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Goals for today • • • • • Share “what works” with each other Possibly learn some new things about 2nd and 3rd grade math Get new resources and do some planning Go inside other’s classrooms Think about how to provide intervention support as needed (“Daily 5”) 3 sessions 1) Addition and subtraction (today) 2) Multiplication and division (Mar. 20) 3) Measuring length, geometric shapes, working with time and money (Apr. 17) Big Ideas about 2-3 Math 1) Addition and subtraction fluency builds on a strong foundation of solving structured problems and using mental math. 2) Multiplication fluency is rooted in 2nd grade “skip counting” and 3rd grade mental math. 3) Great resources are available for early numeracy activities. 4) Diagnostic assessments can help students who are behind their classmates. Key Teaching Strategies 1) Teach the underlying structure of word problems to promote problem solving ability 2) Use visual representations often 3) Verbalize your thought processes when explaining procedures 4) Have students verbalize their thinking to each other by explaining what they’re doing and why 5) Provide lots of practice with feedback 6) Use frequent cumulative review 7) Emphasize reasoning, looking for patterns, and problem solving Framework • Cognitively Guided Instruction • Add/Subtract Situations – Strategies – Fluency • Tens and Ones – Place Value – Multi-digit Operations Resources • • • • • Van de Walle and Lovin Kathy Richardson Grayson Wheatley Elementary Math Resources wiki Georgia and New York lesson plans Classroom Visits • Teaching Channel • Number Talks • Eventually your own classrooms?? Sharing and Planning • Always share • Keep track of “What I Want to Try” • Make and take – Daily 5 CCSS Collaborative Cards Carla is packing treat bags for Halloween. Each treat bag has to have exactly the same items in it or the neighborhood kids will complain! She has 99 Hershey’s Kisses and 108 Jelly Beans. a) What is the greatest number of treat bags she can make if she wants to use all the treats she has? Explain. b) How many of each kind of treat is in one bag? In your class, how many can… • Add and subtract single digit numbers fluently? • Add multi-digit numbers using strategies? • Solve word problems involving single-digit or multi-digit numbers? • Fluently add two- or three-digit numbers without regrouping? • Fluently add two- or three-digit numbers with regrouping? Purpose of these sessions • Purpose: Develop professional knowledge – deep content understanding and ability to use effective teaching strategies • Outcome: Responsive teaching, leading to greater readiness for the next grade 2nd grade critical areas (1) extend understanding of base-ten notation; (2) build fluency with addition and subtraction; (3) use standard units of measure; and (4) describe and analyze shapes. 3rd grade critical areas (1) develop understanding of multiplication and division and strategies for multiplication and division within 100; (2) develop understanding of fractions, especially unit fractions (fractions with numerator 1); (3) develop understanding of the structure of rectangular arrays and of area; (4) describe and analyze two-dimensional shapes. CCSS – Add and subtract K: Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. K: Fluently add and subtract within 5. 1: Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations. 1: Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. 2: Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers. Children’s Mathematics Cognitively Guided Instruction Carpenter, Fennema, Franke, Levi and Empson What operation is this? • Steven had 4 toy cars. He wanted 9. How many more toy cars would Steven need to have 9 altogether? • Show how a kindergarten or 1st grade student might solve this. Modeling the Action • Liz had 8 cookies. She ate 3 of them. How many cookies does Eliz have left? • Liz has 3 marbles. How many more marbles does she need to buy to have 8 marbles? • Liz has 3 fish. Tom has 8 fish. How many more fish does Tom have than Liz? Rachel’s Problems • Try each of the problems. Think about how students might model the action in the problem. • Discuss your solutions with a partner. • As you watch the video, think about which problems seem harder for Rachel. Basic assumptions about children’s learning of mathematics • Very young children know how to solve math problems. • Children develop mathematical understanding and acquire fluency with whole number computation by solving a variety of problems in any way that they choose. • Children learn more advanced computational and problem solving strategies by watching their classmates solve problems. Problem Types • solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions CCSS CGI Adding to, putting together Join Taking from, taking apart Separate Comparing Compare Part-Part-Whole Where is the unknown? 1. Lucy has 8 fish. She wants to buy 5 more fish. How many fish would Lucy have then? 3. Janelle has 7 trolls in her collection. How many more does she have to buy to have 11 trolls? 2. TJ had 13 chocolate chip cookies. At lunch she ate 5 of those cookies. How many cookies did TJ have left? 6. 11 children were playing in the sandbox. Some children went home. There were 3 children still playing in the sandbox. How many children went home? 4. Max had some money. He spent $9 on a video game. Now he has $7 left. How much money did Max have to start with? Problem Types - Action Join Separate Result Unknown Change Unknown Start Unknown 5+2= 8–3= 5+=7 8–=5 +2=7 –3=5 No-action problems Part-part-whole problems • 6 boys and 4 girls were playing soccer. How many children were playing soccer? • 10 children were playing soccer. 6 were boys and the rest were girls. How many girls were playing soccer? 10 6 No-action problems Comparison problems • Willy has 12 crayons. Lucy has 7 crayons. How many more crayons does Willy have than Lucy? Are some more difficult? • There are 14 hats in the closet. 6 are red and the rest are green. How many green hats are in the closet? • 14 birds were in a tree. 6 flew away. How many birds were left? Try this twice a week • Present a problem to the whole class, let them work on it individually, then have several students present their approaches. • For older children, use some two-digit problems that don’t require regrouping and some that do. • Keep track of students’ solutions. Solution Strategies • • • • Direct modeling of the action in the problem Counting strategies Derived facts Fluency The structure of a problem determines how difficult it is for children to solve and determines their initial solution strategies. Video of children using counting and derived facts. Effects of CGI • Teachers prepared in CGI spend more time having their students solve problems, listen more to their students, and are more likely to expect students to find multiple solution strategies to problems than teachers who are not prepared in CGI (Carpenter, Fennema, Peterson, Chiang, & Loef, 1989). • CGI also results in improved performance by primarygrade students on both standardized and problemsolving tests (Carpenter et al., 1989; Fennema, Carpenter, & Peterson, 1989; Peterson et al., 1989). Fluency with “math facts” • The use of manipulatives, counting and derived-fact strategies eventually grows into knowledge of most math facts. • Explicit instruction on strategies can be helpful for building math facts that haven’t come naturally through problem solving. Explicit strategy instruction For addition and subtraction, children can derive unknown facts using a combination of • anchor sums to 10 (6 + 4, 7 + 3, 8 + 2) – (7 + 6 can be derived by knowing that 7 + 3 is 10, and 3 more is 13) • doubles (7 + 6 can be derived by knowing that 7 + 7 is 14, so the answer is one less) Number Talks 6+8 sums to 5 If you know the sums to 5, like 3+2, you can find other sums like 3+4, because 3+4 = 3+2 plus 2 more sums to 10 If you know that 6+4 = 10, then you can figure out 6+5, because it’s 1 more than 6+4 doubles 6+7 = (6+6) + 1 more plus one doubles 7+9 = (7+7) + 2 more plus two nines Add ten then subtract 1 Fluency – Practice and Drill “Practice” refers to lessons that are problembased and that encourage students to develop flexible and useful strategies that are personally meaningful. “Drill” is repetitive non-problem-based activity to help children become facile with strategies they know already in order to internalize (remember) the fact combinations. From Van de Walle, Elementary and Middle School Mathematics: Teaching Developme Practice Sum Search Math Squares • Always follow the small group work with a whole class discussion where students explain their methods. Number Talks 8+6 Kathy Richardson, Developing Number Concepts Bk. 2, p. 136-137 Drill Once strategies are learned, students can often be induced into remembering combinations by playing competitive games that require quick recall of number combinations, such as “addition war” or “subtraction war” or games where pieces are moved around game boards by rolling dice and calling out sums or differences. Vary the drill http://www.fun4thebrain.com/addition.html Print triangle flash cards Summarize and Plan • What are the critical elements of 2nd grade math instruction (and 3rd grade practice and intervention) that support children’s development of single-digit adding and subtracting? • Word problems – various types that grow in complexity • Children’s use of strategies to solve them – developmental • Number Talks to build mental math and further develop computation strategies • Games for practice that reward quick recall Does your curriculum emphasize these things? What are your favorite games for developing fluency? CCSS – Multi-digit add/subt 1: Add within 100, including adding a two-digit number and a one digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten. 2: Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. CCSS – Multi-digit add/subt 2: Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds. 3: Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. (A range of algorithms may be used.) A classroom view • Counting Collections to 100 Place value K.NBT.1 Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (such as 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones. 1.NBT.2 Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases: a. 10 can be thought of as a bundle of ten ones — called a “ten.” b. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones. c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones). 2.NBT.1 Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases: a. 100 can be thought of as a bundle of ten tens — called a “hundred.” b. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones). Another classroom • Making Math Fun with Place Value Games Multi-digit Progression 1. Mental strategies 2. Place value representations (e.g. base 10 blocks, pictures) 3. Algorithms Multi-digit Problems 1. Separating, result unknown Peter had 28 cookies. He ate 13 of them. How many did he have left? Write this as a number sentence: 28 – 13 = ____ There were 51 geese in the farmer’s field. 28 of the geese flew away. How many geese were left in the field? 2. Comparing two amounts (height, weight, quantity) There are 18 girls on a soccer team and 5 boys. How many more girls are there than boys on the soccer team? Multi-digit Problems 3. Part-whole where a part is unknown There are 23 players on a soccer team. 18 are girls and the rest are boys. How many boys are on the soccer team? 4. Distance between two points on a number line (difference in age, distance between mileposts) How far is it on the number line between 27 and 42? Multi-digit Problems • There were 51 geese in the farmer’s field. 28 of the geese flew away. How many geese were left in the field? • There were 28 girls and 35 boys on the playground at recess. How many children were there on the playground at recess? • Misha has 34 dollars. How many dollars does she have to earn to have 47 dollars? • Strategies? Counting single units. Direct modeling with tens and ones. Invented algorithms: Incrementing by tens and then ones, Combining tens and ones, Compensating. Mental Math Base Ten Concepts Using objects grouped by ten: • There are 10 popsicle sticks in each of these 5 bundles, and 3 loose popsicle sticks. How many popsicle sticks are there all together? • Students’ strategies? • The extension: The teacher puts out one more bundle of ten popsicle sticks and asks students “Now how many popsicle sticks are there all together?” What strategies would students use to answer this? Development of Algorithms • The C-R-A approach is used to develop meaning for algorithms. • Without meaning, students can’t generalize the algorithm to more complex problems. Practice? Drill? • What would be good kinds of practice for double digit work? • Good kinds of drill? Typical Learning Problems Create your own • You can differentiate instruction appropriately for your students by adjusting the numbers in the problems. • Try this twice a week before our next session. • Come back with stories to tell! • You can also create learning centers around these problems, number strategy games and fluency practice. When children work at centers during “Math Daily 5” you can bring a small group together who needs additional support from you. • http://elementary-math-resources.wiki.inghamisd.org/2-3 Summarize and Plan • What are the critical elements of 2nd grade math instruction (and 3rd grade practice and intervention) that support children’s development of multi-digit adding and subtracting? • • • • Word problems – same types, larger numbers Children’s use of strategies to solve them – developmental Place value understanding C-R-A Does your curriculum emphasize these things? Do your students have both conceptual understanding and procedural skill?