Why the Common Core? Preparation: For college and career Competition: internationally benchmarked and globally competitive. Equity: consistent for all Clarity: focused, coherent, and clear Collaboration: a foundation to work collaboratively across states and districts
communicate their strategies and justify their solutions. 21st Century Mathematics Skills Our goals for our students in mathematics Students need to: be flexible with numbers adapt their skills to multiple mathematical situations make sense of problems and persevere in solving them
Key Elements of the Common Core for Math The standards are fewer, but more rigorous Procedural skills and conceptual understanding are equally important Students go beyond computation Application of higher order thinking skills are an emphasis
The Six Shifts of the Common Core Focus Coherence Fluency Deep Understanding Applications Dual Intensity
Shift #1: Focus Narrow and deepen the scope of how time is spent in the classroom - spend more time on fewer concepts Focus deeply on only the concepts that are prioritized in the standards The goals: To reach a strong foundational knowledge and deep conceptual understanding Students can transfer the mathematical skills and understanding across concepts and grades
Focus Area by Grade Band Grade K - 2 3 5 6-8 High School Focus Addition and subtraction, measurement using whole number quantities Multiplication and division of whole numbers and fractions Ratios and proportional reasoning; early expressions and equations; arithmetic of rational numbers; Linear algebra Algebra, Geometry, Functions, Probability and Statistics, Numbers and Quantities
Shift #2: Coherence Teachers carefully connect the learning within and across grades Students build new understanding onto foundations built in previous years Each standard is an extension of previous learning Teachers can count on deep conceptual understanding of core content and build upon it
Shift #3: Fluency Students are expected to have speed and accuracy with simple calculations Teachers structure class and homework time for students to become fluent with the core functions/fluencies These fluencies will make them better able to understand and manipulate more complex
Shift #4: Deep Understanding Teachers demand that students can do more than get the answer They push students to see math as more than a set of mnemonics or discrete procedures Students demonstrate deep conceptual understanding They can apply concepts to new situations They can write and speak about their understanding They can solve and demonstrate problems in more than one way
Shift #5: Application Teachers provide opportunities at all grade levels for students to apply math concepts in real world situations They will know which math to use for which situation They will apply math as it comes up in their daily lives
Shift #6: Dual Intensity Students need both practice and understanding Must be able to use math facts fast, as well as apply them in the real world Not just a balance of both - but both are occurring with INTENSITY Teachers should implement fact practice routines
Design and Organization The Common Core State Standards are organized into two sections: Standards for Mathematical Practice Standards for Mathematical Content
Standards for Mathematical Practice Carry across all grade levels Describe habits of mind of a mathematically expert student
Mathematical Practices Make sense of problems and persevere in solving them. STUDENTS SHOULD BE ABLE TO: Use multiple representations (verbal descriptions, symbolic, tables, graphs,etc.) Check their answers using different methods Continually ask, Does this make sense? Understand the approaches of others and identify correspondences between different approaches
Mathematical Practices Reason abstractly and quantitatively. STUDENTS SHOULD BE ABLE TO: Make sense of quantities and their relationships in problem situations Create a coherent representation of the problem, consider the units involved, and attend to the meaning of quantities
Mathematical Practices Use appropriate tools strategically. STUDENTS SHOULD BE ABLE TO: Consider the available tools when solving mathematical problems Know the tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful Identify relevant external mathematical resources and use them to pose and solve problems Use technological tools to explore and deepen their understanding of concepts
Standards for Mathematical Content K- 8 standards presented by grade level Organized into domains that progress over several grades Organized into domains that progress over Grade introductions give 2 44 focal points at each grade level Grade introductions give 2 High school standards presented by conceptual theme (Number & Quantity, Algebra, Functions, Modeling, Geometry, Statistics & Probability) High school standards presented by conceptual
Elementary Grades In studying other high-achieving countries, and through research done in the US the following ideas have come forth Early mathematics learning should focus on number sense (whole number, operations, and relations) Another focus should be geometry and measurement Less time spent on data analysis and algebra
The South Brunswick K-5 Math Curriculum Deep learning of concepts is stressed Time is spent on a topic and on learning it well. This counters the mile wide, inch deep criticism leveled at most previous U.S. standards Majority of time is spent on number sense, increasing in complexity, with only limited time on geometry, measurement, and data analysis Multiple strategies for computation are required
K-2 Focus and Fluencies Focus on Number Sense How numbers correspond to quantities How to put numbers together and take them apart Develop an understanding of addition and subtraction using multiple strategies Moving beyond the traditional algorithms and procedures Building fluency fast and accurate Place Value concepts and number relationships
Begin an understanding of decimal operations 3-5 Focus and Fluencies Building an understanding of multi-digit multiplication and division Build fluency and understanding with multiple strategies for each Develop an understanding of fractions, fraction equivalence, and fluency with fraction operations Demonstrate understanding with fractional models Develop an understanding of rectangular arrays, area, volume, and geometric properties
Fluency by Grade Grade Required Fluency K Add/subtract within 5 1 Add/subtract within 10 2 3 Add/subtract within 20 Add/subtract within 100 (pencil and paper) Multiply/divide within 100 Add/subtract within 1000 4 Add/subtract within 1,000,000 5 Multi-digit multiplication 6 Multi-digit division Multi-digit decimal operations
K-2 Sample Questions Max has 16 balloons. He gives some to a friend. Now he has 9. How many did he give to his friend?
K-2 Sample Questions When I emptied my pocket, I found 9 coins, including pennies, nickels, and dimes. The value of my money is 58 cents. What coins did I have in my pocket? Rodrigo made 15 pancakes for his family. Some friends came for breakfast, so Rodrigo made 4 more pancakes. After Rodrigo s family and friends ate, 5 pancakes were left. How many pancakes were eaten in all? If you have 23 tens, 4 hundreds, and 18 ones, how much do you have?
3-5 Sample Questions Write a number that uses the same 6 digits as 901,735 but where the digit 3 represents 10 times what it represents in 901,735. A school auditorium has 32 rows of seats. Each row has 15 seats. The letter k represents the total number of seats. Write an equation that can be used to find k.
3-5 Sample Questions How many hundreds equal 100,000? The place value of the 3 in 0.3 is how many times the place value of the 3 in 30? Explain your answer.
Flexibility with Numbers Solving mathematical problems based on the numbers before them, not based on one algorithm Build a repertoire of efficient strategies Investigate and apply mathematical relationships What is the best way to solve 199 + 199? Number Talks in the Classroom
Grade 6 Focus and Fluencies Ratios and Proportional Relationships Understand ratio concepts and use ratio reasoning to solve problems. The Number System Apply and extend previous understandings of multiplication and division to divide fractions by fractions. Compute fluently with multi-digit numbers and find common factors and multiples. Apply and extend previous understandings of numbers to the system of rational numbers. Geometry Solve real-world and mathematical problems involving area, surface area, and volume.
Grade 6 Focus and Fluencies Expressions and Equations Apply and extend previous understandings of arithmetic to algebraic expressions. Reason about and solve one-variable equations and inequalities. Represent and analyze quantitative relationships between dependent and independent variables. Statistics and Probability Develop understanding of statistical variability. Summarize and describe distributions.
Sample 6 th Grade Math Problem Triangle ADE is inside rectangle ABCD. Point E is halfway between points B and C on the rectangle. Side AB is 8 cm and side AD is 7 cm. Part A: What is the area of triangle ADE? Show your work. Part B: What is the ratio of the area of triangle ABE to the area of triangle ADE? Part C: What is the ratio of the area of triangle CDE to the area of rectangle ABCD? B E C A D
Solve real-life and mathematical problems using numerical and algebraic expressions and equations. Grade 7 Focus and Fluencies Ratios and Proportional Relationships Analyze proportional relationships and use them to solve real-world and mathematical problems. The Number System Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. Expressions and Equations Use properties of operations to generate equivalent expressions.
Grade 7 Focus and Fluencies Statistics and Probability Use random sampling to draw inferences about a population. Draw informal comparative inferences about two populations. Investigate chance processes and develop, use, and evaluate probability models. Geometry Draw, construct and describe geometrical figures and describe the relationships between them. Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
Sample 7 th Grade Math Problem At a discount furniture store, Chris offered a salesperson $600 for a couch and a chair. The offer includes the 8% sales tax. If the salesperson accepts the offer, what would be the price of the furniture, to the nearest dollar, before tax? A $552 B $556 C $592 D $648
Grade 8 Focus and Fluencies The Number System Know that there are numbers that are not rational, and approximate them by rational numbers. Work with radicals and integer exponents. Expressions and Equations Understand the connections between proportional relationships, lines, and linear equations. Analyze and solve linear equations and pairs of simultaneous linear equations. Functions Define, evaluate, and compare functions. Use functions to model relationships between quantities.
Grade 8 Focus and Fluencies Geometry Understand congruence and similarity using physical models, transparencies, or geometry software. Understand and apply the Pythagorean Theorem. Solve real-world and mathematical problems involving volume of cylinders, cones and spheres. Statistics & Probability Investigate patterns of association in bivariate data.
Sample 8 th Grade Math Problem A trainer for a professional football team keeps track of the amount of water players consume throughout practice. The trainer observes that the amount of water consumed is a linear function of the temperature on a given day. The trainer finds that when it is 90 F the players consume about 220 gallons of water, and when it is 76 F the players consume about 178 gallons of water. Part A: Write a linear function to model the relationship between the gallons of water consumed and the temperature. Part B: Explain the meaning of the slope in the context of the
High School Focus and Fluencies Numbers and Quantities Extend the Real Number to include work with rational exponents and study of the properties of rational and irrational numbers Required for higher math and/or STEM Compute with and use the Complex Numbers Represent and use vectors Compute with matrices
High School Focus and Fluencies Algebra Creating, reading, and manipulating expressions Solving equations and inequalities Required for higher math and/or STEM Expand a binomial using the Binomial Theorem Represent a system of linear equations as a matrix equation Find the inverse if it exists and use it to solve a system of equations
Sample Algebra 1 Problem A company wants to purchase two types of lightbulbs, CFL (compact fluorescent) and LED (light emitting diode). The cost of each CFL bulb is $2, and the cost of each LED bulb is $20. The company must purchase a total of 800 lightbulbs, must spend at most $5,000 on the lightbulbs, and wants to purchase as many LED bulbs as possible. Let L represent the number of LED lightbulbs purchased, and C represent the number of CFL lightbulbs purchased. Which of the following systems models the situation described? a. 2C + 20L = 800 C + L < 5000 b. 20C + 2L = 800 C + L < 5000 c. 2C+20L < 5000 C + L = 800 d. 20C+2L < 5000 C + L = 800
High School Focus and Fluencies Functions Emphasis is on linear and exponential models Extends trigonometric functions to functions defined in the unit circle and modeling periodic phenomena Required for higher math and/or STEM Graph rational functions and identify zeros and asymptotes Prove the addition and subtraction formulas for trigonometric functions and use them to solve problems Inverse functions
Sample Algebra 2 Problem The graph of the fourth-degree polynomial function is shown in the coordinate plane below. Based on the graph, list all linear factors of f(x).
High School Focus and Fluencies Geometry Understanding congruence Using similarity, right triangles, and trigonometry to solve problems Circles Expressing geometric properties with equations Required for higher math and/or STEM Non-right triangle trigonometry Derive equations of hyperbolas and ellipses
Sample Geometry Problem Quadrilateral PQRS is shown. Which of the following transformations of triangle PTS could be used to show that triangle PTS is congruent to triangle QTR? a) A reflection over segment QS b) A reflection over segment PR c) A reflection over line m d) A reflection over line l
High School Focus and Fluencies Statistics and Probability Analyze single and two variable data Understand the role of randomization in experiments Make decisions, use inference and justify conclusions from statistical studies Use the rules of probability
Overview of High School Mathematics Standards The high school mathematics standards: Call on students to practice applying mathematical ways of thinking to real world issues and challenges Require students to develop a depth of understanding and ability to apply mathematics to novel situations, as college students and employees regularly are called to do Emphasize mathematical modeling, the use of mathematics and statistics to analyze empirical situations, understand them better, and improve decisions Identify the mathematics that all students should study in order to be college and career ready. High school standards are organized around five conceptual categories: Number and Quantity, Algebra, Functions, Geometry, and Statistics and Probability
The South Brunswick Curriculum Documents All math curriculum documents are located on the South Brunswick Schools website under Departments: Curriculum/Instructions, Parent Guides
Resources & Contact Information Resources & Websites are included in your handouts Elementary Stacey Ta stacey.ta@sbschools.org Middle School Teresa Valentin teresa.valentin@sbschools.org High School Anna Alfieri anna.alfieri@sbschools.org