Fractions

Lesson 1

Objective

Partition a whole into equal parts, identifying and counting unit fractions using concrete area models.

Common Core Standards

Core Standards

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  • 3.G.A.2 — Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape.

  • 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.

Foundational Standards

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  • 2.G.A.3

Criteria for Success

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  1. Understand that a fraction is an equal share of a whole.
  2. Understand that a fractional unit is the unit that the whole is partitioned into, which could be halves, thirds, fourths/quarters, etc.
  3. Understand that the fractional unit when a whole is partitioned into 6 parts is sixths. (Note that students will have seen all other fractional units (including halves, thirds, and fourths/quarters) in Grade 2) (2.G.3).
  4. Given a shape made from identical pattern blocks, determine its fractional unit.
  5. Given the unit fraction of a shape, use pattern blocks to construct a whole (MP.5).
  6. Count unit fractions up to a whole.

Tips for Teachers

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  • In Lesson 1, students will count the unit fractions up to a whole in the Anchor Tasks but are not asked to do so on the Problem Set or Homework. When they identify unit fractions on the Problem Set and Homework, they are not expected to use fraction notation yet. In Lesson 2, students will count the unit fractions up to a whole on the Anchor Tasks as well as the Problem Sets and Homework, but they are still not expected to use fraction notation. In Lesson 3, students will use fraction notation to represent both unit and non-unit fractions.
  • “In whole-number learning, counting precedes and helps students compare the size of numbers and later to add and subtract. This is also true with fractions. 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). Thus, to reinforce this idea of counting, it is recommended to give students daily practice with counting fractions so students can understand the relationship between the part and the whole. Fractions can be counted in a variety of ways, including without regard to whole numbers (e.g., 1 third, 2 thirds, 3 thirds, 4 thirds, etc.) or by replacing fractions equivalent to whole numbers with the whole numbers themselves (e.g., 1 third, 2 thirds, 1, 4 thirds, etc. Note: you could also have students say “1 whole” so that students are always saying the unit involved.). Eventually, students could even count by whole numbers (e.g., 2 halves, 4 halves, 6 halves, etc.). You can use the same routines as you did for skip-counting in units 2 and 3, as well as a third new one, below. (Source for these routines: Jessica Shumway, Number Sense Routines: Building Numerical Literacy Every Day in Grades K–3, pp. 55–67, 2011).
    • Choral Counting: Choral counting is simply counting out loud as a whole class. You can involve movement, as well, by having students count on their fingers, do jumping jacks with every count, etc. You can ask some basic questions about the count sequence, but because this routine really just helps introduce students to new sequences or ones that they struggle with, more rigorous questions can be saved for the other routines (which are in bullets below).
    • Count around the Circle: Have one child start with the first number in the counting sequence, and go around the circle having each child say one number. You could decide to write the numbers on the board as students write them, either as a scaffold for them or to encourage a discussion of patterns afterward. Some questions you can ask before/during/after this routine to encourage sense-making include:
      • If we count by twos around the circle starting with Student A, what number do you think Student Z will say? If you didn’t count to figure that out, how did you solve?
      •  If we count around the circle by fives and we go around twice, what will Student X say?
      •  Why did you choose ____ as an estimate?
      • Why didn’t anyone choose ____ as an estimate?
      • How did you know what comes next?
      • (After a child gets stuck but figures it out): What did you do to figure it out?
      • (If you’ve written the numbers on the board as students counted): What do you notice? What do you wonder?
    • Organic Number Line: You could introduce this routine anytime after the beginning of Topic B. While this task is not explicitly a counting routine, many students may refer to the constructed number line to help them when counting in either of the routines above. Present students with a number line that starts at 0 and ask someone to place the number 1, then ask where to place $$\frac{1}{2}$$. You could move the 1 benchmark and ask if it moves the placement of $$\frac{1}{2}$$ and how much. Then add more fractions to the number line. It is best to start your discussions of the organic number line with benchmarks such as $$\frac{1}{2}$$ and $$\frac{1}{4}$$, and think of ways these benchmarks can be represented to display underneath their placement. Then add more fractions as they come up or students seem ready for them, including adding fractions equivalent to 1, greater than 1, and equivalent to whole numbers greater than 1.

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Anchor Tasks

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Problem 1

George made his younger brother, Neil, hexagonal cookies to celebrate his 6th birthday. One of the cookies is shown below.

  1. Two of Neil’s friends want to share a cookie. They decide to split the cookie in the following way:

Is this a fair share of the cookie? Why or why not?

  1. Neil and his friends Shantelle and Oscar want to share a cookie. They decide to split the cookie in the following way:

Is this a fair share of the cookie? Why or why not?

  1. Four of Neil’s friends want to share a cookie. They decide to split the cookie in the following way:

Is this a fair share of the cookie? Why or why not?

Guiding Questions

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Problem 2

Identify the fractional units of each of the following pattern block shapes. You can use your pattern blocks if it will help you.

a. 

b.

c.

d.

Guiding Questions

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Problem 3

Create pattern block shapes where one piece represents each of the following fractions.

  1. 1 half

  2. 1 fourth

  3. 1 third

  4. 1 sixth

Guiding Questions

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Problem Set & Homework

Discussion of Problem Set

  • How did you know your answer was correct in #2?
  • What pattern block did you use to create your shape in #6a? How many of those blocks did you use? Was there another way you could have arranged those blocks?
  • #7 through #9 were similar to tasks that we used in Unit 4. How is an area unit similar to a fractional unit? How is it different?

Target Task

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Problem 1

Build the shaded shape below with your pattern blocks. Then, name the fraction that each triangle represents.

Problem 2

Build a shape with pattern blocks whose fractional unit is fourths. Then trace the shape below.

Mastery Response

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