Fractions

Lesson 8

Objective

Draw the whole when given the unit fraction.

Common Core Standards

Core Standards

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  • 3.NF.A.1 — Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

Criteria for Success

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  1. Given a unit fraction, construct the whole by making a certain number of copies of the unit fraction adjacent to one another (e.g., to create the whole given $$ \frac{1}{5}$$, make copies of the unit fraction so that there are 5 equal-sized pieces adjacent to one another). (Note: adjacent shapes is not completely necessary, since non-adjacent copies of the unit fraction can come to resemble a set model. However, since students will not work with set models in Grade 3, avoid discussing these representations as a whole group.)
  2. Understand that there is more than one way to represent a whole given a unit fraction (and thus, wholes that are different shapes but the same size are equivalent).
  3. Given a non-unit fraction, construct the whole by partitioning the given fraction into the appropriate number of equal-sized pieces so that they represent the unit fraction, and then add copies of the unit fraction so that they make a whole (e.g., to create the whole given $$\frac{3}{5}$$, partition the given part into 3 equal-sized pieces so that they each represent $$\frac{1}{5}$$, then make copies of that unit fraction so that there are 5 equal-sized pieces adjacent to one another). (Note: this Criteria for Success and its corresponding Anchor Task and problems on the Problem Set and Homework are optional.)

Tips for Teachers

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  • As mentioned in Lesson 1, “counting fractional parts, initially unit fractions, to see how multiple parts compare to the whole helps students understand the relationship between the parts (the numerator) and the whole (the denominator)” (Van de Walle, Teaching Student-Centered Mathematics, Grades 3–5, vol. 2, p. 213). This lesson extends students’ understanding of the relationship between the part and the whole by having students iterate a unit fraction to represent the whole, which “helps children conceive of a whole as a multiple of a unit fraction, consisting of a certain number of copies of a same-size unit (Steffe and Olive, 2010) that draws their attention to the number of times a unit fraction fits within a whole” (Tzur and Hunt, “Iteration: Unit Fraction Knowledge and the French Fry Tasks,” Teaching Children Mathematics, Vol. 22 No. 3, October 2015). While the idea that a whole is a multiple of a unit fraction is not an explicit one, the idea of iteration and copies will certainly come in handy when students begin their work with number lines, which is constructed by iterating a whole (or a unit fraction) of equal length.
  • This lesson provides many opportunities for students to make sense of problems and persevere in solving them (MP.1). Each task pushes students to make sense of the meaning of unit fractions (and optionally non-unit fractions) and their numerators and denominators, and use that understanding to relate them to the whole.
  • As a supplement to the Problem Set, students can practice their iteration skills using the applet called “JavaBars” created by John Olive. After downloading the applet, do the following:
    1. To make a bar (whole, or an estimate to be iterated), click on the BAR icon (upper left).
    2. When the button is blue, it is operative. To draw a bar, click anywhere on the screen and drag while holding down your mouse button. (Note: You can only increase a bar; you cannot make it smaller.)
    3. As long as the BAR button is highlighted (blue), you can continue drawing more and more bars on the screen.
    4. Now, draw a long, skinny bar and consider it to be one whole.
    5. To construct an estimate of, say, the size of $$\frac{1}{4}$$, draw a new bar underneath the unit whole.
    6. Next, you will need to practice “testing” the size of your estimate.
    7. To iterate or “repeat” an estimate, click on REPEAT at the top menu line, click on your estimate (this is the bar you want to iterate), and click to the side of that bar (right, left, up, or down) in the direction you want the iteration to go.
    8. We want to go to the RIGHT because we are checking to see if (in this case) four copies of our estimate will produce the unit whole exactly. (Note: After the computer determines the direction, all subsequent clicks will continue the iteration in the same direction.)
    9. Was your estimate too long or too short? How much longer or shorter will you make your next estimate? Keep practicing.
    10. After you have successfully estimated the size of the unit fraction, try again with another unit fraction and/or a different-sized whole.

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Anchor Tasks

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Problem 1

Mr. Strader brings in a chocolate bar and decides to share it with Mr. Silver, Mrs. Ingall, and Ms. Kosowsky. Here is what each of their equal-sized pieces looks like:

How big was the chocolate bar before Mr. Strader broke it up?

Guiding Questions

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Problem 2

Each shape represents the unit fraction. Draw a picture representing a possible whole.

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Guiding Questions

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Problem 3

Each shape represents the fraction. Draw a picture representing a possible whole.

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Problem Set & Homework

Discussion of Problem Set

  • Which wholes had the most equal parts?
  • Which wholes had the least equal parts?
  • What did you notice about size of the whole when given a unit fraction with a larger denominator? What about with a smaller denominator?
  • Which fractions and/or shapes were the most difficult to create the whole for? Why?
  • What if all the wholes were the same size? What would happen to the equal parts?
  • Were there any unit fractions for which the wholes seemed to actually be the same? Which ones? How can you prove that their wholes are the same?
  • Why are both Aileen’s and Jack’s drawings correct? How many other possible unique drawings could there be?

Target Task

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Each shape represents the unit fraction. Draw a picture representing a possible whole.

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References

EngageNY Mathematics Grade 3 Mathematics > Module 5 > Topic C > Lesson 12Exit Ticket, Questions #1 and 2

Grade 3 Mathematics > Module 5 > Topic C > Lesson 12 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by The Match Foundation, Inc.

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