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Vacaville USD November 4, 2014 AGENDA • Problem Solving, Patterns, Expressions and Equations • Math Practice Standards and High Leverage Instructional Practices • Number Talks – Computation Strategies • Fractions Expectations • We are each responsible for our own learning and for the learning of the group. • We respect each others learning styles and work together to make this time successful for everyone. • We value the opinions and knowledge of all participants. Cubes in a Line How many faces (face units) are there when: 6 cubes are put together? 10 cubes are put together? 100 cubes are put together? n cubes are put together? Questions? What do I mean by a “face unit”? Do I count the faces I can’t see? Cubes in a Line How many faces (face units) are there when: 6 cubes are put together? 10 cubes are put together? 100 cubes are put together? n cubes are put together? Cubes in a Line Cubes in a Line Cubes in a Line Cubes in a Line Cubes in a Line Cubes in a Line We found several different number sentences that represent this problem. • What has to be true about all of these number sentences? 5.OA.2. Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product. 5.OA.3 Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so. Math Practice Standards • Remember the 8 Standards for Mathematical Practice • Which of those standards would be addressed by using a problem such as this? 1. Make sense of problems and persevere in solving them 6. Attend to precision OVERARCHING HABITS OF MIND CCSS Mathematical Practices REASONING AND EXPLAINING 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others MODELING AND USING TOOLS 4. Model with mathematics 5. Use appropriate tools strategically SEEING STRUCTURE AND GENERALIZING 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning High Leverage Instructional Practices High-Leverage Mathematics Instructional Practices An instructional emphasis that approaches mathematics learning as problem solving. 1. Make sense of problems and persevere in solving them. An instructional emphasis on cognitively demanding conceptual tasks that encourages all students to remain engaged in the task without watering down the expectation level (maintaining cognitive demand) 1. Make sense of problems and persevere in solving them. Instruction that places the highest value on student understanding 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively Instruction that emphasizes the discussion of alternative strategies 3. Construct viable arguments and critique the reasoning of others Instruction that includes extensive mathematics discussion (math talk) generated through effective teacher questioning 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning Teacher and student explanations to support strategies and conjectures 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others The use of multiple representations 1. Make sense of problems and persevere in solving them. 4. Model with mathematics 5. Use appropriate tools strategically Number Talks What is a Number Talk? • Also called Math Talks • A strategy for helping students develop a deeper understanding of mathematics – Learn to reason quantitatively – Develop number sense – Check for reasonableness – Number Talks by Sherry Parrish What is Math Talk? • A pivotal vehicle for developing efficient, flexible, and accurate computation strategies that build upon key foundational ideas of mathematics such as – Composition and decomposition of numbers – Our system of tens – The application of properties Key Components • • • • • Classroom environment/community Classroom discussions Teacher’s role Mental math Purposeful computation problems Classroom Discussions • What are the benefits of sharing and discussing computation strategies? • Students have the opportunity to: – Clarify their own thinking – Consider and test other strategies to see if they are mathematically logical – Investigate and apply mathematical relationships – Build a repertoire of efficient strategies – Make decisions about choosing efficient strategies for specific problems 5 Goals for Math Classrooms • • • • • Number sense Place Value Fluency Properties Connecting mathematical ideas Clip 5.6 – 5th Grade Subtraction: 1000 – 674 • Before we watch the clip, talk at your tables – What possible student strategies might you see? – How might you record them? • What evidence is there that the students understand place value? • How do the students’ strategies exhibit number sense? • How does fluency with smaller numbers connect to the students’ strategies? • How are accuracy, flexibility, and efficiency interwoven in the students’ strategies? Clip 5.1 – 5th Grade Multiplication: 12 x 15 • Before we watch the clip, talk at your tables – What possible student strategies might you see? – How might you record them? • What evidence is there that students understand place value? • How do student strategies exhibit number sense? • How do the teacher and students connect math ideas? • What questions does the teacher use to facilitate student thinking about big ideas? Clip 5.5 – 5th Grade Division String: 496 ÷ 8 • Before we watch the clip, talk at your tables – What possible student strategies might you see? – How might you record them? • What evidence is there that students understand place value? • How do students build upon their understanding of multiplication to divide? • How does the teacher connect math ideas throughout the number talk? Solving Word Problems 3 Benefits of Real Life Contents • Engages students in mathematics that is relevant to them • Attaches meaning to numbers • Helps students access the mathematics. A crane operator unloaded the following cargo: • 5 pallets of lumber. Each pallet weighs 7.3 tons. • 9 pallets of concrete. Each pallet weighs 4.8 tons. a) How many pounds of cargo were unloaded? b) Which load of cargo was heavier, the lumber or the concrete? How many pounds heavier? Ava is saving for a new computer that costs $1,218. She has already saved half of the money. Ava earns $14.00 per hour. How many hours must Ava work in order to save the rest of the money? Mrs. Onusko made 60 cookies for a bake sale. She sold 2/3 of them and gave 3/4 of the remaining cookies to the students working at the sale. How many cookies did she have left? Equivalent Fractions 5th Grade Use equivalent fractions as a strategy to add and subtract fractions. CaCCSS • Fractions are equivalent (equal) if they are the same size or they name the same point on the number line. Fraction Families 1 2 3 4 5 6 ....... 2 4 6 8 10 12 1 2 3 4 5 6 ....... 3 6 9 12 15 18 3 6 9 12 15 18 ....... 5 10 15 20 25 30 Equivalent Fractions • Fraction Family Activity • Equivalent Fraction Activity 5th Grade CCSS-M 5.F.1 Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.) Adding Fractions 1 10 8 1 0 • Add 3 7 8 8 2 1 8 1 1 4 2 2 5 3 8 4 5 3 8 4 5 3 5 6 8 4 8 8 5 3 5 6 11 3 1 8 4 8 8 8 8 3 5 3 6 4 5 3 6 4 5 3 10 9 6 4 12 12 5 3 10 9 19 7 1 6 4 12 12 12 12 Subtracting Fractions Subtracting Fractions Possible sequence of instruction • Subtracting 2 fractions less than 1 3 1 4 8 Subtracting Fractions • Subtracting when 1 fraction is between 1 and 2 and 1 fraction is less than 1 3 5 1 4 8 1 5 1 4 8 Subtracting Fractions • Subtracting mixed numbers 3 2 4 2 4 3 1 5 5 2 3 6 Subtracting Fractions • Strategies: Change to improper fractions 1 3 2 19 8 57 32 25 4 2 2 12 4 3 4 3 12 12 12 1 5 16 17 32 17 15 3 1 5 2 2 2 3 6 3 6 6 6 6 6 2 Subtracting Fractions • Strategies: Borrow 9 8 1 3 2 4 2 4 2 2 12 12 12 4 3 2 5 2 5 1 5 3 1 5 2 5 2 4 1 2 2 2 6 6 6 6 3 6 6 2 Subtracting Fractions • Strategies: Shift (Compensate) 1 1 3 5 5 3 6 6 5 1 2 3 6 6 3 1 2 2 6 2 Multiplying Fractions 5.F.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.) 5.F.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side