Harnessing the Power of Modeling Tasks through the Lens of a Math - - PowerPoint PPT Presentation

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Harnessing the Power of Modeling Tasks through the Lens of a Math - - PowerPoint PPT Presentation

Feb 08, 2023 39 likes •1.47k views

Harnessing the Power of Modeling Tasks through the Lens of a Math Progression Graham Fletcher gfletchy@gmail.com @gfletchy www.gfletchy.com/shelton18 Broken Squares Designate a shape keeper 6 congruent squares No

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Graham Fletcher gfletchy@gmail.com @gfletchy www.gfletchy.com/shelton18

Harnessing the Power of Modeling Tasks through the Lens of a Math Progression

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Broken Squares

  • Designate a shape keeper
  • 6 congruent squares
  • No shapes left over
  • Everyone is encouraged to OFFER. No one may TAKE.

They may ACCEPT a puzzle piece to use if OFFERED.

  • NO TALKING
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Procedural Fluency Application Conceptual Understanding

http://www.corestandards.org/other-resources/key-shifts-in-mathematics/
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Procedural Fluency Application Conceptual Understanding

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Procedural Fluency Application Conceptual Understanding

http://www.corestandards.org/other-resources/key-shifts-in-mathematics/
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SLIDE 6 @RobertKaplinsky
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In a particular condominium community 2/3 of all of the men are married to 3/5 of all of the women. What fraction of the entire condominium community are married?

The Condominium Problem

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SLIDE 9 Instructor’s Manual for Elementary and Middle School Mathema7cs Teaching Developmentally Sixth Edi7on - John A. Van de Walle (Virginia Commonwealth University)

students doing the solving and the sense making teacher showing a very conceptual approach top-down, rule oriented approach

3

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?

3 questions

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Billion Circles

1

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  • 100 circles : minute
  • 144,000 circles : day
  • 1,000,000,000 would take 6944 days
  • 19+ years with no sleep

How long to draw 1 billion circles:?

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Where does 1 billion go on the number line? 1 trillion

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Where does 1 billion go on the number line? 1 trillion

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5 8 6 x

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6 x 5 8

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Use equivalent fractions as a strategy to add and subtract fractions. 5.NF.1 Add and subtract fractions and mixed numbers with unlike denominators by finding a common denominator and equivalent fractions to produce like denominators.

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4 1

3 7 8

1

Pause I I

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13 3 15 8

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104 24 45 24

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59 24

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59 24 11 24

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Subtracting Mixed Numbers with Regrouping

C C S A C

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Subtracting Mixed Numbers with Regrouping

Chubby Crocodiles Swim Against Current

C C S A C

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Subtracting Mixed Numbers with Regrouping

Change mixed numbers to improper fractions Cross multiply to find common denominators Subtract only the numerators ALWAYS simplify Change improper fraction back to a mixed number

C C S A C

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1 3

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Today’s Goals

  • Understand the structure of 3-act task and see how

they fit into the scope and sequence of a unit.

  • Explore the importance of progressional

understanding and how a good task can be used as formative assessment.

  • Numbers and Operations in Fractions
  • Understand the importance of an effective closing and

the role it plays in deciding our next move.

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How many orange wedges are in the bowl? Estimate

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How many orange wedges are in the bowl? What information do you need to know?

?

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Each orange wedges is a quarter.

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Graham had 5 oranges and cut them into quarters. How many orange wedges did Graham have?

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3-Act Tasks

Act 1:

  • Real world problem or scenario presented
  • What do you notice? What do you wonder?
  • Make estimates

Act 2:

  • Identify missing variables and missing variables to solve
  • Define solution path using variables

Act 3:

  • Solve and interpret results of the solution
  • Validate answer
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Most asked questions:

  • How often should we use 3-Act Tasks?

How do they fit into the scope of a unit?

  • How long does one task usually take?
  • What if we don’t have the time?
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SLIDE 53 MTMS: Vol. 14, No. 9, May 2009-5 Prac7ces for Orchestra7ng Produc7ve Mathema7cs Discussions
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  • 1. Anticipating student responses to challenging mathematical tasks;
  • 2. Monitoring students’ work on and engagement with the tasks;
  • 3. Selecting particular students to present their mathematical work;
  • 4. Sequencing the student responses that will be displayed in a specific
  • rder and;
  • 5. Connecting different students’ responses and connecting the responses
to key mathematical ideas.

5

The practices are:

MTMS: Vol. 14, No. 9, May 2009-5 Prac7ces for Orchestra7ng Produc7ve Mathema7cs Discussions
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5 oranges Each wedge is a quarter

?

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  • 1. Anticipating student responses to challenging mathematical tasks;
  • 2. Monitoring students’ work on and engagement with the tasks;
  • 3. Selecting particular students to present their mathematical work;
  • 4. Sequencing the student responses that will be displayed in a
specific order and;
  • 5. Connecting different students’ responses and connecting the
responses to key mathematical ideas.

5

The practices are:

MTMS: Vol. 14, No. 9, May 2009-5 Prac7ces for Orchestra7ng Produc7ve Mathema7cs Discussions
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  • 1. Anticipating student responses to challenging mathematical tasks;
  • 2. Monitoring students’ work on and engagement with the tasks;
  • 3. Selecting particular students to present their mathematical work;
  • 4. Sequencing the student responses that will be displayed in a
specific order and;
  • 5. Connecting different students’ responses and connecting the
responses to key mathematical ideas.

5

The practices are:

MTMS: Vol. 14, No. 9, May 2009-5 Prac7ces for Orchestra7ng Produc7ve Mathema7cs Discussions
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SLIDE 66 1a-Counting Up 1b-Counting Up
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SLIDE 67 1-Counting Up 2a-Skip Counting 2b-Skip Counting 1b-Counting Up
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SLIDE 68 1-Counting Up 2a-Skip Counting 2b-Skip Counting 1b-Counting Up 3a-Multiplicative 3b-Multiplicative
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Group 1 Group 2 Group 3

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Unit Fractions

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Tangram Challenge

4 16 1 16 2 16 2 16 2 16 1 16 4 16

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Representation of a Fraction

1 a

unit fraction —

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3 4

Say this fraction

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3 4

Say this fraction three one-fourths

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3 = 1 + 1 + 1

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3 4 3 = 1 + 1 + 1 = + + 1 4 1 4 1 4

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What’s the Sum?

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What’s the Sum?

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What’s the Sum?

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random dice roller

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SLIDE 96 Directions: Using the whole numbers 1-9 no more than one time each, create and place 4 fractions on the number line in the correct order. A is less than 2. Fractions B, C, and D equal 2.

Open Middle

A C B D

2 = =

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SLIDE 97 Directions: Using the whole numbers 1-9 once each, create and place 4 fractions greater than 1 on the number line in the correct order. (fractions B & C are equal) CCSS.MATH.CONTENT.4.NF.A.2

Open Middle

A C B D
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Equivalent Fractions

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= = = 6 2 3 3 2

Equal Fraction

4

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W T F ?

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hat’s his raction

W T F ?

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What about “the test”

?

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3 6 + is equal to which of the following? 1 6 4 12 a. b. c.

  • d. None of the above

8 12 3 6

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It is possible to over-emphasize the importance of simplifying fractions in this way. There is no mathematical reason why fractions must be written in simplified form, although it may be convenient to do so in some cases.

http://commoncoretools.me/wp-content/uploads/2011/08/ccss_progression_nf_35_2013_09_19.pdf
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Equivalence Simplifying

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Comparing Fractions

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Which girl ate more apple?

?

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Which girl ate more apple? What information do you need to know?

?

Pause I I

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twelfths eighths

Pause I I

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Apple Eat Off Act-3

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Big sister ate of an apple and little sister ate

  • f an apple. Which sister ate more apple?

10 12 7 8

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  • 1. Anticipating student responses to challenging mathematical tasks;
  • 2. Monitoring students’ work on and engagement with the tasks;
  • 3. Selecting particular students to present their mathematical work;
  • 4. Sequencing the student responses that will be displayed in a
specific order and;
  • 5. Connecting different students’ responses and connecting the
responses to key mathematical ideas.

5

The practices are:

MTMS: Vol. 14, No. 9, May 2009-5 Prac7ces for Orchestra7ng Produc7ve Mathema7cs Discussions
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S1 S6 S5 S4 S3 S2

Pause I I

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S3 S4 S1

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What We Know Misconceptions… Understanding…

  • counting the number of pieces leftover to compare
  • seeing numerator & denominator as an individual

quantity

  • tricks and procedures for comparing fractions
  • equally partitioning a number line
  • common denominator understanding without

multiplying to find a common factor

  • equivalent fractions
  • comparison of unit fractions
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It Takes 3 to Prove it to Me

3

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1 3 4 4

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1 3 4 4

Common Denominator

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1 3 4 4

Common Denominator

5 3 6 4

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1 3 4 4

Common Denominator

5 3 6 4

Missing Parts

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1 3 4 4

Common Denominator

5 3 6 4

Missing Parts

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1 3 4 4

Common Denominator

5 3 6 4

Missing Parts

2 4 4 6

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1 3 4 4

Common Denominator

5 3 6 4

Missing Parts

2 4 4 6

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1 3 4 4

Common Denominator

5 3 6 4

Missing Parts

2 4 4 6

Benchmark

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1 3 4 4

Common Denominator

5 3 6 4

Missing Parts

2 4 4 6

Benchmark

2 2 3 5

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1 3 4 4

Common Denominator

5 3 6 4

Missing Parts

2 4 4 6

Benchmark

2 2 3 5

Common Numerator

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1 3 4 4

Common Denominator

5 3 6 4

Missing Parts

2 4 4 6

Benchmark

2 2 3 5

Common Numerator

Unit Fraction Understanding Unit Fraction Understanding Unit Fraction Understanding
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8 4 11 7

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8 8 11 14

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SLIDE 142 CCSS.MATH.CONTENT.3.NF.A.3.D Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. CCSS.MATH.CONTENT.4.NF.A.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Comparing Fractions

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