TEACHING FRACTIONS WITH LEARNING PROGRESSIONS

Kristin Frang, Mathematics Consultant

MAISD Regional Math & Science Center

Laura Wilson, Teacher Leader

Vicksburg Community Schools

SERIES GOALS Explore a framework that balances conceptual

understanding, procedural fluency and application.

Identify critical components to understanding fractions and instructional strategies that support their development.

Understand students’ development of core concepts and ways of reasoning about fractions

YOUR QUESTIONS

How do I get in so much higher order thinking?MAISD CCSSM wiki

Overwhelming, but every journey begins with a step.

TODAY’S AGENDA

Review last session & homework Major Emphasis in CCSS SBAC Performance Task Organizing Instruction Levels 0-2 Student Reasoning

LUNCH

Levels 3-4 Student Reasoning Matching Instruction to Student Reasoning

MIND STREAMING

Find a partner at your table: Partner A talks for 1 min about the topic Partner B listens and encourages Partner A

The role reverses: Partner B talks for 1 min about the topic Partner A listens and encourages Partner B

Equal Partitioning

Iterating

Unit Fractions

Fraction Comparison

Big Ideas of Fraction

Concepts

Chapter 1 Thoughts

It is difficult for students to move from working with whole numbers to working with fractions because a fractional quantity is described with two numbers, not one, and understanding fraction requires one to explicitly comprehend a relationship between two quantities – the whole and its parts.

Battista

By focusing on relationships, children come to see fractions as connected to other things they know about number and operation Empson & Levi

CCSS CRITICAL AREA

3rd grade – page 21

4th grade – page 27

5th grade- page 33

CLUSTER HEADINGS

Cluster headings function like topic sentences in a paragraph in that they state the point of, and lend additional meaning to, the individual content standards that follow…Materials do not simply treat the Standards as a sum of individual content standards and individual practice standards.

K-8 Publishers’ Criteria

for the CCSS for Mathematics

Connections at a single grade level can be used to improve focus, by tightly linking secondary topics in the major work of the grade. K-8 Publishers’ Criteria

for the CCSS for Mathematics

Focus on Major Work: In any single grade, students and teachers using the materials as designed spend the large majority of their time, approximately 3/4 , on the major work of each grade level… In addition, major work should especially predominate in the first half of the year

K-8 Publishers’ Criteria

for the CCSS for Mathematics

NUMBER WORK (10 – 15 min)

Develops

• number sense without story context

• communication• connections• algebraic reasoning

Allows• multiple entry points / answers• access for wide range of skills &

abilities

Can be• single problem• series of problems

NUMBER WORK EXAMPLE

Differences of (Purpose: differences)

Ask students to write pairs of numbers with a given difference (e.g. difference of 7) for a predetermined number of minutes

Share responses and repeat several times. (e.g. 32/39, 54/61, 1009/1016)

Variation: Use decimals and fractional differences

NUMBER WORK EXAMPLES

Number work is …

• commonly done whole class

• beginning of math block

• can also be tied into ‘calendar’ routines for younger students

• work was done independently, but chosen specifically for individual students.

Balanced Mathematics

Kristin Frang, MAISD Regional Mathematics & Science CenterAdapted from Cathy Seely, Balance is Basic and Larry Ainsworth, Five Easy Steps to a Balanced Math Program

Formative

Assessment Skills, Facts, Procedures

Conceptual

UnderstandingProblem

Solving

Flexible Set of Thinking& Reasoning

Instruction

Problem Solving (30 – 45 min)

Number Work or Inspecting

Equations (10 – 15

min)

Practice

Fluency & Maintenance

(10 – 15 min)

PROBLEM SOLVING

All content domains

Large & small group settings

May serve multiple purposes, such as: Introducing new conceptsChallenging students to develop and apply

effective strategiesProviding a context for improving skills

PROBLEM SOLVING

Utilizes problems that: Emerge from the students’ environment

Can be found in the school’s curricular materials

Allow for multiple solution strategies

Lead to justification and generalization

INSPECTING EQUATIONS

Focus is on learning about how the equal sign expresses equality relationships.

Students:

Explore equality & meaning of equal sign

Analyze conventional symbols of #, operations & relationships

Look for & use #

relationships.

Use computation to confirm equality of

relationships.

INSPECTING EQUATIONS EXAMPLE

Can students accurately solve equations with repeated variables?

g + g + 4 = 16

h + h – 3 = 11

a + a + a = 15

Do students take advantage of familiar number relationships?

25 + 47 + 75 = y

98 + 69 + 2 = y

FACT FLUENCY

Illuminates a child’s sense of number relationships and the thinking strategies he/she uses to calculate single-digit computation

Identify size of numbers a student

can compute mentally

Defines numbers sizes to use in other blocks

Defines numbers sizes to work on for individual fluency &

maintenance

3-2-1 REFLECTION 3 things you want to remember from our

time this morning

Highlight 2 things you want to focus on implementing first

Choose 1 and create a plan to help you implement in your current unit

Problem solving should be the central focus of the mathematics curriculum. Problem solving is not a distinct topic, but a process that should permeate the entire program and provide the context in which concepts and skills can be learned

ACCESSIBLE COGNITIVE JUMPS

LEVELS OF SOPHISTICATION IN STUDENT REASONING

LEVEL 0 (SUBLEVEL 0.1 & 0.2)

Tell what fraction is shaded.

Response: 1

LEVEL 0 (SUBLEVEL 0.1 & 0.2)

Level 0.1 Level 0.2

Three people want to share the pizza below equally. Show how much each person gets.

LEVEL 1Student recognizes only familiar pictures of fractions

Shape A looks like a half. The rest are not halves

LEVEL 1

I think A,B,C and D look like ¾ . E and F don’t look like ¾.

LEVEL 1

LEVEL 2 Count all parts, count shaded parts

Don’t explicitly focus on whole Difficult time finding fractions of sets

Don’t use iteration/partitioning into equal parts

Can make comparisons by drawing pictures

A, B and F are halves because one part is shaded and one part is not.

LEVEL 2

LEVEL 2 Count all parts, count shaded parts

Don’t explicitly focus on whole Difficult time finding fractions of sets

Don’t use iteration/partitioning into equal parts

Can make comparisons by drawing pictures

A, B, C and D are ¾ because they all have 4 parts and 3 are shaded. E and F are not ¾ because they have to many pieces.

LEVEL 2

LEVEL 1 VS. LEVEL 2

LEVEL 3 Partition into equal parts only when whole

is explicitly specified

Difficulty find fractions of sets, improper fractions and performing arithmetic operations because whole is more difficult to maintain

Reasoning is restricted to shape, do not reason about quantities

Use splitting and iterating

• A is one-half because there are 2 equal pieces and 1 is shaded. Same thing on B.

• C, move square in bottom row to empty spot so the gray & white parts are the same (see red arrow)

• D, move the bottom shaded squares to the left to make one half.

• E, move all shaded squares to the left.

• F is not one half because the gray and white parts aren’t equal.

LEVEL 3

LEVEL 3 Partition into equal parts only when whole

is explicitly specified

Difficulty find fractions of sets, improper fractions and performing arithmetic operations because whole is more difficult to maintain

Reasoning is restricted to shape, do not reason about quantities

Use splitting and iterating

• B and D because they both have 4 equal parts and 3 shaded.

• The parts aren’t equal in A and C.

• E is ¾ because you can divide it so that 3 out of 4 equal parts are shaded.

• F is not because there are 6 squares shaded.

LEVEL 3

LEVEL 3 Partition into equal parts only when whole

is explicitly specified

Difficulty find fractions of sets, improper fractions and performing arithmetic operations because whole is more difficult to maintain

Reasoning is restricted to shape, do not reason about quantities

Use splitting and iterating

Read pages 26-27 and add to your mind-maps

LEVEL 3Other Characteristics & Illustrations

LEVEL 4 (SUBLEVEL 4.1 & 4.2)

Students have created a new link between number & shape

Flexibly go back and forth between actions on shape and actions on number

Generalize understanding to multiple contexts

Keep track and maintain the whole

When students reason about the numerical quantities

A,B,C,D and E are each one-half because 8 squares are shaded and 8 are not shaded.

LEVEL 4 (SUBLEVEL 4.1 & 4.2)

• B and D because they both have 4 equal parts and 3 shaded.

• The parts aren’t equal in A and C.

• E is ¾ because you can divide it so that 3 out of 4 equal parts are shaded.

• F is not because there are 6 squares shaded.

LEVEL 3

• B and D because they both have 4 equal parts and 3 shaded.

• The parts aren’t equal in A and C.

• E is ¾ because you can divide it so that 3 out of 4 equal parts are shaded.

• F is not because there are 6 squares shaded.

LEVEL 3 VS 4

• E: each row has 6 squares, so the rows are equal; and 3 of the 4 rows are shaded.

• F: 3 groups of 2 squares shaded, and 1 group of 2 squares unshaded (see circled groups above)

LEVEL 4

Read pages 33 and add to your mind-maps

LEVEL 4Other Characteristics & Illustrations

LEVELS OF SOPHISTICATION IN STUDENT REASONING

FRACTIONS

AND

OPERATIONS

ACCESSIBLE COGNITIVE JUMPS

[I]f algorithms are taught to early in students’ development of reasoning about fractions, students cannot understand the algorithms conceptually, so they learn them by rote…Students who learn computational algorithms for fractions before achieving Level 5 reasoning will learn them rotely. Battista, pg.10

Level 0.1Help students develop understanding of partitioning Focus on equal

partitions not naming fractions

TEACHING STUDENTS AT LEVEL 0.1 & 0.2

Level 0.2Help students develop initial understanding of fractions Link Unit fractions to equal

partitions

Ask questions such as “How many thirds are in a whole?

Introduce Non-Unit Fractions

Ask questions such as “How many thirds are in two-thirds?

NUMBER LINES

The focus on fractional parts is an important beginning, but number sense with fractions demands more – it requires that students have some intuitive feel for fractions.

John Van de Walle

PARTITIONING WITH LENGTH MODELS

Give students number lines partitioned such that only some of the partitions are showing.

Nicole walks 1 mile to school everyday. Using the number line below, how far has Nicole gone? How do you know?

FRACTION NUMBER SENSE

• More and less than ½ or 1

• Distance from ½ or 1

• Same # of parts,

• Different size parts

• More of same-size parts

   

  

CLOSE TO

HOMEWORK

1. Read Chapter 2, pages 12-40

2. Choose several assessment tasks from pages 105-111 to try with students.

Record Student Strategies Remember you goal is to learn about your

students thinking

At session 3 on April 9th, we will explore operating with fractions.