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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
A Story of Units

Progression of Algorithms


Solve using strategies other than the standard 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
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
AGENDA

Introduction to the Algorithms

Addition and Multiplication

Subtraction and Division


What is an algorithm
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
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.


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?


Why a system of algorithms
Why a system of algorithms? onions weighed 1,050 grams less than the potatoes. How much did the potatoes and onions weigh together?

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?


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.

Rational

Number

System 

Counting Numbers 

The Real Number System 

Fractions

Complex Numbers 

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


Agenda1
AGENDA long division…

Introduction to the Algorithms

Addition and Multiplication

Subtraction and Division


8 4 examples of addition algorithms in grade 1 counting all and completing a unit
8 + 4 long division…Examples 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 + long division…58 The standard addition algorithm begun in Grade 2 with the language of units


24 58 changing the unit in grades 4 and 5
24 + 58 long division…—changing the unit in Grades 4 and 5



Units of six the foundation of the standard m ultiplication a lgorithm in grade 3
Units of Six long division…The foundation of the standard multiplication algorithm in Grade 3


Units of six the distributive p roperty in grade 3
Units of long division…SixThe distributive property in Grade 3


6 24 the one digit by multi digit m ultiplication a lgorithm in grade 4
6 × 24 long division…The one-digit by multi-digit multiplication algorithm in Grade 4


6 24 with the language of units
6 × 24 long division…with the language of units


6 24 alternate algorithms in grade 4
6 × 24 long division…Alternate algorithms in Grade 4


30 24 the algorithm for multiplying by multiples of ten
30 × 24 long division…The algorithm for multiplying by multiples of ten


30 24 the algorithm for multiplying by multiples of ten1
30 × 24 long division…The algorithm for multiplying by multiples of ten


30 x 24 with the language of units
30 x 24 long division…with the language of units


30 24 alternate a lgorithms for m ultiplying by multiples of ten
30 long division…× 24 Alternate algorithms for multiplying by multiples of ten


36 24 the t wo digit by two digit m ultiplication a lgorithm in grades 4 and 5
36 long division…× 24 The two-digit by two-digit multiplication algorithm in Grades 4 and 5


36 x 24 an alternate algorithm
36 x 24 long division…An alternate algorithm


36 2 4 the multi digit m ultiplication a lgorithm in grade 5 with decimals
36 long division…× 2.4 The multi-digit multiplication algorithm in Grade 5 with decimals



Agenda2
AGENDA units.

Introduction to the Algorithms

Addition and Multiplication

Subtraction and Division


12 8 a subtraction a lgorithm in grade 1
12 – 8 units. A subtraction algorithm in Grade 1


82 24 the subtraction a lgorithm in grade 2
82 – 24 units. The subtraction algorithm in Grade 2


82 24 with the language of units
82 – 24 units. with the language of units.


600 24 subtraction with zeros in the minuend
600 – 24 units. Subtraction with zeros in the minuend


600 24 with the language of units
600 – 24 units. with the language of units.



Units of six a foundational division algorithm in grade 3
Units of Six units.A foundational division algorithm in Grade 3


Units of six the distributive property with division in grade 3
Units of units.SixThe distributive property with division in Grade 3


42 3 the long division a lgorithm in grade 4
42 ÷ 3 units.The long division algorithm in Grade 4


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


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


4287 29 the division algorithm in grade 5
4287 ÷ 29 units.The division algorithm in Grade 5



A progression of algorithms
A Progression of Algorithms units.

  • 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
Key Points units.

  • 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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