Projects in Mathematics

1. Projects in Mathematics Adaptable For All Classes Presented by: Marianne Ilmanen Joe Henderson

2. Why Projects ? • Projects give students a longer period of time to solve a problem than is available in a class period. • Projects allow students to deepen their understanding of math with hands-on, problem-solving activities. • Projects help to improve student reasoning, problem-solving, communication and make math connections to real world situations • Projects can be done in groups that require students to contribute to a group effort and to accept group consensus.

3. Sample Projects • Projects that Incorporate Literacy into Mathematics: Career Investigation • In Class Group Projects: Review for Final Exam • Research Projects: Standardized Test Question Analysis • Real World Topics that Reinforce the curriculum: Recipe for MathPyramid PowerMapping Distance in a Plane • Projects Adapted for Different Course Levels: Function Families Volume of a Box

4. Career Investigations Project Project Description: Students investigate a career that has strong mathematics base from a list of careers. - Students are given a list of careers that have a mathematics base, a set of questions to be addressed and the project grading rubric. • - Students choose a career from the Career List. • - Students research their chosen career using www.bridges.com at the career center or at home.

5. Career Project Questions • What are the duties of the career? • What academic path is required for the career? • What math classes do you need to take during your education to prepare for this career? • How is math important for this career? • Which schools have respected programs in this field? • Are there jobs available in this career in your area?

6. Career List Actuary Administrator Bank financial managerBank loan officer Business consultant Cash flow analystCommunications consultant Computer aided designer Computer network designerComputer software designer Computer technician ConstructionConsumer behavior analyst Financial analyst Corporate plannerCost account Customer service rep Demographic analystEconomic analyst Engineer Environmental forecasterFactor analyst/medical research Insurance analyst Factor analyst/ social systemsHuman resources manager Investment analyst Industrial cost controllerManagement consultant Marketing consultant Modeler of genetic systemsPension fund controller Policy change analyst Portfolio managerProduct designer Production planner Program analystResearch data analyst Researcher Resource analystSafety coordinator Salary and benefits analyst Statistical consultantStock and bond analyst Tax consultant Taxation systems consultantTeacher/professor Urban planner Veterinary Medicine Agricultural Management Biomedical Research Petroleum Engineering Construction Management Independent Music Production CartographyAerospace Medicine Astronomical Research Meteorology Food Management Operations Research Government Finance Special Education Public Accounting Computer Consulting Manufacturing Engineering Architect Structural Engineer Navigator Physicist Seismologist Surveyor Product Manager Statistician Math Textbook Editor Opinion Researcher Actuary Math TV Content Director U.S. Navy Officer Airplane PilotHelicopter Pilot Casino Manager

7. Rubric

8. Final Exam Review Project Description: Students identify the maintopics that are covered on the final exam, using post-it tabs.

9. Linear Equations X and Y intercepts Slope Slope-intercept form Point-slope form Standard form Labeled Graph Final Exam Review Project Directions • Provide a sheet of butcher paper for each general topic. • Title each sheet with a topic. You may add any key questions that you want addressed.

10. 3. Pre-select groups of 4 students for each group. • Arrange your room into clusters of four desks. Place a poster on each table and assign a group to each. WARNING DO NOT let the students pick their own groups! • Give each group a different color marker so that their contributions to each poster can be identified.

11. 6. Have the groups rotate to each poster at one minute intervals. • At each stop, the group will provide the definition of one item on the poster. They can write out a definition or draw a picture with labels. Each group should havea person assigned to each of the following job assignments: • One writer/recorder • Two journal researchers • One person to check for accuracy

12. Example of Rotation

13. When the groups have rotated through all of the stations, they are assigned to check and edit the poster at their last station. • Each student is given a copy of the Review Outline. • 10. Each group presents their posters while the other students complete their individual outlines.

14. Standardized Test Question Analysis Project Purpose: To allow students to assess their own math knowledge, to determine if the test is trying to confuse them in the question or with the answers and to discover if the answers contain any “boobie traps”.

15. Preparation: • Select a set of standardized test questions, so that each student will have a unique question. • Prepare a large scale version of the questions for use on the overhead projector or for use in power point. • Give each student one of the selected standardized test questions and a copy of the rubric.

16. Independent Student Work Students answer a set of questionsabout their multiple choice problem. Students type their answers in paragraph form…or make a poster. Students present their answers in class to the other students.

17. Activities and Questions • Draw a diagram or picture. • What math knowledge do you have to have to answer the question? • How are they trying to confuse you in the question? • Show your work and answer the math question. • How are they trying to confuse • you with the answers? • 6. Which answers are “boobie traps” ?

18. Rubric

19. Recipe for Math Project Project Description: Using the recipe for HERSHEY'S "PERFECTLY CHOCOLATE" CHOCOLATE CHIP COOKIES, students will find algebraic relationships of the ingredients.

20. Student Investigations 1. Students write an equation relating the number of cups of brown sugar (S) to the number of cups of flour (F ) in the cookie recipe. Using their equation, students determine the amount of flour or brown sugar would be needed in different situations: For a double recipe of cookies   For a recipe that increases the amount of flour For a recipe that increases the amount of brown sugar

21. Using a Graph 2. Students graph the equation relating the amount of brown sugar to the amount of flour and use the graph to answer these questions: - How much brown sugar will be needed if 10 cups of flour are used. - How much flour will be needed if 6 cups of brown sugar are used.

22. Making Connections Students are asked to re-write the cookie recipe for a bakery that makes 24 dozen cookies in each batch.

23. Pyramid Project Project Description: Students will make a scale model of the Great Pyramid of Khufu. From the scale model, they will determine estimates of the actual measures.

24. Model • Using the pattern provided, each student creates a scale model of the Great Pyramid • 2. Given the scale of the model, students determine the actual measurements of the Great Pyramid. • 3. Students determine the slope of each face of the Great Pyramid from the measurements.

25. Making Connections 4. Students draw a representation of the Great Pyramid on a coordinate System, determinethe coordinates of the cornersand the slope of the edge of the face. 5. Students construct their own scale models of the other two pyramids of the Giza Plateau, Khafre and Menkaure.

26. Distance on a Plane Project Description: Using maps of Sacramento and Long Beach, students investigate different ways to determine the distances between different landmarks.

27. Sacramento Students investigate the drivable distance and direct distance between different buildings

28. Long Beach Students use a map of Long Beach with a coordinate grid to investigate relationships of lines that connect schools.

29. Function Families Project Description: Function Families are sets of functions with similar properties. The functions used are those studied in each course and enhance the concepts that the student has learned. Functions: Linear Equations Quadratic Equations Various Functions Course: Algebra Int Algebra Advanced Math

30. Linear Function Families The purpose of the project is to investigate the properties of linear equations and their graphs.Students are expected to prepare a report for 10 different equations that contains the following data.  1) The equation in slope-intercept form2) The equation in standard form3) An x-y chart of at least 5 coordinate pairs4) The graph on a coordinate system5) The Domain 6) The Range7) The x-intercept8) The y-intercept9) The slope10) A description of the line that includes its direction, path and how it compares to the graph of y = x

31. Sample Page for Linear Equations • y = 2x + 4 • The given equation, y = 2x + 4, is in slope-intercept form • The equation in standard form is 2x – y = - 4 • 3) An x-y chart of coordinate pairs for this equation • 4) The graph on a coordinate system  • 5) The Domain for this equation is x = { all real numbers}  • 6) The Range for this equation is y = { all real numbers}  • 7) The x-intercept is at ( -2, 0 )  • 8) The y-intercept is at ( 0, 3 )  • 9) The slope is m = 2  • 10) The graph is a line that is slanted up to the right. The path between any two points is one space to the right and two spaces up. The line is steeper than the line y = x

32. Quadratic Equation Families The purpose of the project is to investigate the properties of quadratic equations and their graphs.Students are expected to prepare a report for 10 different equations that contains the following data. 1) The equation in general form2) The equation in vertex form3) The equation in factored form4) The graph on a coordinate system5) The Domain6) The Range7) The x-intercepts, if any 8) The y-intercept, if it exists9) The location of its vertex, and identify it as a maximum or minimum10) A description of how the parabola compares to the graph of y = x2 that includes its direction, width, location of the vertex and the number of real roots.

33. Sample Page for Quadratic Equations y = x2 + 2x  15 1) The given equation is in general form:: y = x2 + 2x  15 2) The equation in vertex form: y = ( x – 1 ) 2 – 16 3) The equation in factored form: y = (x + 5) ( x  3) 4) The graph on a coordinate system 5) The Domain: x = { all real numbers } 6) The Range: y ≥ -16 7) The x-intercepts are at ( - 5, 0 ) and ( 3, 0 ) 8) The y-intercept is at ( 0, -15 ) 9) The vertex is at ( -1, -16 ). The vertex is the minimum value of y. 10) The parabola is the same size and opens upward the same as y = x2 . The vertex has been moved to the point ( -1, -16 ). It has two real roots.

34. Function Families The purpose of the project is to investigate the properties of a variety of functions and their graphs.Students are expected to prepare a report for 10 different equations that contains the following data.  a) The name of the functionb) The format of the equation of the functionc) An example of the function 1) The equation 2) The graph on a coordinate system 3) The Domain, including any undefined values, if they occur 4) The Range 5) The roots (if, any exist) 6) The y-intercept 7) The intervals where it increases, decreases or is constant 8) The inverse function, and any restrictions that may apply so that it is a function

35. Functions Used 1) The Constant Function: y = k 2) The Linear Function: y = mx + b or ax + by = c 3) The Quadratic Function: y = ax2 + bx + c or y = a(x  x1)(x  x2) 4) The Cubic Function: y = ax3 + bx2 + cx + d or y = a(x  x1)(x  x2)(x  x3) 5) The Radical (Square Root) Function: 6) The Cubic Root Function: 7) The Greatest Integer (Step) Function: y = a [[ x  k ]] + h 8) The Absolute Value Function: y = a | x  k | + h 9) The Piecewise Function: 10) The Rational Function:

36. Sample Page for Functions Family The Quadratic Function The Standard Format of the function is y = ax2 + bx + c or y = a(x  x1)(x  x2) Example: y = x2 16 or y = (x + 4) ( x  4) Domain: all real numbers Range: y  -16 Roots: x = -4 and x = 4 Y-intercept: ( 0, -16) The function increases for x > 0, and decreases for x < 0 The inverse function is , for x  0

37. Volume of a Box Project Description: Students in Geometry, Precalculus and Calculus investigate the volume of a box made by cutting out congruent squares from each corner of a sheet of cardboard and folding up the sides. Course: Geometry Precalculus Calculus Project: Volume and Surface Area Volume as a function Minimize the volume function

38. x 8-2x 10-2x Volume and Surface Area Project Description: Students make a box and calculate the volume and the surface area of the box. Then they compare the size of the square that was cut from the cardboard to the surface area and the volume of eight different boxes.

39. x 8-2x 10-2x Volume as a Function V(x)=x(10-2x)(8-2x) Project Description: Students make a box andcalculate the volume. They also write the function V(x) to determine the volume, with x as the measure of the side of the squares. Using the graph of V(x), the students analyze the function

40. x 8-2x 10-2x Volume Function and Derivative Project Description: Students make a box and calculate the volume. They also write the volume as the function V(x). They find the derivative V ’(x) to determine the critical values of x and the maximum volume. V(x)=x(10-2x)(8-2x) V(x)=4x3- 36x2+80x V(x)=12x2 – 72x + 80