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A Story of Units. Progression of Algorithms. Solve using strategies other than the standard algorithms. 298 + 357 656 – 298 4527 + 3219$10 - $3.68 5 x 2481240 ÷ 5 25 x 34 850 ÷ 25 6 x 24 4281 ÷ 3. Session Objectives.

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A Story of Units

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A Story of Units

Progression of Algorithms


Solve using strategies other than the standard algorithms.

298 + 357 656 – 298

4527 + 3219$10 - $3.68

5 x 2481240 ÷ 5

25 x 34 850 ÷ 25

6 x 24 4281 ÷ 3


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.


AGENDA

Introduction to the Algorithms

Addition and Multiplication

Subtraction and Division


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


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.


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?


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?


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

Number

System 

Counting Numbers 

The Real Number System 

Fractions

Complex Numbers 

Grade: K 1 2 3 4 5 6 7 8 High School


AGENDA

Introduction to the Algorithms

Addition and Multiplication

Subtraction and Division


8 + 4Examples of addition algorithms in Grade 1: counting all and completing a unit


24 + 58 The standard addition algorithm begun in Grade 2 with the language of units


24 + 58—changing the unit in Grades 4 and 5


Practice the addition algorithm with the language of units.


Units of SixThe foundation of the standard multiplication algorithm in Grade 3


Units of SixThe distributive property in Grade 3


6 × 24 The one-digit by multi-digit multiplication algorithm in Grade 4


6 × 24 with the language of units


6 × 24 Alternate algorithms in Grade 4


30 × 24 The algorithm for multiplying by multiples of ten


30 × 24 The algorithm for multiplying by multiples of ten


30 x 24with the language of units


30 × 24 Alternate algorithms for multiplying by multiples of ten


36 × 24 The two-digit by two-digit multiplication algorithm in Grades 4 and 5


36 x 24An alternate algorithm


36 × 2.4 The multi-digit multiplication algorithm in Grade 5 with decimals


Practice the multiplication algorithm with the language of units.


AGENDA

Introduction to the Algorithms

Addition and Multiplication

Subtraction and Division


12 – 8A subtraction algorithm in Grade 1


82 – 24The subtraction algorithm in Grade 2


82 – 24with the language of units.


600 – 24 Subtraction with zeros in the minuend


600 – 24with the language of units.


Practice the subtraction algorithm using the language of units.


Units of SixA foundational division algorithm in Grade 3


Units of SixThe distributive property with division in Grade 3


42 ÷ 3 The long division algorithm in Grade 4


42 ÷ 3An alternate model for the long division algorithm in Grade 4.


420 ÷ 30 = 42 tens ÷ 3 tens= 42 ÷ 3Strategies for dividing by multiples of ten.


4287 ÷ 29The division algorithm in Grade 5


Practice the long division algorithm using the language 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?


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.


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