Math.Content.HSA.REI.C.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. CCSS.Math.Content.HSS.ID.B.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. CCSS.Math.Content.HSS.ID.B.6.a Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. CCSS.Math.Content.HSS.ID.B.6.b Informally assess the fit of a function by plotting and analyzing residuals. CCSS.Math.Content.HSS.ID.B.6.c Fit a linear function for a scatter plot that suggests a linear association CCSS.Math.Content.HSS.ID.C.7

Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

ISTE Standards:

1.a,c: Students demonstrate creative thinking, construct knowledge, and develop innovative products and processes

using technology by applying existing knowledge to generate new ideas, products, or process; and use models and

simulations to explore complex systems and issues.

2.a,b,d: Students use digital media and environments to communicate and work collaboratively, including a distance, to

support individual learning and contributing to the learning of others by: interacting, collaborating, and publishing with

peers, experts, or others employing a variety of digital environments and media; Communicating information and ideas

effectively to multiple audiences using a variety of media and formats; Contributing to project teams to produce original

works or solve problems.

3.b,c,d: Students apply digital tools to gather, evaluate, and use information by: locating, organizing, synthesizing, and

ethically using information from a variety of sources and media; Evaluating and selecting information sources and digital

tools based on appropriateness to specific tasks; Processing data and reporting results.

4.a,b,c: Students use critical thinking skills to plan and conduct research, manage projects, solve problems, and make

informed decisions using appropriate digital tools and resources by: Identifying and defining authentic problems and

significant questions for investigation; Planning and managing activities to develop a solution or complete a project;

Collecting and analyzing data to identify solutions and/or make informed decisions.

College and Career Readiness Standards:

I.C.1. Students shall use estimation to check for errors and reasonableness of solutions

II.B,D. Students shall use algebraic field properties to combine and transform expressions; interpret multiple

representations of equations and relations; translate among multiple representations of equations and relationships.

IV.A,B,C. Students shall select the appropriate type of unit for the attribute being measured; Convert between

measurement systems, determine indirect measurements of figures using scale drawing and similar figures.

VII.B,C. Students shall understand, analyze, and algebraically construct the features of a function and new functions;

Apply function models and develop a function to model a situation.

VIII. A,B,C. Students shall analyze information, formulate a plan, determine a solution, justify the solution, and evaluate

their problem solving process; develop convincing arguments and use various types of reasoning to do so; formulate a

solution to a real world situation based on the solution to a mathematical problem, use a function to model a real world

situation, and evaluate their problem solving process.

IX.ABC. Students shall use mathematical terminology to represent given and unknown information; use mathematical

language to represent the concepts in a problem, and use mathematics as a language for reasoning, problem solving,

making connections, and generalizations; model mathematical ideas using multiple representations and summarize their

interpretations of information provided orally, visually, or in written form. Communicate, explain, display, justify their

ideas.

X.AB. Students shall connect mathematics to the study of other disciplines; use multiple representations to demonstrate

links between mathematical and real world situations, understand and use appropriate mathematical models in the

natural and physical sciences and know and understand the use of mathematics in a variety of careers and professions.

Procedures

1) It is assumed by the project that students have been introduced to the following topics: Vertex Form of a

Quadratic Equation, Completing the Square method of solving Quadratic Equations. It is recommended that this

project is started on a Wednesday so that the preceding Monday and Tuesday may be used to introduce the

students to the concepts if they have not already at the teacher’s discretion.

2) What the “Weapons Testing” sheet lacks in authenticity, it makes up for in practice of skills needed for students

to succeed in this project. The problems are specifically designed to have mostly integer solutions for the

convenience of both the teacher and the students. It is recommended to allow the use of a calculator to

accelerate this drill.

3) The ”Trouble-Shooting” sheet is to be used to gauge student comfort and mastery levels with completing the

square and vertex form of quadratic equations. If students struggle on this more abstract level of these topics, it

is recommended to spend the rest of the period or the next day reviewing the concepts further before diving

into the project.

4) The Project is designed to suggest Catapults to students as their war-machine, however some creative students

may attempt to create Ballistae or Trebuchets. It is the teacher’s discretion to allow these machines to be built

but they do, admittedly, fire in a quadratic manner which fulfills the project’s objectives. The teacher is

recommended to caution students that Trebuchets are difficult to both make and calibrate as they contain TWO

independent variables that require calibration (counterweight mass, angle of hook).

5) Students are to select a castle or town to siege from the list provided on “Quadratic War-Machines” as these

locations have been checked to ensure that wall-heights can be found without too much difficulty from online

sources.

6) It is recommended that students build their own war-machine before constructing wall or distance models as

they can scale their model to a real war-machine and use similarity laws from there to construct other models.

7) This project is designed to put the creativity and problem solving capabilities of students to the test. As such, it is

recommended that the teacher only check in on groups that seem to be struggling and otherwise act as board

for students to bounce ideas off of as reasonability of assumptions, assumptions made, etc are all a part of the

project. Encourage students that often time in the sciences (and humanities, and life in general) we have many

smoking craters of what was once a workshop before we got a working invention and that failure is all a part of

the design process.

8) Completed Copies of “Weapons Testing” and “Trouble-Shooting” may be found before the Authorship and

Works Referenced pages.

Throughout the history of weaponry, throwing things has always been very popular. Throwing things, of

course, led to launching things and projectiles which have achieved flight via either method achieved motion

which may be modeled by quadratic equations. In order to prepare us for our own catapult testing, we will

look at the following weapons and their hypothetical associated quadratic equations. We will determine if the

weapons can clear a wall 50 meters away and 6 meters high. As Reference:

Weapon Model Vertex (How high?)

Roots (How far?)

Clears Wall?

Ballista

𝑓(𝑥) = −.01𝑥2 + .6𝑥

( , ) At ______ meters away, the projectile is ______ meters high.

( , ) ( , )

The projectile will land _______ meters from the weapon.

f( ) =

Yes No

Bombard

𝑓(𝑥) = −1

200(𝑥 − 40)2 + 8

( , ) At ______ meters away, the projectile is ______ meters high.

( , ) ( , )

The projectile will land _______ meters from the weapon.

f( ) =

Yes No

Culverin

𝑓(𝑥) = −1

405𝑥2 +

2

11 𝑥

( , ) At ______ meters away, the projectile is ______ meters high.

( , ) ( , )

The projectile will land _______ meters from the weapon.

f( ) =

Yes No

Mangonel

𝑓(𝑥) = −1

90(𝑥 − 30)2 + 10

( , ) At ______ meters away, the projectile is ______ meters high.

( , ) ( , )

The projectile will land _______ meters from the weapon.

f( ) =

Yes No

Mortar

𝑓(𝑥) = −1

30𝑥2 + 2𝑥

( , ) At ______ meters away, the projectile is ______ meters high.

( , ) ( , )

The projectile will land _______ meters from the weapon.

f( ) =

Yes No

Trebuchet

𝑓(𝑥) = −1

13(𝑥 − 26)2 + 52

( , ) At ______ meters away, the projectile is ______ meters high.

( , ) ( , )

The projectile will land _______ meters from the weapon.

f( ) =

Yes No

Vertex Form f(x) = a ( x – h )2 + k

Standard Form f(x) = ax2 + bx + c

Deciding that they’d rather work with the vertex form of an equation, students named Anthony, Brittany, and

Catherine convert a quadratic equation given in standard form to vertex form by completing the square and

then stated the vertex equation. Convert the equation yourself first and determine which, if any, of the

students performed the conversion correctly. If a student’s solution is incorrect, then identify where the first

error in his or her work occurred.

Given: 𝒇(𝒙) = 𝟔𝒙𝟐 − 𝟖𝟒𝒙 + 𝟐𝟗𝟕

Your Work:

Anthony’s Work Brittany’s Work Catherine’s Work

𝒇(𝒙) = 𝟔𝒙𝟐 − 𝟖𝟒𝒙 + 𝟐𝟗𝟕

𝒇(𝒙) = 𝟔(𝒙𝟐 − 𝟏𝟒𝒙) + 𝟐𝟗𝟕 𝒇(𝒙) = 𝟔(𝒙𝟐 − 𝟏𝟒𝒙 + 𝟒𝟗) + 𝟐𝟗𝟕 − 𝟒𝟗

𝒇(𝒙) = 𝟔(𝒙 − 𝟕)𝟐 + 𝟐𝟒𝟖

Vertex: (7 , 248)

𝒇(𝒙) = 𝟔𝒙𝟐 − 𝟖𝟒𝒙 + 𝟐𝟗𝟕

𝒇(𝒙) = 𝟔(𝒙𝟐 − 𝟏𝟒𝒙) + 𝟐𝟗𝟕

𝒇(𝒙) = 𝟔(𝒙𝟐 − 𝟏𝟒𝒙 + 𝟒𝟗) + 𝟐𝟗𝟕 − 𝟐𝟗𝟒

𝒇(𝒙) = 𝟔(𝒙 − 𝟕)𝟐 + 𝟑

Vertex: (3 , 7)

𝒇(𝒙) = 𝟔𝒙𝟐 − 𝟖𝟒𝒙 + 𝟐𝟗𝟕

𝒇(𝒙) = 𝟔(𝒙𝟐 − 𝟏𝟒𝒙) + 𝟐𝟗𝟕 𝒇(𝒙) = 𝟔(𝒙𝟐 − 𝟏𝟒𝒙 + 𝟗𝟖) + 𝟐𝟗𝟕 − 𝟓𝟖𝟖

𝒇(𝒙) = 𝟔(𝒙 − 𝟕)𝟐 − 𝟐𝟗𝟏 Vertex: (7 , -291)

Anthony was (right / wrong) Because…

Brittany was (right / wrong) Because…

Catherine was (right / wrong) Because…

For this project, you and your colleagues will be assuming the role of a siege engineer in the 12th century or

earlier, prior to the use of gunpowder as a weapon of war on a large scale. Below, you will find a list of famous

castles and walled cities from around the world and you will select one to lay siege.

A Town Protected by Hadrian’s Wall

The Walled City of Handan

The Theodesian Walls of Constantinople

The Walled City of Tulum

The Aurelian Walls of Rome

The Castle of Edo

The Fortress of Agra

Framlingham Castle

Once a selection has been made your group will need to do the following:

Research your target: How tall are its walls? Where is it located?

Build a Siege Engine: Some materials will be provided by the teacher, groups are welcome to bring

additional materials from home. Students will be given three trials to clear the wall.

Determine how to scale a real siege engine and walls to the model which you are creating.

o Useful note: Life Sized catapults have standard dimensions of (13’4” x 5’ x 10’) (L x W x H)

Determine a reasonable distance to place your catapult from the city walls (The occupants are likely to

defend themselves with bows and arrows and you, the engineer, would like to live through the siege).

Completion of the above criteria will earn group members a 90% for their grade, showing a reasonable

mastery of the subject at hand.

For an additional 10%, groups may complete a weapons calibration sheet found in this packet to help “fine

tune” their catapult.

For an additional 10% groups may write a letter to the monarch or General of the Army for whom they are

building this weapon. Groups must bear in mind that the monarch or general are not mathematicians or

engineers, hence they must explain their model and defend their assumptions to these individuals in clear,

unambiguous language to gain their client’s support and funding.

Target:

Wall Height (in meters):

Location and Current Occupant:

Defensible Range (How far away would a real catapult need to be from the walls?):

Determine the scale of Defensible Range to Wall Height

Remember that your model catapult will need to be to scale!

Additional Information:

Before any machine is built, it must go through a design phase. Below, sketch your catapult from at least two different angles including both measurements and desired supplies. Have your design approved by your teacher before beginning any construction. Remember not to go overboard with the parts needed, or as Antoine de Saint Exupéry said, “perfection is achieved not when there is nothing more to add, but when there this nothing less to take away.”

Sketch from angle 1: Sketch from angle 2:

Parts Required:

We will now determine the quadratic equation that represents your siege engine. To do so, you will need to

acquire a stopwatch and a tape measure.

Place your catapult on the floor and fire a projectile. You will need to measure the distance between the

launching point and the approximate landing point (in meters) as well as record the time from the stopwatch

to see how long the projectile was in the air (in seconds). Record the gathered information from at least three

reliable trials in the table below. Then, sketch the resulting quadratic equations in the space below

Trial 1 Trial 2 Trial 3 Trial 4 (optional)

Time (in seconds)

(‘t’)

Distance (in meters)

(‘d’)

Acceleration due to Gravity (in m/s2) (‘g’)

-9.8 -9.8 -9.8 -9.8

Maximum Height (in meters)

y =|( 1

2𝑔𝑡2)| ‡

Three points gathered. Launch point (0,0) Landing point (d,0)

Vertex ( 𝑑

2 , y)

( , )

( , )

( , )

( , )

( , )

( , )

( , )

( , )

( , )

( , )

( , )

( , )

Quadratic Equation (you can use a graphing calculator and a system of equations to gain this information)

Y = Y = Y = Y =

‡ Teacher’s note: since the object starts at rest and on ground level, 𝑣𝑦0 𝑡 + 𝑦0 is omitted

Pre-work:

What strategies will you use to create a siege engine that will launch a projectile the farthest?

Why do you think your design will allow this?

Post-work:

Were your design choices successful?

How could your design be improved?

What are the potential sources of error in this project?

How could these sources of error be minimized if this project were performed again?

Below are three pairs of aspects of projectile motion, based on your experiment, tell whether each pair

experience direct variation, inverse variation, or are unrelated.

Maximum Height/Distance Traveled | Time in Flight/Maximum Height | Distance Traveled/Time in Flight

| |

Each team is responsible for compiling their data into a PowerPoint, I-Movie, Prezi, or other multimedia

program. The presentation should focus on the project, design process, and research into your chosen siege

location. The presentation should include, but is not limited to: original design ideas, items used in the design

and justification for the items being chosen, changes made during testing of the design, data, and a reflective

analysis of the project (What would change? What did you learn?)

Category 4 3 2 1

Presentation

Smooth, confident, and easy to follow delivery

Fairly smooth delivery. Confidence matters.

Choppy delivery, difficult to hear or understand.

Delivery was difficult to follow.

Content (Math)

Covers topic in-depth and displays mastery over subject area

Covers topic and displays comfort with subject knowledge

Content is minimal but the facts are sound

Content is minimal and factual errors are present

Content (History)

Knowledge of siege location is excellent. Design and materials reflect this.

Knowledge of siege location is appropriate

Knowledge of siege location is minimal

Knowledge of siege location is not displayed or is very vague

Mechanics

No misspellings or grammatical errors

3 or fewer misspellings and/or grammatical errors

6 or fewer misspellings and/or grammatical errors

7+ misspellings and/or grammatical errors

Attractiveness

Makes excellent, professional use of font, color, graphics, etc. to enhance presentation

Makes good use of font, color, graphics, etc. to enhance presentation

Makes unattractive use of font, color, graphics, etc. to enhance presentation

Use of font, color, graphics, etc. makes presentation difficult to view

Requirements

All requirements are met or exceeded

All but one requirement are met.

All but two requirements are met

Three or more requirements are not met

Originality

Product shows a large amount of original, creative, and inventive thought

Product shows some original thoughts, new ideas or insights

Little evidence of original thinking.

Plagiarism

For an additional 10%, please complete the following steps to calibrate your war-machine in order to make it

both more accurate and precise during your demonstration. Both accuracy and precision will factor into your

demonstration grade regardless of this page’s completion.

We will be completing a table and a graph to determine the function that relates the independent variable of

your war machine to the dependent variables of distance traveled and maximum height attained.

If your war machine is basically like a…. Then your independent variable is…

Ballista Draw Distance (how far back you pull the projectile, most likely in centimeters)

Catapult Angular Draw (how far back you pull the projectile, in degrees)

Trebuchet Counterweight Mass (in grams)

After identifying your war-machine, your group will complete the following table (remember, the more data,

the better!) For Ballista groups, you will need to carefully measure your draw distance with a ruler. For

Catapult groups, you will need a protractor. For Trebuchet groups, you will need a scale or premeasured

weights.

Independent Variable Distance Max Height Independent Variable Distance Max Height

After completing at least the table above, a group member will need to enter the information onto a graphing

calculator by selecting the STAT edit mode, entering the data columns of data into the L1, L2, and L3 lists

respectively. The group member will then need to go to [2nd] [Y=] and access the STAT PLOT editor. L1 is the

independent variable and L2 is the dependent variable. Once the scatter plot is displayed, perform a linear (or

possible a quadratic) regression on the data. This will tell you how to calibrate your machine for certain

distances. Repeat this process for L1 and L3 to find a calibration function for desired maximum heights.

Carefully sketch your scatter plots and the line or quadratic (both if you’d like) on the graphs on the next page.

Independent Variable (units): Dependent Variable (units):

X-Scale: Y-Scale:

Line/Quadratic of Best Fit:

Independent Variable (units): Dependent Variable (units):

X-Scale: Y-Scale:

Line/Quadratic of Best Fit:

Group Members:

Metric Explanation Total Worth Total Earned

Multimedia Presentation

See Individual Rubric

28%

The War-Machine

Construction

Ability to Fire Projectile

Precision

(Able to hit in reasonably tight cluster during demonstration)

10%

10%

5%

Design

War-Machine, Distance, and Wall

to scale

Able to clear wall

Reasonable Design/Materials

15%

2%

5%

Research

Siege target was adequately

researched, group shows working knowledge of region, people, and

target

10%

Citations

All information gathered from

online, textbooks, etc is properly and consistently cited

5%

Calibrations Worksheet See Additional Worksheet 10%

Letter to the Monarch Letter deemed to be appropriate and convincing

10%

FINAL GRADE

Throughout the history of weaponry, throwing things has always been very popular. Throwing things, of

course, led to launching things and projectiles which have achieved flight via either method achieved motion

which may be modeled by quadratic equations. In order to prepare us for our own catapult testing, we will

look at the following weapons and their hypothetical associated quadratic equations. We will determine if the

weapons can clear a wall 50 meters away and 6 meters high. As Reference:

Weapon Model Vertex (How high?)

Roots (How far?)

Clears Wall?

Ballista

𝑓(𝑥) = −.01𝑥2 + .6𝑥

(30 , 9) At 30 meters away, the projectile is 9 meters high.

(0,0) (60, 0)

The projectile will land 60 meters from the weapon.

f(50) = 5

Yes No

Bombard

𝑓(𝑥) = −1

200(𝑥 − 40)2 + 8

(40, 8) At 40 meters away, the projectile is 8 meters high.

(0, 0) (80 , 0)

The projectile will land 80 meters from the weapon.

f(50) = 7.5

Yes No

Culverin

𝑓(𝑥) = −1

405𝑥2 +

2

9 𝑥

(45,5) At 45 meters away, the projectile is 5meters high.

(0 ,0) (90,0)

The projectile will land 90 meters from the weapon.

f(50) = 4.94

Yes No

Mangonel

𝑓(𝑥) = −1

90(𝑥 − 30)2 + 10

(30 ,10) At 30 meters away, the projectile is 10 meters high.

(0,0) (60,0)

The projectile will land 60meters from the weapon.

f(50) = 5.56

Yes No

Mortar

𝑓(𝑥) = −1

30𝑥2 + 2𝑥

(30 ,30) At 30 meters away, the projectile is 30meters high.

( 0 , 0) (60,0)

The projectile will land 60 meters from the weapon.

f(50) = 16.67

Yes No

Trebuchet

𝑓(𝑥) = −1

13(𝑥 − 26)2 + 52

(26, 52) At 26 meters away, the projectile is 52meters high.

(0, 0) (52, 0)

The projectile will land 52 meters from the weapon.

f(50) = 7.69

Yes No

Vertex Form f(x) = a ( x – h )2 + k

Standard Form f(x) = ax2 + bx + c

Deciding that they’d rather work with the vertex form of an equation, students named Anthony, Brittany, and

Catherine convert a quadratic equation given in standard form to vertex form by completing the square and

then stated the vertex equation. Convert the equation yourself first and determine which, if any, of the

students performed the conversion correctly. If a student’s solution is incorrect, then identify where the first

error in his or her work occurred.

Given: 𝒇(𝒙) = 𝟔𝒙𝟐 − 𝟖𝟒𝒙 + 𝟐𝟗𝟕

Your Work:

𝒇(𝒙) = 𝟔𝒙𝟐 − 𝟖𝟒𝒙 + 𝟐𝟗𝟕

𝒇(𝒙) = 𝟔(𝒙𝟐 − 𝟏𝟒𝒙) + 𝟐𝟗𝟕

𝒇(𝒙) = 𝟔(𝒙𝟐 − 𝟏𝟒𝒙 + 𝟒𝟗) + 𝟐𝟗𝟕 − 𝟐𝟗𝟒

𝒇(𝒙) = 𝟔(𝒙 − 𝟕)𝟐 + 𝟑 ; Vertex (7 , 3)

Anthony’s Work Brittany’s Work Catherine’s Work

𝒇(𝒙) = 𝟔𝒙𝟐 − 𝟖𝟒𝒙 + 𝟐𝟗𝟕

𝒇(𝒙) = 𝟔(𝒙𝟐 − 𝟏𝟒𝒙) + 𝟐𝟗𝟕

𝒇(𝒙) = 𝟔(𝒙𝟐 − 𝟏𝟒𝒙 + 𝟒𝟗) + 𝟐𝟗𝟕 − 𝟒𝟗

𝒇(𝒙) = 𝟔(𝒙 − 𝟕)𝟐 + 𝟐𝟒𝟖 Vertex: (7 , 248)

𝒇(𝒙) = 𝟔𝒙𝟐 − 𝟖𝟒𝒙 + 𝟐𝟗𝟕

𝒇(𝒙) = 𝟔(𝒙𝟐 − 𝟏𝟒𝒙) + 𝟐𝟗𝟕

𝒇(𝒙) = 𝟔(𝒙𝟐 − 𝟏𝟒𝒙 + 𝟒𝟗) + 𝟐𝟗𝟕 − 𝟐𝟗𝟒

𝒇(𝒙) = 𝟔(𝒙 − 𝟕)𝟐 + 𝟑

Vertex: (3 , 7)

𝒇(𝒙) = 𝟔𝒙𝟐 − 𝟖𝟒𝒙 + 𝟐𝟗𝟕

𝒇(𝒙) = 𝟔(𝒙𝟐 − 𝟏𝟒𝒙) + 𝟐𝟗𝟕 𝒇(𝒙) = 𝟔(𝒙𝟐 − 𝟏𝟒𝒙 + 𝟗𝟖) + 𝟐𝟗𝟕 − 𝟓𝟖𝟖

𝒇(𝒙) = 𝟔(𝒙 − 𝟕)𝟐 − 𝟐𝟗𝟏 Vertex: (7 , -291)

Anthony was (right / wrong) Because… He didn’t multiple the 49 by 6 before balancing the square

Brittany was (right / wrong) Because… She confused what h and k represented

Catherine was (right / wrong) Because… She squared b then divided by two rather than the other way around

Authorship Page and Works Referenced

This packet was created to fulfill the “Going from ‘Good’ to Better” project requirement of the Spring 2016

session of EDUC 3302 at Trinity University. This packet was created, in full, by Dayton King using the

“Quadratic Catapult” packet provided by Lisa Hindley of the NEISD STEM Academy as both inspiration and a

blueprint. Images used on the cover were created via photoshop by Dayton King using images considered

within the public domain and protected by U.S. Section Code 7.107 as they are intended for classroom, non-

profit use.

Works Referenced

National Governors Association Center for Best Practices, Council of Chief State School Officers. (2010). In Common Core

State Standards Mathematics.

NEISD STEM Academy. (2011). Quadratic Catapult Project (pp. 1-18).

Texas Education Agency. (2010, September). Chapter 111 Mathematics. In Texas Essential Knowledge and Skills.

Texas Education Agency. (2012, May). Subchapter 2, Mathematics Standards. In Texas Career and College Readiness

Standards.

The Internation Society for Technology in Education. (2007). ISTE Standards for Students. In ISTE Standards for Students.

Retrieved from http://www.iste.org/standards/iste-standards/standards-for-students