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Handout 4.1a:

Meaning of Fractions

What are the different ways fractions can be represented and interpreted? Try to name three ideas. Likely, shading a region of a shape popped in your mind first. Can you think of any other representations/interpretations of fractions? You can use number, words, and math models to support your reflection.

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Math | Module 4.1 (Gr 4)

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Handout 4.1b:

Video: Interpretation of Fractions

Now, watch the video “Interpretation of Fractions” two times.

The first time you watch and listen, take a mental note your reactions.
The second time you watch and listen, jot down some of your reflections and/or key ideas from the video.

You can use the questions below and any type of math modeling to support your thinking:

What three interpretations of fractions does the video offer?
How are the interpretations the same and/or different?
What model might each interpretation suggest?

Math | Module 4.1 (Gr 4)

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Handout 4.1c:

Correct Shares, Fraction Interview

(A)

(B)

(C)

(D)

(E)

(F)

(G)

Students learning about fractional parts should be able to tell which of these figures are correctly partitioned in fourths. They should also be able to explain why the other figures are not showing fourths.

Correct Shares

Draw regions like the ones shown in the above figures, showing examples and non-examples (which are very important to use with students with disabilities) of fractional parts. Have students identify the wholes that are correctly partitioned into requested fractional parts, and those that are not. For each response, have students explain their reasoning.

Elementary and Middle School Mathematics, Teaching Developmentally by John A. Van de Walle, Karen S. Karp, Jennifer Bay-Williams. 7th edition 2010.

Math | Module 4.1 (Gr 4)

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Handout 4.1d [pg. 1 of 2]:

Possible Responses and What They Mean

Possible Student Responses
Different shape, Equivalent Size: Students may select A and F as the only shapes correctly partitioned into fourths. This student has likely had many learning experiences where wholes are pre-partitioned and pre-labeled and fewer learning opportunities involving the need for partitioning fair shares. These students are missing the critical idea that fractional parts do not have to be the same shape to be equivalent. In other words, fraction pieces can be equivalent in size and different in shape.
Part-Whole Relationship. Students may select B, C and/or D as representing correct shares. Likely they are missing the fundamental idea that fractions can represent a part-whole relationship. In this instance, the student is only considering the number of parts and disregards how the part is related to the whole.
A student who selects A, E, F, and G understand, when it come to partitioning area and the relationship between the piece and the whole, equivalent pieces only need to be the same size, in relationship to the same whole, not the same shape.

Consider how the thinking outlined below is similar or different from what you anticipated your own students would say when completing the Correct Shares Interview.

Math | Module 4.1 (Gr 4)

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Handout 4.1d [pg. 2 of 2]:

Possible Responses and What They Mean

Grade Level Foundational Standards	Connections to Big Ideas
Develop understanding of fraction equivalence and operations with fractions; Recognize that two different fractions can be equal (e.g., 15/9 = 5/3), and develop methods for generating and recognizing equivalent fractions	Fractions represent a part and whole relationship and the size of the whole matters In the context of division, fractions may have a quotient less than 1 Equivalent fraction pieces must be the same size however, can differ in shape For equivalence, the ratio must remain constant If numerators are the same, only comparing the denominators matters
Extend previous understandings about how fractions are built from unit fractions, composing fractions from unit fractions, decomposing fractions into unit fractions, and using the meaning of fractions and the meaning of multiplication to multiply a fraction by a whole number	Fractions can be decomposed into unit fractions and unit fractions can be combined into larger fraction amounts as long as the whole is the same Multiplication is related to fractions (e.g., ⅗ = 3 x ⅕) This is also related to repeated addition (e.g., 3 x ⅕ = ⅕ + ⅕ + ⅕) Fractions can be thought of as an operator
Use 0-2 number lines to represent relative fraction magnitude for fractions less than 1, equal to one, and greater than 1 as well as mixed numbers	When comparing fractions a common whole is needed Fractions can represent a number on the number line With unit fraction the greater the denominator the smaller the piece Fractions can be less than, equal to or greater than 1 whole

Math | Module 4.1 (Gr 4)