Transitions in the Conception of Number: From Whole Numbers to Rational Numbers to Algebra

Appendix on Implementing District Standards

In second grade the geometry standard is

2.G.A.2 Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

This relates to standard 2.OA.4 where students arrange objects in an array of rows and columns. Students partition a rectangle into rows and columns of same-size squares and use repeated addition to find the total number of squares. This work provides background for understanding area, for viewing multiplication as repeated addition, and for connected multiplication to arrays and area.

2.G.A.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc. and describe the whole as two halves, three-thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape. In this standard, students will be able to partition rectangles and circles of various sizes into two, three, or four equal shares. Through experiences with multiple representations, students should use the words, halves, thirds and fourths, and the phrases half of, third of and fourth of (or quarter of) to describe their thinking and solutions. Working with the “the whole”, students understand that “the whole” is composed of two halves, three thirds, or four fourths. This standard also addresses the idea that equal shares of identical wholes may not have the same shape. Students should understand that fractional parts may not be symmetrical.

Students divide 2-dimensional shapes into halves, thirds, fourths, and they learn that equal shares of the same whole can have different shapes. The big idea that students need to understand is that the name of each share refers not only to the number shares, but also to the fact that each of those shares represents an equal portion of the whole. This helps students learn that equal shares of the same whole can have different shapes. This content is a foundation for understanding fractions as numbers.

Students should be able to use Mathematical Practice 3 when working on standard 2: G.A.3. The student should construct viable arguments and critique the reasoning of others.

There are 8 mathematical practices that students should use regularly. Some of the practices that students will be using doing this fraction unit are as follows.

1. Reason abstractly and quantitatively.

2. Student reason abstractly when they identify that a shape can be partitioned in different ways and still result in equal parts.

3. Use appropriate tools strategically.  For the purpose of this unit, this means that students must know how to use square tiles to identify equal parts in a share.

4. Look for and express regularity in repeated addition when they analyze and create repeated design using equal shares.

According to the Common Core State Standards in grade 3, fractions are a major emphasis, with attention to using fraction symbols, exploring unit fractions (fractions with numerator 1), and comparing fractions. Students need to master the second-grade standard in order to be prepared for the standards in third grade.

This unit will start with introducing the key vocabulary needed to understand fractions. Starting the unit with vocabulary is important. We will start with vocabulary such as fractions, whole, partitions, half, halves, thirds, thirds, quarters, and fourths. Once students understand the vocabulary, they have a better understanding of the skills you plan to teach.

I will start by showing a whole sheet of paper. Next, I will show students shares/pieces of the whole by folding paper. Studies show that when teaching fractions students should be taught concrete and representational methods. This will help lay the foundational skills needed.  My method of teaching vocabulary is by introducing a game where I describe the definition and the student gives the correct vocabulary word for the definition.

Once the students understand what fractions are, they will be able to identify how big or small they are. I think that using fractions strips will help the students with a visual representation of the fractions. They will be able to describe the whole as being a multiple of a unit fraction.

Here is a list of Effective Teaching Practices published by The National Council of Teachers of Mathematics.

1. Implement tasks that promote reasoning and problem solving.
2. Support productive struggle in learning mathematics.
3. Pose purposeful questioning.
4. Facilitate meaningful mathematical discourse.
5. Establish mathematics goals to focus learning.
6. Use and connect mathematical representations.
7. Build procedural fluency from conceptual understanding.
8. Elicit and use evidence of student thinking.