Grade 5 Math Module 3: Addition and Subtraction of Fractions

During this morning’s grade 5 math session, participants worked through the Module 3: Addition and Subtraction of Fractions.  The first two lessons (topic A) review grade 4 standards and the concept of equivalent fractions.  Students use paper folding activities to demonstrate equivalent fractions, which helps make the concept very concrete. They use visual modeling via arrays and number lines to help make meaningful connections and the relationships that fractions have to one another.

Topic B is where students encounter fractions with un-like denominators.  They know the language of 1 apple +1 apple, 1 third + 1 third, so when they encounter 1 third +1 half, they know the units are not the same and that they need to make the units the same (common units).  Lots of work with arrays is involved here and there are some struggles, but overall the visual model helps the transition from concrete to abstract.

The group spent a lot of time on a two-step word problem from Lesson 7 that involved subtracting from a whole with uncommon units.  The solution was presented two different ways using visual models (arrays).

For topic C, participants took a look at three student solutions to the problem 3 3/5 – 2 ½.  It was interesting and exciting to see students use unbundling and decomposition that they saw in previous grades with whole numbers, and applying those concepts to their work automatically with fractions.

Topic D focuses on the problem-solving practice and application of the concepts learned. Multi-step problems are tackled with various strategies. The module does a nice job of showing the progression of equivalent fractions, making units pictorially, making units numerically, and then being able to apply knowledge to solve problems that involve addition and subtraction of unlike fractional units.

Living an Elementary ELA Lesson

During Thursday’s afternoon ELA session, teachers took on the role of the student and started with a reading of the non-fiction article “Characteristics of a Multinational Company” in order to come up with the gist of the article. This activity led into another where the session participants could then read and find the gist of another text around which the lesson is based, “The Red Cross.” The participants then used a graphic organizer (3-column Note Catcher) to complete the tasks associated with the Taking Notes Task card protocol. Through close reading of the article, everyone was able to identify key vocabulary terms and find text-based evidence to justify their definitions, just as their students would do. The graphic organizer is a tool that helps students to define a new term/concept based on the information gleaned from the two texts. In this case, the two articles were used to identify what a multinational aid organization is.

Through a Chalk Talk, the participants discussed their own notes based on the articles they read. Then, through the Think Pair Share protocol, ideas were exchanged about the best way to respond when a community is struck by a natural disaster. Participants shared their ideas with the rest of the group in order to explain key ideas from the texts they read. After “removing their student hats,” the teachers discussed the benefits of the protocols and how they are an effective way to address the Speaking and Listening standards. One major benefit was that the protocols helped to grow their thinking through the sharing of mutual ideas and they can offer help to students who need their thinking redirected.

Session participants read the lesson that was the source of the day’s activities and looked for evidence of collaboration, protocols, and conversations. This lesson is a good example of incorporating the Speaking and Listening standards into a Common Core-aligned curriculum. A question was asked about the appropriateness of the protocols for various lessons. Each protocol in the list provided on EngageNY is prefaced with a purpose so that a teacher can decide which protocol to use depending on the goals of the lesson. The session concluded with teachers reflecting on what kinds of classroom management practices need to be in place in order to foster a collaborative environment.

The remainder of the afternoon was spent looking at how protocols can improve literacy skills and be effective in a collaborative classroom environment. Four Corners and Fishbowl were put into practice in this session. The teachers again played the role of the student for the Four Corners activity. Here, the teacher/students had to choose (from 4 options) the most important thing the Red Cross does when communities are struck by natural disaster. Conversations were text-based and gave the teachers an opportunity to see this protocol in action through the eyes of their students. They then participated in the Fishbowl protocol where volunteers discussed the purposes of the Red Cross based on the notes they had taken throughout the “lesson.” The emphasis here is on relevant notes, relevant information, and relevant evidence. Observers of the Fishbowl conversation had to look for evidence of relevance in the conversations along with other qualities including how the participants in the discussion behaved throughout the exercise. Everyone spent the remainder of the session developing a rubric for Speaking and Listening standards.

The activities and texts in these sessions came from Grade 5, Module 4, Unit 3, Lesson 3.

A Deeper Look at Grade 4 Math Module 4

Thursday’s grade 4 math afternoon session covered lessons from the Grade 4 Module 4: Angle Measure and Plane Figures. This module introduces students to seven new geometry terms in the first lesson that set the foundation for their progression to high school geometry. They draw points, lines, segments, rays, and angles and learn how to identify them with geometrical notation. By using a right angle template, students are taught to easily identify right, acute and obtuse angles in Lesson 2.

Physiometry fluency tasks are introduced in Lesson 2 and involve using arms to construct types of angles, which is fun and engaging. Continuing with the use of the right angle template, students study the relationships of perpendicular and parallel lines. A fun activity involves using the letters of the alphabet to identify segments that are parallel and perpendicular. Students will come to realize that points are needed in order to name segments. Attending to precision and communicating/critiquing the reasoning of others play a key role here.

Students are introduced to the concept of degree after visualizing the progression of ¼ turns, ½ turns, and ¾ turns using the two-circle manipulative that makes angles pop out and come alive. Students transition from using the two-circle manipulative to using a circular protractor, and then to using the 180-degree protractor when identifying angle types and measuring.

A hard-to-grasp concept is angle measure vs. arc length measure. When is an angle larger? Students will explore this concept using two circles, one smaller than the other. Each circle is folded into quarters, showing a right angle. Students will notice that the arc-length of the larger circle is larger than the arc-length of the smaller circle, even though the right angles forming the arcs are the same. The circles are excellent visual aids in this example.

Paper folding exercises that students will be using to understand angle addition and subtraction were demonstrated. These exercises will lead classrooms into the discussion of complementary, supplementary, and vertical angles and students will be able to solve missing angle problems based on these relationships that they can visually recognize. The vocabulary here is not part of the 4th grade, but provides great exposure and provides students with the skill of being able to visually recognize these relationships leaving 4th grade.

The remainder of the module focuses on classifying two-dimensional figures and lines of symmetry. Classifying quadrilaterals is always challenging (i.e., is a rectangle a square, or is a square a rectangle?). A nice presentation on the hierarchy of the “trapezoid” family was drawn out to help the audience. Students/teachers should start with drawing trapezoids based on the fact that they have at least one set of parallel sides. From there, progress to drawing just parallelograms, then just rectangles, then squares. This is a great activity to help students see the progression. The module is loaded with fun, engaging, and solid activities that will build the strong foundation needed for the geometry to come in later modules and grades.

Building a Change-Focused Culture

The first morning English Language Arts session, hosted by Expeditionary Learning, focused on the protocols included in the grades 3-8 modules and collaboration between educational professionals. Participants started by doing the World Café protocol, which appears in several of the modules. Groups read two different case studies that would be the source of conversation and idea exchange during the activity.

The case studies were about implementation of the Common Core and school district use of the modules. Participants discussed how to build positive change focus within different types of schools and districts. This activity served multiple purposes: teachers saw the protocol from the student perspective, they talked about its effectiveness, and they were able to exchange ideas about how to implement the modules and make the shift in instructional practices a positive experience for teachers.

The second morning session focused on another protocol used in the modules around the topic of the Speaking and Listening standards. This protocol, Text Rendering, includes close reading, collaboration, and note-taking to reinforce speaking and listening skills and using text-based evidence in discussions with peers. This protocol would be useful when students need to summarize different texts by emphasizing key points from the reading. In this case, teachers looked at the Speaking and Listening standards for their own grade(s) and discussed things they noticed, like how the standards change over the grade levels and the emphasis on making students responsible users of electronic sources like social media.

Participants then chose an article to read that would be the source for the discussion in the text rendering protocol. The articles read were “Collaborative Conversations” (Fisher and Frey), “Habits improve classroom discussions” (Bambrick-Santoyo), and “Structuring the Talk: Ensuring Academic Conversations Matter” (Frey and Fisher).

Module Focus: Grade 4 Math Module 3

The morning session for grades 4 and 5 math focused on Grade 4 Module 3: Multi-Digit Multiplication and Division, where students see multiplication and division in action.  The standard algorithm is introduced in grade 4, but it is not a fluency for this grade level.  Multiplication is a fluency in grade 5 and division is a fluency in grade 6, so the intent of the module is to allow for the deep conceptual understanding of the process through the use of modeling techniques.

Participants took a look at division through the use of number bonds, array/area models, and place value/number disk charts.  This exercise provided a great visualization of the process.  Language plays a key role, with the correct use of the terms, “whole,” “quotient,” and “remainder” (how many are left). Phrases like “distributing evenly” and “decomposing” are important in explaining the process.

Teachers need to pick fluency activities that tie into the lesson well.  Participants looked at an example:

How many groups of ____ are in _____? Prove it by counting by ______.

The fluency activity will lead into the concept of “remainders,” an important link to the lesson.  Teachers need to make sure there is a connection with the fluency to the lesson objective.

Strategies for Adaptation: Providing Support to Diverse Learners

The intent of the afternoon ELA session was to help teachers plan curriculum adaptations that maintain the rigor and alignment to the Common Core. Participants started with a discussion about the common learning barriers that students may have. Students reading well below grade level, English levels of ELLs, and scaffolding for processing time were highlighted as some concerns teachers have.

We need to be flexible in our educational goals and teaching methods in order to help all learners succeed. One effective strategy to address the needs of these students is to keep some of these guiding questions in mind when adapting curriculum:

  • Are there ways I can promote learners’ ability to monitor their own learning?
  • Where might my students benefit from different methods of presenting the lesson?
  • Have I identified potential barriers to learning that may be present in the instructional methods and materials?

Other guiding questions may be found within the NTI materials on

Participants also examined potential barriers to learning and adaptations that may inform their own classroom practice. Two student profiles were provided so participants could discuss strategies for adapting Unit 1, Lesson 8 of the Grade 9 Module 1. By first deconstructing the standard and the assessment, teachers could identify three things that the students should be able to do. Participants then talked about the strategies that could be used to adapt the lesson and assessment to meet the needs of the students in the example profiles. The protocols provided in the NTI materials will be very helpful for teachers using the modules who need to adapt to meet the needs of diverse learners.

The second part of the afternoon session was called “Office Hours.” The much-anticipated second ninth grade module was previewed. Kate Gerson introduced the texts:

  • Unit 1- Dickinson’s “I felt a funeral in my brain” and Poe’s “Tell-Tale Heart”;
  • Unit 2- Oedipus the King by Sophocles;
  • Unit 3 contains all non-fiction: “True Crime,” a piece by Walter Mosley, and two articles relating to the crimes of Bernie Madoff.

The final performance task for the module was also introduced: two options for identifying and writing about a central theme in both an informational and literary text that was studied in the module.  Kate also highlighted the standards that are addressed and assessed in the module as well as how the questioning throughout the module is spiraled and leads to a series of standards-related Quick Writes. Feedback from the first module was taken into consideration and had an impact in the development of this module. Formatting and pacing are two major changes between the first two modules. The new module will be released soon!

“Bridging the Gap” in Mathematics

The afternoon mathematics session, “Bridging the Gap,” led by Jill Dinitz from Common Core Inc., was a great, powerful presentation on how to assess and remediate students’ prerequisite knowledge and conceptual understanding of key mathematical ideas and how to further develop their mathematical fluency.  The remediation strategy presented was in three parts:  assess, discuss, and repeat.

Jill introduced a great instructional tool that helps to assess where a student is in terms of their conceptual knowledge by having them create their own word problem to model a computational one.
Example: Write a problem that would require you to compute 12 divided by ½ to solve the problem.

Participants looked at how visual models like tape diagrams and arrays/area models can be used to help students transition from the concrete, pictorial to the abstract.  They covered the four basic operations, focusing especially on division.  How we ask the division question is important (i.e., twelve divided by ½ vs. how many ½’s are in twelve).  The answer, “24,” jumps out in the second questioning technique.

Participants spent time going over various fractional situations that students encounter through grade 8. They did a very cool paper folding exercise that involved finding two shapes, each with an area of ½, but different shapes, utilizing one post-it note.  Participants were highly motivated and involved, so one can just imagine the potential for the engagement in a classroom doing this exercise.

Participants also took a look at operations with negatives, which is another tricky area for students. Jill demonstrated how to prove that the product of two negatives is always positive through the use of mathematical properties, as well as through area models, both excellent instructional ideas.

The session ended with a presentation of a systems of equations problem found in pre-calculus.  Due to timing, the session had to quickly end, but it would have held interest for a much longer time frame if allowed.  Lots of great ideas were presented here this afternoon, and participants went home with excellent strategies to use in their districts/classrooms.