Download 1 - STEM Georgia

Name of Unit: Quadratics(math) Catapulting(engineering):
Properties of Matter(chemistry) or Newtonian Relations(physical science)
(Approximate Time Frame: 8-10 Days)
OVERVIEW:
This unit introduces students to the topics of the engineering problem-solving process, properties of matter, and
characteristics of quadratics. Students will conduct a STEM project in which they work in collaborative teams to
build a catapult which launches 1cm3 masses of different materials. With data collection and mathematical analysis
of quadratic functions utilizing technology, students will optimize their catapults and evaluate properties and
characteristics of atomic structure and understandings of Newtonian relations, through collaboration and the iterative
process.
CTAE Standards
STANDARDS ADDRESSED IN THIS UNIT
Science Standards
Math Standards
ENGR-FET4 – Students will apply
mathematics and science to the
solution of a technological
problem.
(a) Describe the role of mathematics
and science in technological
development.
(b) Construct a mathematical model
for a known technological system.
(c) Explain the scientific principles
behind a basic machine.
ENGR-FET6 – Students will use
visual and verbal communication
to express basic design elements.
(a) Demonstrate fundamentals of
technical sketching.
ENGR-EC2 – Students will
demonstrate the engineering
design process.
(a) Describe the role of problem
identification, problem definition,
search, constraints, criteria,
alternative solutions, analysis,
decision, specification, and
communication as activities
Chemistry
SC1 Students will analyze
the nature of matter and its
classifications.
b. Identify substances based
on chemical and physical
properties.
SC3 Students will use the
modern atomic theory to
explain the characteristics of
atoms.
a. Discriminate between the
relative size, charge, and
position of protons, neutrons,
and electrons in the atom.
c. Explain the relationship of
the proton number to the
element’s identity.
Physical Science
SPS1. Students will investigate our
current understanding of the
atom. a. Examine the structure of
the atom in terms of
MCC9‐12.F.IF.4 For a function that
models a relationship between two
quantities, interpret key features of
graphs and tables in terms of the
quantities, and sketch graphs showing
key features given a verbal
description of the relationship. Key
features include: intercepts; intervals
where the function is increasing,
decreasing, positive, or negative;
relative maximums and minimums;
symmetries; end behavior; and
periodicity.
MCC9‐12.F.IF.5 Relate the domain
of a function to its graph and, where
applicable, to the quantitative
relationship it describes. For
example, if the function h(n) gives the
number of person‐hours it takes to
assemble n engines in a factory, then
the positive integers would be an
comprising the engineering design
process.
(b) Organize the iterative processes
necessary to develop and optimize a
design solution.
(c) Apply engineering design to the
solution of a problem.
ENGR-EC3 – Students will solve
problems using basic engineering
tools and resources.
(d) Create an Excel spreadsheet to
perform basic arithmetic and
algebraic computations on data
related to an engineering design
problem.
(e) Use laboratory tools and
equipment to determine the
properties of materials.
ENGR-EC4 – Students will
demonstrate a whole systems
approach to engineering and
problem solving.
(b) Apply leadership skills to
participation in design team
activities.
(c) Demonstrate a team approach in
applying engineering design to the
solution of a technological problem.
(d) Apply continuous process
improvement principles in designing
a problem solution.
(e) Demonstrate concurrent
communication skills in developing
a design solution.
ENGR-EC5 – Students will apply
engineering graphics and technical
writing to communication of an
engineering design.
(f) Prepare a report of engineering
design activities including a
•
proton, electron, and neutron
locations.
• atomic mass and atomic
number.
• explain the relationship of the
proton number to the element’s
identity.
SPS2. Students will explore the
nature of matter, its
classifications, and its system for
naming types of matter.
a. Calculate density when given a
means to determine a substance’s
mass and volume.
SPS8. Students will determine
relationships among force, mass,
and motion.
a. Calculate velocity and
acceleration.
b. Apply Newton’s three laws to
everyday situations by explaining
the following:
Inertia
Relationship between force, mass
and acceleration
Equal and opposite forces
c. Relate falling objects to
gravitational force
d. Explain the difference in mass and
weight.
e. Calculate amounts of work and
mechanical advantage using simple
machines.
Characteristics of Science:
SCSh5. Students will
demonstrate the computation
and estimation skills
appropriate domain for the function.
MCC9-12.F.IF.7 Graph functions
expressed symbolically and show key
features of the graph, by hand in
simple cases and using technology
for more complicated cases.
MCC9-12.F.IF.7a Graph linear and
quadratic functions and show
intercepts, maxima, and minima.
MCC9-12.F.IF.8 Write a function
defined by an expression in different
but equivalent forms to reveal and
explain different properties of the
function.
MCC9-12.F.IF.8a Use the process of
factoring and completing the square
in a quadratic function to show zeros,
extreme values, and symmetry of the
graph, and interpret these in terms of
a context.
description of analysis, optimization,
and selection of a final solution.
ENGR-EA2 – Students will
develop and follow a detailed plan
for the solution of a design
problem.
(b) Apply mathematical models and
calculations necessary to complete
predictive analysis.
(c) Modify a design plan to
accommodate unforeseen
constraints.
(d) Assess the effectiveness of
design plans.
ENGR-EA3 – Students will
demonstrate prototype
development.
(a) Identify appropriate modeling
techniques.
(b) Select and apply appropriate
materials, tools, and processes for
prototype development.
(c) Evaluate effectiveness of
prototyped solution and modify as
needed.
necessary for analyzing data
and developing reasonable
scientific explanations.
a. Trace the source on any large
disparity between estimated and
calculated answers to problems.
b. Consider possible effects of
measurement errors on calculations.
e. Solve scientific problems by
substituting quantitative values,
using dimensional analysis and/or
simple algebraic formulas as
appropriate.
ENDURING UNDERSTANDINGS
Engineering:
1. Engineers solve problems using the engineering design process.
Science:
1. Different pure substances have different densities because the atoms that they are composed of have
different numbers of subatomic particles.
2. Different pure substances will have different masses if they have equal volumes.
3. Density is a physical characteristic property of matter which can be used to identify a pure substance.
Math:
1.
2.
3.
4.
Quadratic equations have even symmetry.
The domain and range of a quadratic function provide contextual meanings to the graphical representation.
The vertex form of a quadratic equation reveals many characteristics of the graph of the equation.
Standard form of a quadratic equation can be utilized in conjunction with the quadratic formula to find the
5.
6.
7.
zeros of the function.
Zeros of a graph and x-intercepts are equivalent expressions.
The extrema of a quadratic equation is related to the direction the graph opens and reflections
Transformations of a quadratic equation move the parent function graphically in specific manners.
ESSENTIAL QUESTIONS
Engineering Essential Questions:
1. What roles do math and science play in technological development?
2. What are the basic scientific principles behind a basic machine?
3. What are the six steps of the problem solving process?
4. How does the iterative process optimize design solutions?
Science Essential Questions:
5. What is density and how is density calculated?
6. Why does the density of a substance remain the same regardless of sample size?
7. Why do different pure substances have different densities?
8. How is the density of an element related to the number of protons and neutrons in the nucleus of an atom of
an element?
9. What is the relationship between force, mass, and acceleration based upon Newton’s 2nd Law of Motion?
10. How does the mass of an object affect the acceleration of an object with a constant force?
Math Essential Questions:
11. What function does the flight path of the object model?
12. What key characteristics of the quadratic function are distinguishable when modeling this through the flight
path of the object?
13. What information can you gain from converting the flight path of the object between standard, vertex, and
slope intercept form of a quadratic equation?
14. Compare and contrast the graphs and equations of different elements flight paths taking into consideration
the key features.
15. What do the domain and range of the functions represent when relating them to the flight paths?
16. What is the end behavior of the functions of the flight path? Will it change based on the data you have
collected?
CONCEPTS
Engineering:
Understand and apply the problem solving process. Apply the iterative process to optimize the solution. Statistically
analyze data and relate this data to a theoretical concept. Predictive analysis using date input in an Excel spreadsheet
and converted into a scatter graph.
Science:
Atomic theory and the composition of the atoms of the elements. The density equation. The relationship between
atomic mass and the number of subatomic particles in the nucleus of an atom of a specific element. Equal volumes
of different pure substances will have different masses due to having different densities. Physical vs. chemical
properties. Newton’s Laws of Motion.
Math:
Students will understand and analyze the characteristics of quadratic equations taking into account the domain and
range values, convert from vertex to standard form through gathered data, and graph quadratic equations utilizing
appropriate technologies and Cartesian coordinate grids. Students will also solve quadratic equations to determine the
zeros of the function algebraically using the quadratic formula.
MISCONCEPTIONS
PROPER CONCEPTIONS
Engineering:
Engineering:
Math:
Math:
Science:
Science:
LANGUAGE:
Engineering:
Problem solving process (state the problem, collect information, develop possible solutions, select best solution,
implement solution, evaluate solution), Iterative Process, Optimization
Science:
Mass, volume, density, atom, proton, neutron, electron, pure substance, element, compound, physical property,
chemical property, characteristic property, nucleus, electron cloud, force, acceleration, Newton’s 2nd Law of Motion
Math:
Extrema (maxima, minima), zeros of a function, vertex, intervals of increase, intervals of decrease, domain, range,
quadratic function, standard form of a quadratic equation, vertex form of a quadratic equation, reflection across the xaxis, axis of symmetry, end behavior, vertical shift, horizontal shift
EVIDENCE OF LEARNING
Assessments:
1. Thinking Maps
2.
3.
Open-ended higher order thinking discussions embedded throughout the lesson
Iterative process assessments based upon data generated from testing catapult and applying data to Catapult
Excel File spreadsheet
Engineering
1. Engineering Problem Solving Process Rubrics
2. Problem Solving Quiz (provided by teacher)
3. Design and build of the catapult
Science:
1. Science Density PowerPoint Thought Questions
2. Density Cubes Lab
3. Density Quiz
4. Science Newton’s Laws of Motion PowerPoint Thought Questions
5. Newton’s Laws of Motion Quiz
Math:
1. Quadratic Equations Vocabulary Quiz (use student response system, if available)
2. Mathematical Summary of Catapult Data
Culminating Activity:
Student teams will design and build a catapult, according to specifications provided in the Launcher Design Brief,
which will launch 1 cm3 metals of different densities at a specified target (see Height Banner and Bulls Eye Target
images). Students will collect data from launches using the Launch Data Sheet and input data into the Catapult
Excel File spreadsheet. The data input will be converted into a scatter plot to apply predictive analysis. By
understanding predictive analysis and the iterative process, students can increase or decrease the amount of force
applied to the lever arm or increase or decrease the distance from the catapult to the target based upon which metal is
being launched. Within this activity, students’ understanding in each discipline will be assessed through appropriate
questioning, discussion, and analysis strategies.
TASKS
The collection of the following tasks represents the level of depth, rigor, and complexity
expected of all students to demonstrate evidence of learning.
Task(s):
Engineering:
 Learn and understand the problem solving process.
 Understand and apply the iterative process.
 Analyze data with the use of spreadsheets and scatter plots.
Science:
 Measure mass and volume in a laboratory setting and express measurements with proper units.
 Calculate density of unknown substances using empirical data and relate the calculated density to accepted
values for pure substances to identify the unknown substances.
 Describe why two different pure substances with the same volume have different masses with respect to the
atoms that make up the substances in a short paper using appropriate technical language.
Math:
 Understand and apply related vocabulary.
o Mathematics Quadratic Equations Vocabulary and Guided Notes PowerPoint
o Guided student note-taking strategies
o Quizlet
o Quadratic Equations Vocabulary Quiz (use student response system, if available)
 Convert between standard and vertex form of quadratic equations and apply the quadratic formula to find
solve for the zeroes of the equation.
o Mathematics Quadratic Equations Converting and Formula PowerPoint
o Individual/group practice
STEM Activity
See Culminating Activity description above.
UNIT RESOURCES
Suggested 21st Century Technology to be used in this unit
Engineering:
Math:
Science:
Learning Experiences/ Sequence of Instruction
Delivery Mechanism Suggestions:
Suggestion 1: The following is a suggested plan for teaching the content of this unit. The subjects (CTAE, Science, and
Mathematics) can be taught in any order. Students will be broken into three groups and receive 2 days of instruction in each
content area before completing the culminating activity.
Suggestion 2:
Day 1
CTAE Instruction
Day
1
Science Instruction
(Chemistry or Physical Science)
Math Instruction
GROUP A
GROUP B
GROUP C
Opening: Students will be asked to
solve a simple problem. After being
given time to solve the problem,
students are asked to reflect upon the
process in which they solved the
problem.
Chemistry Lesson
Opening: As students enter the room,
they will see a sketch of the graph of
a quadratic equation on the board
with the question, “Where in the real
world do you see this? Think about
sports.” As students are responding,
the teacher will help students relate
their responses to the Essential
Questions and Standards. The teacher
will direct the discussion to sports
and the flightpath of the ball by
asking, “Where is the highest point of
the flightpath of the ball found?”
Students should discuss that it is the
middle. The teacher will then explain
that this is the middle of the graph
and the axis of symmetry. The
teacher will also explain that this is
the maximum value. Ask, “What
would the lowest value be called if
Work Session: Students are
introduced to the Problem Solving
Process. With the aid of the
Engineering Problem Solving
Process PowerPoint, the teacher
leads a discussion on the first 4 steps
of the problem solving process.
Closing/Assessment: Students will
perform a brainstorming activity. Put
students in groups of 4 – 6. Provide
them with an idea or concept to
Opening: To introduce this unit, provide
students with the following materials:
• 3 – 150 mL beakers
• 1 – 600 mL beaker
• water
• corn syrup
• vegetable oil
• food coloring
• several small objects (raisins,
paperclips, pennies, small corks,
rubber stoppers, plastic bottle caps,
etc.)
Have students navigate to the following
website and follow the directions for
experiments 1 and 2. Create a data table for
your observations and record what you
observe.
brainstorm, for example a new design
for a coffee mug. Make sure each
student is following procedural steps
for brainstorming as previously
discussed.
http://www.hometrainingtools.com/explo
ring-liquid-density-newsletter/a/1309/
Work Session: Provide guided lecture on
density using the Science Density
PowerPoint. Show the opening essential
questions slide. Have students work in
groups to discuss the essential questions and
develop possible answers to these questions.
Present the guided lecture. After the guided
lecture, have students relate the opening
investigation to the guided lecture
information.
Closing/Assessment: Review and revise
the answers to the essential questions.
Assign the thought questions in the Science
Density PowerPoint for homework.
Physical Science Lesson
Opening: Provide students with the
following materials:
1. 3 different masses with hooks
2. a rubber band
3. a ruler
Have students hang each mass from the
rubber band and measure the length that the
rubber band stretches. Have students create
a data table to record their observations.
Have students answer the following
questions:
the maximum is the highest point?”
(Answer: minimum)
Work Session: Students will take
notes on the vocabulary found in the
Mathematics Quadratic Equations
Vocabulary and Guided Notes
PowerPoint.
Students will take the Quadratic
Equations Vocabulary Quiz using
their notes individually. The teacher
will review the quiz and answer any
questions.
Closing/Assessment: Students will
find the characteristics of the same
sketch of the graph used as the bell
ringer. Students are encouraged to
study the vocabulary for a quiz the
next day.
1. Which mass stretched the rubber
band the furthest?
2. Why do you think this is so?
Work Session: Guided lecture over
Science Newton’s Laws of Motion
PowerPoint. Introduce Newton’s 2nd law
of Motion and provide examples for
solving. Guide the discussion. Break
students up into groups to work out the
thought problems.
Day
2
Closing/Assessment: Have students relate
the laws of motion to the introductory
activity. (Be sure that they understand that
the acceleration due to gravity is constant at
9.8m/s2). Discuss the solutions to the
thought questions in the Science Newton’s
Laws of Motion PowerPoint.
GROUP A
GROUP B
GROUP C
Opening: Teacher and students do a quick
review of the 4 steps covered the previous
day.
Chemistry Lesson Plan
Opening: Students will complete the
Quadratic Equations Vocabulary Quiz
without notes.
Work Session: With the aid of the
Engineering Problem Solving Process
PowerPoint, the teacher leads a discussion
of the last 2 steps of the problem solving
process. As an exemplary activity, teacher
will guide students through a simple problem
solving activity, the Paper Platform. The
teacher provides students with the paper
Opening: Open with a bell-ringer that reviews the
concepts learned the previous day. Students will
solve for density, mass and/or volume given 2 out of
the 3 variables. Students will discuss with the
instructor as well as the class the relationship
between density of a pure substance and the atoms
that compose that substance. Students will review
essential vocabulary.
Work Session: Working in collaborative pairs,
students will complete the Density Cubes Lab.
Work Session: Students will take notes and
understand examples of finding the zeroes of
a quadratic function using the quadratic
formula and converting between standard
and vertex form. Students will take notes
over the Mathematics Quadratic Equations
Converting and Formula PowerPoint,
Students will work examples of each concept
platform design brief and 5 index cards. The
teacher also provides each student with a
problem solving rubric packet. This packet
needs to be completed by each student as the
teacher leads the paper platform problem
solving activity.
21st Century Skills Application:
Slideshow Software, Interactive Whiteboard, Student
Response System
21st Century Skills Application:
Slideshow Software, Interactive Whiteboard
Closing/Assessment: Students will complete the
analysis section of the density cubes lab and submit
to the instructor for grading. Students will be given
the Density Quiz.
Closing/Assessment: Student teams present
their brainstorming ideas from the previous
day.
in pairs as the teacher guides and helps
students understand the processes for each.
Physical Science Lesson Plan
Opening: Open with a bell-ringer that reviews the
essential vocabulary and calculations for Newton’s
second law, solving for Force, mass, and/or
acceleration given 2 out of the 3 variables.
21st Century Skills Application:
Slideshow Software, Interactive Whiteboard
,Student Response System, Thinking Maps,
Calculator
Closing/Assessment: Students will
complete a Ticket Out The Door by finding
the zeroes of the function given in vertex
form. In order to do this, students would first
need to convert the equation from vertex to
standard form and then use the quadratic
formula to find the zeros.
Work Session: Students will design and draw a
machine in the style of Rube Goldberg’s style – see
http://www.rubegoldberg.com/ for examples and
details – that tests Newton’s laws of motion.
Students will write an explanation of how the
machine illustrates Newton’s laws of motion.
21st Century Skills Application:
Slideshow Software, Interactive Whiteboard, Student
Response System
Day
3
Repeat 2 day Instructional Cycle with
GROUP C
Closing/Assessment: Students will submit the
completed assignment (drawing and explanation) to
the instructor for assessment and the Newton’s Laws
of Motion Quiz.
Repeat 2 day Instructional Cycle with GROUP A
Repeat 2 day Instructional Cycle with
GROUP B
Day
4
Day
5
Day
6
Days
7+
Repeat 2 day Instructional Cycle with
GROUP C
Repeat 2 day Instructional Cycle with GROUP A
Repeat 2 day Instructional Cycle with
GROUP B
Repeat 2 day Instructional Cycle with
GROUP B
Repeat 2 day Instructional Cycle with GROUP C
Repeat 2 day Instructional Cycle with
GROUP A
Repeat 2 day Instructional Cycle with
GROUP B
Repeat 2 day Instructional Cycle with GROUP C
Repeat 2 day Instructional Cycle with
GROUP A
Completion of Culminating Activity
Work session: Student teams will design and build a catapult, according to specifications provided in the Launcher Design Brief, which will launch
1 cm3 metals of different densities at a specified target (see Height Banner and Bulls Eye Target images). Students will collect data from launches
using the Launch Data Sheet and input data into the Catapult Excel File spreadsheet. The data input will be converted into a scatter plot to apply
predictive analysis. By understanding predictive analysis and the iterative process, students can increase or decrease the amount of force applied to
the lever arm or increase or decrease the distance from the catapult to the target based upon which metal is being launched. Within this activity,
students’ understanding in each discipline will be assessed through appropriate questioning, discussion, and analysis strategies.
Engineering Problem Solving Process PowerPoint
Slide 1
Problem Solving
After studying this content, students will be
able to:

Explain and carry out the Problem Solving Process

Understand Problem Solving as an Iterative Process
Slide 2
Problem Solving Process
Used to develop workable solutions to problems!
6 Steps of the Problem Solving Process
1)
2)
3)
4)
5)
6)
State the Problem Clearly
Collect Information
Develop Possible Solutions
Select the Best Solution
Implement the Solution
Evaluate the Solution
Slide 3
State the Problem Clearly
Why is this important?
Solving any problem starts here. You have to know what
the problem is.
Sometimes simply stating the problem offers a simple
solution.
Some people say that stating the problem is half the job of
solving it.
For Example, what about the problem with garbage. Is the
problem (1) to design new ways of removing the garbage,
(2) to design ways of reusing the garbage, (3) to design
ways of making products that create less garbage, or (4) a
combination of all 3.
Slide 4
Collect Information
Why is collecting information important?
Once the problem is thoroughly stated, information that can
be used to develop a good solution must be gathered.
Where can we collect information?
Libraries, Internet, Books, Media or Magazines, Other
People
Slide 5
Develop Possible Solutions
Why is this important?
Most problems have more than one possible solution.
At first, the more possible solutions there are, the better.
That way, there are more options from which to choose.
There are 2 ways of coming up with possible solutions:
Trial and Error
Brainstorming
Slide 6
Trial & Error

What is trial & error?
• Used when there is no information or data available
• Thomas Edison
Slide 7
Brainstorming

What is Brainstorming?
• Thinking of as many possible solutions as possible.

Steps for Brainstorming
• Present ideas in an open forum.
• Generate and record ideas.
• Keep the mind alert through rapidly paced sessions.
• Develop preliminary ideas.
Slide 8
Select the Best Solution
Why is important to Select the Best Solution?
In order to select the best solution, all the possible
solutions need to be evaluated.
What is evaluating?
Evaluating involves looking at all the advantages and
disadvantages of each possible solution to decide which one
best solves the problem.
Part of being a good problem solver is being able to
recognize which factors are most important.
Rarely is there a perfect solution to any problem!!!!
Slide 9
Implement the Solution
Why is it important to Implement the Solution?
During the implementation process, models are made and
ideas are tested to make the solution workable.
What is a simulation?
During a simulation, equipment is set up in a lab or testing
area in a way in which it simulates or imitates as closely as
possible the real life circumstances for which it was
designed.
Simulations are a good way to implement a possible
solution.
Slide 10
Evaluate the Solution
Why is it important to Evaluate the Solution?
The information that is obtained from implementing the
solution or doing a simulation helps refine the solution.
Often times, the evaluation process determines new
problems that were never thought of or discussed. When
this happens, the problem solving process starts all over
again.
PAPER PLATFORM
Bridges have it. Spider webs have it. Houses have it. A skeleton has it. The chair you are sitting in has it. They all have structure.
Structure is how materials work together for strength. Since technologists first started building and producing structures to solve many of their day
to day problems, there has been a constant effort to make less material do more work. Technologists take what materials are available, process
them and assemble them in such a way that they will perform work efficiently. Limited supply, excessive weight, limited resources or access to
those resources have always been problems to overcome in the building of a structure. With that in mind you are going to build a structure to
solve a specific problem.
OBJECTIVE
Develop and construct a platform that will support the weight of 25 pounds.
MATERIALS
1. Five - 8” x 5” Index Cards
2. Masking tape - 3 inches
3. White glue
TOOLS
1. Scissors
2. Ruler
LIMITATIONS
1. Students may only use the materials provided.
2. The platform must be within these specifications or it will be disqualified:
Height - 1" to 1 1/2" tall
Width - 5" wide
Length - 8" long
INSTRUCTIONS
1. Students will work individually
2. Students must use the problem solving process.
3. Students will construct a platform using the provided materials.
4. The instructor will test the platform at the end of the allotted time frame.
5. The structure must support at least 25 lbs.
6. All steps of the problem solving process must be listed on a separate sheet of paper. All information under each step of the PSP must be
recorded and turned in. Before construction, at least 3 ideas must be listed and sketched to express your ideas.
7. Step 6 of the PSP must be in essay form.
Mathematics Quadratic Equations Vocabulary and Guided Notes PowerPoint
Slide 1
Quadratic Equations
Vocabulary Guided Notes Presentation
STEM Integrated Unit
Slide 2
Quadratic
Function
f(x) = x²
parent function
Parabola
The shape the quadratic function creates.
Slide 3
Standard Form
of a
Quadratic Equation
This makes it quadratic – it is raised to the second power.
ax² + bx + c = 0
This is the form needed for
using the quadratic formula.
Slide 4
Vertex Form
of a
Quadratic Equation
This makes it quadratic – it is raised to the second power.
y= a(x-h)²+ k
This form is the most useful when graphing
because it shows the
transformations of the graph.
Slide 5
Vertex
The point where the parabola crosses its axis of
symmetry. It is the highest or the lowest point on
the graph.
In the vertex form of a quadratic equation the
vertex is (h,k).
y= a(x-h)²+ k
When finding the vertex, remember to read it as opposite of h. If h = -2, the value for
the vertex would be 2. Likewise, it the value of h was 2, the x-value of the vertex
would be -2 … think the opposite of !
Slide 6
Extrema
Maxima: the highest point on the graph. (the yvalue)
Minima: the lowest point on the graph. (the yvalue)
Slide 7
Zeroes of a Function
A value of x which makes a function f(x) equal to
zero.
Where the function crosses the x-axis.
Slide 8
Intervals of Increase
Domain of a function where its values are getting
smaller.
Intervals of Decrease
Domain of a function where its values are getting
bigger.
Slide 9
Domain
The set of all possible input values.
The x-values.
Range
The set of all possible output values.
The y-values.
Slide 10
Axis of Symmetry
The line that divides the
parabola in half,
creating two equal
sides. It is the x-value
of the vertex (x,y) of
the graph of the
function.
http://www.mathwarehouse.com/geometry/parabola/axis-of-symmetry.php
Slide 11
Reflection across the x-axis
y = -f(x)
For every point (x,y), there is a point (x, -y)
Like a mirror image.
Slide 12
Horizontal Shift
Shift in the parent graph left or right along the xaxis. This is found in the value of h in the vertex
form of a quadratic equation.
Vertical Shift
Shift in the parent graph up or down along the yaxis. This is found in the value of k in the vertex
form of a quadratic equation
Quadratic Equations Vocabulary Quiz
Match the definition to the correct word.
Name_______________________
b. The set of all possible input values. The xvalues
c. Where the function crosses the x-axis.
1) _____ Standard form of a quadratic
equation
2) _____ Quadratic Function
3) _____ Vertex form of a quadratic
function
d. a mirror image where for every point (x,y)
there is a point (x, -y)
e. Shift in the parent graph left or right along the
x-axis. It is the h value in the vertex
f. The line that divides the parabola in half.
4) _____ Vertex
g. The highest y-value of a graph.
5) _____ Maxima
h. The lowest y-value of a graph.
6) _____ Minima
i. f(x) = x2
7) _____ Zeroes of a function
j. ax2 + bx + c = 0
8) _____ Intervals of increase
k. y = a(x-h)2 + k
9) _____ Intervals of decrease
l. The point where the parabola crosses its
axis of symmetry. (h,k)
10) _____ Domain
11) _____ Range
12) _____ Axis of Symmetry
13) _____ Reflection across the x-axis
14) _____ Horizontal shift
15) _____ Vertical shift
a. shift in the parent graph up or down along the
yaxis.
m. the set of all possible output values. (the
y-values)
n. domain of a function where its values are
getting bigger.
o. domain of a function where its values are
getting smaller.
Quadratic Equations Converting and Formula PowerPoint
Slide 1
Quadratic Equations
Converting
Standard to Vertex
Vertex to Standard
Quadratic Formula
Guided Notes Presentation
STEM Integrated Unit
Slide 2
Standard Form
→
Vertex Form
ax² + bx + c → a(x-h)² + k
STEPS
1) a = the same “a” in each equation
2) find h. h = -b/2a
3) plug in “h” for x into the original equation. The
answer is k.
4) put in the values of a, h, and k into the vertex
form. Remember to use the opposite of h.
Slide 3
Standard Form
→
Vertex Form
ax² + bx + c → a(x-h)² + k
Example
a=1
+ 71
y = x² + 16x + 71
h = -b/2a
k = (-8)² + 16(-8)
= -16/2(1)
= 64 – 128
+ 71
= -8
(
=7
8)²
7
Slide 4
Vertex Form
→
Standard Form
a(x-h)² + k → ax² + bx + c
STEPS
1) Foil
2) Distribute
3) Combine Like Terms
Slide 5
Vertex Form
→
Standard Form
a(x-h)² + k → ax² + bx + c
Example
y = 1(x – 8)² + 7
y = 1(x² + 16x + 64) + 7
y = x² + 16x + 64 + 7
y = x² + 16x + 71
y = x² + 16x + 71
Slide 6
Quadratic Formula
Use the quadratic formula to find the solutions to
the equations. These are also know as the xintercepts.
Slide 7
Find the Solutions
Using the
Quadratic Formula
Example x² – 5x – 14
5
√(-5)² – 4(1)(-14)
2(1)
5
√81
2
5 + 9 and 5 – 9
2
2
= 7 and = -2
Science Density PowerPoint
Slide 1
Density
1. What makes elements different?
2. What is density and how is it calculated?
3. Why do different elements have different
densities?
4. How can density be used to identify a pure
substance?
5. What is the relationship between density and
the composition of the atomic nucleus for an
element?
Slide 2
Vocabulary
•
•
•
•
•
•
•
•
Mass
Volume
Density
Atom
Proton
Neutron
Electron
pure substance
•
•
•
•
•
•
•
•
Element
Compound
Physical property
Chemical property
Characteristic property
Nucleus
Electron cloud
Mass number
Slide 3
What makes elements different?
Different elements have different numbers
subatomic particles (protons, neutrons, and
electrons).
The number of protons in the nucleus of the
atom (atomic number) defines the identity of an
element.
The mass of an element is determined by the
number of protons and neutrons in the nucleus
of the atom (mass number) , as electrons
contribute very little mass to the atom.
Slide 4
Relative masses of subatomic
particles
http://www.education.com/study-help/article/structure-atom/
Slide 5
What is Density?
Density is a physical property of matter that
describes the amount of mass per volume for a
substance.
Density for a substance can be calculated
using the formula Density = mass ÷ volume
D=m/v
The density of pure substances, elements and
compounds, are unique. (characteristic
property)
Slide 6
How can density be used to identify a
pure substance?
The mass of a pure substance is dependent on
the elements that make up that substance.
The masses of the elements are determined by
the number of protons and neutrons in the
nucleus of their atoms.
Since the identity of a pure substance is
determined by the composition of the atoms that
make up the pure substance, a given volume of
any pure substance will have a different mass.
Slide 7
By calculating the density of an unknown
pure substance, you can identify the pure
substance by comparing the calculated density
to the known densities of pure substances.
The known densities of pure substances can
be found in the Handbook of Chemistry and
Physics
Slide 8
What is the relationship between density and
the composition of the atomic nucleus for an
element?
Since each element has a unique number of
protons, each element has a unique mass.
Some atoms of the same element have different
numbers of neutrons (isotopes).
This means that the masses of the different
elements in a given volume will be different.
Since density is a ratio between mass and volume
each element will have a unique density that is
related to the number of protons and neutrons in the
nuclei of the atoms of that element.
Slide 9
Thought Questions
1. What makes an atom of gold different from an
atom of silver?
2. Aluminum has a density of 2.7 g/cm3. If a
sample of metal has a mass of 55.0g and a
volume of 20.4 cm3, is this metal aluminum?
Explain.
3. Metal A has a volume of 1.5cm3 while metal B
has a volume of 1.0cm3. Their masses are
equal. Which metal is more dense? Explain.
Science Newton’s Laws of Motion PowerPoint
Slide 1
Mass
The mass of an object is a measure
of the inertia of the object. Also, it is
the amount of or quantity of matter.
Inertia is the tendency of a body at
rest to stay at rest, and of a body in
motion to stay in motion with
unchanged velocity.
Slide 2
Force
What is force?
Force is that which changes the velocity of
a object.
Force is a vector quantity. What is vector
quantity?
Slide 3
Net External Force
NET external force causes the object to
accelerate in the direction of that force.
The acceleration is directly proportional to the
force and inversely proportional to the
mass of the object.
Slide 4
The Newton
The SI unit of force is the Newton.
1N is that resultant force which will give a 1kg
object an acceleration of 1 m/s2
1 N = 1 kg · m/s2
1 pound = 4.45 N
Slide 5
Newton’s 1st Law
An object at rest will remain at rest at rest; an
object in motion will continue in motion with
constant velocity except when acted upon
by an external force. (Law of Inertia)
Slide 6
Newton’s 2nd Law
If a net force acts on a object, the object will
accelerate in the direction of the net force.
The acceleration is proportional to the force and
inversely to the mass of the object. (Law of
Acceleration)
F=m·a
F = force
m = mass
a = acceleration
Slide 7
Newton’s 3rd Law
For each force exerted on one body, there is
an equal, but oppositely directed, force on
some other body interacting with it. (Law of
Opposites)
Slide 8
Weight
What is weight?
It is the gravitational force acting downward
on an object.
It is measured in newtons or pounds.
Slide 9
Tensile Force
What is tensile force?
The force tending to stretch a string, cable,
etc.
The magnitude of the tensile force is the
tension.
Slide 10
Friction Force
A tangential force (what is this?) acting on an
object that opposes the sliding of the object
on an adjacent surface with which it is
contact.
The friction force is parallel to the surface
and opposite to the direction of motion.
Only when the applied force exceeds the
static friction will the object slide.
Slide 11
Normal Force
The normal force on an object that is being
supported by a surface is the component of
the supporting force that is perpendicular to
the surface.
Slide 12
Coefficient of Kinetic Fricion
This is used to define what we call friction
when one surface is sliding across another
at constant speed.
It is identified as friction force/normal force.
Slide 13
Coefficient of Static Friction.
Static Friction is defined as when one surface
is on the verge of sliding across another.
Slide 14
Thought questions:
1.
2.
3.
Calculate the force of an object that has a
mass of 2.5 kg and an acceleration of 10
m/s2? 25 N
Calculate the mass of an object that has a
force of 15 N and an acceleration of 5
m/s2?
75 kg
Calculate the acceleration of an object
that has a force of 10 N and a mass of 20
kg?
2 m/s2
Density Cubes Lab
Name:
Purpose: To explore the physical property of density of different pure substances and relate the density of
materials to the identity of a pure substance.
Essential questions:
1.
2.
3.
4.
What is density and how is density calculated?
Why does the density of a substance remain the same regardless of sample size?
Why do different pure substances have different densities?
How is the density of an element related to the number of protons and neutrons in the
nucleus of an atom of an element?
GPS covered in this lab:
Physical Science:
SPS1. Students will investigate our current understanding of the atom.
a. Examine the structure of the atom in terms of
•
•
•
proton, electron, and neutron locations.
atomic mass and atomic number.
explain the relationship of the proton number to the element’s identity.
SPS2. Students will explore the nature of matter, its classifications, and its system for naming types
of matter.
a. Calculate density when given a means to determine a substance’s mass and volume.
Chemistry:
SC1 Students will analyze the nature of matter and its classifications.
b. Identify substances based on chemical and physical properties.
SC3 Students will use the modern atomic theory to explain the characteristics of atoms.
a. Discriminate between the relative size, charge, and position of protons, neutrons, and electrons in
the atom.
c. Explain the relationship of the proton number to the element’s identity.
Characteristics of Science:
SCSh5. Students will demonstrate the computation and estimation skills necessary for analyzing
data and developing reasonable scientific explanations.
b. Consider possible effects of measurement errors on calculations.
e. Solve scientific problems by substituting quantitative values, using dimensional analysis and/or
simple algebraic formulas as appropriate.
Materials:
• Electronic balance
• 6 cubes of metal (1cm x1cm x 1cm) Pb, Cu, Zn, Al, Fe, Brass
• Metric ruler
Procedure:
1. Obtain a sample of each different metal. Observe the physical characteristics of each
metal and record this in the data table.
2. Measure each dimension, length, width, and height, and record this in the data table.
3. Place each sample of metal on the balance and record the mass in grams in the data table.
4. Using the formula D = m/v calculate the density of each sample of metal and record this
on the data table.
5. Using the accepted values for each metal determine the identity of each metal.
Data Table
Physical Description
Length
Width
Height
Volume
Mass
Density
cm
cm
cm
LxWxH
(cm3)
g
D=m/v
g/cm3
Identity
Accepted Values:
Pb = 11.3 g/cm3
Cu = 8.9 g/cm3
Fe = 7.9 g/cm3
Brass = 8.6 g/cm3
Al = 2.7 g cm3
Zn = 7.1 g/cm3
Analysis Questions:
1.
2.
3.
4.
What is density and how is density calculated?
Why does the density of a substance remain the same regardless of sample size?
Why do different pure substances have different densities?
How is the density of an element related to the number of protons and neutrons in the
nucleus of an atom of an element?
5. Did your calculations exactly correlate with the accepted values for each metal? Explain
why or why not?
Density Quiz
1. Density is a ratio between:
a. Volume and number of electrons in a substance
b. Mass and number of protons in a substance
c. Mass and volume for a substance
d. The number of each subatomic particle in a substance
2. The number of _____ in the nucleus of an atom of an element defines the element
a. protons
b. neutrons
c. electrons
d. particles
3. A proton is found _____ and has a charge of _____.
a. outside the nucleus; 1+
b. outside the nucleus; 1c. in the nucleus; 1d. in the nucleus; 1+
4. The mass of a pure substance is dependent upon
a. the number of protons in the atoms that make up the substance
b. the number of neutrons in the atoms that make up the substance
c. the number of protons and neutrons in the atoms that make up the substance
d. the number of electrons in the atoms that make up the substance
5. Which of the following properties can be used to identify an unknown sample of a pure
substance?
a. weight
b. mass
c. volume
d. density
6. The number of protons and neutrons in the nucleus of an atom is called:
a. mass number
b. atomic weight
c. atomic number
d. average atomic mass
7. If two samples of substances are analyzed and found to have different masses and volumes, but
the same density, which of the following is true:
a. they are different substances
b. they are the same size
c. they will float in water
d. they are the same substance
8. A substance has a mass of 25.0g and a volume of 5.0mL. Calculate density/
a. 125g/mL
b. 5.0g.mL
c. 0.20g/mL
d. 30.0g/ml
9. What volume would an object occupy that has a mass of 30.0g and a density of 6.0g/cm3?
a. 5.0cm3
b. 6.0cm3
c. 180cm3
d. 0.20cm3
10. What is the mass of a substance that has a density of 0.75g/mL and occupies a volume of 2.5mL?
a. 3.33g
b. 0.30g
c. 5.25g
d. 1.88g
Newton’s Laws of Motion Quiz
Name: ___________________
Directions: Place the letter of the term/statement that best fits the question on the line provided.
1. ______ Which of the following describes the amount or quantity of matter in an object?
a. Force
b. Mass correct
c. Acceleration
d. Inertia
2. ______ Force is best described as:
a. The velocity times time for an object in motion
b. The quantity of matter in an object
c. The tendency of an object in motion to remain in motion
d. That which changes the velocity of an object correct
3. ______ Which describes the relationship between acceleration, force, and mass?
a. Acceleration is directly proportional to the force and inversely proportional to the mass
of an object correct
b. Acceleration is directly proportional to the mass and inversely proportional to the force
on an object
c. Acceleration is directly proportional to both the force and the mass of an object
d. There is no relationship between acceleration, force and mass for an object
4. ______ Which of the following represents Newton’s 2nd law of motion?
a. The law of Inertia
b. The law of opposites
c. The law of Acceleration correct
d. The concept of weight
5. ______ Which of the following terms best describes “an object at rest will remain at rest and
an object in motion will remain in motion unless acted upon by an external force”?
a. Inertia correct
b. Force
c. Acceleration
d. Weight
Sample Image of Catapult Excel File
State/Define the Problem
1
1
State/Define
the Problem
Does Not
Meet
Expectations
(0-25% of
points)
Offers an unclear
statement of the
problem. Little or no
work is evident.
2
Attempted
to Meet
Expectations
(25-50% of
points)
Problem is vaguely
stated and does not
lead to collecting
information.
Meets
3
Expectations
(50-75% of
points)
Problem is stated but
lacks specific
information and does
not lead to collecting
information.
4
Surpasses
Expectations
(75-100% of
points)
States the problem
correctly and
thoroughly, leading to
collecting information.
In the space provided below, define the problem. Follow the rubric above for guidance.
Collect Information Through Brainstorming
2
COLLECT
INFORMATION
AND
BRAINSTORM
1
Does Not
Meet
Expectations
(0-25% of
points)
Little research and
brainstorming
accomplished.
Ideas generated
are not original.
2
Attempted
to Meet
Expectations
(25-50% of
points)
Research is
evident as an
outcome of
brainstorming.
Ideas generated
are a result of the
brainstorming
process and not
original.
Surpasses
Expectations
Meets
3
Expectations
(50-75% of
points)
Ideas generated
are new and
original as an
outcome of
brainstorming and
research. Little
suggestions are
offered for the rest
of the design
process if any.
4
(75-100% of
points)
Many new ideas
are generated as
an outcome of
brainstorming and
research.
Suggestions and
details are given
for design
constraints of the
product leading to
developing possible
solutions.
In the space provided below, research, brainstorm, and develop your ideas. Be sure to write down any
relevant information as evidence of you thoughts. Follow the rubric above for guidance.
Develop Possible Solutions
3
CONCEPTUAL
DESIGN AND
SKETCHING
1
Does Not
Meet
Expectations
(0-25% of
points)
Only one thumbnail
sketch is offered
Additionally,
accurate design
specifications and
thorough
annotations are
clearly noted on
the sketches,
exemplifying the
brainstorming
process.
Constraints are
also considered and
noted.
2
Attempted
to Meet
Expectations
(25-50% of
points)
Two thumbnail
sketches are
offered
Additionally,
accurate design
specifications and
thorough
annotations are
clearly noted on
the sketches,
exemplifying the
brainstorming
process.
Constraints are
also considered
and noted.
Meets
3
Expectations
(50-75% of
points)
Three or Four
thumbnail sketches
are offered
Additionally,
accurate design
specifications and
thorough
annotations are
clearly noted on
the sketches,
exemplifying the
brainstorming
process.
Constraints are
also considered and
noted.
4
Surpasses
Expectations
(75-100% of
points)
Multiple thumbnail
sketches are
offered (minimum
of 5). Additionally,
accurate design
specifications and
thorough
annotations are
clearly noted on
the sketches,
exemplifying the
brainstorming
process.
Constraints are
also considered
and noted.
The following pages provide you with the space needed to create brainstorming thoughts and ideas
and to develop a minimum of five thumbnail sketches. Be sure to use the rubric above for guidance.
Select the Best Solution
4
1
Does Not
Meet
Expectations
(0-25% of
points)
Very vague sketch is
drawn
And/Or
DEVELOPING
THE DESIGN
Very vague reasoning
as to why solution was
chosen.
2
Attempted
to Meet
Expectations
(25-50% of
points)
Selects best solution
and sketch is vague
but offers no
reasoning as to why
solution was chosen.
Meets
3
Expectations
(50-75% of
points)
Selects best solution
and offers a reason,
but shows no
sketches of solution.
Or
Or
Selects best solution
offering reasoning as
to why solution was
chosen but no sketch
is drawn.
Shows a rough sketch
but does not offer
reasoning behind
solution.
4
Surpasses
Expectations
(75-100% of
points)
Student provides
reasoning as to why
solution was chosen.
Student provides a
detailed sketch,
labeling parts,
providing dimensions,
and offers detailed
notes as to how
device will work.
The following area provides you with the space needed to create a detailed drawing of your
prototype. Be sure to use the rubric above for guidance.
Implement Solution
5
1
Does Not
Meet
Expectations
(0-25% of
points)
Student only tests
solution
2
Attempted to
Meet
Expectations
(25-50% of points)
Student tests solution but
no documentation of test is
recorded
Or
IMPLEMENT
SOLUTION
Does not make changes to
improve solution
Meets
3
Expectations
(50-75% of
points)
Student tests solution,
records results,
makes changes, but
does not provide
documentation as to
why the changes were
made.
4
Surpasses
Expectations
(75-100% of
points)
Student tests solution,
records results,
makes changes to
improve solution, and
provides thorough
documentation as to
why the changes
were made.
Or
Provides documentation as
to why changes were
made.
In the space provided below, write down any observations or notes about the test performed and
recorded data. Describe what changes are necessary and provide any revisions.
Be sure to use the rubric above for guidance.
Evaluate Solution
6
1
Does Not
Meet
Expectations
(0-25% of
points)
Student offers no
evaluation of the
solution.
EVALUATE
SOLUTION
2
Attempted
to Meet
Expectations
(25-50% of
points)
Student shows little
understanding as to
why the solution did
or did not work and
does not use any key
terms discussed in
class.
Meets
3
Expectations
(50-75% of
points)
Student shows
understanding as to
why the solution did or
did not work and uses
only a few key terms
discussed in class.
4
Surpasses
Expectations
(75-100% of
points)
Through analytical
reasoning, student shows
excellent understanding as
to why solution did or did
not work using all key
terms discussed.
The following area provides you with space to write a detailed evaluation of this project.
Be sure to use the rubric above for guidance.
Mathematical Summary of Catapult Data
1. What are the axis of symmetry and the vertex of the graph, and what do they mean?
2. What are the domain and the range of the graph? Explain how you found the domain and the range.
3. At what points is the graph increasing and decreasing? Explain how you determine this information.
4. Based on the information of each subgroup tests, how did your graph and equation change?
Launcher Design Brief
With the materials below, you must build a device that will launch a ping pong ball a distance of 10 feet
to the center of a target.
You cannot build a slingshot!!!!!!!
Materials
1 ea Paper Cup
1 ea Straw
1 ea 1’’ x 2’’ x 6’’ piece of wood
1 ea ¼’’ x 3’’ dowel rod
1 ea ¼’’ x ¼’’ x 24’’ bass wood
2 ea eye screws
2 ea Rubber Bands
2 ea Push Pins
Glue gun to apply hot glue.
You may only use the materials provided. If you lose or break any material, you will not be given
more materials.
The dimensions of the target are the following:
Black center bull’s eye is 10” diameter.
White part of target is 20” diameter.
Red part of target is 33” diameter.
The following points will be awarded for hitting the target:
Black bull’s eye = 10 points
White area of target = 7 points
Red area of target = 5 points
Height Banner Image
Bulls Eye Target Image
Launch Data Sheet
Launch the catapult and record X,Y measurements
Point 2
(X2, Y2)
(____, ____)
Y2
Point 1
Point 3
(X1, Y1)
(0, 0)
(X3, Y3)
(____, 0)
X2
X3
Point 1
X1 =
Y1=
(
,
)
Point 1
X1 =
Y1=
(
,
)
Point 2
X2 =
Y2=
(
,
)
Point 2
X2 =
Y2=
(
,
)
Point 3
X3 =
Y3=
(
,
)
Point 3
X3 =
Y3=
(
,
)
Point 1
X1 =
Y1=
(
,
)
Point 1
X1 =
Y1=
(
,
)
Point 2
X2 =
Y2=
(
,
)
Point 2
X2 =
Y2=
(
,
)
Point 3
X3 =
Y3=
(
,
)
Point 3
X3 =
Y3=
(
,
)
Point 1
X1 =
Y1=
(
,
)
Point 2
X2 =
Y2=
(
,
)
Point 3
X3 =
Y3=
(
,
)