SERGEI ABRAMOVICH SUNY POTSDAM

1. HOW MANY WAYS AND WHAT ARE THE CHANCES ? COUNTING AND REASONING WITH MANIPULATIVES AT THE ELEMENTARY LEVEL. SERGEI ABRAMOVICHSUNY POTSDAM

2. COMMON CORE STATE STANDARDS FOR MATHEMATICS PRIMARY SCHOOL

3. Mathematics experiences in early childhood settings should concentrate on (1) number (which includes whole number, operations, and relations) and (2) geometry, spatial relations, and measurement, with more mathematics learning time devoted to number than to other topics. Mathematical process goals should be integrated in these content areas. —Mathematics Learning in Early Childhood, National Research Council, 2009

5. 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. (Kindergarten students should see addition and subtraction equations, and student writing of equations in kindergarten is encouraged, but is not required.) Operations and Algebraic Thinking: Kindergarten

6. What is a big idea? NCTM (2000, p. 19): “... worthwhile tasks should be intriguing, with a level of challenge that invites speculation and hard work.” Such tasks should enable their use at different grade levels and be associated with any of the 25 elements of the content-process standards matrix. Big ideas and worthwhile tasks

7. Worthwhile Task 1. Provide students with four two-color counters. Have students create different combinations of red and yellow counters, record each combination they find and stop when they believe they found all of them. [Adapted from New York State Education Department. (1998). Mathematics Resource Guide with Core Curriculum. Albany, NY: Author.]

8. How many ? WHO ARE THE BROTHERS ? 2&5: 4 = 2 + 2 4 = 2 + 2 10&11: 4 = 3 + 1 4 = 3 + 1

10. Four red/yellow counters can be arranged in 16 ways. Four can be partitioned in positive integers in 8 = 24 - 1 ways. Generalization: Positive integer N can be partitioned in positive integers in 2N – 1 ways. CONNECTING VISUAL TO SYMBOLIC

11. What are the chances that A combination of four counters does not include a red one? No two red counters appear in a row? A combination of four counters begins with a red one? A combination of four counters ends with a red one? Why do we need to know the total?

12. 2. Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape. GEOMETRY GRADE 1(Common Core Standards)

13. 8. Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters. GEOMETRIC MEASUREMENTGRADE 3 (Common Core Standards)

14. 3. Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry. GEOMETRY GRADE 4(Common Core Standards)

15. Using 10 linking cubes, construct 5 towers and arrange them from the lowest to the highest. There may be more than one way to do that. Construct all such combinations of 5 towers using the 10 cubes. Record your combinations (the sets of towers). Describe what you have found. Worthwhile Task 2

16. 10=1+1+2+3+3 16=5+3+2+1+1+1+2+1 2 3 5 1 1 2 3 3 Area = 10 Perimeter = 16 1 4 5 10=1+2+2+2+3 16=5+3+1+1+3+1+1+1 Area = 10, Perimeter = 16 1 2 2 2 3

17. 1 1 1 2 5 10=1+1+1+2+5 20=5+5+1+3+1+1+3+1 1 1 1 1 1 5 Area = 10, Perimeter = 20 10=1+1+1+1+6 22=5+6+1+5+4+1 1 1 1 2 5 Area = 10, Perimeter = 22 1 1 1 1 6

18. 10=1+1+1+3+4 18=5+4+1+1+1+2+3+1 1 2 2 5 Area = 10, Perimeter = 18 10=1+1+2+2+4 18=5+4+1+2+2+1+2+1 1 1 3 5 Area = 10, Perimeter = 18 1 1 1 3 4 1 1 2 2 4

19. Ferrers-Young diagrams in the primary school. 10=2+2+2+2+2 14=5+2+5+2 Area = 10, Perimeter = 14 5 5 2 2 2 2 2

20. COMMON CORE STATE STANDARDS Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades.

21. Hidden mathematics curriculum: The number of partitions on 10 (n) into 5 (m) parts is equal to the number of partitions of 10 (n) with the largest part 5 (m). Rotation as a mathematical demonstration Partition in three parts turns into partition with the largest part three

22. Mathematically proficient students can ... draw diagrams of important features and relationships ... Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” Standards for Mathematical Practice

23. Why is perimeter always an even number? • Why can Ferrers-Young diagrams with the same perimeter and area be created? • May it be the case for partition into two parts? Questions(for “mathematically proficient” students)

24. HOW MANY SYMMETRIES ? 10=2+2+2+2+2

25. SYMMETRY ? A=10=2+1+4+1+2 P=22=5+2+1+1+1+3+1+3+1+1+1+2 10=1+2+4+2+1 P=18=5+1+1+1+1+2+1+2+1+1+1+1

26. area and perimeter are the same (numerically) • mean, median, and mode height are the same • the set is symmetrical • the set can be rearranged to be symmetrical • perimeter is an odd number • perimeter is an even number What are the chances thatusing 10 linking cubes Jonny constructed one set of 5 towers, arranged them from the lowest to the highest and

27. Manipulative materials for challenging tasks • Extending CCSS • Using worthwhile tasks • Connecting mathematics across grades • Preparing students for upper grades • Making mathematics learning fun CONCLUDING REMARKS

28. Thank you! abramovs@potsdam.edu