Attending to Fractions in the Common Core

with Measurement Connections MiCTM 2012

Traverse City, MI

Introductions

Funda – 4th yr doctoral student, working on the STEM project all 4 years, formerly taught middle school mathematics for 4 years, taught content courses for elementary teachers, and now also working on CMP Project

Jia- 4th yr doctoral student, working on the STEM project 3 years, taught content courses for elementary teachers, and now also working on the Preparing Teaching Algebra Project

Agenda

• Reflecting on fraction Performance with connections to measurement

• Fractions in CCSSM

• Measurement in CCSSM

• Problem 1

• Problem 2

• Fractions measurement connections

• Discussion

NAEP , 2003, Grade 4

Jim has 3/4 of a yard of string which he wishes to divide into pieces, each 1/8 of a yard long. How many pieces will he have?

A. 3

B. 4

C. 6

D. 8

19%

30%

27%

21%

3% omitted

Fractions and Rational Number in CCSSM

K.G.6

Kindergarten

1.G.3

1.G.2

Grade 1

2.G.3

Grade 2

3.MD.4

3.G.2

3.NF.3

3.NF.2

3.NF.1

Grade 3

4.MD.5

4.MD.4

4.MD.2

4.NF.7

4.NF.6

4.NF.5

4.NF.4

4.NF.3

4.NF.2

4.NF.1

Grade 4

5.MD.2

5.NF.7

5.NF.6

5.NF.5

5.NF.4

5.NF.3

5.NF.2

5.NF.1

Grade 5

Elementary

Source: https://atmim.wildapricot.org/

Fractions in CCSSM

• Grade Focus: – Meaning of fractions and different representations: Grades 3 and 4 – Operations with fractions: Grades 4 and 5, 6, 7 – Ratio and Proportionality: Grades 6 and 7

• Progressions: – Fractions as numbers – Equivalent fractions, comparing fractions – Addition, subtraction and multiplication – Division

• Solving problems involving different operations

– Ratio – Proportionality

• Solving problems analyzing proportions

The Meaning of Fractions Understand fractions as numbers on the

number line

• Grades 1 and 2: Use fraction language to describe partitions of shapes into equal shares

• Grade 3: Start to develop the idea of fractions more formally as numbers as they use a number line (starts with unit fractions and build fractions by using these unit fractions).

• Grade 4: Extend understanding of fractions equivalence and ordering

Operations with Fractions Relating fractions operations to whole number operations e.g. Addition: Putting together; Division: not a/b but as a÷b.

Grade 4: Addition and subtraction with like denominators composing and decomposing fractions with the same denominator.

Multiplication with whole numbers

a/b is equivalent to a x 1/b

Grade 5: Addition and subtraction with unlike denominators

Use equivalent fraction strategy

Multiplication and Division

Represent fraction products as areas of rectangles

Interpret multiplication as scaling (resizing)

Division of a fraction to a whole number

Grade 6: Division of a fraction by fraction

Grade 7: Apply understanding of operations with fractions to rational numbers

Ratio and Proportionality Apply proportional reasoning in analyzing equal ratios, unit rate, and to solve problems. Multiplication as scaling (resizing) will help work in ratio and proportionality.

Two perspectives on Ratio:

– Ratio as a composed unit or batch: Two quantities are in a ratio of A to B if for every A units present of the first quantity there are B units present of the second quantity.

– Ratio as fixed numbers of parts: Two quantities which have the same units, are in a ratio of A to B if there is a part of some size such that there are A parts present of the first quantity and B parts present of the second quantity

Grade 6: Understand and use ratio concepts

Concept of unit rate associated with ratio

Use ratio reasoning to convert measurement units

Grade 7: Analyze proportional relationships

Test for equivalent ratios to determine proportional relationships

Find constant of proportionality (unit rate) in these proportional relationships

Measurement in CCSSM

• Grade focus – Length: grade 1& 2 – Area: grade 3 – Volume: grade 5

• Progression – Pre-measurement

• Qualitative comparison • Decompose and compose shape

– Meaning of length, area and volume measurement (unit, unit iteration) – Solving problems involving length, area and volume. E.g., use moving and

additivity principle to find the area of an irregular shape.

• Educator’s Companion to Measurement in CCSSM

Length in CCSSM

• Grade K: direct comparison of length • Grade 1:

– Meaning and processes of measurement by iterating – Transitivity principle for indirect comparison

• Grade 2: – Standard units – Use rulers and understand the rational (unit iteration) – Unit-measure compensation

• Grade 3: perimeter • Grade 4: unit conversion (larger units to smaller units) • Grade 5: unit conversion (smaller units to larger units) • Grade 6: involve length in ratio, rate and proportional reasoning • Grade 7: finding the circumference of circles • Grade 8: distance

Area in CCSSM

• Grade K: recognize , describe shapes • Grade 1 & 2: compose and decompose shapes • Grade 3:

– Area as an attribute of two-dimensional regions – Area measure – Row and column structure of rectangle – Connect area to multiplication

• Grade 4: – Application of formulas for area and perimeter for rectangles – Use rectangular arrays to understand multiplication

• Grade 5: use rectangles with fraction side length to support fraction multiplication

• Grade 6 & 7: find the area and surface area of particular shapes by composing and decomposing shapes

• Grade 8: not the focus

Volume in CCSSM

• Grade K: recognize , describe shapes • Grade 1 & 2: compose and decompose shapes • Grade 3: Measure and estimate liquid volume • Grade 4: unit conversion of liquid volume (e.g., liter) • Grade 5:

– Volume as an attribute of three-dimensional space. – Volume measurement – Standard unit for measuring volume: 1-unit by 1-unit by 1-unit – Solving real world problems involving volume. – Find the volume of rectangular prisms by attending to its layers structures.

• Grade 6: – Find the volume of particular shapes by composing and decomposing shapes – Extend formulas for the volume of a right rectangular prism to fractional side lengths.

• Grade 7: – Relate 2d shape to 3d shape by examining cross sections – Solve real-world problems involving volume

• Grade 8: solve volume problems involving cones, cylinders, and spheres.

Connections?

• How do fractions might help understanding measurement?

• How does measurement might help understanding fractions?

• Which topics in fractions/measurement can be helpful in practicing concepts in fractions/measurement?

Problem 1

• Find 2 1/3 x 1 1/6

• Explain by using different representations

Task 1 – Sample Answer

49/18 = 2 13/18

CCSSM Connection

5NF4b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.

Problem 2

Given

A: B = 12: 1

B: C = 5:2

Find A:C.

Explain by using different representations.

Problem 2 –Sample Answer

CCSSM Connection

6RPd. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities

Fractions Measurement Connections Fraction/ratio helps to solve measurement problem (explicit) Use ratio and rate reasoning to solve measurement related real-world problem: e.g., constant speed (6RP3b) Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities (6RP3d)

Measurement as a tool to learn fraction/ratio (implicit) Use measurement model, e.g., length or area model, to make sense of certain procedures on fraction (e.g., prove fraction equivalence) and ratio. Use measurement as a context to make sense of the pure procedure: understand fraction division in a real problem context

Resources

• https://www.msu.edu/~stemproj/

• http://commoncoretools.me/tools/

• http://www.turnonccmath.com/

• http://nces.ed.gov/nationsreportcard/