Course title
METPre-requisite
Algebra I, Algebra II Must have demonstrated excellence in the first two years of high school math and science. Prerequisites: MET Engineering I and MET Engineering I/ MET Professional Internship Co-requisite: MET Engineering II and MET Engineering II/ MCourse description
MET Engineering II is the second course of the MET Professional Academy Engineering Strand. The MET Engineering strand is designed for students who are interested in a combination of rigorous science and engineering fundamentals; entrepreneurship; and innovation. The student learning experience can be individually tailored to engineering interests such as; electrical; mechanical; aeronautics; etc. and other principles including renewable energy and biomedical engineering. College-level classroom activities and lab work will advance MET students’ understanding of the fundamentals of engineering; entrepreneurship and innovation. Students continue to work on EPICS/high and Seed Spot Next projects. Students are introduced to problem modeling; heuristics and algorithms. Students learn mathematical modeling with Excel. Students learn to questioning and interviewing techniques; how to evaluate customer experiences and interfaces; and onboarding. Students learn how to develop value propositions; cost structuring and revenue streams for their chosen project. Students learn to improve their social media presence; public speaking and presentation skills. Students have the opportunity for OSHA-10 certification. Students will have the opportunity to participate in the Phoenix Youth Startup Weekend; EPICS Showcase and the Seed Spot Next Showcase. Dual credit may be obtained from Glendale Community College through the MET Professional Academy
MET Engineering II: This course is designed to introduce students to the real world of Engineering Design; problem solving and mathematical modeling through the design thinking process. The course is designed for any student who intends to pursue future training in any engineering program or technical industry. Students use problem-solving models to improve existing products while utilizing development processes to create new products. They will learn how to apply these models to solve problems in and out of the classroom setting. Students will apply mathematical and scientific concepts and models when being introduced to the fundamentals of mechanical; structural (civil; architectural and green technology); aeronautical; robotic; computer and electrical concepts. Students learn mathematical modelling; multi-criteria decision making; design thinking; human centered design; Engineering Projects in Community Service (EPICS/high); business model canvas; social entrepreneurship (through Seed Spot Next); experiment design; and making. Students will use multiple software packages; engineering poster sessions; and presentations to develop technical communication literacy skills. Project based instruction; lab activities and classroom discussions will be used to create a foundation for problem solving and engineering concepts. Dual credit may be obtained from Glendale Community College for this program. Honors credits are awarded with this course. Students mathematically model mechanical and electrical systems; truss analysis; and geometric; predicative and growth analysis. This class incorporates various technological processes and manufacturing technologies. Students will develop a sense of the interdependency of the various engineering disciplines. Students will also develop an understanding that engineering is a human endeavor intended to address the needs of a global society. Utilizing activities; projects and problems; students will develop the skills to solve problems using math; science; and technology in engineering processes to benefit society. The use of CAD/CAM industry based software and equipment is an extensive part of this class. Students will be introduced to state of the art microprocessors and software. Students will apply concepts of mechanical; electrical and control systems in various design problems. This course will continue to develop technical communication literacy (reading; writing; and speaking. Students utilize three dimensional geometric relationships as they develop their skills in three dimensional modeling. Sample problems presented in this class include:
I. When an oil company drills a new oil well they are only capable of extracting approximately 35% of the oil available. Your job is to develop a way that is economically viable and environmentally responsible; to extract at least 50% more of the oil. Requires the use of volumes; mathematical functions; statistics; and defending a mathematical argument. Students need to consider flow rates; viscosity; temperature; pump pressures; and pipe diameters.
II. The Ogallala Aquifer occupies the High Plains of the United States; extending from western Texas to South Dakota. The Ogallala Aquifer is used to water the crops and cattle that we use for food. We will no longer be able to get water from the Ogallala Aquifer sometime in next 50 years. Your job is to develop a way to continue to water the crops and cattle of the American Midwest without using the aquifer. Your solution must be economically viable and environmentally responsible. Students need to consider the possibility of geometric regression for water usage (from the aquifer); geometric growth for water usage; flow rates; volume; mathematical functions; statistics and be able to defend a mathematical argument.
III. The current EPA regulations are making it increasingly difficult to use the most abundant energy source in the United States. With the current technology; alternative energy is unreliable. This leaves nuclear energy as the best option at this time. When nuclear fuel rods are removed as “used” from the reactor; 99% of the uranium has not been used. Your job has two parts. First; you must develop a way to recycle the fuel rods so that they may be reinserted into the reactor. Second; develop a way to utilize the radiation energy being emitted from the waste material. Your solution must be economically viable and environmentally responsible. Note: The issue with storage of nuclear waste in the United States is a political issue; not one of science or technology. Students need to use exponential functions; geometric series; geometry; combinational number theory; functions and be able to defend a mathematical argument.
IV. The average family in the United States deposits 4.3 pounds of solid waste into landfills every day. This creates a large impact on open spaces and land resources. Your job is to develop a way (other than recycling) to reduce the impact of the amount of waste that is placed in these landfills. Your solution must be economically viable and environmentally responsible. Students need to consider exponential growth; geometry; and statistics and be able to defend a mathematical argument.
V. The current scientific theories claim that global climate change is the result of human activity in the form of releasing carbon dioxide into the atmosphere. It is necessary to continue using the energy that creates the carbon dioxide in order to maintain the global economy. Your job is to develop an economically viable and environmentally responsible way to sequester the carbon dioxide that is being emitted into the atmosphere. Students need to consider geometric series; geometry; statistics and algebraic functions as well as being able to defend a mathematical argument.
VI. One sixth of the world’s population does not have access to potable drinking water. Your job is to develop a way to provide a sustainable method to provide safe; clean drinking water to these populations. You may assume that a water source is available. Your solution must be economically viable and environmental responsible. Students need to consider population growth; statistics; flow rates; algebraic functions and be able to defend a mathematical series.
VII. In 2005; Hurricane Katrina devastated the city of New Orleans. Despite many overtones; not much has been done to prevent a repeat of the event. Your job is to develop an economically viable and environmentally responsible way to safeguard the people and the property of New Orleans against a repeat of a Katrina-like disaster. Students need to use geometry; statistics; probability; and algebraic functions as well as be able to defend a mathematical argument.
VIII. Approximately 12.5% of the global population suffers from chronic malnourishment. While there are many reasons for this; the primary reason is access to food. Your job is to develop a way to provide healthy food to these people. Your solution should be economically viable; environmentally responsible and sustainable. Students need to consider population growth; geometry; and statistics and be able to defend a mathematical function.
Students spend approximately 6 hours per week working on projects/labs.
Engineering III: This course applies engineering technology and skills to the manufacturing processes while recognizing that engineering is a human endeavor intended to address the needs of a global society. This course will examine the relationship of manufacturing and process development to the world of engineering. Students will use advanced engineering design; production; and programming techniques for Mobile robotics (VEX); Robotic Arm; 3 Dimensional Modeling; and Computer Numerical Control (CNC). Students will incorporate mathematical and scientific modeling and processes in order to solve real world manufacturing and production problems. Students will identify the impact of various engineering disciplines on manufacturing processes. Students will continue to utilize and improve technical communication skills (reading; writing; and speaking). Students will investigate the impact of manufacturing and robotics on both local and global societies. Career and Technical Student Organization (CTSO) standards will be an integral part of this class. Dual credit may be obtained from Embry Riddle Aeronautical University for this program at high schools offering this option. An honors option is available to all students enrolled in the course.
IX. to consider population growth; statistics; flow rates; algebraic functions and be able to defend a mathematical series.
XV. In 2005; Hurricane Katrina devastated the city of New Orleans. Despite many overtones; not much has been done to prevent a repeat of the event. Your job is to develop an economically viable and environmentally responsible way to safeguard the people and the property of New Orleans against a repeat of a Katrina-like disaster. Students need to use geometry; statistics; probability; and algebraic functions as well as be able to defend a mathematical argument.
XVI. Approximately 12.5% of the global population suffers from chronic malnourishment. While there are many reasons for this; the primary reason is access to food. Your job is to develop a way to provide healthy food to these people. Your solution should be economically viable; environmentally responsible and sustainable. Students need to consider population growth; geometry; and statistics and be able to defend a mathematical function.
Course Outline and Standard Alignment:
Introduction to Engineering Design Engineering Course Unit 1.0—Career/Portfolio
1. Career/Portfolio
A. Objective: Create electronic portfolio
1. Why it is important and content
a. Multi-media
b. Samples of work
c. Resume
d. Cover letter
e. Networking - Myspace/Facebook/LinkedIn/etc.
f. Internships
2. Create portfolio
a. Online tutorial (Front Page/Publisher/Other)
b. Upload samples of work
3. Bridges
a. Interest inventory
4. Cover letter
a. Resources
5. Resume
Arizona State Mathematical Standard or Benchmark with Engineering Standard Connections:
HS.N-Q.1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
Connections:
SCHS-S1C4-02;SSHS-S5C5-01HS.N-Q.2. Define appropriate quantities for the purpose of descriptive modeling.
Connection: SSHS-S5C5-01
HS.N-Q.3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
HS.N-VM.1. Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments; and use appropriate symbols for vectors and their magnitudes (e.g.; v; |v|; ||v||; v).
HS.N-VM.3. Solve problems involving velocity and other quantities that can be represented by vectors.
Connections: 11-12.RST.9 SCHS-S5C2-01;SCHS-S5C2-02;SCHS-S5C2-06;11-12.WHST.2d
a. Understand vector subtraction v – w as v + (–w); where –w is the additive inverse of w; with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order; and perform vector subtraction component-wise.Connection: ETHS-S6C1-03
HS.N-VM.5. Multiply a vector by a scalar.
a. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise; e.g.; as c(vx; vy) = (cvx; cvy). Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0; the direction of cv is either along v (for c > 0) or against v (for c
HS.N-VM.8. Add; subtract; and multiply matrices of appropriate dimensions.
b. Connections: 9-10.RST.3;ETHS-S6C2-03
c. HS.N-VM.9. Understand that; unlike multiplication of numbers; matrix multiplication for square matrices is not a commutative operation; but still satisfies the associative and distributive properties.
Connections: ETHS-S6C2-03;9-10.WHST.1e
HS.A-SSE.1. Interpret expressions that represent a quantity in terms of its context.
a. Interpret parts of an expression; such as terms; factors; and coefficients.
Connection: 9-10.RST.4
nterpret complicated expressions by viewing one or more of their parts as a single entity. For example; interpret P(1+r)n as the product of P and a factor not depending on P.
HS.A-SSE.2. Use the structure of an expression to identify ways to rewrite it. For example; see x4 – y4 as
(x2)2 – (y2)2; thus recognizing it as a difference of squares that can be factored as
(x2 – y2)(x2 + y2).
HS.A-SSE.3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
Connections: 9-10.WHST.1c;
11-12.WHST.1c
Factor a quadratic expression to reveal the zeros of the function it defines.
Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.
HS.A-SSE.4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1); and use the formula to solve problems. For example; calculate mortgage payments.
Connection: 11-12.RST.4
HS.A-CED.1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions; and simple rational and exponential functions.
HS.A-CED.4. Rearrange formulas to highlight a quantity of interest; using the same reasoning as in solving equations. For example; rearrange Ohm’s law V = IR to highlight resistance R.
HS.A-REI.2. Solve simple rational and radical equations in one variable; and give examples showing how extraneous solutions may arise.
HS.A-REI.3. Solve linear equations and inequalities in one variable; including equations with coefficients represented by letters.
HS.A-REI.8. Represent a system of linear equations as a single matrix equation in a vector variable.
HS.A-REI.9. Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 ´ 3 or greater).
Connection: ETHS-S6C2-03
HS.A-REI.10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane; often forming a curve (which could be a line).
HS.A-REI.11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately; e.g.; using technology to graph the functions; make tables of values; or find successive approximations. Include cases where f(x) and/or g(x) are linear; polynomial; rational; absolute value; exponential; and logarithmic functions.
Connection: ETHS-S6C2-03
HS.A-REI.12. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality); and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
HS.F-IF.1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain; then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
HS.F-IF.2. Use function notation; evaluate functions for inputs in their domains; and interpret statements that use function notation in terms of a context.
Connection: 9-10.RST.4
HS.F-IF.3. Recognize that sequences are functions; sometimes defined recursively; whose domain is a subset of the integers. For example; the Fibonacci sequence is defined recursively by f(0) = f(1) = 1; f(n+1) = f(n) + f(n-1) for n ‚â• 1.
HS.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.
Connections: ETHS-S6C2.03;9-10.RST.7; 11-12.RST.7
HS.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 appropriate domain for the function.
Connection: 9-10.WHST.2f
HS.F-IF.6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
Connections:ETHS-S1C2-01; 9-10.RST.3
HS.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.
a. Graph linear and quadratic functions and show intercepts; maxima; and minima.
Connections: ETHS-S6C1-03;ETHS-S6C2-03
b. Graph square root; cube root; and piecewise-defined functions; including step functions and absolute value functions.
Connections:ETHS-S6C1-03;ETHS-S6C2-03
c. Graph polynomial functions; identifying zeros when suitable factorizations are available; and showing end behavior.
Connections:ETHS-S6C1-03;ETHS-S6C2-03
d. Graph rational functions; identifying zeros and asymptotes when suitable factorizations are available; and showing end behavior.
Connections:ETHS-S6C1-03;ETHS-S6C2-03
e. Graph exponential and logarithmic functions; showing intercepts and end behavior; and trigonometric functions; showing period; midline; and amplitude.
Connections:ETHS-S6C1-03;ETHS-S6C2-03
HS.F-IF.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
Connection: 11-12.RST.7
a. 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.
Connection: 11-12.RST.7
b. Use the properties of exponents to interpret expressions for exponential functions. For example; identify percent rate of change in functions such as y = (1.02)t; y = (0.97)t; y = (1.01)12t; y = (1.2)t/10; and classify them as representing exponential growth or decay.
c.
Connection: 11-12.RST.7
HS.F-IF.9. Compare properties of two functions each represented in a different way (algebraically; graphically; numerically in tables; or by verbal descriptions). For example; given a graph of one quadratic function and an algebraic expression for another; say which has the larger maximum.Connections:ETHS-S6C1-03;ETHS-S6C2-03;
9-10.RST.7
HS.F-BF.1. Write a function that describes a relationship between two quantities.
Connections:ETHS-S6C1-03;ETHS-S6C2-03
a. Determine an explicit expression; a recursive process; or steps for calculation from a context.
Connections: ETHS-S6C1-03;ETHS-S6C2-03;9-10.RST.7; 11-12.RST.7
b. Combine standard function types using arithmetic operations. For example; build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential; and relate these functions to the model.
Connections:ETHS-S6C1-03;ETHS-S6C2-03
c. Compose functions. For example; if T(y) is the temperature in the atmosphere as a function of height; and h(t) is the height of a weather balloon as a function of time; then T(h(t)) is the temperature at the location of the weather balloon as a function of time.
Connections:ETHS-S6C1-03;ETHS-S6C2-03
HS.F-BF.2. Write arithmetic and geometric sequences both recursively and with an explicit formula; use them to model situations; and translate between the two forms.
HS.F-BF.3. Identify the effect on the graph of replacing f(x) by f(x) + k; k f(x); f(kx); and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
Connections: ETHS-S6C2-03;11-12.WHST.2eHS.F-BF.4. Find inverse functions.
Connection: ETHS-S6C2-03
Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example; f(x) =2 x3 or f(x) = (x+1)/(x-1) for x ≠ 1. Verify by composition that one function is the inverse of another. Read values of an inverse function from a graph or a table; given that the function has an inverse. Produce an invertible function from a non-invertible function by restricting the domain.
HS.F-BF.5. Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
Connection: ETHS-S6C2-03
HS.F-LE.1. Distinguish between situations that can be modeled with linear functions and with exponential functions.
Connections: ETHS-S6C2-03;SSHS-S5C5-03
a. Prove that linear functions grow by equal differences over equal intervals; and that exponential functions grow by equal factors over equal intervals.
Connection: 11-12.WHST.1a-1e
b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
Connection: 11-12.RST.4
c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
Connections:ETHS-S6C1-03; ETHS-S6C2-03; 11-12.RST.4
HS.F-LE.2. Construct linear and exponential functions; including arithmetic and geometric sequences; given a graph; a description of a relationship; or two input-output pairs (include reading these from a table).
Connections: ETHS-S6C1-03;ETHS-S6C2-03;11-12.RST.4; SSHS-S5C5-03
HS.F-LE.3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly; quadratically; or (more generally) as a polynomial function.
HS.F-LE.4. For exponential models; express as a logarithm the solution to abct = d where a; c; and d are numbers and the base b is 2; 10; or e; evaluate the logarithm using technology.
Connections:ETHS-S6C1-03; ETHS-S6C2-03; 11-12.RST.3
HS.F-LE.5. Interpret the parameters in a linear or exponential function in terms of a context.
Connections: ETHS-S6C1-03;ETHS-S6C2-03;SSHS-S5C5-03; 11-12.WHST.2e
HS.F-TF.1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
HS.F-TF.2. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers; interpreted as radian measures of angles traversed counterclockwise around the unit circle.
Connections: ETHS-S1C2-01;11-12.WHST.2b;11-12.WHST.2e
HS.F-TF.3. Use special triangles to determine geometrically the values of sine; cosine; tangent for π /3; π/4 and π/6; and use the unit circle to express the values of sine; cosine; and tangent for π-x; π+x; and 2π-x in terms of their values for x; where x is any real number.
Connection: 11-12.WHST.2b
HS.F-TF.4. Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
Connections: ETHS-S1C2-01;11-12.WHST.2c
HS.F-TF.5. Choose trigonometric functions to model periodic phenomena with specified amplitude; frequency; and midline.
Connection: ETHS-S1C2-01
HS.F-TF.6. Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
Connections:ETHS-S1C2-01;11-12.WHST.2e
HS.F-TF.7. Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology; and interpret them in terms of the context.
Connections: ETHS-S1C2-01;11-12.WHST.1a
HS.F-TF.8. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it find sin(θ); cos(θ); or tan(θ) given sin(θ); cos(θ); or tan(θ) and the quadrant of the angle.
Connection:11-12.WHST.1a-1e HS.F-TF.9. Prove the addition and subtraction formulas for sine; cosine; and tangent and use them to solve problems.
Connection:
11-12.WHST.1a-1eHS.G-CO.1. Know precise definitions of angle; circle; perpendicular line; parallel line; and line segment; based on the undefined notions of point; line; distance along a line; and distance around a circular arc.
Connection: 9-10.RST.4
HS.G-CO.2. Represent transformations in the plane using; e.g.; transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g.; translation versus horizontal stretch).
Connection: ETHS-S6C1-03HS.G-CO.3. Given a rectangle; parallelogram; trapezoid; or regular polygon; describe the rotations and reflections that carry it onto itself.
Connections:ETHS-S6C1-03;9-10.WHST.2c
HS.G-CO.4. Develop definitions of rotations; reflections; and translations in terms of angles; circles; perpendicular lines; parallel lines; and line segments.
Connections:ETHS-S6C1-03;9-10.WHST.4
HS.G-CO.5. Given a geometric figure and a rotation; reflection; or translation; draw the transformed figure using; e.g.; graph paper; tracing paper; or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
Connections:ETHS-S6C1-03;9-10.WHST.3
HS.G-CO.6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures; use the definition of congruence in terms of rigid motions to decide if they are congruent.
Connections: ETHS-S1C2-01;9-10.WHST.1e
HS.G-CO.7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
Connection: 9-10.WHST.1e
HS.G-CO.8. Explain how the criteria for triangle congruence (ASA; SAS; and SSS) follow from the definition of congruence in terms of rigid motions.
Connection: 9-10.WHST.1e
HS.G-CO.9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines; alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.
Connections:
ETHS-S1C2-01;
9-10.WHST.1a-1e
HS.G-CO.10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
Connections: ETHS-S1C2-01;9-10.WHST.1a-1e
HS.G-CO.11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent; opposite angles are congruent; the diagonals of a parallelogram bisect each other
School country
United StatesSchool state
ArizonaSchool city
GlendaleSchool / district Address
6330 West Thunderbird RoadSchool zip code
85306Requested competency code
Lab ScienceDate submitted
Approved
YesApproved competency code
- LPHY
- Physics