Module Focus: Grade 4 – Module 4 Sequence of Sessions

Overarching Objectives of this November 2013 Network Team Institute Module Focus sessions for K-5 will follow the sequence of the Concept Development component of the specified modules, using this narrative as a tool

for achieving deep understanding of mathematical concepts. Relevant examples of Fluency, Application, and Student Debrief will be highlighted in order to examine the ways in which these elements contribute to and enhance conceptual understanding.

High-Level Purpose of this Session Focus.  Participants will be able to identify the major work of each grade using the Curriculum Overview document as a resource in preparation for

teaching these modules. Coherence: P-5.  Participants will draw connections between the progression documents and the careful sequence of mathematical concepts that

develop within each module, thereby enabling participants to enact cross- grade coherence in their classrooms and support their colleagues to do the same .  (Specific progression document to be determined as appropriate for each grade level and module being presented.)

Standards alignment.  Participants will be able to articulate how the topics and lessons promote mastery of the focus standards and how the module addresses the major work of the grade in order to fully implement the curriculum.   

Implementation.  Participants will be prepared to implement the modules and to make appropriate instructional choices to meet the needs of their students while maintaining the balance of rigor that is built into the curriculum.    Instructional supports.  Participants will be prepared to utilize models appropriately in promoting conceptual understanding throughout A Story of Units.

Related Learning Experiences● This session is part of a sequence of Module Focus sessions examining the Grade 4 curriculum, A Story of Units.

Key Points• Points, lines, segments, rays, and angles of the first topic support the rest of the module as students construct angles and figures.

• A degree is not a length measure.

• Analyzing attributes of figures leads to more formalized definitions.

• Spend time physically manipulating angles and figures. It will lead to a deeper understanding.

Session Outcomes

What do we want participants to be able to do as a result of this session?

How will we know that they are able to do this?

Focus.  Participants will be able to identify the major work of each grade using the Curriculum Overview document as a resource in preparation for teaching these modules.

Coherence: P-5.  Participants will draw connections between the progression documents and the careful sequence of mathematical concepts that develop within each module, thereby enabling participants to enact cross- grade coherence in their classrooms and support their colleagues to do the same .  (Specific progression document to be determined as appropriate for each grade level and module being presented.)

Standards alignment.  Participants will be able to articulate how the topics and lessons promote mastery of the focus standards and how the module addresses the major work of the grade in order to fully implement the curriculum.   

Implementation.  Participants will be prepared to implement the modules and to make appropriate instructional choices to meet the needs of their students while maintaining the balance of rigor that is built into the curriculum.   

Instructional supports.  Participants will be prepared to utilize models appropriately in promoting conceptual understanding throughout A Story of Units.

Participants will be able to articulate the key points listed above.

Session Overview

Section Time Overview Prepared Resources Facilitator Preparation

Introduction to the Module

13 minOverview of the instructional focus of Grade 4 Module 4.

Grade 4 Module 4 Grade 4 Module 4 PPT

Concept Development

118 min

Examination of the development of mathematical understanding across the module using a focus on Concept Development within the lessons.

Grade 4 Module 4 Grade 4 Module 4 PPT

Review Grade 4 Module 4 Overview, Topic Openers, and Assessments.

Module Review 4 minArticulate the key points of this session.

Grade 4 Module 4 Grade 4 Module 4 PPT

Session Roadmap

Section: Introduction to the Module Time: 13 minutes

[3 minutes] In this section, you will…Provide an overview of the instructional focus of Grade 1 Module 3.

Materials used include:PPT and Participant materials

Time Slide

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1 1 NOTE THAT THIS SESSION IS DESIGNED TO BE 135 MINUTES IN LENGTH.Welcome! In this module focus session, we will examine Grade 4 – Module 4.

1 2 Our objectives for this session are to:• Examination of the development of mathematical

understanding across the module with a focus on the Concept Development within the lessons.

• Introduction to mathematical models and instructional strategies to support implementation of A Story of Units.

1 3 We will begin by exploring the Module Overview to understand the purpose of this module. Then we will dig in to the math of the module. We’ll lead you through the teaching sequence, one concept at a time. Along the way, we’ll also examine the other lesson components and how they function in collaboration with the Concept Development. Finally, we’ll take a look back at the module, reflecting on all the parts as one cohesive whole.Let’s get started with the Module Overview.

1 4 Module 4 of Grade 4 is Angle Measure and Plane Figures. The module includes 16 lessons and is allotted 20 instructional days.This module builds on understandings established in Kindergarten where students analyze and compare 2-dimensionsal shapes and prepares students for classifying 2-D figures in G4-M4 based on specific attributes. Angle measure is brand new to students, but they are prepared to think about this new form of measurement from their previous measurement experiences in G1 and G2. This module prepares students for the middle grades where these concepts gain sophistication.

9 5 Allow 8 minutes for participants to read through the Module Overview. Encourage them to underline components that are familiar to them and/or their students. Highlight what is new or unfamiliar for teachers and/or students.This session will strive to cover many of the unfamiliar components in the module, including the vocabulary and models used. Further information about geometry can be found in the Progressions. Keep in mind that there are two Progressions documents that cover the content in this module, the Geometry Progression and the Geometric Measurement Progression. Both documents are helpful to read prior to instruction. To allot more time to the models of this module, we will not be reading the Progressions.

Section: Concept Development Time: 118 minutes

[ minutes] In this section, you will…Examine of the development of mathematical understanding across the module using a focus on Concept Development within the lessons.

Materials used include:PPT and Participant materials

Time

Slide #

Slide #/ Pic of Slide Script/ Activity directions GROUP

1 6 Allow 8 minutes for participants to read through the Module Overview. Encourage them to underline components that are familiar to them and/or their students. Highlight what is new or unfamiliar for teachers and/or students.This session will strive to cover many of the unfamiliar components in the module, including the vocabulary and models used. Further information about geometry can be found in the Progressions. Keep in mind that there are two Progressions documents that cover the content in this module, the Geometry Progression and the Geometric Measurement Progression. Both documents are helpful to read prior to instruction. To allot more time to the models of this module, we will not be reading the Progressions.

1 7 Topic A consists of four lessons where students will learn several new terms by drawing representations of them and identifying them in 2D figures. Students will build a right angle template from paper and will use a straightedge in these lessons.

8 8 Lesson 1 introduces seven terms. That is daunting, but each term is provided within a context to give strong meaning. Memorizing a definition is not the goal. Using the term appropriately is a goal. Students will broaden and deepen their understanding of these terms as the lessons progress. The direction of this and many lessons will lead students to creating figures. All will look different. Celebrate the difference and use the many different figures to allow students time to critique each other’s work. “Are you sure that is a ray?” “Does a segment end?” “Prove the two rays form a 90 degree angle.”(Switch to document camera.)MODEL: Lead the participants through the Concept Development, Problems 1-3. Adjust the script to be about 5 minutes long. Participants should use pencil and paper and a straightedge as you model with them.Focus on the delivery of the terms. Do not front the model with terms. Instead, allow participants the chance to discover the terms just as the students will.Example: T: You drew these dots, or points, to mark specific locations.Adjust the script as you see fit for the participants. Remember to allow time for the participants to interact with each other and with you as if they are students.New terms are: point, line segment, line, ray, angle, arc, and figure.If time permits:Debrief Why is it hard to find real world examples of lines, rays, and points?

6 9 MODEL:Directions for making a right angle template:

1. Fold any piece of paper in half.2. Fold it again so that the folded side aligns with itself.3. Trace the 2 segments with your fingers that form the angle.4. The point where the segments meet is call the vertex. 5. Identify the right angle at this vertex by drawing a small square

at the vertex.*Consider having participants cut their 8x11 paper into a rough circle first. This will help students to see the right angle and not to confuse the right angle with other angles on the folded template.CLICK TO ADVANCE BULLETS 1-3. A right angle template can be made from any piece of paper and is a tool used throughout the module. Students build one and use it to identify angles around the room and on a practice sheet. The teacher directly tells students the angle created is called a right angle. The teacher also directs students to label right angles with a small square symbol at the angle, or where the two rays meet at the vertex. They discuss the different angles found in relation to the right angle. In using expressions like “This angle is less than a right angle” and “My template fits inside this angle,” the teacher introduces the terms acute and obtuse.Next, students align their right angle templates with rulers to draw angles, including a straight angle.There are deliberate terms that must be introduced within the curriculum, even if the terms are not widely used throughout the Module. The term interior of an angle is helpful in certain lessons. If a term is introduced in a Debrief, but not in a Concept Development, we do not expect students to use that term widely. Students should know, however, that it will come up in later grades, specifically the middle grades where geometry is taught more in depth.CLICK TO ADVANCE BULLET 4. Finally, participants complete the Lesson 2 Problem Set for 3 minutes.

2 10 Inform participants about a fluency across many lessons called Physiometry (geometry and physical). Do a few samples with them like right/acute/obtuse angles, lines, rays, segments, and parallel and perpendicular lines.Sample script is below from Lesson 4. It uses parallel and perpendicular. We are modeling with these terms so participants are familiar with parallel and perpendicular lines before we practice drawing them. Reminder: Vocabulary is not introduced in fluency.

5 11 Switch to document camera.MODEL part of Problem 1 (as shown below):MODEL: Draw perpendicular lines using a ruler and a right angle template.See Problem 4.

4 12 Switch to document camera.MODEL: Draw parallel lines using a ruler and a right angle template.See Problem 4.

3 13 Partner Debrief.Give the directions and allow 3 minutes for partners to complete. Look for partners using their straightedge and right angle template for proof.Encourage participants to use terms.Listen for use of terms (parallel, perpendicular, segment, point, vertex, right angle).

1 14 Topic B consist of four lessons where students will learn that angles turn and are measured in degrees. Students will measure angles with protractors and will use protractors, including a 360 degree circular protractor, to draw angles of a specified size.

10 15 MODEL the Application Problem.• Using the right angle template, determine how many right angles

fit around the center point of the circle.• If necessary, team up with partners at your table to share

templates. Overlaps are not allowed.• Debrief: 4 angles fit. Students will use that to divide the total

degrees of a circle by 4, discovering a right angle is 90 degrees.CLICK TO ADVANCE THE SLIDE: photo of paper protractors.Construct a Paper Protractor.MODEL the Lesson.See Problem 1: Make ¼ and 1/8 turns. Debrief about 360 turns in a circle, 1/360 of a turn is called 1 degree. Use a protractor to measure

angles.CLICK TO ADVANCE THE SLIDE: photo of 1/8 turns.CLICK TO ADVANCE THE SLIDE: photo of 360 degree protractorWalk through Lesson 2: Identify quarter turns as their degrees (0, 180, 270, 360). Align the 360 degree protractor for proof.Discuss Problem 3: Explain that students use each circle as a benchmark angle template, one showing 45 degree angles, the other 30 degree angles.

5 16 Pose the question.Answer is neither. They are the same size angle.Why?Length of segments does not determine the angle measure or the number of turns around a center point.MODEL:

1. Fold each circle into fourths to make 4 right angles. Trace the folds.

2. Label the larger circle as A, the smaller as B.3. Trace along the outside arc of the same section of a 90 degree

angle on each triangle.MODEL a revised Problem 1:Trace with your finger along each arc. Which is longer? (A)I thought right angles are 90 degree angles. Why are the arcs different lengths? (Different sized circles.)How many degrees in a circle? (360 degrees.)If I divide Circle A into 360 degrees along the outer edge, each arc will be larger than if I did that to Circle B.Place Circle B on top of Circle A. See how both arcs measure 90 degrees? The length of the segments, or where on the segments I take a measurement, doesn’t determine the arc. The arc will always be in reference to the 360 total degrees around a central point in a circle.

10 17 Allow 10 minutes for participants to complete Problem Sets 6 and 7.Instruct on Problem Set 6 to extend the length of segments using a ruler. This reinforces that the length doesn’t matter. Everyone will measure the same (within a margin of error) number of degrees no matter how long the segments or which protractor is used.Encourage participants to share their protractors. Likely there are many different sizes and types at the tables. Use the circular protractor too!Point out Problem 3 on Problem Set for Lesson 6. The term collinear is a new term that is only mentioned in the Debrief. (Lesson 8 is not included in the presentation. It covers angles as turns, such as quarter- or half- turns, and recognizing them in various contexts. “The skateboarder did a 180!”)(If needed, complete a Fluency of Physiometry. Follow the script below. Add 3 minutes to presentation.)

1 18 Following the Mid-Module Assessment, Topic C consists of 3 lessons where students decompose and compose angles. Pattern blocks serve as a tool for understanding how to solve for unknown angles. Students progress to using subtraction and addition with variables to solve for adjacent angle measures.

5 19 Switch to document camera.MODEL four squares around a center point. Label 3 points to identify a right angle. Determine it is 90 degrees. Model that 90 degrees takes four turns around a center point. 360 degree s÷ 4 = 90 degrees, confirming that the angle is 90 degrees. Write 90+90+90+90=360.(For all participants, if time and materials allow. Otherwise quickly demonstrate.) Have participants use triangle blocks to arrange around a center point. Follow the same procedure as above, reaching the addition sentence.Show a triangle and square aligned adjacent to each other. Draw an angle at their vertex. Do I know the value of angle ABC? (No, but you can use a protractor.)I’m fresh out of protractors. But I know this square is 90, and this triangle is 60. Let’s add!Write an addition sentence using degree symbols. 60 + 90 = 150.

10 20 MODEL Problem 1 and 2 using the folded paper activity.CLICK TO ADVANCE, after the models , 3 samples of students solving to find an unknown angle using a variable. Students are also introduced to complementary and supplementary angles in the Debrief of Lesson 10.

10 21 CLICK TO ADVANCE THE SLIDE.In Lesson 11, students focus on the use of 360 degrees for measuring and finding unknown angles. In the first example, students relate this work to their previous work with pattern blocks. They remember that four square pattern blocks around a point measures 360 degrees. They remember six triangles around a point measures 360 degrees. Using 2 known angle measurements surrounding a central point, students create an equation, using a variable, to solve for the unknown angle measurement.CLICK TO ADVANCE THE SLIDE.As you can see, the next activity asks students to draw any two intersecting lines and measure every angle. Here they see that opposite angles are equal to each other. The term vertical angles is introduced only in the Debrief. MODEL (using document camera for all).

• Draw 2 intersecting lines where one angle is about 20 degrees, using 2 different colors.

• Measure and label the smallest angle. Label the other 3 angles x, y, and z.

• Discuss at your table. What is the measure of x, without using a protractor? (160°)

• How did you find that? (Using the red line, 20° and an unknown angle must equal 180°.)

• Find the measure of y and z without using a protractor. Discuss at your table.

Allow 5 minutes for participants to complete the Problem Sets for Lessons 10 and 11.

1 22 The final topic, Topic D, consists of 5 lessons where students analyze, classify, and draw 2D figures. Students will use all of their knowledge of lines and angles to do so. This will prepare students for classification of figures using a hierarchy in Grade 5. The End-of-Module Assessment follows the final lesson.

2 23 CLICK TO ADVANCE THE SLIDE.Students study lines of symmetry in various objects, first by folding paper squares, rectangles, pentagons, parallelograms, rhombuses, and circles. They count how many lines there are and reason why related shapes like a square and a rhombus did not have the same amount of lines of symmetry. They discuss how a circle has an infinite amount of lines of symmetry because a circle is a set of points the same distance away from a center point. (When measuring from the center of a square to the points on the side, you get different lengths.) This is a circle’s special attribute.CLICK TO ADVANCE THE SLIDE.Students identify lines of symmetry in familiar figures like these:CLICK TO ADVANCE THE SLIDE, showing shapes.CLICK TO ADVANCE THE SLIDE.Students complete the other half of a figure by reasoning about the other half and its line of symmetry. Drawings done on graph paper.CLICK TO ADVANCE THE SLIDE, showing picture of drawing on graph paper.

5 24 Allow participants 5 minutes to follow the directions on the slide.Assist them by considering sorting triangles by side lengths or angles measurements.Note some triangles may fit into one or more categories.

5 25 A teacher would extract from the students two classifications: Side length and Angle Measure. CLICK TO ADVANCE THE SLIDE.Label those on the Practice Sheet for the 3rd and 4th column.Students explain their findings and which triangles fit into each category. CLICK TO ADVANCE THE SLIDE.Side length answers appear. Participants check their work.CLICK TO ADVANCE THE SLIDE.Teacher also discusses using tick marks to show same size length sides. On the sketch in the 1st column, draw the tick marks.CLICK TO ADVANCE THE SLIDE.Side length answers appear. Participants check their work.CLICK TO ADVANCE THE SLIDE.Teacher also discusses using small squares to mark a right angle, as done in earlier angle measurement work. Record them on Triangles D and E.MODEL: (From Problem 3)Fold Triangle B on its line of symmetry. Same length sides = isosceles triangle Same angle measurement = isosceles triangle, not a 4th grade standard, but likely something students will pick up on. (Measurement of angles in triangles is not G4 material.)Fold Triangle A Same side lengths and angles = equilateral triangle.

Equilateral triangle is a special isosceles triangle.Fold Triangle D’s non-right angles to the right angle The 2 other angles add up to 90 degrees.

8 26 At the end of Lesson 13, students plot 3 points and connect the points with segments. Some create triangles and can be classified. (Others will result in only 2 segments because students plotted 3 collinear points.) This work prompts students to think about a more formalized definition of a triangle in relation to their earlier work with points and lines on a plane.Allow participants about 5 minutes to construct triangles and to answer T/F questions in the Lesson 14 Problem Set. If needed, walk participants through constructing each triangle, or offer participants to come “teach” for practice.

15 27 Use Document Camera to MODEL.Use the Problem Set to guide participants through the Concept Development. (Step-by-step instructions are found in the script.) Construct trapezoids, parallelograms, rectangles, and squares.Make participants aware that there are multiple answers for all drawings, with the exception of the square which will be the same except for the scale to which it is drawn. (This precedes the hierarchy work of grade 5.)

Section: Module Review Time: 4 minutes

[minutes] In this section, you will…Facilitate as participants articulate the key points of this session and clarify as needed.

Materials used include:PPT and Participant materials

Time Slide

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0 28

3 29 Take two minutes to turn and talk with others at your table. During this session, what information was particularly helpful and/or insightful? What new questions do you have?Allow 2 minutes for participants to turn and talk. Bring the group to order and advance to the next slide.

1 30 Let’s review some key points of this session.

Use the following icons in the script to indicate different learning modes.

Video Reflect on a prompt Active learning Turn and talk

Turnkey Materials Provided

● Grade 4 Module 4 PPT

● K-5 Geometric Measurement Progression

Additional Suggested Resources

● A Story of Units Year Long Curriculum Overview