Math Three Shifts - Alaska Department of Education & Early ...
Alaska Mathematics Standards Overview Shifts in Mathematics 1. Focus: 2-3 topics focused on deeply in each grade. 2. Coherence: Concepts logically connected from one grade to the next and linked to other major topics within the grade. 3. Rigor: Fluency with arithmetic, application of knowledge to real-world situations, and deep understanding of mathematical concepts. K-8 Standards for Mathematical Content Grade Span Instructional Focus
Standards for Mathematical Content Grade Level Domains Clusters Standards 3 Domains Counting, Cardinality and Ordinality Operations and Algebraic Thinking Number and Operations in Base Ten Measurement and Data
Number and Operations Fractions Geometry Ratios and Proportional Relationships The Number System Expressions and Equations Functions Statistics and Probability Shift #1 Focus Critical Areas Instructional Focus: Kindergarten through Second Grade 4 Kindergarten Instructional Focus Instructional time should focus on two critical areas:
(1) representing, relating, and operating on whole numbers, initially with sets of objects; (2) describing shapes and space. More learning time in Kindergarten should be devoted to number than to other topics. 5 Examining a critical area further For each critical area for K-8 there is an further explanation and often examples
to clarify. 6 Critical Areas by Grade Levels Grade K2 35 6 7 8 7
Addition and subtraction, measurement using whole number quantities Multiplication and division of whole numbers and fractions Ratios and proportional reasoning; early expressions and equations Ratios and proportional reasoning; arithmetic of rational numbers Linear algebra Shift #2 Coherence - Domains Domains large groups of related standards. They may begin and end in different grades. 8
Coherence - Clusters Clusters groups of closely related standards inside domains, subsets of domains. Lets look at the grades 3-5 clusters. 9 Operations and Algebraic Thinking Grade 3 Represent and solve problems involving multiplication and division. Understand properties of
multiplication and the relationship between multiplication and division. Multiply and divide up to 100. Solve problems involving the four operations, and identify and explain patterns in arithmetic. 10 Grade 4 Use the four operations with whole numbers to
solve problems. Gain familiarity with factors and multiples. Generate and analyze patterns. (OA) Grade 5 Write and interpret numerical expressions. Analyze patterns and relationships. Number and Operations in Base Ten
(NBT) Grade 3 Use place value understanding and properties of operations to perform multi-digit arithmetic. 11 Grade 4 Generalize place value understanding for multidigit whole numbers. Use place value understanding and
properties of operations to perform multi-digit arithmetic. Grade 5 Understand the place value system. Perform operations with multi-digit whole numbers and with decimals to hundredths. Number and OperationsFractions (NF) Grade 3 Develop understanding
of fractions as numbers. Grade 4 Extend understanding of fraction equivalence and ordering. Grade 5 Use equivalent fractions as a strategy to add and subtract fractions. Build fractions from unit fractions by applying and extending previous
understandings of operations on whole numbers. Apply and extend previous understandings of multiplication and division to multiply and divide fractions. Understand decimal notation for fractions, and compare decimal fractions. 12
Measurement and Data (MD) Grade 3 Grade 4 Grade 5 Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects. Represent and interpret data. Geometric measurement:
understand concepts of area and relate area to multiplication and to addition. recognize perimeter as an attribute of plane figures and distinguish between linear and area measures. Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
Represent and interpret data. Geometric measurement: understand concepts of angle and measure angles. Convert like measurement units within a given measurement system. Represent and interpret data. Geometric
measurement: understand concepts of volume and relate volume to multiplication and to addition. 13 Geometry (G) Grade 3 Grade 4 Reason with shapes and Draw and identify lines their attributes. and angles, and classify
shapes by properties of their lines and angles. Grade 5 Graph points on the coordinate plane to solve real-world and mathematical problems. Classify two-dimensional figures into categories based on their properties. 14 Examining a cluster
Clusters Generalize place value understanding for multi-digit whole numbers. Use place value understanding and properties of operations to preform multi-digit arithmetic. The standards grouped under the clusters are even more closely related. 15 Mathematical Content Standards
Overview Provides the domains and clusters by each of the grade span 16 Closer Look at Standards (6-8) Domain Clusters 17 Closer Look at Standards (6-8) Example
Subparts 18 A Strong Foundation for Algebra Through the K-8 Standards Progression Focus on number, operations, and fractions in early grades Increased attention to proportionality, probability and statistics in middle grades In depth study of linear algebra and introductions of functions in Grade 8 Provide good preparation for high school mathematics 19
High School Standards for Math Content Organized in six Conceptual Categories These crosses course boundaries Span all the high school years 9-12 Narrative highlights key information 20 Conceptual Categories Number & Quantity Algebra
Functions Modeling Geometry Statistics & Probability High School Standards for Math Content Standards Majority are core for all students + Additional standards for advanced courses to prepare students to take courses such as calculus, discrete mathematics, or advanced statistics. * Standards indicate connection to Modeling threaded throughout the other domains.
21 Shift #1 Focus - Modeling Each conceptual category has a narrative Highlights key information 22 9-12 Conceptual Categories and Domains Modeling Number and Quantity Algebra The Real Number System Seeing Structure in Integrated in the other
Expressions Conceptual Categories Quantities Arithmetic with Specific modeling The Complex Number Polynomials and standards appear System Rational Expressions throughout the high school standards Vector and Matrix Creating Equations* indicated by an Quantities
asterisk (*). Reasoning with Equations and Inequalities 23 9-12 Conceptual Categories and Domains Functions Geometry Statistics and Probability* Interpreting
Functions Congruence Interpreting Categorical and Quantitative Data Building Functions Linear, Quadratic, and Exponential Models* Trigonometric Functions Similarity, Right Triangles,
and Trigonometry Circles Expressing Geometric Properties with Equations Geometric Measurement and Dimension Modeling with Geometry 24 Making Inferences and Justifying Conclusions Conditional Probability and the Rules of Probability Using Probability to Make
Decisions Number and Quantity Build on and informally extend their understanding of integer exponents to consider exponential functions; Reason with the units in which those quantities are measured when functions describe relationships between quantities arising from a context; Explore distinctions between rational and irrational numbers in preparation for work with quadratic relationships; Identify zeros of polynomials, including complex zeros of quadratic polynomials, and make connections between zeros of polynomials and solutions of polynomial equations; and Work with quantities and the relationships between them to provide grounding for work with expressions, equations, and functions.
25 Algebra Analyze and explain the process of solving an equation, develop fluency writing, interpreting, and translating between various forms of linear equations and inequalities, and using them to solve problems, and master the solution of linear equations and apply related solution techniques and the laws of exponents to the creation and solution of simple exponential equations; Explore systems of equations and inequalities to find and interpret their solutions; Strengthen their ability to see structure in and create quadratic and exponential expressions, create and solve equations, inequalities, and systems of equations involving quadratic expressions; and
Connect multiplication of polynomials with multiplication of multidigit integers, and division of polynomials with long division of integers. 26 Functions Learn function notation and develop the concepts of domain and range, explore many examples of functions, including sequences, interpret functions given graphically, numerically, symbolically, and verbally, translate between representations, and understand the limitations of various representations; Build on and informally extend their understanding of integer exponents to consider exponential functions, compare and contrast linear and exponential functions, distinguishing between additive and multiplicative change, and interpret arithmetic sequences as linear functions and geometric sequences as exponential functions;
27 Functions (continued) Consider quadratic functions by comparing key characteristics, select from among these functions to model phenomena, learn to anticipate the graph of a quadratic function by interpreting various forms of quadratic expressions, identify the real solutions of a quadratic equation, and expand their experience with functions to include more specialized functions; Use the coordinate plane to extend trigonometry to model periodic phenomena; and Extend their work with exponential functions to include solving exponential equations with logarithms, explore the effects of transformations on graphs of diverse functions, and identify
appropriate types of functions to model a situation adjusting parameters and analyzing appropriateness of fit and making judgments about the domain. 28 Geometry Establish and use triangle congruence, prove theorems and solve problems about triangles, quadrilaterals, and other polygons, and apply reasoning to complete geometric constructions and explain why they work; Build a formal understanding of similarity, use similarity to solve problems, and apply similarity to understand right triangle trigonometry and the Pythagorean theorem, and develop the Laws of Sines and Cosines; Extend experience with two-dimensional and three-dimensional
objects to include informal explanations of circumference, area and volume formulas, apply knowledge of two-dimensional shapes to consider the shapes of cross-sections and the result of rotating a two-dimensional object about a line; 29 Geometry (continued) Use a rectangular coordinate system to verify geometric relationships and continue their study of quadratics by connecting the geometric and algebraic definitions of the parabola; and Prove basic theorems about circles and study relationships as an application of similarity, use the distance formula to write the equation of a circle, draw the graph in the coordinate plane, and apply techniques for solving quadratic equations to determine
intersections between lines and circles or parabolas and between two circles. 30 Statistics and Probability Students encounter: more formal means of assessing how a model fits data including using regression technique and make judgments about the appropriateness of linear models; the languages of set theory to expand their ability to compute and interpret theoretical and experimental probabilities for compound events, attending to mutually exclusive events, independent events, and conditional probability; geometric probability models and use probability to make informed
decisions; and different ways of collecting dataincluding sample surveys, experiments, and simulationsand the role that randomness and careful design play in the conclusions that can be drawn. 31 Comparison Tool for Standards Transition Provided for each H.S. Conceptual Category 32 Closer Look at Standards (9-12) Conceptual Category - Geometry Domains
33 Closer Look at Standards (9-12) Geometry Geometric Measurement and Dimension Clusters 34 Closer Look at Standards (9-12) Standards Core (+) Additional
Modeling (*) 35 Shift #3 Rigor Modeling Links classroom mathematics and statistics to everyday life, work and decision-making. In Grades K-8, modeling skills are developed as one of the Standards for Mathematical Practice. Modeling is both a High School Domain and a Standards for Mathematical Practice Domains, Strands, Clusters or Individual Standards have been identified by * 36
Standards for Mathematical Practice Both a goal and a vehicle The goal is to move students along a continuum always deepening their use of the mathematical practices. They are a vehicle for learning the content standards. 37 Background Information National Council of Teachers of Mathematics Principal and Standards (2000)
38 Problem Solving Reasoning and Proof Connections Communication Representation Background Information The National Research Councils Report
Adding It Up (2001) 39 Strands of Mathematical Proficiency Conceptual Understanding comprehension of mathematical concepts, operations, and relations Procedural Fluency skill in carrying out procedures flexibly, accurately, efficiently, and appropriately Strategic Competence ability to formulate, represent, and solve mathematical problems Adaptive Reasoning capacity for logical thought, reflection, explanation, and justification Productive Disposition habitual inclination to see mathematics as sensible, useful, and worthwhile,
coupled with a belief in diligence and ones own efficacy. 40 Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning 41
Mathematical Practice Descriptions Read the paragraph description for Practice #5. Notice the verbs used to describe actions students will take. 42 Grade-Span Proficiency Descriptors 5. Use appropriate tools strategically Read the proficiency descriptors for TWO grade spans. Notice the continuum of development. 43
Mathematical Practice Task In October, Johns PFD check for $1100 arrives. His parents give him 2 choices so the rest can be saved for post-secondary options. Choice 1: Spend 1/5 of the PFD Choice 2: Spent 15% of the PFD Which would give him the most spending money? Justify your answer 44
Standards for Mathematical Practice Graphic A B C 45 Thank You Your time is much appreciated! Questions about the math standards Deborah Riddle, Math Content Specialist
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