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Standards for Mathematics Standards for Mathematical Practice • • Apply across all grade levels Describe habits of mind of a mathematically proficient student Standards for Mathematical Content • • • • K-8 standards presented by grade level High school standards presented by topic (Number & Quantity, Algebra, Functions, Modeling, Geometry, Statistics & Probability) Organized into domains that progress over several grades Two to four “critical areas” at each grade level Mathematical Practices 1. Make sense of problems 5. Use appropriate tools and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Design and Organization Critical areas at each grade level Design and Organization Grade Level Overviews Domain, Cluster and Standards Ratios and Proportional Relationships Understand ratio concepts and use ratio reasoning to solve problems. 1. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.” 2. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” Reading the Standards Standards define what students should understand and be able to do. • Specific advice is often given about instructional approaches. • The actual language can be dense, and probably requires “unpacking.” K-6 “I Can” statements are available on our website. I understand what it means to divide by a fraction, and I can solve word problems that involve division with fractions. I can create equivalent fractions by scaling up, and I can show this process using visual fraction models. 1 4 2 2 2 8 High School The high school standards specify the mathematics that all students should study in order to be college and career ready. Additional mathematics that students should learn in order to take advanced courses such as calculus, advanced statistics, or discrete mathematics is indicated by (+), as in this example: (+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers). All standards without a (+) symbol should be in the common mathematics curriculum for all college and career ready students. Standards with a (+) symbol may also appear in courses intended for all students. Modeling Standards Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). Course topics • Appendix A shows which standards fit into Algebra I, Algebra II, and Geometry. Major Shifts in Mathematics Working like a mathematician: • Problem solving • Reasoning (proportional reasoning, geometric reasoning, etc. • Creating viable arguments • Modeling with mathematics Reasoning and Sense-making In square ABCE shown below, D is the midpoint of CE. Which of the following is the ratio of the area of triangle ADE to the area of triangle ADB? F. 1:1 G. 1:2 H. 1:3 J. 1:4 K. 1:8 Videos of classrooms From the Inside Mathematics web resource http://InsideMathematics.org Properties of Quadrilaterals, Grades 9-10, Tuesday Group Work, Pt. A Major Shifts in Mathematics Focus and coherence • • Focus on key topics at each grade level. Coherent progressions across grade levels. Balance of concepts, skills and problem-solving • Content standards require a blend of deep understanding, procedural fluency and application. Mathematical practices • Foster reasoning and sense-making in mathematics. Focus: Critical Areas Mathematics Grade 3 In Grade 3, instructional time should focus on four critical areas: (1) developing understanding of multiplication and division and strategies for multiplication and division within 100; (2) developing understanding of fractions, especially unit fractions (fractions with numerator 1); (3) developing understanding of the structure of rectangular arrays and of area; and (4) describing and analyzing two-dimensional shapes. Coherence: Learning progressions 2nd grade Students connect arrays and the area of rectangles with repeated addition 3rd grade Students solve word problems involving equal groups, arrays and area; relate division to multiplication; use strategies for multiplying and dividing; and become fluent with one-digit multiplication and division. 4th grade Students solve word problems involving multiplication as a comparison; solve multi-step multiplication and division word problems; continue to use equations to represent situations; and multiply multi-digit numbers (4x1 and 2x2) Coherence: Learning Progressions Fractions, Grades 3-6 3rd grade: Develop an understanding of fractions as numbers. 4th grade: Extend understanding of fraction equivalence and ordering. 5th grade: Use equivalent fractions as a strategy to add and subtract fractions. 5th grade: Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 6th grade: Apply and extend previous understandings of multiplication and division to divide fractions by fractions. On-line Learning Trajectories http://turnonccmath.net/ Balance of concept, skill & application Multiplication, single and multi-digit Fluently multiply and divide within 100, using strategies… By the end of Grade 3, know from memory all products of one-digit numbers.3.OA.7 Solve two-step word problems using the four operations. 3.OA.8 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies… Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. 4.NBT.5 Solve multistep word problems posed with whole numbers, including problems in which remainders must be interpreted. 4.OA.3 Fluently multiply multi-digit whole numbers using the standard algorithm. 5.NBT.5 Mathematical Practices • Make sense of problems and • • • • • • • persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Marty made two types of cookies. He used 2/3 cup of sugar for one recipe and 1/4 cup of sugar for the other. He only has 1 cup of sugar. Is that enough? Explain your reasoning. Some people might say that A) Why might they say this? B) Do you agree? C) How would you solve this problem?