Professional development designed for groups preparing to review K-8 Math instructional materials. The modules are designed for groups to collaboratively build a deep understanding of the Instructional Materials Evaluation Tool (IMET) alignment criteria and metrics and how they reflect the expectations of the Standards in advance of conducting a full review of materials. Through learning about the IMET structure, the Non-Negotiable and Alignment Criteria metrics, and seeing examples and non-examples, participants will gain a solid understanding of what alignment might look like in instructional materials.
Also included is "Conducting a Shared Review" which provides support for facilitators as they lead a shared review of a full set of instructional materials.
There are many steps involved with conducting a materials evaluation review. Professional development modules are available to support building knowledge of the Instructional Materials Evaluation Tool (IMET), planning a review, preparing a review team, and in-depth professional development for the review team using the IMET.
Introduction to the IMET: Give a high-level overview of the IMET to various stakeholders with ELA/Literacy and math specific modules
Understanding the IMET: Introduction to the Criteria and Metrics of the IMET: Provide in-depth professional development on the IMET with ELA/Literacy & math specific modules
We continue to learn from educators and ongoing research about both the critical features of instructional materials and the ways in which educators enact instruction. Check back for updates to these training materials.
Module 101 is designed to lead a group of reviewers using the IMET through the non-negotiables of the resource: Focus and Coherence.
Module 103 is designed to lead a group of reviewers using the IMET through the Alignment Criteria of the resource: Standards for Mathematical Practice and Access for All Students.
Math specific module designed to provide an additional level of guidance for teams who are beginning a full review of materials using the IMET. Module includes a facilitator guide, participant guide, and a resource packet.
Concerning the third method, using a benchmark to compare two fractions is explicitly mentioned in 4.NF.2. Because the meaning of a whole is fundamental to understanding a fraction, it is appropriate to use 1 as a benchmark in the third grade. The teacher may, however, choose to remove those cards containing pairs of fractions where one is larger than a whole and the other is less than a whole.
Two different sets of cards are provided as attachments, one with a picture of the two fractions being compared and one without. The pictures allow students to make a visual comparison of the fractions which is important. However, the teacher may wish for students to provide these pictures as one means of explaining their decision. Similarly, the teacher may also wish to remove cards having equivalent fractions if the goal is to work exclusively on inequalities.
Question 2 is intended to motivate a classroom discussion after students have completed the activity. In order to better prepare them for this, the teacher may suggest that students think about the strategies they are using to compare fractions as they play the game. Some methods, like drawing pictures or using fraction strips, can be used for all of the pairs of fractions. But other methods such as looking for a common numerator and common denominator are conceptually important and the teacher will want to make sure that these methods are discussed.
Solution
a. There are four types of fractions which students will need to compare:
Fractions having the same numerator. The denominator tells us how many equal pieces are in the whole, determining the size of each piece, and the numerator tells us how many of those pieces we have. For example, to compare and , there are more fifths in the whole than thirds so fifths are smaller. This means that $\frac{2}{5} < \frac{2}{3}$.
Fractions having the same denominator. For example, we see that $\frac{2}{3} > \frac{1}{3}$ because $\frac{2}{3}$ is $\frac{1}{3}$ and an additional third so it is bigger. This relates to the reasoning described in the common numerator situation: the denominator tells us there are the same number of pieces in the whole, however one fraction has more of those pieces than the other.
One fraction is less than 1 and the other fraction is larger than 1. For example,$\frac{2}{3} < \frac{3}{2}$ because $\frac{2}{3}$ is one third short of a whole while $\frac{3}{2}$ is an entire whole with an additional half added.
Simple equivalent fractions such as $\frac{1}{2}$ and $\frac{2}{4}$. One way to show that these fractions represent the same quantity is with a picture:
Here the two large squares are equally sized wholes which have been divided into two equal parts (on the left) and four equal parts (on the right). The same fraction of the whole is shaded in each picture so $\frac{1}{2}$ is equivalent to $\frac{2}{4}$.
b. There are many important lessons to be learned from comparing these fractions including:
If I draw a picture of the two fractions, the larger fraction will have more shaded than the smaller fraction. If the two fractions are equal, the same amount will be shaded in both.
The denominator tells me how many pieces to cut my whole into. When the whole is cut into more pieces, the pieces are smaller (this is why $\frac{1}{3}$ is less than $\frac{1}{2}$).
The numerator tells me how many equal sized pieces I have. So $\frac{3}{5}$ is more than $\frac{2}{5}$ because I have one extra piece.
Fractions are built from the unit fractions so it is important to understand and be able to represent the unit fractions.
If using the fraction cards with pictures, equal sized wholes are important when comparing fractions.
Equivalent fractions have different sized pieces, but the same total amount shaded.
When the numerator is a bigger number than the denominator, the fraction is greater than one whole.
When doing mathematics, patterns emerge. These patterns support students in making conjectures, supporting their reasoning, and proving mathematical claims.
This task was created as part of the Adapting Materials Project. The goal of this project was to create a replicable process for teachers intending to adapt their materials, and to help create an environment of trust, where teachers felt empowered with the knowledge, confidence, and authority to change their own instructional materials in a way that better reflects the standards. To learn more about the work of these districts, read the “Collaborative Learning and Updating Materials” article from Aligned or access the complete case study.
For more information on the specific expectations for students working with fractions in grade 3, including the need for fractions to be referring to the same whole, read pages 3–5 in the progression document, Number and Operations–Fractions, available at www.achievethecore.org/progressions.
