This module was developed by Carrie Ziegler, Nathan Auck, and Steve Jackson. They are the three principle designers of the course, Principles to Actions, in the UEN and Utah State Office of Education Professional Learning Series.

USOE Professional Learning Series

Principles to Actions

Learning Progressions

Objective: Understand how core standards and grade level teaching and learning connect to student development in mathematics.

Find a Half TaskDirections: Which of the following cards show a half? Sort them into two groups: those that show a half, and those that do not show a half. As you place the cards into each group record your thinking: Why is this card a half? Or: Why isn't this card showing a half?

Adapted from Treacy, K., & Cairnduff, J. (2009). Revealing what students think: Diagnostic tasks for fractional numbers (pp. 21-34). Ascot, W.A.: STEPS Professional Development.

Find a HalfMathematics Learning Goals

Students will understand that:

Two halves (equal shares) equal one whole.

Halves must be the same size: have an equal area or an equal number of pieces.

A fraction describes the division of a whole region or area into equal parts.

A fraction is relative to the size of the whole unit.

If the numerator is half the quantity in the denominator then the fraction is equal to a half.

Fractions have an infinite number of equivalent forms, regardless of whether or not the pieces of the whole unit are adjacent.

Connections to the CCSSM Grade 3Standards for Mathematical Content

Number and Operations – Fractions (NF)

Develop understanding of fractions as numbers.

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.

3.NF.A.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

(a) Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.

(b) Recognize and generate simple equivalent fractions (e.g., 1/2=2/4, 4/6=2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model.

National Governors Association Center for Best Practices (NGA Center) and Council of Chief State School Officers (CCSSO). (2014). Common core state standards for mathematics. http://www.corestandards.org/Math/Content/3/NF

Connections to prior learning in the CCSSM

GeometryReason with shapes and their attributes.1.G.3 Partition circles and rectangles into two and four equal

shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

2.G.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.

K.G.3 Decompose numbers…into pairs in more than one way…

Overview

Read the part of the progressions documents related to your grade level learning: previous, current, and future grade levels.

Chart and discuss how fractions develop throughout the CCSSM.

Discuss connections between grade level learning.

Learning Progressions

Read previous, current, and next grade parts of the Learning Progressions.

Geometry, Grades K-3

Number and Operations-Fractions, Grades 3-5

Ratios and Proportional Relationships, Grades 6-7

The Number System, Grades 6-8, High School

These documents can be found at http://ime.math.arizona.edu/progressions/

Learning Progressions

What are the important fraction concepts for your grade level/band? (K-2, 3-5, 6-8, 9-12)

Create a chart of concepts: include models and strategies.

Establish Mathematics Goals to Focus Learning

Learning Goals should:

• clearly state what it is students are to learn and understand about mathematics as the result of instruction;

• be situated within learning progressions; and

• frame the decisions teachers make during a lesson.

“Formulating clear, explicit learning goals sets the stage for everything else.”

(Hiebert, Morris, Berk, & Janssen, 2007, p.57)

Learning Progressions

Where are the connections?

Effective Mathematics Teaching Practices

Effective Mathematics Teaching Practices

1. Establish mathematics goals to focus learning.

2. Implement tasks that promote reasoning and problem solving.

3. Use and connect mathematical representations.

4. Facilitate meaningful mathematical discourse.

5. Pose purposeful questions.

6. Build procedural fluency from conceptual understanding.

7. Support productive struggle in learning mathematics.

8. Elicit and use evidence of student thinking.