finding the missing pieces: middle grades fractions

FINDING THE MISSING PIECES:

MIDDLE GRADES FRACTIONSSolve using a visual fraction model

and give a context:

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FINDING THE MISSING PIECES

Middle Grades FractionsJeanne Simpson

NCTM 2014

FOCUSStrategies and resources to embed fractional reasoning while maintaining the rigor

and relevance required in the middle grades Common Core

Standards

TOPICSEssential fraction conceptsConnections to 6-8 contentStrategies to fit the two together

fractionprogression.wikispaces.com

THE TROUBLE WITH FRACTIONS…

rhymes

clear steps

KCFsongs basic facts practice

practice

practicepractice

review

review

review

mixed practice

quizzes

notes

posters tutoringreteaching

WHY CAN’T STUDENTS REMEMBER?

“Students who are asked to practice the algorithm over and over…stop thinking. They sacrifice the relationships in order to treat the numbers simply as digits.”

Imm, Fosnot, Uittenbogaard (2012)

WHY ARE FRACTIONS IMPORTANT?

Difficulty with learning fractions is pervasive and is an obstacle to further progress in mathematics and other domains dependent on mathematics, including algebra. It has also been linked to difficulties in adulthood, such as failure to understand medication regimens.

National Mathematics Panel Report, 2008

The Alabama Department of

Education’s initiative to improve math and

science teaching (K-12)

GOALS FOR AMSTI MATH

Students should:Use multiple and alternative strategies to solve problems.

Communicate mathematical understanding of content and process standards, both orally and in writing.

Become mathematical problem solvers.Learn to reason mathematicallyBe able to make real world connections through the application of mathematical knowledge.

rhymes

student writing

inquirysongs basic facts practice

projects

practicemanipulatives

review

review

review

mixed practice

quizzes

discourse

practice in context

tutoringreteaching

WHERE ARE THE FRACTIONS?

6.NS.1 – Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.

THE STRUCTURE IS THE STANDARDSDaro, McCallum, Zimba

The natural distribution of prior knowledge in classrooms should not prompt abandoning instruction in grade level content, but should prompt explicit attention to connecting grade level content to content from prior learning. To do this, instruction should reflect the progressions on which the CCSSM are built. For example, the development of fluency with division using the standard algorithm in grade 6 is the occasion to surface and deal with unfinished learning with respect to place value. Much unfinished learning from earlier grades can be managed best inside grade level work when the progressions are used to understand student thinking.

http://commoncoretools.me/2012/02/16/the-structure-is-the-standards/#more-422

THE STRUCTURE IS THE STANDARDSDaro, McCallum, Zimba

This is a basic condition of teaching and should not be ignored in the name of standards. Nearly every student has more to learn about the mathematics referenced by standards from earlier grades. Indeed, it is the nature of mathematics that much new learning is about extending knowledge from prior learning to new situations. For this reason, teachers need to understand the progressions in the standards so they can see where individual students and groups of students are coming from, and where they are heading. But progressions disappear when standards are torn out of context and taught as isolated events.http://commoncoretools.me/2012/02/16/the-structure-is-the-standards/#more-422

PROGRESSIONS – THE MISSING PIECE?

PROGRESSION FOR 3-5 NUMBER AND OPERATIONS - FRACTIONSThe meaning of fractions PartitioningUnit fractionsModels

Number linesEquivalent fractionsComparing fractionsOperations

ONGOING ASSESSMENT PROJECT (OGAP)

Developed as a part of the Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF EHR-0227057 and the US DOE (S366A20002).

Understand research on how students learn specific math concepts.

Strengthen content knowledge in these areas.

Understand how research is reflected in math programs and CCSS.

Increase ability to analyze evidence in student work to inform instruction.

CONTINUOUS ASSESSMENT

Preassess before each topic and continuously engineer discussions, activities, and tasks all along that are purposefully designed to elicit specific student understandings.

Use data to differentiate Tier 1 instruction and to ensure that 80% of students are thriving in Tier 1.

Provide interventions via small groups.

SAMPLE PREASSESSMENT

RTI AND MORE…

FRACTION RESOURCES

THE MEANING OF FRACTIONS

DEVELOPING EFFECTIVE FRACTIONS INSTRUCTION IN KINDERGARTEN

THROUGH 8TH GRADERecommendations1. Build on students’ informal understanding of sharing and proportionality to

develop initial fraction concepts.2. Help students recognize that fractions are numbers and that they

expand the number system beyond whole numbers. Use number lines as a central representational tool in teaching this and other fraction concepts from the early grades onward.

3. Help students understand why procedures for computations with fractions make sense.

4. Develop students’ conceptual understanding of strategies for solving ratio, rate, and proportion problems before exposing them to cross-multiplication as a procedure to use to solve such problems.

5. Professional development programs should place a high priority on improving teachers’ understanding of fractions and of how to teach them.

THE MEANING OF FRACTIONS

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What does this

number mean?

CIRCLE OF THE TRIANGLES

8TH GRADE STUDENTS

THE MEANING OF FRACTIONS

The top number counts.

The bottom number tells what is counted.

Together they represent one number.

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MODELS FOR FRACTIONSArea Length Set

Whole = area of defined regionParts = equal areasFraction = part of area covered

Whole = Unit of distance or lengthParts = equal distances or lengthsFraction = location in relation to other points on the line

Whole = one setParts = equal number of objectsFraction = the count of objects in the subset

Petit, Laird, Marsden (2010)

PARTITIONINGThe Act of Dividing

More sharers means smaller shares.Measures must be equal, but shares don’t have to be identical.

WHICH OF THE FOLLOWING SHOW FOURTHS? WHY?

(Small, 2014, p. 14)

CIRCLE THE FRACTION THAT IS CLOSEST TO

PATTERN BLOCK FRACTIONSCynthia Lanius

6TH GRADE PARTITIONING

6.G.1 – Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes…

6.G.4 – Represent 3D figures using nets…find the surface area.

6.RP.1 – Understand a ratio..

Begin the school year with a study of area.

Assess partitioning skills and understandings.

Focus on area model for fractions. Differentiate instruction based on assessments.

Use Number Talks for whole class discussions on items that <80% of class understands.

Work with small groups on other concepts.

8TH GRADE PARTITIONING

Pythagorean Theorem – Circle Sandwich A square is inscribed in a circle

which is inscribed in a square as shown. Note that the vertices of the inner square meet the midpoints of the outer square’s sides.

Consider the area of the region left by removing the interior of the small square from the interior of the big square. Is the area of the blue region more or less than of that?

NUMBER LINESHelp students recognize that fractions are numbers and that they expand the number system beyond

whole numbers. Use number lines as a central representational tool in teaching this and other fraction concepts from the early grades onward.

Place these fractions on the number line below in their correct position.

Place these fractions on the number line below in their correct position.

6TH GRADE NUMBER LINES

6.NS.5 - Understand that positive and negative numbers are used together to describe quantities having opposite directions or values…

6.NS.6 – Understand a rational number as a point on the number line… 6.NS.7 – Understanding ordering and absolute value of rational numbers. 6.NS.8 – Solve real-world and mathematical problems by graphing points in all

four quadrants of the coordinate plane… 6.G.3 - Draw polygons in the coordinate plane given coordinates for vertices; use

coordinates to find the length of a side joining points with the same first or second coordinate.

7TH AND 8TH GRADE NUMBER LINES

7th Grade Integers Graphing proportional

relationships

8th Grade

Irrational numbers Graphing equations and

functions Transformations Statistical graphs

EQUIVALENT FRACTIONS

reducing

THE EXPERTS SAY….

“Do not tell students that their answer is incorrect if not in the simplest or lowest terms.

This also misinforms students about the equivalence of fractions. If you want the

answer in simplified form, provide feedback to the student that the answer is correct but must

be simplified.”

Van de Walle, Bay-Williams, Lovin, Karp (2014, p. 119)

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EQUIVALENT AREAS

44

EQUIVALENT LENGTHS

EQUIVALENCE IN 6TH - 8TH GRADE

ExpressionsEquationsExponents Irrational numbersSlope

8.EE.6

Illustrative Mathematics

COMPARING FRACTIONS

WHICH FRACTION IS GREATER?

a. or More of the same-size partsb. or Same number of parts, different sizes

c. or More or less than one wholed. or More or less than one halfe. or Distance from one whole

OPERATIONS

DO YOUR STUDENTS REMEMBER PROCEDURES?

MAYBE A DIFFERENT PROCEDURE WOULD HELP?

3. HELP STUDENTS UNDERSTAND WHY PROCEDURES FOR COMPUTATIONS WITH

FRACTION MAKE SENSEHelp students understand why procedures for computations with fractions make sense.1. Use contextual tasks.2. Explore each operation with a variety of

models.3. Let estimation and informal methods play

a big role in the development of strategies.4. Address common misconceptions

regarding the operations.

WHICH OPERATION SHOULD YOU USE?

Ashley bought 6 pounds of candy. She put the candy into bags that each hold pound. How many bags of candy did she fill?

ADDING AND SUBTRACTING WITH LIKE DENOMINATORS

Depending on your students’ abilities, you may need to stop

here and teach fraction operations to the whole class. Each time you add a type of operation, practice solving

equations that apply that skill.

MINILESSONS OR NUMBER TALKS/STRINGS

10-15 minutes 6-8 carefully crafted problems Whole class or small group using mental

math Work one at time and allow students to

share strategies Structured with developmental

progressions in mind Helps students develop a variety of

mental math strategies

ADDING AND SUBTRACTING WITH LIKE DENOMINATORS (GRADE 4)

IMPROPER FRACTIONS (GRADE 4)

ADDING AND SUBTRACTING FRACTIONS WITH UNLIKE DENOMINATORS (GRADE 4) Begin with fractions in which one denominator is a multiple of the

other so that only one fraction has to be changed.

GRID MODEL FOR ADDING

Uncomplicating Fractions , p. 71 (Small, 2014)

MULTIPLYING A FRACTION BY A WHOLE NUMBER (GRADE 4)

DIVIDE WHOLE NUMBERS THAT GIVE FRACTIONAL ANSWERS (GRADE 5)

MULTIPLY A WHOLE NUMBER BY A FRACTION (GRADE 5)

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MULTIPLY BY A FRACTION (GRADE 5)

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MULTIPLY BY A FRACTION (GRADE 5)

MULTIPLY BY A FRACTION (GRADE 5)

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DIVIDE WHOLE NUMBERS BY UNIT FRACTIONS AND UNIT FRACTIONS BY WHOLE NUMBERS (GRADE 5)

4

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5TH GRADE DIVISION

12 ÷3

MULTIPLICATION DIVISION 𝟏𝟑×

𝟏𝟐=

𝟏𝟔 =

70

5TH GRADE DIVISION PROBLEMS

How much chocolate will each person get if 3 people share ½ pound equally?

How many 1/3 cup servings are in 2/3 cups of raisins?

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FRACTION DIVISION IN GRADE 6

6.NS.1 – Interpret and compute quotients of fractions, and solve word problems involving division of fractions, e.g., by using visual fraction models and equations to represent the problem.Examples: Create a story context…Use a visual fraction model to show the quotient…

Explain division using its relationship with multiplication

CREATE A STORY CONTEXT FOR

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6TH GRADE DIVISION

23 ÷34

INVERT AND MULTIPLY…WHY?6

My way… Rebecca’s way…

CONTACT INFORMATION

Jeanne SimpsonJeanne.Simpson@uah.edu

Alabama Math, Science, and Technology Initiative

www.amsti.org

Virtual Handoutfractionprogression.wikispaces.com

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