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# A Story of Units - PowerPoint PPT Presentation

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 248 1240 ÷ 5 25 x 34 850 ÷ 25 6 x 24 4281 ÷ 3. Session Objectives.

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### A Story of Units

Progression of Algorithms

298 + 357 656 – 298

4527 + 3219 \$10 - \$3.68

5 x 248 1240 ÷ 5

25 x 34 850 ÷ 25

6 x 24 4281 ÷ 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.

Introduction to the Algorithms

Subtraction and Division

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? 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?

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 long division…

Introduction to the Algorithms

Subtraction and Division

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

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

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

Units of Six long division…The foundation of the standard multiplication algorithm in Grade 3

Units of long division…SixThe distributive property in Grade 3

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

6 × 24 long division…with the language of units

6 × 24 long division…Alternate algorithms in Grade 4

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

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

30 x 24 long division…with the language of units

30 long division…× 24 Alternate algorithms for multiplying by multiples of ten

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

36 x 24 long division…An alternate algorithm

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

AGENDA units.

Introduction to the Algorithms

Subtraction and Division

12 – 8 units. A subtraction algorithm in Grade 1

82 – 24 units. The subtraction algorithm in Grade 2

82 – 24 units. with the language of units.

600 – 24 units. Subtraction with zeros in the minuend

600 – 24 units. with the language of units.

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

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

42 ÷ 3 units.The long division algorithm in Grade 4

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

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

4287 ÷ 29 units.The division algorithm in Grade 5

• 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 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.