1. The development of the Algebra Strand from 6 th grade through Math III John Pritchett, Athens Drive HS jpritchett@wcpss.net Triangle High Five High School Math Summit 2016 Panther Creek high School July 20, 2016

2. The impetus I asked the questions: What can I expect them to know? What do I need to be sure they have before going to the next course?

3. The approach Mathematical thinking is the focus of instruction. Mathematics is to take much more of a modelling approach than an operational approach.

4. Threads in the Algebra and Function strands Functions as mathematical models Operations Equations & Inequalities Systems OperationsSystems Equations & Inequalities Models Construct a model to represent the problem Determine when a model fulfills a given condition Solve more complex equations Determine when two or more models fulfill the same condition

5. Mathematical models – Planting roots Students in 6 th grade begin to analyze relationships between quantities. How much does it cost to buy 6 tomatoes if tomatoes cost $0.69 each? Grows to the total cost of tomatoes is $0.69 times the number of tomatoes. Students in 6 th grade convert between units These unit conversions can become standardized How many groups of 12 inches are in a given number of feet? Grows to

6. Models growing vertically - Slope 6 th grade – students use ratio and rate to solve problems 7 th grade – students find unit rate 8 th grade – students recognize unit rate as slope Math I – students calculate and interpret average rate of change over an interval

7. Building a toolkit of mathematical models 7 th grade – recognize proportional relationships as equations (direct variation) 8 th grade – interpret y = mx + b as a linear relationship Math I – calculate and graph models to represent linear, exponential, & quadratic relationships Math II – calculate and graph models to represent quadratic, square root, & inverse variation relationships Math III – calculate and graph models to represent absolute value, polynomial, exponential, & rational relationships

8. Graphs and other representations Students should be able to describe features of graphs Math I – intercepts, intervals of increase/decrease, intervals of positive/negative range, maximum/minimum Math II – AND domain/range, symmetries, rate of change, end behavior Math III – discontinuities, periodicity, effect of parameters on sine/cosine Functions analyzed: Linear, exponential, quadratic Quadratic, square root, inverse variation Absolute value, polynomial, rational, sine/cosine

9. Solving equations/inequalities in one variable 6 th grade – understand a solution as the value(s) that make a math sentence true, solve one-step linear equations, interpret x c 7 th grade – solve two-step linear equations and inequalities in one variable 8 th grade – solve linear equations with rational coefficients, solve x 2 = p and x 3 = p with roots Math I – understand the relationship between factors of a quadratic and the roots of the equation Math II – solve quadratic, square root, inverse variation, right-triangle trig Math III – absolute value, polynomial, exponential, and rational relationships.

10. Operations to support solving equations 6 th grade – write expressions using variable, translate words to algebraic expressions, write & evaluate expressions with whole number exponents 7 th grade – simplify and expand linear expressions 8 th grade – operate with scientific notation Math I – use function notation to evaluate linear, exponential, and quadratic expressions; operate with polynomial expressions Math II – explain how expressions with radical exponents can be re-written using radicals, re-express quadratics by completing the square, add/subtract/multiply polynomials Math III – divide polynomials, analyze piecewise-defined functions

11. Interpret expressions with multiple parts as a combination of entities Math I – linear, exponential, quadratic Math II – quadratic, square root

12. Systems of equations/inequalities 8 th grade – model real-world problems with systems of linear equations, solve systems of linear equations Math I – represent the solution to a linear inequality/system of linear inequalities as a shaded region of the plane, connect the intersection point to the solution of f(x) = g(x), approximate solutions to systems of quadratic/exponential functions using technology/tables of values Math II – create systems of linear, quadratic, square root, inverse variation to model situations in context, solve with technology/tables/algebra

13. New standards of 2016 The following changes were included in the NC Math Standards for Math I – III Middle school math and the 4 th math course are to be reviewed beginning in 2016 - 2017

14. Math I Moved In Solve quadratic equations by square root and factoring only Systems involving quadratics and exponentials using tech/tables Geometric sequences Translating between explicit and recursive Moved out Rational exponents “Use structure to re-write” Rational functions No more “constraints” Transformations

15. Math II Moved In Introduction of radical exponents Examine the closure of rational numbers under +, , ,  Introduce  as in Introduce inverse variation Complete the square to derive the quadratic formula Moved Out Advanced functions – cube root, step function, absolute value, piecewise-defined functions

16. Math III Moved In Introduce absolute value function and piecewise-defined Write equivalent forms of exponential functions Review of solving radical equations Moved Out Closure of rational numbers (II) Operations on complex numbers (IV) Complete the square(II) Sequences & series (I & IV) Proving polynomial identities (IV) Pythagorean Identity (IV)

17. Placement For each question on the following slides, choose the course in which you would use the question As an enrichment question As a regular practice or test question As a remedial question

18. 1. Which expression is equivalent to ? a) b) c) d)

19. 1. Rational Exponents Enrichment – Math II Practice – Math II Review – Math III or beyond Rational exponents are not introduced until Math II

20. 2. Find the x-coordinate of the solution to this system: x + 2y = 11; 3x – 4y = 3 a)x = 5 b)x = 3 c)x = 3.5 d)No solution

21. 2. System of linear equations Enrichment – 8 th grade Practice – 8 th grade Review – Math I and beyond

22. 3. Construct a quadratic function with zeros at x =  3 and x = 2. a) y = x 2 + 5x – 6 b) y = x 2 + x – 6 c) y = x 2 – x – 5 d) y = x – 1

23. 3. Zeros of quadratic functions Enrichment – Math I Practice – Math I Review – Math I and beyond

24. 4. Which of these makes 3x + 11 = 5x + 7 true? a)x =  2 b)x = 2 c)x = 2.25 d)x = 0.5

25. 4. Solution to a linear equation Enrichment – 6 th grade Practice – 7 th or 8 th grade Review – 8 th grade and beyond

26. 5. Convert 96 inches to feet. a)8 feet b)288 feet c)16 feet d)1152 feet

27. 5. Convert units Enrichment – 6 th grade Practice – 6 th grade Review – 6 th grade and beyond

28. 6. Find the average rate of change for f(x) = 3x 2 – 10x + 5 from x = 2 to x = 6. a)3 b)½ c)14 d)12.5

29. 6. Average rate of change Enrichment – Math I Practice – Math I Review – Math I and beyond

30. 7. Create a mathematical model for an initial deposit of $2000 invested at 4% compounded annually. a)y = 2000(1.4) x b)y = 2000(1.04) x c)y = 2000 + 0.04x d)y = 2000 + 1.04x

31. 7. Construct an exponential model Enrichment – Math I Practice – Math I Review – Math II and beyond

32. 8. Which of these functions will be largest in the long run? Function 1: y = x 2 + 5x – 4 Function 2: a)Function 1 will be larger b)Function 2 will be large c)They will be equal d)Not enough information xy = 0.1(2) x 20.4 41.6 66.4 825.6

33. 8. Compare two functions in different representations. Enrichment – Math I Practice – Math I Review – Math II and beyond

34. Questions? Comments? I would greatly appreciate any feedback at jpritchett@wcpss.net Thank you!!!!