Please sit by grade level.

1. Please sit by grade level. Did you try something during the last few weeks from our first session? If so, how did it go? Prepare to share a little with others at your table.

2. Goals for Today • Understand the details of learning to add and subtract in K and 1st grade, based on the Common Core and research. • Connect adding and subtracting to the foundation in early number concepts from last session. • Share methods for engaging children in activities that promote learning of adding and subtracting.

3. Teaching Math to Young Children Cognitively Guided Instruction

4. In a first grade class… • Three children successfully solved addition and subtraction tasks for two digit numbers. • Fourteen of the twenty-one children used their fingers to count all or count on as they solved such problems as 6 + 3 = ___ and 8 + 9 = ___. • Three of the children needed cubes to solve such problems and counted all the cubes. • One child had difficulty counting more than seven cubes accurately.

5. Kindergarten: By the end of the kindergarten year, some children will be ready to begin acting out addition and subtraction story problems and to describe a number according to its parts. • First Grade: Addition and subtraction are important mathematical topics for first grade and should be a central part of the math program for several months.

7. Number sense first • The first priority is to develop early number sense in all children: • Counting • Comparing Sets • Composing and Decomposing Numbers It is very important to give your children enough time to make sense of these beginning number relationships. Do not feel pushed to have them memorize relationships they don’t really understand. A strong foundation will serve them in the long run.

8. 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.

9. 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 Eliz?

10. 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.

11. CCSS Kindergarten • K.OA.1 Represent addition and subtraction with objects, fingers, mental images, drawings (drawings need not show details, but should show the mathematics in the problem), sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations. • K.OA.2 Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

12. CCSS 1st Grade • 1.OA.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 with a symbol for the unknown number to represent the problem.

13. 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.

14. Where is the unknown?

15. Problem Types - Action Result Change Start Unknown Unknown Unknown Join 5 + 2 =  5 +  = 7  + 2 = 7 Separate 8 – 3 =  8 –  = 5  – 3 = 5

16. CCSS – 1st “unknowns” • 1.OA.8 Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11 5 = ＿ – 3 6 + 6 = ＿.

17. No-action 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? • Mark has 3 mice. Joy has 7 mice. Joy has how many more mice than Mark?

18. No-action problems • Equalizing: Abby has read 6 books so far this summer. Mollie has read 4. In order to have read as many books as Abby, how many more does Mollie need to read?(number sentence?) • Missing part:There are 14 hats in the closet. 6 are red and the rest are green. How many green hats are in the closet?(number sentence?) • Comparative subtraction: Jessica found 6 rocks on the trail and Tia found 7. How many more rocks did Tia find than Jessica?(number sentence?)

19. 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?

20. Try this once a week • Present a problem to the whole class, let them work on it individually, then have several students present their approaches. • Keep track of their solutions • Use problems with numbers that are appropriate for your students.

21. Solution Strategies • Children learn more advanced computational and problem solving strategies by watching their classmates solve problems. • How many different strategies so you see in this video? • As you watch these children solve simple joining and separating problems, think about who in your class might be at each stage.

22. Solution Strategies • Direct modeling of the action in the problem • Counting strategies • Derived facts • Fluency

23. CCSS K – Strategies • K.OA.3 Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1). • K.OA.4 For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.

24. CCSS 1st Grade – Strategies • 1.OA.3 Apply properties of operations as strategies to add and subtract(e.g. 5+3 = 3+5) • 1.OA.4 Understand subtraction as an unknown-addend problem. (12-8 can be “what do I add to 8 to get 12?) • 1.OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten; decomposing a number leading to a ten; using the relationship between addition and subtraction; and creating equivalent but easier or known sums, e.g. (7+6 = 6+6 + 1).

25. Solution Strategies • Try “How Would Children Solve These Problems?” using each of the types of strategies. • Then try “Finding a Problem for a Strategy” – the Jeopardy Game of CGI.

26. Use problems like these often in class and record students’ progress through the strategies. • The packet has so many problems that you shouldn’t run out, but if you do, they’re easy to make up. • The teacher’s role is to guide student’s learning by knowing each child’s cognition. Elementary Math Resources wiki

27. There were seven apples on the tree. A farmer came along and picked five of the apples. How many apples are still on the tree? • Four ladybugs were crawling in the grass. Three more came to join them. How many ladybugs were there then?

30. Which Problems are Harder? • The structure of a problem determines how difficult it is for children to solve and determines their initial solution strategies.

31. 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, but that isn’t necessary until 2nd grade.

32. Math Stations/Small Group Work • Magic Box Cards – Making Trains by adding and subtracting

35. What do we have to do to get to 3? Do we add or subtract?

37. Number Talks to Develop Fluency • K.OA.5 Fluently add and subtract within 5. • 1.OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10.

38. Number Talks and Math Centers • A different approach for helping children see number relationships and move toward fluency involves work with dot cards and five- and ten-frame cards. • Not to take the place of problem-solving, but to supplement itand provide more opportunities for learning and practicing strategies.

39. Numbers within Numbers • K.OA.4 For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation. As each number talk is shown, ask students, “How many dots do you see? How do you see them?”

40. K.OA.3 Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).

41. There are many of these in the Number Talks book.

43. 1st Grade Addition Number Talks • Counting All/Counting On… • Doubles/Near Doubles… • Making Tens… • Landmark or Friendly Numbers…

45. Place Value in CCSS 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. Activities: Counting objects in the teens, grouping onto a place value mat, linking 10 cubes to make 10. Writing and saying the number. What else? Nat’l Library of Virtual Manipulatives

46. Place Value in CCSS 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). 1.NBT.3 Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.

47. 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? • 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?

48. Adding Within 100 1.NBT.4 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. Counting Collections to 100

49. 1.NBT.5 Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used. 1.NBT.6 Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), 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.

50. Hundreds Chart Games