Bobson Wong Bayside H.S. Math for America Master Teacher Exploring Fractions from Elementary to High School Math TMC in NYC Conference Wed., Aug. 22, 2018 Bobson Wong Bayside H.S. Math for America Master Teacher

DO NOW Think about one of the math courses that you teach. Make a short list of the big themes or topics that you think are important in your course. Introduction: Served on NYS committee to revise CCSS As Ed Specialist, write and edit Regents questions

About me @bobsonwong, http://bobsonwong.com Bayside High School (in NE Queens) Math for America Master Teacher Worked on NY State Ed. Dept. committee that revised Common Core standards NYSED Educational Specialist: write and edit HS Regents (state exam) questions

Common Core math What themes do you notice? Grade 3 Grade 4 Grade 5 Algebra I Geometry Algebra II

Grade 3-5 Common Core Standards (OA) Operations and Algebraic Thinking (NBT) Number and Operations in Base Ten (NF) Number and Operations—Fractions (MD) Measurement and Data (G) Geometry

Grades 6-8 Common Core Standards (RP) Ratios & Proportional Relationships (NS) The Number System (EE) Expressions and Equations (G) Geometry (S) Statistics & Probability (F) Functions (Grade 8 only)

Algebra I Standards Number and Quantity Algebra Functions (N-RN) The Real Number System Algebra (A-SSE) Seeing Structure in Expressions (A-APR) Arithmetic with Polynomials and Rational Expressions (A-CED) Creating Equations (A-REI) Reasoning with Equations and Inequalities Functions (F-IF) Interpreting Functions (F-BF) Building Functions (F-LE) Linear, Quadratic, and Exponential Models Statistics and Probability (S-ID) Interpreting Categorical and Quantitative Data

Geometry Standards (G-CO) Congruence (G-SRT) Similarity, Right Triangles, & Trigonometry (G-C) Circles (G-GPE) Expressing Geometric Properties with Equations (G-GMD) Geometric Measurement and Dimensions (G-MG) Modeling with Geometry

Algebra II Standards Number and Quantity Algebra Functions (N-RN) The Complex Number System Algebra (A-SSE) Seeing Structure in Expressions (A-APR) Arithmetic with Polynomials and Rational Expressions (A-CED) Creating Equations (A-REI) Reasoning with Equations and Inequalities Functions (F-IF) Interpreting Functions (F-BF) Building Functions (F-LE) Linear, Quadratic, and Exponential Models (F-TF) Trigonometric Functions Statistics and Probability (S-ID) Interpreting Categorical and Quantitative Data (S-IC) Making Inferences and Justifying Conclusions (S-CP) Conditional Probability and the Rules of Probability

Major Themes in Common Core Math Grades K-5: A Story of Units Grades 6-8: A Story of Ratios Grades 9-12: A Story of Functions

Does all Common Core math lead to functions? What about: What’s the big theme? Does all Common Core math lead to functions? What about: Geometry? Statistics and probability? Number and quantity?

The Importance of proportional reasoning Fractions appear in every Common Core course from Grade 3 to Algebra II Fractions and ratios appear in all major themes of Common Core math: Geometry Statistics and probability Number and quantity Algebra Functions

Major Themes in Common Core Math Grades K-5: A Story of Units Grades 6-8: A Story of Ratios Grades 9-12: A Story of Functions Understanding fractions is critical at all levels!

Grade 6 (6.RP.A.3a) The accompanying table shows different possibilities for the number of games a team would need to win to maintain a certain percentage of wins. Which ratio of the number of games won to the number of games played could also be included in this table? (A) 18:20 (B) 30:20 (C) 18:30 (D) 50:30 Important to see the RELATIONSHIP between two quantities: 3/5 vs. 5/3 – which quantity is in numerator?

Grade 7 (7.RP.A.2c) A company ordered 10 boxed lunches from a deli for $74.50. Each boxed lunch cost the same amount. Which equation represents the proportional relationship between y, the total cost of the boxed lunches, and x, the number of boxed lunches? (A) 7.45x = y (B) 7.45/x = 10/y (C) 74.50x = y (D) 74.50/x = 10/y

Grade 8 (8.EE.B.5) The accompanying graph shows the relationship between the distances run and the time for three people in a 100-yard race. The relationship between the distance run and the time for Kofi can be represented by the equation y = 15.55x, where he ran y yards in x seconds. Which two equations could be used to represent this relationship for Bella and Elsie? (A) Bella: y = 15.15x; Elsie: y = 15.85x (B) Bella: y = 15.85x; Elsie: y = 15.65x (C) Bella: y = 15.45x; Elsie: y = 15.15x (D) Bella: y = 15.85x; Elsie: y = 15.15x

Algebra I (N-Q.A.1) A typical marathon is 26.2 miles. Allan averages 12 kilometers per hour when running in marathons. Determine how long it would take Allan to complete a marathon, to the nearest tenth of an hour. Justify your answer. What do you need to know in order to answer? 1 mi = 1.609 km or 1 km = 0.62 mi

Algebra I (F-LE.B.5) The cost of belonging to a gym can be modeled by C(m) = 50m+ 79.50, where C(m) is the total cost for m months of membership. State the meaning of the slope and y-intercept of this function with respect to the costs associated with the gym membership. SOLUTION: Initial cost of membership = $79.50 Monthly membership fee = $50

Geometry (G-C.B.5) Determine and state, in terms of π, the arc length that intercepts a 40º arc of a circle with a radius of 4.5. STUDENT PITFALLS: Memorize a formula: s = rθ. θ must be in radians! Which conversion should I use?

Geometry (G-C.B.5) Determine and state, in terms of π, the area of a sector that intercepts a 40º arc of a circle with a radius of 4.5. STUDENT PITFALLS Memorize the formula s = rθ s = arc length, not area θ must be in radians! Which conversion should I use?

Geometry (G-C.B.5) Determine and state, in terms of π, the area of a sector that intercepts a 40º arc of a circle with a radius of 4.5. Use proportional reasoning! Let x = area of sector

Geometry (G-C.B.5) Determine and state, in terms of π, the arc length that intercepts a 40º arc of a circle with a radius of 4.5. Use proportional reasoning! Let x = arc length

Algebra II Last year, the total revenue for Home Style, a national restaurant chain, increased 5.25% over the previous year. If this trend were to continue, which expression could the company’s chief financial officer use to approximate their monthly percent increase in revenue? [Let m represent months.]

Algebra II Last year, the total revenue for Home Style, a national restaurant chain, increased 5.25% over the previous year. If this trend were to continue, which expression could the company’s chief financial officer use to approximate their monthly percent increase in revenue? [Let m represent months.] SOLUTION: Let y = number of years. Then 12y = m, so y = m/12.

Algebra II The results of a poll of 200 students are shown in the accompanying table. For this group of students, do these data suggest that gender and preferred music styles are independent of each other? Justify your answer.

Conclusions Fluency with fractions is required in Common Core Only a few topics don’t require fractions E.g. inference, sampling, geometric proofs Understand the meaning of fractions in context is critical! Understand what units are represented (K-5) Understand meaning of ratios in functions (9-12)

Thank you! Questions? Bobson Wong Email: bwong3@schools.nyc.gov Web: http://bobsonwong.com Twitter: @bobsonwong Thanks to Math for America for allowing me to develop many of these ideas for a mini-course that I co-facilitated last year!