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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= ( , )