1. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Progression of Algorithms

2. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Solve using strategies other than the standard algorithms. 2 298 + 357 656 – 298 4527 + 3219$10 - $3.68 5 x 2481240 ÷ 5 25 x 34 850 ÷ 25 6 x 24 4281 ÷ 3

3. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Session Objectives 3 Examine and practice the algorithms employed in A Story of Units. Understand the coherence within and across grades in order to promote conceptual understanding.

4. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units AGENDA 4

5. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units What is an algorithm? 5 An algorithm is a systematic step by step procedure to solve a class of problems. Parker and Baldridge, Elementary Mathematics for Teachers, pg. 57

6. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Why do we want standard algorithms? 6 If no simplification is obvious within a problem, we want students to have an easily accessible tool they can use efficiently. Why? So that the higher level relationships within a problem can be addressed.

7. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units 7 The potatoes Beth bought weighed 3 kilograms 420 grams. Her onions weighed 1,050 grams less than the potatoes. How much did the potatoes and onions weigh together?

8. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Why a system of algorithms? 8 Each algorithm must provide a coherent link to the subsequent algorithm. Addition sets the foundation for subtraction and multiplication. All three set the foundation for the division algorithm. Why is the division algorithm an essential culminating goal of the Pre-Kindergarten to Grade 5 curriculum?

9. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Grade 7.NS.2(d) Convert a rational number to a decimal using long division… Grade 8.NS.1 Know that numbers that are not rational are called irrational. 9 Counting Numbers  Fractions  Rational Number System  The Real Number System  Complex Numbers  Grade: K 1 2 3 4 5 6 7 8 High School

10. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units AGENDA 10

11. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units 8 + 4 Examples of addition algorithms in Grade 1: counting all and completing a unit 11

12. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units 24 + 58 The standard addition algorithm begun in Grade 2 with the language of units 12

13. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units 24 + 58 —changing the unit in Grades 4 and 5 13

14. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Practice the addition algorithm with the language of units. 14

15. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Units of Six The foundation of the standard multiplication algorithm in Grade 3 15

16. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Units of Six The distributive property in Grade 3 16

17. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units 6 × 24 The one-digit by multi-digit multiplication algorithm in Grade 4 17

18. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units 6 × 24 with the language of units 18

19. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units 6 × 24 Alternate algorithms in Grade 4 19

20. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units 30 × 24 The algorithm for multiplying by multiples of ten 20

21. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units 30 × 24 The algorithm for multiplying by multiples of ten 21

22. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units 30 x 24 with the language of units 22

23. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units 30 × 24 Alternate algorithms for multiplying by multiples of ten 23

24. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units 36 × 24 The two-digit by two-digit multiplication algorithm in Grades 4 and 5 24

25. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units 36 x 24 An alternate algorithm 25

26. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units 36 × 2.4 The multi-digit multiplication algorithm in Grade 5 with decimals 26

27. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Practice the multiplication algorithm with the language of units. 27

28. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units AGENDA 28

29. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units 12 – 8 A subtraction algorithm in Grade 1 29

30. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units 82 – 24 The subtraction algorithm in Grade 2 30

31. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units 82 – 24 with the language of units. 31

32. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units 600 – 24 Subtraction with zeros in the minuend 32

33. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units 600 – 24 with the language of units. 33

34. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Practice the subtraction algorithm using the language of units. 34

35. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Units of Six A foundational division algorithm in Grade 3 35

36. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Units of Six The distributive property with division in Grade 3 36

37. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units 42 ÷ 3 The long division algorithm in Grade 4 37

38. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units 42 ÷ 3 An alternate model for the long division algorithm in Grade 4. 38

39. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units 39 420 ÷ 30 = 42 tens ÷ 3 tens = 42 ÷ 3 Strategies for dividing by multiples of ten.

40. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units 4287 ÷ 29 The division algorithm in Grade 5 40

41. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Practice the long division algorithm using the language of units. 41

42. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units A Progression of Algorithms 42 What did you notice about the sequence of algorithms? In what ways will students benefit from this sequence? How does it change or develop your understanding of the algorithms?

43. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Units Key Points 43 All algorithms involve the manipulation of units. Each algorithm builds towards the next, culminating in the long division algorithm. The long division algorithm is foundational to an understanding of the real number system and advanced mathematics.