Running head: PEER TEACHING MODULE

Peer Teaching Module:

Chapter 14- Algebraic Thinking: Generalizations, Patterns, and Functions

Emily Bass

St. Thomas University

PEER TEACHING MODULE 2

Textbook Chapter Summary

Chapter 14: Pg. 253-283

Chapter 14 is titled “Algebraic Thinking: Generalizations, Patterns, and Functions”. The

chapter opens by recognizing the stereotype of algebra being challenging, and suggests this may

be a result of how the subject is being taught. It is stressed that algebra is embedded in all areas

of mathematics, and there are three strands of algebraic reasoning that are focused on. The first

strand is the study of structures in the number system, which includes using algebra as a useful

tool for generalizing arithmetic. Activities and examples are given in how students can make

their own generalizations when computing or exploring number relationships. The second strand

of algebraic reasoning is the study of patterns, relations, and functions. When looking at both

repeating and growing patterns, the text explains how students can look for an algebraic

relationship that will tell them what the pattern will be at any point along the way. Different

patterns and suggestions are shown, such as using pattern blocks, dot patterns, and finding

patterns in graphs. The third strand is the process of mathematical modelling, including the

meaningful use of symbols. The text considers that students not having a strong understanding of

symbols may be the reasoning behind them being unsuccessful. In reality, symbols are described

to represent real events and should be seen as useful tools. Activities are given to strengthen

student’s knowledge of symbols, one example being comprehending the equal sign.

Mathematical models are described as involving the use of mathematics and statistics, which

allows us to analyze situations, test ideas, make predictions, and make informed decisions. The

chapter closes by noting the importance of emphasizing the appropriate vocabulary when

teaching and discussing algebra.

PEER TEACHING MODULE 3

Peer Teaching Focus Concept

Study of Patterns and Functions: Repeating and Growing Patterns – Pg. 266-270

The focus of the following lesson plan will be a basic introduction of repeating and

growing patterns. This concept is designed to be taught early in grade two. The purpose of this

lesson is to engage students in numerous activities that promote searching for patterns, finding

the pattern core, successfully predicting elements in a pattern, and being able to extend a pattern.

This lesson is intentionally set at a fast pace with changing activities in order to keep the

attention of young students, while showing them the many different ways a pattern can be

created. This includes sounds, actions, shapes, music, and more. In order for the lesson to fully

resonate, the students are always using physical manipulatives as well as their imaginations to

create and build their own patterns, either with their classmates or on their own.

To understand this concept, students will learn that a pattern core is a “sequence” or

“unit” in a pattern that continues to repeat itself. Students will use manipulatives to create a

pattern core and experiment with extending the pattern.

PEER TEACHING MODULE 4

Lesson Plan: Study of Patterns and Functions: Repeating and Growing Patterns

Subject: Mathematics Grade: 2 Length: 1 Hour Date: January 18, 2016

Stage 1: Pre-Lesson Preparation

Lesson Title: Introduction to Core and Repeating Patterns

Rationale: To introduce and familiarize students of algebraic thinking by encouraging them to search for patterns, to find the pattern core, and to teach the appropriate methods of describing, extending, and generalizing patterns.

New Brunswick Curricular Outcomes: GCO: Patterns and Relations (PR): Use patterns to describe the world and solve problems. SCO: PR1: Demonstrate an understanding of repeating patterns (three to five elements) PR2: Demonstrate an understanding of increasing patterns: (for PR1 and PR2) by • describing • extending • comparing • creating patterns using manipulatives, diagrams, sounds and actions (numbers to 100)

“The foundation of algebraic thinking is investigating patterns and their representations. Contextual, open ended and situation specific problem solving should be an integral part of everyday mathematics instruction, activities and assessment. Students need to recognize and extend many different forms of the same pattern, including those constructed or in their environment. They must identify the core or sequence, and be able to predict an element in repeating and increasing patterns using many strategies within a variety of contexts (e.g., hundred charts, calendars, number lines, tiling patterns). These predictions should be verified by extending the pattern concretely, pictorially and symbolically” (p. 50).

NCTM Pre-K – 2 Expectations: -Understand patterns, relations, and functions. -sort, classify, and order objects by size, number, and other properties;  -recognize, describe, and extend patterns such as sequences of sounds and shapes or simple numeric patterns and translate from one representation to another;  -analyze how both repeating and growing patterns are generated.

Content: Information gathered from Elementary and Middle School Mathematics: Teaching Developmentally by Van de Walle, Karp, Bay-Williams, McGarvey, & Folk, p. 266-270.

Materials: -Whiteboard and markers for instructional use -4 large plastic bags

PEER TEACHING MODULE 5

Items Used for Creating Patterns: -4 different types of candy separated into large plastic bags -Various coloured snap-cubes -Construction paper cut outs of four different shapes -circle, square, triangle, and rectangle -have around 8 of each shape cut out to be prepared, only use two different colours. Half of the shapes will be a different colour than the rest -Patterns Activity Sheet -Smartboard to display the following two videos: -BrainPop Jr. Video, “Patterns” https://jr.brainpop.com/math/geometry/patterns/ -Youtube Video, “When I’m Gone by Sophia and Bellia” https://www.youtube.com/watch?v=ezKrZiMmvTU

Stage 2: Lesson Planning and Implementation

Introduction/Hook: (10-15 minutes)-Give each student one shape (either circle, triangle, square, or rectangle). Try to evenly distribute the shapes among the students. For example, is there are 20 students in your class, every 5 students will have the same shape. -Prompt Question: Ask the students, “Using these shapes, how can you organize yourselves into a pattern? Show me”. Allow the students 1 or 2 minutes to try and figure out how to arrange themselves. -Ask the students “Why don’t we know how to arrange these shapes? How do we know what order they go in?”-Assess how easy or difficult the students are finding the task. To further assist them, ask if anyone knows the first step of building a pattern that would make arranging themselves easier. -After listening to some of the students’ answers, suggest that they need to choose the first 3 or 4 shapes of the pattern. Ask the students to collectively decide which shapes are going to be at the first of the pattern. -Once the students have decided the arrangement of the first 4 shapes, introduce the vocabulary. Tell the students that they have created the core of the pattern. The core is made up of elements. Explain that the core of the pattern is the part that repeats. Once they are able to identify the core, they are able to extend the pattern. -Ask the rest of the students to extend the pattern by placing themselves in the appropriate sequence. -Have the students read aloud their shapes, so they can see as well as hear the pattern (ex: circle, triangle, square, rectangle / circle, triangle, square, rectangle. -Try to challenge the students by asking them to create different patterns. You can make certain requests such as having the same shape appear twice in a row, having a triangle be every third element, or to only use two of the shapes. When the students are trying these different patterns, they may require a different shape. Have extra easily available to exchange for them. Have the students verbally express the pattern each time once they are done. -After having the students try making a couple different patterns, ask them to base their pattern off of colour rather than shape. Ask the students to say their colour once they have arranged themselves in sequence (ex: brown, white, white/ brown, white, white).

PEER TEACHING MODULE 6

Activities: Short Discussion (3 minutes) -Inform the students that we will be doing two more fun activities in order to practice creating patterns, finding the core in patterns, and extending patterns. Remind students that finding the core of the pattern and extending it makes the pattern “repeating”.

Brain Pop Jr. Video: (5 minutes + small discussion) -Have smartboard ready to show the students the above video (URL in materials list). -After the video, ask the following questions to engage their thinking: Ask the students about the vocabulary Annie used, such as finding the “unit, or section” in the pattern. Remind students of Annie saying the section needs to repeat in order to make a pattern. Ask the students what word we used instead of “section”, being the part that repeats itself (they should reply with “core”)-Ask about the other ways the video said we can make a pattern (ex: sound, actions)-Tell the students that we will be making a “different” kind of pattern using sounds.

Making a Pattern with Sounds and Actions (5 minutes) -Clap/stomp out a pattern for everyone to listen. Ask the students to join in. Make sure the sequence is repeated at least two or three times so everyone can hear the pattern (ex: clap, clap, stomp/ clap, clap, stomp)-Half way through the third or fourth time of making the pattern, suddenly stop and ask the students what action comes next. Invite them to describe and explain the pattern that was created.-After seeing the how students answer, ask them how they knew what action was to come next (ex: was something missing from the core? How could you tell what the pattern core was?). Discuss their answers. -Have a student create their own pattern by clapping and stomping. Have the class join in. -Have a discussion about how we know when to start repeating the pattern, how to identify the core, and how would we know how to continue the pattern if it stopped at different times.

Creating Candy Patterns (8-10 minutes)-Place a plastic baggy of each type of candy at each student group (4 baggies per group)-Write down a pattern on the white board using the letters A-D (ex: ABCD, or ABAB)-Before the students are allowed to work with the candy, inform them that they are going to be allowed to eat the candy at the end of class if they stay on task and be good listeners -Ask the students to use the candy to represent the written pattern core- using the different type of candy for each letter. Tell them to choose one type of candy to signify A, another type to signify B, and so on. -Watch as the students build the core pattern with the candy, and then ask them to repeat the pattern with the candy. -Select a couple students to read aloud their pattern by calling out the candy (ex: smartie, gummie, m&m, skittle/ smartie, gummie, m&m, skittle/.. -Write another pattern core on the board (ex: ABCC, ABBC)-Ask the students to recreate the pattern core using their candy and to extend the pattern. -Have the students check each other’s patterns.-Have a discussion about the types of patterns created – how could they repeat the pattern without me writing it on the board?

PEER TEACHING MODULE 7

Making Patterns Using Snap-Cubes (8-10 minutes) -Instruct each student to create their own repeating pattern using snap cubes-Once their pattern is complete, ask them to randomly take out one cube near the middle of their pattern-Ask the students to find a partner beside them. Allow a group of three if there is an odd amount of students. -Have the students guess which cube of their partner’s pattern is missing. Discuss how they know this. See if the students are answering using vocabulary such as “pattern core” and “repeating” -Have the students create a different pattern using their snap-cubes and repeat the activity. -A variety of questions can be asked throughout this activity: How many times did you repeat your core pattern? Can you easily separate the pattern core from the rest of the pattern? Was it easy or difficult to identify your partner’s missing piece?

“Cups” Video: (2 minutes + small discussion) -Ask the students to sit in front of the smartboard -Ask the students if they have heard the popular “Cups” song featured in the movie Pitch Perfect. -Tell the students we are going to watch a video of two young girls performing the song. Indicate that there is a pattern being made in the video. Ask the students to pay close attention and to watch for a pattern, as we are going to discuss it after the video. -After watching the video, ask the students what pattern they could see in the video responses should describe the cups making the pattern- ask the students if any of them can imitate the core pattern with their water bottle or object.

ClosingSmall Discussion (last 5 minutes of class) + Handout Activity -While still sitting, ask the students what they have learned today-Reward the students by handing out the bangs of candy (if they listened) -Hand out an activity sheet for the children to complete at home -Replay the “Cups” Song and allow the children to sing and dance along

Evaluation: -Take anecdotal notes throughout the lesson when students are doing the activities. -A pre-made chart will be made with spaces to check for each student. This sheet will be carried around during the activities in order to get a general idea of the students’ initial understanding of the concept. -Review the activity sheets once they have been handed in. See if the students generally understood the concepts from the sheet. -As this lesson is an introduction, assessment will be formative and is in place to see the students primary comprehension of searching for patterns, finding the pattern core, and extending patterns.

Differentiation: -For students with learning disabilities that have difficulties with reading, most of the instructions for the activities are given orally with this in mind. All of the activities are hands-on and instructed from the teacher, so anyone with reading difficulties will not fall behind or be

PEER TEACHING MODULE 8

discouraged. -For students with autism that require more structure, they can be given a print out of the lesson schedule with the designated times for when each activity is going to end and start. -For students with ADD/ADHD, this lesson is designed to be fast paced and have the students moving. There are many different activities included to have the students constantly moving and changing task. If the students with ADD/ADHD require more movement, the snap-cube activity allows them to make as many patterns as they like. Also with the candy activity, the students can make as many repeating patterns as they want as long, as they have completed the one on the board. Extra “ABCD” patterns can be written down on the board in advance if they wish to follow a set pattern. They can also be provided with the special job of handing out materials such as the candy and snap-cubes, only if they wish. -For students that struggle with motor skills, sitting in their groups of 4 or 5 will allow students to work in partners and participate. Students can verbally express how to build the patterns while another student physically builds them with the snap-cubes or candy. -For students with hearing impairments, a microphone will be used throughout the entire lesson. -For students with visual impairments, the students are often instructed to read their pattern aloud (ex: circle, square, triangle, rectangle). This way, students can be made aware of the patterns if they cannot be seen clearly. For the snap-cube activity, students can also read the colours of their pattern out loud for the visually impaired to hear (ex: blue, blue, green/blue, blue green, blue, ?) and be asked what comes next). -For gifted students, they will be encouraged to make as many patterns as possible within each activity. They will not be limited to creating a pattern with a pattern core of 3-5 elements, but rather be encouraged to create a core with more elements.

Stage 3: Post-Lesson Activities

Lesson Evaluation and Revision -Record what activities in the lesson went well and which ones have the opportunity for improvement in the future. -Finish recording any additional comments and the chart concerning the students’ understanding.

PEER TEACHING MODULE 9

Student NameShapes Activity Candy Activity Snap-Cube Activity

Was able to show understanding when:

Sequencing themselves in class pattern

Equate candy with letters A-D

Create a pattern core

Create a pattern core

Extend a repeating pattern

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Anecdotal Comments:

PEER TEACHING MODULE 10

Student Name Notes1.

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Activity Sheet on Patterns

PEER TEACHING MODULE 11

Colour in the pattern! Can you see which colours belong in the last two blocks?

A= Red B= Blue C=Yellow

A B B C A B B C ? ?

Look at this pattern:

Can you draw the pattern core?

_______________________________________________________________

Make your own pattern core and repeat the pattern using these shapes:

Circle the mistake in this pattern!

References

PEER TEACHING MODULE 12

BrainPOP Jr. (2016). Math: Patterns. Accessed January 5th, 2016, from https://jr.brainpop.com/math/geometry/patterns/

BrainPOP Educators. (2016). Pattern activities for kids. Accessed on January 9th, 2016 from https://educators.brainpop.com/lesson-plan/patterns-activities-for-kids/

Van de Walle, K., Folk, S., Karp. K., Bay-Williams, J., & McGarvey, L.M. (2015). (4th Canadian

ed). Elementary and middle school mathematics: Teaching developmentally. Toronto, ON: Pearson.

Youtube. (2016). Mugglesam: Cups "when I'm gone" from pitch perfect Anna Kendrick | Sophia & Bella (2013). Accessed on January 7th, 2016, from https://www.youtube.com/watch?v=ezKrZiMmvTU

PEER TEACHING MODULE 13

Teacher Resources

Websites

NLVM has several resources, including Attribute Trains, Block Patterns, Colour Patterns, Pattern Blocks, and Space Blocks (http://nlvm.usu.edu) (National Library of Virtual Manipulatives)

For very young children, see PBS Kids (www.pbs.org/teachers/connect/resources/7661/preview)

Inside Mathematics (www.insidemathematics.org – This site contains sample lessons, formative assessment items, rubrics, student work samples, and professional development resources that support various levels and topics such as patterns, functions, and algebraic properties.

Articles

McGarvey, L.M. (2013). Is it a pattern? Teaching Children Mathematics, 19 (9), 564-571.

Books

Blanton, M.L. (2008). Algebra and the elementary classroom. Portsmouth, NH: Heinemann.

Greenes, C.E., & Rubenstein, R. (Eds.). (2008). Algebra and algebraic thinking in school mathematics: NCTM 70th Yearbook. Reston, VA: NCTM.

Books for children that have patterns/repetition:

Anno’s Magic Seeds (Anno, 1994)

Bats on Parade (Appelt and Sweet, 1999)

Old Lady Who Swallowed a Fly (Taback, 1997)

If You Give a Mouse a Cookie (Numeroff, 1985)