# A Story of Units

N YS CO MMO N COR E MAT H E MAT I C S C U RR I C U LU M A Story of Units Progression of Algorithms 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Units A Story of Units N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Solve using strategies other than the

standard algorithms. 298 + 357 4527 + 3219 656 298 \$10 - \$3.68 5 x 248 25 x 34 6 x 24 1240 5 850 25

4281 3 2 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units Session Objectives Examine and practice the algorithms employed in A Story of Units.

Understand the coherence within and across grades in order to promote conceptual understanding. 3 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units AGENDA

Introduction to the Algorithms Addition and Multiplication Subtraction and Division 4 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units What is an algorithm?

An algorithm is a systematic step by step procedure to solve a class of problems. Parker and Baldridge, Elementary Mathematics for Teachers, pg. 57 5 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units Why do we want standard algorithms?

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. 6 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units

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? 7 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units

Why a system of algorithms? 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? 8 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Units

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M 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. Rational Counting Numbers Number Fractions

Grade: K 1 2 3 4 5 6

7 System 8 The Real Number System Complex Numbers High School 9

2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units AGENDA Introduction to the Algorithms Addition and Multiplication Subtraction and Division 10

2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units 8+4 Examples of addition algorithms in Grade 1: counting all and completing a unit 11

2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units 24 + 58 The standard addition algorithm begun in Grade 2 with the language of units 12 2012 Common Core, Inc. All rights reserved. commoncore.org

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units 24 + 58 changing the unit in Grades 4 and 5 13 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M

A Story of Units Practice the addition algorithm with the language of units. 14 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units

Units of Six The foundation of the standard multiplication algorithm in Grade 3 15 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units Units of Six

The distributive property in Grade 3 16 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units 6 24 The one-digit by multi-digit multiplication algorithm in Grade 4

17 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units 6 24 with the language of units 18

2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units 6 24 Alternate algorithms in Grade 4 19 2012 Common Core, Inc. All rights reserved. commoncore.org

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units 30 24 The algorithm for multiplying by multiples of ten 20 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M

A Story of Units 30 24 The algorithm for multiplying by multiples of ten 21 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units

30 x 24 with the language of units 22 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units 30 24 Alternate algorithms for multiplying by multiples of ten

23 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units 36 24 The two-digit by two-digit multiplication algorithm in Grades 4 and 5

24 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units 36 x 24 An alternate algorithm 25

2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units 36 2.4 The multi-digit multiplication algorithm in Grade 5 with decimals 26 2012 Common Core, Inc. All rights reserved. commoncore.org

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units Practice the multiplication algorithm with the language of units. 27 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M

A Story of Units AGENDA Introduction to the Algorithms Addition and Multiplication Subtraction and Division 28 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M

A Story of Units 12 8 A subtraction algorithm in Grade 1 29 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units

82 24 The subtraction algorithm in Grade 2 30 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units 82 24 with the language of units.

31 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units 600 24 Subtraction with zeros in the minuend 32

2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units 600 24 with the language of units. 33 2012 Common Core, Inc. All rights reserved. commoncore.org

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units Practice the subtraction algorithm using the language of units. 34 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M

A Story of Units Units of Six A foundational division algorithm in Grade 3 35 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units

Units of Six The distributive property with division in Grade 3 36 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units 42 3 The long division algorithm in Grade 4

37 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units 42 3 An alternate model for the long division algorithm in Grade 4.

38 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units 420 30 = 42 tens 3 tens = 42 3 Strategies for dividing by multiples of ten.

39 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units 4287 29 The division algorithm in Grade 5 40

2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units Practice the long division algorithm using the language of units. 41 2012 Common Core, Inc. All rights reserved. commoncore.org

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units A Progression of Algorithms 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?

42 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Units Key Points

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. 43 2012 Common Core, Inc. All rights reserved. commoncore.org

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