Math Week Day 4: Differentiation and Unit 5 Math

1. This PowerPoint is from day 4 of Math Week. It covers…1. Differentiation2. The math of Unit 53. The math of part 4 of Unit 4

2. High School MathThe Standards Based WayDay 4 Nicole Spiller West Georgia RESA

3. Problem of the Day If f(x) = and g(x) = 2x2 + 2, then f(g(2)) = A) 3 B) 5 C)  7 D) 7 5/9 E) 16 2/3 The graphs of y = f(x) and y = g(x) for 0 to 10 are shown in the figure below. For how many values of x is the product f (x) g( x) = 0 for 0 to 10? A)  Two B)  Four C)  Five D)  Six E)  Seven

4. Housekeeping • Breaks • Cell Phones • Restrooms • Parking Lot

5. Essential Question/Enduring Understandings • What is the Math of Unit Five? • What is the Math of Part 4 of Unit Four? • How can I use differentiation in a HS SBC? • How can the Math Support Class support differentiation?

6. Activator • Quick Talk

7. Differentiation is… • Focused on essential understandings, principles, concepts, and skills • Designed to provide respectful, meaningful, and engaging work • Flexible • Qualitative, not Quantitative • Student centered • Dynamic • Rooted in Assessment

8. A Working Definition • Differentiation is: • A teaching philosophy based on the premise that teachers should adapt instruction to student differences • Students have multiple options for acquiring content, processing and making sense of ideas and developing products so each student can learn successfully. • Handout – Comparing Classrooms

9. Preparing for Differentiation • Begin Small – Do not feel rushed to differentiate every minute off every day • Understand one’s self – Teachers are better able to differentiate if they understand themselves as learners • Start with favorite and familiar topics – It is easier if you are already comfortable with the content • Consider working in teams – Other teachers give support and encouragement and add knowledge as well as an experience base - Applied Differentiation: Making it work on the classroom. School Improvement Network. 2005.

10. Managing Differentiation • Sustain a positive climate – A welcoming, safe, positive climate is essential because differentiating instruction requires that students take risks as well as their teachers • Provide anchor activities – These activities engage students when they finish assigned work or do not know what to do • Establish and practice routines – Routines help students know when to give their attention to the teacher and how to begin and end activities. Applied Differentiation: Making it work on the classroom. School Improvement Network. 2005.

11. Tiered Lessons • A tiered lesson is a differentiation strategy that addresses a particular standard, key concept, and generalization, but allows several pathways for students to arrive at an understanding of these components, based on the students’ interests, readiness, or learning profiles. Cheryll Adams, Ph.D. Ball State University

12. Partner Reading • Team up with a partner • Partner A – Read “What is equity and how is it evident in mathematics classrooms?” • Partner B – Read “How can different learning styles be address with consistent expectations?” • Share your thoughts from your reading with your partner, then we will discuss as a group.

13. We have a choice…We can teach to the middle and hope for the best.ORWe can accommodate the full diversity of academic needs and accept the challenge of diversity by taking student readiness, student learning/personality profiles, and student interests into consideration.-Unknown

14. Paula’s PeachesA Launching Task • This task has, as a major emphasis, the ability to solve simple non-linear equation as well as the concepts of: domain, range, zeros, intercepts, intervals of increase and decrease, maximums and minimums, end behavior, as well as the idea that an equation can be seen as two functions set equal to each other where the ‘answers’ are the intersection points.

15. Paula’s PeachesA Launching Task • Factoring Quadratics with a leading coefficient of 1 • May need to review, functions have not been emphasized since Unit Two

16. Part 4 of Unit Four • Card, Marble, Dice, and Simulation Tasks • Transition from independent to dependent events • Use of histograms and frequency tables to calculate probabilities • Create and compare experimental to theoretical probabilities • Perform simulations on the Graphing Calculator • Transition to large sample spaces

17. Unit 5 – Algebra in Context • Prerequisites • Extensive work with operations on integers, rational numbers, an square roots of non-negative numbers • Concepts of similarity and transformations

18. Overview of Unit 5 • An application and extension all of the algebra standards listed as key standards addressed in Units 1 and 2 • Solving equations via factoring • Viewing solutions as points of intersection of graphs • Solving basic radical equations

19. Overview of Unit 5 • Techniques for solving rational equations with a denominator that is a rational number or a 1st degree polynomial • Connections of graphs and solutions • An emphasis on the concept of finding equivalent equations, exceptions to this concept, and the concept of solution sets

20. Essential Question/Enduring Understandings • There is an important distinction between solving an equation and solving an applied problem modeled by an equation. The situation that gave rise to the equation may include restrictions on the solution to the applied problem that eliminate certain solutions to the equation. • The definitions of even and odd symmetry for functions are stated as algebraic conditions on values of functions, but each symmetry has a geometric interpretation related to reflection of the graph through one or more of the coordinate axes. • For any graph, rotational symmetry of 180 degrees about the origin is the same as point symmetry of reflection through the origin.

21. Essential Question/Enduring Understandings • Techniques for solving rational equations include steps that may introduce extraneous solutions that do not solve the original rational equation and, hence, require an extra step of eliminating extraneous solutions. • Understand that any equation in can be interpreted as a statement that the values of two functions are equal, and interpret the solutions of the equation domain values for the points of intersection of the graphs of the two functions. In particular, solutions of equations of the form f(x) = 0, where f(x) is an algebraic expression in the variable x, correspond to the x-intercepts of the graph of the equation y = f(x).

22. Key Standards MM1A1. Students will explore and interpret the characteristics of functions, using graphs, tables, and simple algebraic techniques. c. Graph transformations of basic functions including vertical shifts, stretches, and shrinks, as well as reflections across the x- and y-axes. d. Investigate and explain characteristics of a function: domain, range, zeros, intercepts, intervals of increase and decrease, maximum and minimum values, and end behavior h. Determine graphically and algebraically whether a function has symmetry and whether it is even, odd or neither. i. Understand that any equation in x can be interpreted as the equation f(x) = g(x), and interpret the solutions of the equation as the x-value(s) of the intersection point(s) of the graphs of y = f(x) and y = g(x).

23. Key Standards MM1A2. Students will simplify and operate with radical expressions, polynomials, and rational expressions. a. Simplify algebraic and numeric expressions involving square root. b. Perform operations with square roots. c. Add, subtract, multiply, and divide polynomials. d. Add, subtract, multiply, and divide rational expressions. e. Factor expressions by greatest common factor, grouping, trial and error, and special products limited to the formulas listed.

24. Key Standards MM1A3. Students will solve simple equations. Solve quadratic equations in the form ax2 + bx + c = 0 where a = 1, by using factorization and finding square roots where applicable. Solve equations involving radicals such as, using algebraic techniques. Use a variety of techniques, including technology, tables, and graphs to solve equations resulting from the investigation of . Solve simple rational equations that result in linear equations or quadratic equations with leading coefficient of 1.

25. Related Standards • MM1P1: Students will solve problems (using appropriate technology) • MM1P2: Students will reason and evaluate mathematical arguments • MM1P3: Students will communicate mathematically • MM1P4: Students will make connections among mathematical ideas and to other disciplines • MM1P5: Students will represent mathematics in multiple ways

26. Concepts/Skills to Maintain • Operations on integers, rational numbers, and square roots of non-negative numbers • Basic properties of basic quadratic, cubic, absolute value, square root, and rational functions • Adding, subtracting, multiplying, and dividing elementary polynomial, rational, and radical expressions

27. Tasks • Task 1: A launching task that formalizes factoring and properties of graphs • Tasks 2-4: Techniques for solving rational equations where the denominator is limited to rational numbers and first degree polynomials. Graph symmetry, odd and even functions, solving radical and rational equations. • Task 5: Culminating task that used applications of geometry, distance formula and topics learned in this lesson.

28. Logo SymmetryA Learning Task • Key Points: • Review of line and rotational symmetry • Introduction of point symmetry • Introduction of even and odd functions • Reflection through the x and y axis

29. Resistance A Learning Task • Key Points • Finding a more complex denominator • Elimination of solutions due to physical constraints (domain restrictions) • Extraneous solutions

30. Shadows and Shapes A Learning Task • Key Points: • Use idea of similar triangles • Apply Pythagorean theorem • Solutions to radical equations

31. Fairfield AviationA Culminating Task • This task incorporates all of the material learned in this unit in an applied setting. • It may be appropriate for students to work on this task throughout the unit with periodic deadlines

32. End of Day 4