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The Power of Progressions: Untangling the Knotty Areas of Teaching - - PowerPoint PPT Presentation

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The Power of Progressions: Untangling the Knotty Areas of Teaching and Learning Mathematics Graham Fletcher gfletchy@gmail.com @gfletchy www.gfletchy.com Mornings Goals Understand the structure of 3-act task and see

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

The Power of Progressions: Untangling the Knotty Areas of Teaching and Learning Mathematics

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Morning’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.

  • Early Number and Counting
  • Addition and Subtraction
  • Understand the importance of an effective closing

and the role it plays in deciding our next move.

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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

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Procedural Fluency Application Conceptual Understanding

http://www.corestandards.org/other-resources/key-shifts-in-mathematics/
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SLIDE 7 @RobertKaplinsky
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Demetrius has 17 Skittles which is 12 fewer than Alicia. How many Skittles does Alicia have?

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Demetrius has 17 Skittles which is 12 fewer than Alicia. How many Skittles does Alicia have?

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Demetrius has 17 Skittles which is 12 fewer than Alicia. How many Skittles does Alicia have?

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Demetrius has 17 Skittles which is 12 fewer than Alicia. How many Skittles does Alicia have?

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Demetrius has 17 Skittles which is 12 fewer than Alicia. How many Skittles does Alicia have?

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17 12 fewer

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17 12

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17 12

W T F ?

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17 12

W T F ?

hat’s he ive

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Current Research

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?

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The Big Reveal

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Graham had some Skittles. He had 19 yellow, 15

  • range, 19 green, 17 purple, and 21 red. How many

Skittles 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?
  • When 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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  • 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 Practices for Orchestrating Productive Mathematics Discussions
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Identify and name the strategy used, then place the student work in order in terms of efficiency (least to greatest)

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3-Act Tasks 5 Practices Progressions

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1 sheet for 4 people Cut up all the numbers and symbols and create one

  • equation. All the numbers

and symbols must be used in the equation. Nothing should be leftover except for the black square.

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+ + +

Using the digits 1-9 at most one time each, create 4 numbers that have a sum of 91. You can use the 9 & 1 from the cards

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Kindergarten?

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Subitizing Comparison

Hierarchical Inclusion 1-to-1 Correspondence Number Conservation Cardinality Counting

Number Sense Trajectory

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Subitizing Comparison

Hierarchical Inclusion 1-to-1 Correspondence Number Conservation Cardinality Rote Counting

Number Sense Trajectory

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Perceptual & Conceptual

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5 8 6 x

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6 x 5 8

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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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SLIDE 66 MTMS: Vol. 14, No. 9, May 2009-5 Practices for Orchestrating Productive Mathematics 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 Practices for Orchestrating Productive Mathematics Discussions
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5 oranges Each wedge is a quarter

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SLIDE 70 1-Counting Up 2a-Skip Counting 2b-Skip Counting 1b-Counting Up 3a-Multiplicative 3b-Multiplicative
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Unit Fractions

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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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Open Middle

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SLIDE 92 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 93 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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SLIDE 94 http://wodb.ca/numbers.html
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Equivalent Fractions

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= = = 6 2 3 3 2 4

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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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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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Equivalence Simplifying

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Comparing Fractions

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Which girl ate more apple?

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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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S3 S4 S1

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It Takes 3 to Prove it to Me

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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 115 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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A Task

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gfletchy@gmail.com @gfletchy www.gfletchy.com/scoe19