Math at a Glance for April

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1 Audience: School Leaders, Regional Teams Math at a Glance for April The Math at a Glance tool has been developed to support school leaders and region teams as they look for evidence of alignment to Common Core State Standards in Mathematics. The critical areas are designed to highlight the Shifts of Common Core for Mathematics: Focus, Coherence and Rigor at each grade. The Shifts describe the big ideas that educators can use to build their curriculum and to guide instruction. This tool will help school leaders look for coherence among standards as teachers deliver instruction around the grade level Critical Areas of Focus. How students are engaged with the standards is as equally important to the Critical Areas of Focus. Evidence of the applicable Standards for Mathematical Practice should be observed in daily instruction. 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 These Standards should be evident in all classes of mathematics K-12 Standards of Mathematical Practice 1. Make sense of problems and persevere in solving them Highlighted Standards for Mathematical Practice Look-Fors Student Behavior Monitor and evaluate own work, and may report a change of strategy or perspective Teacher Actions In summary, require student justifications and reasonableness, and seek alternative solutions 3. Construct viable arguments and critique the reasoning of others. In examining a proposed solution, ask, Does this make sense? Listen to or read the arguments of others and ask questions for clarification Engage students in developing an exploration or viable argument, which may include graphs, diagrams, and/or constructions Engage students in recalling knowledge, using prior results, assumptions and definitions in building explanations or arguments Additional Resources Instructional Guides Standards for Mathematical Practice Look-Fors If you have any further questions please contact the District Mathematics Leader for your region or your school s ACM.

2 Grade 2 Grade 1 Grade K Critical Area 1a: Understanding Counting Critical Area 1b: Problem Solving with Addition and Subtraction Critical Understandings: Representing and comparing whole numbers Critical Understandings: Developing an understanding of the meaning of addition and subtraction within 10. Counting and Cardinality K.CC.1-7 Number and Operations Base Ten K.NBT.1 Operations and Algebraic Thinking K.OA. 1-5 Measurement and Data K.MD. 3 Numbers can be broken apart and grouped in a different way to make Numbers describe relationships between or among quantities calculations simpler. (compare). Numbers can be composed or decomposed by grouping ten ones and some more ones. The relationship between one quantity and another quantity can be an equality or inequality relationship. Critical Area 1: Problem Solving in Addition and Subtraction Critical Area 2: Place Value Meaning and Computation Critical Understandings: Developing understanding of addition and Critical Understandings: Developing understanding of whole number subtraction, and strategies for addition and subtraction within 20. relationships and place value, including grouping in tens and ones Operations and Algebraic Thinking 1.OA. 1-8 Number and Operations Base Ten1.NBT. 1-6 Number and Operations Base Ten 1. NBT 4-6 Measurement and Data 1.MD.4 Computation strategies for smaller numbers can be extended to addition and subtraction with larger numbers. Some real world problems involving add to, take from, put together/take apart, or comparison can be solved using addition; others subtraction. The relationship between one quantity and another quantity can be an equality or inequality relationship. Critical Area: Building Fluency with addition and subtraction Critical Understandings: Students use their understanding of addition to develop fluency with addition and subtraction within 100. They solve problems within 1000 by applying their understanding of models for addition and subtraction, and they develop, discuss, and use efficient, accurate, and generalizable methods to compute sums and differences of whole numbers in base-ten notation, using their understanding of place value and the properties of operations. They select and accurately apply methods that are appropriate for the context and the numbers involved to mentally calculate sums and differences for numbers with only tens or only hundreds. Number and Operations in Base Ten 2.NBT.5-9 Measurement and Data 2.MD 5,6, & 8 Operations and Algebraic Thinking 2.OA.1-4 Students flexibly apply commutative and associative properties of addition to calculate sums of numbers mentally and in writing. Subtraction is the inverse of addition and solving a subtraction problem can be thought of as finding a missing addend. Students develop efficient strategies using their understanding of place value to solve problems within 1000.

3 5 th grade 4 th grade 3 rd grade Critical Area: Unit Fractions Critical Understandings: Developing understanding of fractions, especially unit fractions (fractions with numerator 1) Number and Operations Fractions 3.NF.2-3 Students will understand that a comparison of the part to the whole can be represented using a fraction beginning by developing an understanding of unit fractions. Students will understand fractions in general as being built out of unit fractions, and they use fractions along with visual models to represent parts of a whole. Students will understand fractions as parts of unit wholes and as locations on number lines and use models, benchmarks, and equivalent forms to judge the size of fractions. Critical Area: Equivalent Fractions and Operations with Fractions Critical Understandings: Developing an understanding of fraction equivalence, addition and subtraction of fraction with like denominators, and multiplication of fractions by whole numbers Number and Operations Fractions 4.NF.5, 6, 7 Measurement and Data 4.MD.2,& 4 Students will understand equivalent fractions. Students will understand and use properties of operations and the relationships between them. Students will understand conversion of measurements. Critical Area: Number and Operations with Fractions Applications (unit fractions) Critical Understandings: Developing fluency with addition and subtraction of fractions and developing understanding of the multiplication of fractions and of division of fractions in limited cases ( unit fractions divided by whole numbers and whole numbers divided by unit fractions) Number and Operations-Fractions 5.NF 3,4, 4a, 4b, 5, 5a, 5b, 6, 7, 7a, 7b Students understand equivalent fractions Students understand multiplication and division are inverse operations

4 8 th grade 7 th grade 6 th grade Critical Area: Fractions Critical Understandings: Complete understanding of division of fractions and extend the notion of number to the system of rational numbers, which includes negative numbers. The Number System 6.NS.4,6, 6a, 6b, 6c, 7, 7c, 7d & 8 Students understand and use positive and negative numbers. Students understand rational numbers (positive and negative) as points on a number line and coordinates on a grid. Students understand ordering and absolute value of rational numbers in real world and mathematical problems. Critical Area: Expressions and Equations Critical Understandings: Write, interpret, and use expressions and equations. Expressions and Equations 6.EE.2, 2a, 2b, 3,4,6-8 Students extend the use of properties of arithmetic to algebraic expressions. Students reason about and solve one variable equations and inequalities. Students represent and analyze quantitative relationships between dependent and independent variables. Critical Area: Statistics and Probability Critical Understandings: Draw inferences about populations based on samples. Statistics and Probability 7.SP.1-4 Key Student Understanding: Students apply proportional reasoning to draw inferences about populations from samples. Critical Area: Geometry Critical Understandings: analyze two- and three-dimensional space and figures using distance, angle, similarity, and congruence, and understand and apply the Pythagorean Theorem. Geometry 8.G.1-9 Students understand informal transformations by exploring size, position and orientation of shapes. Students can explain a proof of the Pythagorean Theorem and its converse. *Beyond the Critical Areas: Geometry and Fluency with multiplication and division of whole numbers. Critical Understanding: Students reason about relationships among shapes to determine area, surface area, and volume. Geometry 6.G.2-4 Students solve real world and mathematical problems involving area, surface area, and volume. *Beyond the Critical Areas: The Number System; Expressions and Equations Critical Understandings: Extend work with expressions and equations wit integer exponents, square and cube roots. Understandings of very large and very small numbers, the place value system, and exponents are combined in representations and computations with scientific notation. Number System 8.NS.1-2 Expressions and Equations 8.EE.2 Students understand the concept of rational and irrational numbers. Students understand properties of square roots, powers of ten, base ten, and exponents.

5 Algebra II Geometry Algebra Critical Area 4: Exponential Relationships and Models Critical Understandings: Students build on and informally extend their understanding of integer exponents to consider exponential functions to model data. They compare and contrast linear and exponential functions, distinguishing between additive and multiplicative change. They interpret arithmetic sequences as linear functions and geometric sequences as exponential functions. Interpreting Functions F-IF.1, 3, 4, 5, 8b, 9 Quantities N-Q.1-3 Building Functions F-BF.1ab, 3 Interpreting Categorical and Quantitative Data S-ID.6a-c Linear Models F-LE.1-3, 5 Making Inferences and Justifying Conclusions S-IC.1, 2 Students will understand that in exponential functions, when the inputs increase by a constant amount, the outputs increase or decrease by the same ratio. Given a real world situation, students will understand and communicate the appropriate mathematical model for the situation. Critical Area 5: Similarity, Right Triangles, and Trigonometry Critical Understandings: Students apply their earlier experience with dilations and proportional reasoning to build a formal understanding of similarity. They identify criteria for similarity of triangles and use similarity to solve problems They apply similarity in right triangles to understand right triangle trigonometry, with particular attention to the Pythagorean Theorem and its relationship to circles. Similarity, Right Triangles, and Trigonometry G-SRT.1-8 Modeling with Geometry G-MG.1-3 Real Number System N-RN.3 Congruence G-CO.11 Expressing Geometric Properties with Equations G-GPE.4, 6, 7 Circles G-C.1 Students will understand that similarity is defined as a dilation transformation and be able to apply scale factors to shapes. Students will understand how to apply theorems about triangles, including proportionality of corresponding sides in similar figures and the Pythagorean Theorem, to solve real-world problems. Students will understand that right triangle trigonometry is based on similarity and how to apply trigonometry in appropriate situations. Critical Area 3: Really Cool New Relationships Critical Understandings: This critical area develops the structural similarities between the system of polynomials and the system of integers. Students draw on analogies between polynomial arithmetic and base-ten computation, focusing on properties of operations, particularly the distributive property. Students connect multiplication of polynomials with multiplication of multi-digit integers, and division of polynomials with long division of integers. Students identify zeros of polynomials, including complex zeros of quadratic polynomials, and make connections between zeros of polynomials and solutions of polynomial equations. Rational numbers extend the arithmetic of integers by allowing division by all numbers except 0. Similarly, rational expressions extend the arithmetic of polynomials by allowing division by all polynomials except the zero polynomial. A central theme of this critical area is that the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers. The Complex Number System N-CN.1, 2, 7 The Real Number System N-RN.1,2 Arithmetic with Polynomials and Rational Expressions A-APR.1-4, 6 Reasoning with Equations and Inequalities A-REI.2,7,11 Interpreting Functions F-IF.7 Building Functions F-BF.4a Linear Models F-LE.4 Expressing Geometric Properties with Equations G-GPE. 1,2 Students will expand the number system to include the complex numbers allowing for the solution of quadratic equations. Students will understand algebraic expressions can be written in infinitely many equivalent forms; different forms may be useful for answering different questions about the context of the problem. Students will understand the relationship between zeros and the factors of polynomials.

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