Promoting Powerful Discussions about Critical...

Promoting Powerful Discussions about Critical Concepts

Joyce Bishop December 13, 2012

[email protected] 3-5

First, a few words about the Common Core State Standards for Mathematics

A few words

45 States + DC Have Adopted the Math Common Core State Standards

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*Minnesota adopted the CCSS in ELA/literacy only www.corestandards.org

The Common Core State Standards

corestandards.org 4

 Have rigorous content and application of knowledge

 Use lessons learned from current state standards

 Are internationally benchmarked  Are based on evidence and research

Isbe website

Common Core State Standards

Fewer, Clearer, Higher Focus Coherence Rigor

Isbe website

Common Core State Standards

Fewer, Clearer, Higher Focus Coherence Rigor

conceptual understanding procedural understanding applications

Isbe website

What the Standards do NOT define: •  The nature of advanced work beyond the

core

•  The interven5ons needed for students well below grade level

•  The full range of support for English language learners and students with special needs

Isbe website

Standards, Clusters, Domains

Organization of the document •  Standards define what students should

understand and be able to do. •  Clusters summarize groups of related

standards. •  Domains are larger groups of related

standards.

**Standards from different domains and clusters may sometimes be closely related.

Isbe website

Domains, Clusters, Standards

Write and interpret numerical expressions. (Domain 5OA) • 5.OA.1. Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. • 5.OA.2. Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.

• (The two standards compose a cluster within the domain.)

Standard:

Standard:

Isbe website

Common Core State Standards for Mathematics

Examine the Common Core Standards for your grade level.

•  Notice the notation for domains, clusters, and standards.

•  Check concepts you already address reasonably well.

•  Underline concepts for which you need more resources or support.

Standards for Mathematical Practice These are K-‐12 standards that describe habits of mind of a mathema5cally proficient student

1.  Make sense of problems and persevere in solving them.

2.  Reason abstractly and quantitatively. 3.  Construct viable arguments and critique the

reasoning of others. 4.  Model with mathematics. 5.  Use appropriate tools strategically. 6.  Attend to precision. 7.  Look for and make use of structure. 8.  Look for and express regularity in repeated

reasoning. 11

Isbe website

Structuring the Practices

NCSM

Student-friendly Language for Mathematical Practices

•  1) I can try many times to understand and solve a math problem.

•  2) I can think about the math problem in my head, first.

•  3) I can make a plan, called a strategy, to solve the problem and discuss other students’ strategies too.

•  4) I can use math symbols and numbers to solve the problem.

J. White & L. Dauksas, TCM, March 2012

Student-friendly Language for Mathematical Practices

•  5) I can use math tools, pictures, drawings, and objects to solve the problem.

•  6) I can check to see if my strategy and calculations are correct.

•  7) I can use what I already know about math to solve the problem.

•  8) I can use a strategy that I used to solve another math problem.

J. White & L. Dauksas, TCM, March 2012

Relational Thinking

Relational Thinking: Making use of the relationships among numbers.

Consider each number sentence. Without calculating, determine whether each number sentence is true or false and explain your reasoning.

What does it mean to solve without calculating?

Number Talk

Elysha Passeggi 4th grade San Carlos Charter School

What mathematical ideas are students considering during this discussion?

What Mathematical Practices are incorporated?

8 + 4 = ☐ + 5

http://insidemathematics.org/index.php/4th-grade-can-this-be-true

Properties of operations and equality

CCSSM

Priorities for Grades 3-5

•  Multiplication and division of whole numbers

•  Fractions – concepts, skills, and problem solving

Source: achievethecore.org

Grade Level Domains 3-5

Grade 3 Operations and Algebraic Thinking •  Represent and solve problems involving

multiplication and division •  Understand properties of multiplication and

the relationship between multiplication and division

•  Multiply and divide within 100 •  Solve problems involving the four operations,

and identify and explain patterns in arithmetic


Grade 4 Operations and Algebraic Thinking •  Gain familiarity with factors and multiples Number and Operations in Base 10 •  Generalize place value understanding for

multi-digit whole numbers •  Use place value understanding and

properties of operations to perform multi-digit arithmetic


Grade 5 Number and Operations in Base 10 •  Understand the place value system •  Perform operations with multi-digit whole

numbers and with decimals to hundredths

To help students develop understanding:

•  Have students represent problem situations with concrete objects,

•  then make pictorial representations, •  then represent abstractly.

A basic tenet of Singapore math Theory by Jerome Bruner

MathWithMeaning: Foundations of Number Sense

Applying Bruner’s C-P-A Approach to Learning Multiplication

You have 4 plates with 3 cookies on each plate. How many cookies in all?

•  Concrete – Students stand in 4 hula hoops to show 3 cookies on each plate.

•  Pictorial – Students draw diagrams of groups •  Abstract – students write number sentences

based on a diagram of the problem shown on a slide or overhead.

•  MathWithMeaning: Foundations of Number Sense

Learning Multiplication Through an Understanding of Place Value

Activities may include: •  Multiplying without renaming •  Multiplying a decimal by a whole number

with renaming •  Multiplying using the distributive method •  Multiplying using the area model

•  MathWithMeaning: Foundations of Number Sense p. 90

Model Multiplication with Place Value Materials without Renaming

3 groups of 13 13 x 3

Using Familiar Base 10 Blocks to Model Decimal Numbers

•  flat = 1

•  Long = 0.1

•  small cube = 0.01

•  small flat = 0.001

Multiplying Decimals Using an Understanding of Place Value

Using Place Value Materials (With Renaming)

1.6 x 2

2 groups of 1.6

Multiplying Decimals Using an Understanding of Place Value

Using Place Value Materials (With Renaming)

1.6 x 2

Trade to make another unit. 1.6 x 2 = 3.2

Distributive Property

a x (b + c) = a x b + a x c

Ex. What is the area of a swimming pool if the shallow end is 12 x 14 feet and the deep end is 12 x 10 feet? 14 ft 10 ft

12 ft

Distributive Property

a x (b + c) = a x b + a x c

14 ft 10 ft

12 ft

12 x (10 + 14) = 12 x 10 + 12 x 14

Multiplication Using the Distributive Property

3 x

(3 x 200) + (3 x 10) + (3 x 3) 600 + 30 + 9 = 639


2

2

1

1

3

3 0 0 0

Multiply Using the Distributive Property

1) 3 x 203 2) 4 x 316

Multplication Using the Area Model

1.  3 x 233 a) 4 x 122

200 30 3 3

600 + 90 + 9 = 699

600 90 9

More Area Model

31 x 16

30 1

10 (300 + 180) + (10 + 6)

6 480 + 16 = 496

300 10

180 6

Multiply Using the Area Model

1) 28 x 14 2) 32 x 45

Lattice Multiplication

1 8

2

6

0 2

0 6

1 6

4 8

Lattice Multiplication

1 8

16 8 (8 x 1) + (16 x 10) + (3 x 100) = 468

2

6

0 2

0 6

1 6

4 8

0

3

Developing Understanding of Multiplication and Division

Draw a picture for the following situation:

Maria saved $24. She saved 3 times as much as Wayne.

Assessing Understanding of Multiplication and Division

Video: Becca Sherman conducts a 4th grade formative assessment of multiplication and division concepts using the same Maria/Wayne question.

•  What misconceptions do you notice? •  How do the teacher’s questions shape children’s

thinking? •  Why do you think the teacher introduced Charlie’s

way?

•  http://insidemathematics.org/index.php/standard-1

Charlie’s Way

$24

Maria’s $ $32

Wayne’s $

Why do you think the teacher introduced Charlie’s Way?

$8 $8 $8

$8

Multiplication and Division Word Problems

•  Write a one-step multiplication word problem.

•  Write a one-step division word problem.

Multiplication and division situations

CCSSM

Multiplication / Division Situation

A playground measures 80 ft x 125 feet. The area is 1000 square feet.

•  Three measurements: any one of the measurements may be missing.

•  Measurement and area connect concepts of geometry with the operations of multiplication and division.

Number Bonds

•  A playground measures 80 ft x 125 feet. The area is 1000 square feet.

80 ft

125 ft

1000 sq ft

Grade 3 Expectations for Multiplication and Division

•  Area problems where regions are partitioned by unit squares are foundational for Grade 3.

•  By end of Grade 3, expect fluency in multiplying and dividing all single-digit numbers and 10.

•  By end of Grade 3, know from memory all products of two one-digit numbers.

•  Begin to use parentheses to indicate Order of Operations.

•  Two-step problems may involve one step addition or subtraction to 1000 and one step of multiplication or division.

Grade 4 Expectation for Multiplication and Division

•  Area problems where regions are partitioned by unit squares serve as a model for multi-digit multiplication and division.

•  Multiplicative comparisons first appear in 4th grade with whole number and (unit) fraction language.

•  Multi-step problems use 4 operations with whole numbers. Number of steps should be no more than 3 and involve only easy and medium + and –.

•  Students interpret and use remainders in context. •  Extend the idea of decomposition to multiplication. •  Begin reasoning about patterns.

Grade 5 Expectations for Multiplication and Division

•  Area problems where regions are partitioned by unit squares serve as a model for multiplication and division of decimals and fractions.

•  Students write expressions to describe calculations.

•  Students work briefly with two number patterns that can be related and examine these relationships in ordered pairs and graphs in the first quadrant, preparing them to study proportional relationships and functions in middle school.

Exploring a Multiplication Sequence to Extend Understand of Place Value

Given the value of the first expression in each of the following, find the value of the second expression without multiplying.

a)  If 3 x 59 is 177, how much is 30 x 59?

b)  If 24 x 60 is 1,440, how much is 240 x 6?

c)  If 30 x 59 is 1770, how much is 300 x 59?

Exploring a Multiplication Sequence to Extend Understand of Place Value

Given the value of the first expression in each of the following, find the value of the second expression without multiplying.

e) If 9 x 72 is 648, how much is 90 x 720?

f) If 400 x 720 is 288,000, how much is 401 x 720?

IMAP video 19 Gr. 3-4

Students have considered the following: •  3 x 4 •  6 x 8 •  3 x 16 •  1½ x 32 •  1½ x 64 •  3 x 64 •  30 x 64 •  30 x 640 •  300 x 640 •  301 x 640 Students are discussing the number of tens in 192,000, the

product of 300 x 640.

Using Place Value Materials to Divide

Without renaming:

1. 2 24 Represent number with base 10 blocks. Use the

divisor to determine how many groups to make.

Twelve in each group.

Using Place Value Materials to Divide

•  With renaming:

•  3 42

•  Use as many of the original pieces as you can to make 3 equal groups and rename what’s left.

Using the Distributive Property to Divide

The Distributive Property

When dividing by a 1-digit divisor, use the distributive property to break the dividend into 2 or more friendly numbers that can be evenly divided by the 1-digit divisor. This method can be used when the dividend is a multiple of the divisor.


Using the Distributive Property to Divide

1.  51 ÷ 3 a) 72 ÷ 6

30 21

(30 ÷ 3) + (21 ÷ 3)

10 + 7 = 17

Partial Quotient

3 35 10 – 30 5

Partial Quotient

3 35 10 a) 6 5 1 9 – 30 5 1 – 3 11 R 2

Fractions Grades 3-5

3rd – develop understanding of fractions (including vocabulary and notation), especially unit fractions (fractions with numerator 1 and denominators 2, 3, 4, 6, 8)

4th – addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers; (denominators 2, 3, 4, 5, 6, 8, 10, 12, 100)

5th – develop fluency with addition and subtraction of fractions, and develop understanding of multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions. Adapted from Joan Barrett

Fraction Concepts: A Priority

A lack of understanding of fraction concepts translates into untold difficulties with fraction computation, decimal and percent concepts, the use of fractions in measurement, and ratio and proportion concepts. John Van de Walle, Elementary and Middle School Mathematics: Teaching Developmentally, 2007.

Fraction Concepts meaning of fractions fraction number sense order of fractions benchmarks: 0, ½, 1 equivalence concept of unit

Fraction Kit

1 whole unit

1/2 1/2

1/3 1/3 1/3

1/4 1/4 1/4 1/4

1/6 1/6 1/6 1/6 1/6 1/6

1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8

Create a number line

•  Add a triangular piece of construction paper to each end of a strip of paper to simulate a number line.

•  Place the number line strip below the two half strips. Mark 1 on the number line. 2

•  Place the number line strip below each strip and mark on the number line hash marks to indicate 1, 1, 1, 1. 3 4 6 8

Adapted from Joan Barrett

Game of Cover Up

The game is for 2 or more players. Each player starts with their whole strip. The goal is to cover up the whole strip completely with other pieces of their fractions strips. No overlapping is allowed.

1.  Players take turns rolling the dice labeled with fractions. 2.  The fraction face up on the cube tells what size piece to

place on the whole strip. 3.  When the game nears the end and a player needs only a

small piece such as , rolling a or won’t do. The student must roll exactly what is needed.

4.  Keep going until all players cover their whole strip keeping track of who comes in first, second, etc.

5.  Each player should record how they covered their whole.

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Introducing Notation

•  After the cover up game, introduce the notation beyond unit fractions by having students report out how they covered their whole strip. Write the equation as dictated by the student.

•  Ex. + + + + + = 1

•  Count up how many one eights you have and how many one-fourths and shorten the equation to + . Adapted from Joan Barrett

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Uncover Game

Provides experience with equivalent fractions. 1.  Before the game begins each player places their two

pieces on top of their whole piece. 2.  A player has 3 options on each turn/roll:

1.  To remove a piece (only if he has a piece the size indicated by the fraction face-up on the cube.)

2.  To exchange any of the pieces left for equivalent pieces. 3.  Pass the cube to the next player. (A player may not

exchange pieces and remove on the same turn, but can do one or the other. All players need to check that others trade correctly.

4.  The person who clears their whole strip first wins. Taken from About Teaching Mathematics 3rd Ediion by Marilyn Burns, pp. 271-272 adapted from Joan Barrett

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Which is more and Why? No cross products, common denominators, no decimals, no percents.

or or or

or or or

or or or €

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Which is more?

Grade 5 Well I know that is two eighths away from a whole. And I know that is and is two tenths away from a whole.

Since tenths are smaller than eighths, is smaller than , so is more.

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Which is more?

I know that is .

And is .

And so I know that 15ths are bigger than 16ths so is more. Marilyn Burns, NCTM Annual, 1999

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Fraction Sort

Use these fraction cards:

Task #1: Sort the fraction cards into two groups—those fractions less than one half and those greater than one half. Justify your thinking.

Task #2: Order the fractions by estimating the size of each.

Task #3: Pair fractions so that each pair has a sum near 1.

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How many names can you find for the shaded area? Explain.

What number patterns do you see in your answers?

Robbie ate half of a pizza. Sophia ate half of another pizza. Sophia says she ate more pizza than Robbie. Robbie says that can’t be true. Who is right? How do you know?

Model Drawing for Fraction Problems

In Mrs. Parr’s 4th grade class, of the students like to play dodge ball during P.E. and the rest like to go on the climbing wall. If Mrs. Parr has 27 students in her class how many of her students like to go on the climbing wall?

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DB CW CW

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Foundations of Model Drawing Resource Book



DB = 27 ÷ 3 = 9 CW = 27 – 9 = 18

18 students like to go on the climbing wall.

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DB CW CW

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Foundations of Model Drawing Resource Book

9 9 9


David spent of his money on books. The books totaled $25. How much money did he have at first?

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$25

books books



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$25

$12.50 $12.50 $12.50 $12.50 $12.50

David had $62.50 at first. Foundations of Model Drawing Resource Book

Adding and Subtracting Fractions

4th grade – addition and subtraction of fractions with like denominators

5th grade – develop fluency with addition and subtraction of fractions.

1)  Model with manipulatives. How can we use the fraction kit to add & subtract fractions?

2)  Draw pictures. 3)  Represent with fraction notation. 4)  Solve word problems with fractional quantities.

Example: Orin has of a candy bar. His brother gave him of a candy bar. How much does he have now?

Use your fraction kit to find like fractions that are equivalent to the numbers in the problem. Draw a picture to show how you found the answer.

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Multiplying Fractions

4th grade – multiplication of fractions by whole numbers – Model with manipulatives and record. Notice patterns in the numbers.

Ex. Model with fraction kit: 4 x , 3 x , 4 x

5th grade – develop understanding of multiplication of fractions – Model contextual problems with manipulatives, draw diagrams, represent with fraction notation.

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Understanding Multiplication of Fractions

Which of the following can be solved by multiplying x ?

a.  Two-fifths of the students in Anya’s fifth grade class are girls. One-eighth of the girls wear glasses. What fraction of Anya’s class consists of girls who wear glasses?

b.  A farm is in the shape of a rectangle of a mile long and of a mile wide. What is the area of the farm?

c.  There is of a pizza left. If Jamie eats another of the original whole pizza, what fraction of the original pizza is left over?

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Illustrative Mathematics 5.NF 0609

Dividing Fractions

5th grade – unit fractions divided by whole numbers and whole numbers divided by unit fractions

Explore patterns: 12 ÷ 4, 12 ÷ 2, 12 ÷ 1, 12 ÷ , 12 ÷ , 12 ÷

Explore with manipulatives and draw diagrams: ÷ 3, ÷ 4, ÷ 2, ÷ 4

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In my mind there are at least four essential components of powerful mathematical discussions: 1.  A classroom climate that respects and

encourages all students to participate 2.  Worthwhile tasks 3.  Effective questions 4.  Thoughtful orchestration of the discussion

I wish you success with all four.

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