# Examining Change

## Unit Summary

How do things grow? How do they change? How can we use mathematics to predict change? What does it mean to grow linearly? In this unit, students explore various growth patterns and make generalizations by reasoning about the structure of patterns. Students use their generalizations to make mathematical conjectures backed by data and reasoning, and they use the Showing Evidence Tool to organize their conjectures. Students conclude the unit by presenting their conjecture, data, and reasoning using a multimedia presentation. Students then self-evaluate their conjectures and presentations for strengths and areas in which they can improve as well as decide which conjecture best represents the problem context.

## Curriculum-Framing Questions

• Essential Question
What happens next?
• Unit Questions
How do we predict growth?
• Content Questions
What is linear growth?
How do you represent and analyze mathematical situations and structures using algebraic symbols?
How do you use mathematical models to represent and understand quantitative relationships?
How does change in one variable relate to change in a second variable?

## Assessment Processes

View how a variety of student-centered assessments are used in the Examining Change Unit Plan. These assessments help students and teachers set goals; monitor student progress; provide feedback; assess thinking, processes, performances, and products; and reflect on learning throughout the learning cycle.

## Instructional Procedures

Introducing the Unit

This unit can be used to introduce students to linear functions. The problems students investigate build on their prior work with patterns and use geometric contexts that allow visual and conceptual access to the concepts of slope and y-intercept.

Begin the unit by writing on the board the Essential Question, What happens next? Ask students to reflect on the question and how it relates to mathematics and patterns. Guide students to think about how they use patterns to help them predict what happens, and ask students to give examples of how they have used patterns to describe, extend, and make generalizations.

Pose the Unit Question, How do we predict growth? Have students propose some ways that things grow. Provide examples of growth patterns. Encourage students to share their thoughts as you record their responses on the board. Try to elicit from students a variety of patterns—pointing out to students the difference between a growing pattern (such as, growing by adding 4 each time) and a repeating pattern (such as, even, odd, even, odd)

Analyzing Change in Various Contexts
Growing By Dots Lesson

• Begin the lesson by posting a sheet of paper with 1 black dot, then a sheet with 5 dots, then one with 9 dots, then one with 13 dots on the blackboard or whiteboard (using tape or magnetic strips).

• Ask students: What is changing, growing? Elicit several student responses (number of dots increases four at a time, squares are increasing, dots are increasing away from the one in the center, and so forth). Ask students to think of increasing patterns in their everyday lives. Then have students think of a specific context in everyday life that the growing dot pattern might represent (such as saving for a purchase, traffic buildup at rush hour, virus increase, and so forth).
• After deciding on a context for the growing dots, ask students to come up with a problem to solve—such as, at 48 minutes, how many viruses are there? Have students work on the problem individually, and then have students discuss the problem with partners or in small groups. Walk around during this time asking probing questions and monitoring the methods students are using to find their solutions, noting if students are approaching the problem using the following two common but distinct methods:
• Choose students to display different methods on the board, encouraging the use of color to distinguish parts of the method. Ask the class how many other students used the same method. Then, ask if each method always works and why. Have students compare and contrast the methods. At this point, students do not need to come up with formalized algebraic rules to represent their generalizations. The important point is to focus on how written and verbal rules correspond to the geometric model of the dots—what is increasing each minute (a square of four dots or a dot in each of the four arms). Understanding how the dots are increasing serves as the basis for conceptualizing the rate of change in the linear function.

Cubes Train Lesson

• Materials needed: cubes for each group
• Begin the lesson by posing the following problem:

How many face units are there when you put 100 cubes together in a train (sharing a face)? How many face units are there when you put any number of cubes together?

Have students work on the problem individually, and then have students discuss the problem with partners or in small groups. Walk around during this time to see what solutions students are using, noting if students are approaching the problem in different ways. Use the observation checklist to note students’ mathematical thinking, processes and behaviors. Decide which students will share their solution methods or reasoning during the whole class discussion (and in what order). Some typical ways that students approach this task are:

• Choose students to display different methods on the board, encouraging the use of color to distinguish parts of their method. Ask the class how many other students used the same method. Then, ask the class if each method always works, and why. Ask the class to compare and contrast the methods. Push for students to translate their words and numeric expressions into more formal algebraic notation, such as, 4(n-2) + 5 + 5, 4n + 2, and 6n- 2(n-1).
• For homework (and assessment of student learning), pose the following problem to students:
 Compare and contrast the dots and cubes patterns—what was the same, what was different? How did you predict growth of these patterns? Create a dots pattern to represent the expression: 4n +  2.

Review homework the next day and modify instruction as necessary.

Practice Using the Tool
Introduce students to the Showing Evidence Tool by exploring the Try the Tool demonstration space together. Discuss the sample argument together, or create a sample project and show students how to add, describe, and rate evidence and claims. Model how student teams peer review each other’s work. Also, show the Comments feature, and agree on how it will be used. Note that the issue of rating sources of evidence, while important in other explorations, is not as important for this unit’s mathematical explorations, because the source is the problem being investigated. In contrast, the mathematical argument students use to back up their conjectures is important, because the conjectures must be based on evidence or data.

Hold a discussion around the idea of a mathematical argument using data as evidence. Ask students to consider the following questions as they work on their mathematical arguments:

• Does my data back up my conjecture or generalization?
• Is my data complete enough to follow a logical mathematical argument?
• Can someone make a counterargument that will weaken my conjecture?
• Will my evidence convince a skeptic?

Distribute the conjecture and evidence checklist to help guide students’ initial work with the tool.

Set Up the Project
Before proceeding with the next activity, click here to set up the Polygon Problem project in your workspace. Provide polygons (using pattern blocks or paper shapes, such as triangles, squares, pentagons, and hexagons) for students to use during the investigation. Then, pose the following problem to students:

 Polygon Investigation If you line up 100 equilateral triangles in a row, what will the perimeter be? Find a rule for any number of triangles. What if you lined up: Squares? Regular pentagons? Regular hexagons? n-sided polygons?

Use the Tool
Have students log into their Showing Evidence team space. Assign students to work with a partner. Instruct students to explore the problem, beginning with the triangles and investigating all of the polygons in order to find a rule for any polygon. Have students create their conjectures in the Showing Evidence workspace. Give parameters regarding the amount of data needed for each conjecture. For example, explain that students must find at least four pieces of evidence or data that supports the conjecture, and the evidence must form a logical mathematical argument. Remind students that evidence must back up their work on the problem and must not be based on the students’ opinions or guesses.

After students complete the initial problem-investigation stage, use the teacher workspace in Showing Evidence to assign each group of partners a peer group to review. Tell peer review groups to read and assess the conjectures of the group assigned to them, and make constructive comments and corrections where needed to the conjectures and evidence. Students can comment on the work by requesting clarification of evidence, pointing out where conjectures are unclear, identifying questionable facts or assumptions, and correcting distortions of opposing points of view. Reviewers can use the conjecture and evidence checklist to guide this work.

Use the Comments feature and notes from the observation checklist to provide feedback, redirect effort, suggest new avenues of study, or ask for clarification about a team's thinking. When the review process is complete, give the students time to make adjustments and corrections based on the comments from the peer review.

Examine the Showing Evidence Activity
The Showing Evidence Tool space below represents two different teams’ investigations in this project. The cases you see are functional. You can double-click on the evidence and comments to read the teams' descriptions.

Prompt:  What is the perimeter of any number of n-sided regular polygons lined up
in a row?

Explore an interactive demo.

Prompt:  What is the perimeter of any number of n-sided regular polygons lined up
in a row?

Explore an interactive demo.

Revisiting Student Conjectures

Ask each group to give a multimedia presentation detailing their conjectures using the project rubric as a guide.  Have students take notes during the presentations. After the presentations, have the class work together to create a list on the board of the different conjectures made by the student groups (students may have overlapping methods). Students can then self-assess their own conjectures and presentations for strengths and areas needing improvement. During their assessments, have students decide which conjecture best represents the problem context.

Reflecting on the Unit

Again, ask students the Essential Question: What happens next?  In small groups have students discuss the question in relation to what they have learned with their investigations and  discussions. Bring the discussion back to the whole group and give students an opportunity to share what they talked about.

## Prerequisite Skills

• Experience in analyzing patterns
• Experience in generalizing from patterns
• Experience in using various mathematical representations

## Differentiated Instruction

Resource Student

• Provide extra time for investigating problems
• Reduce the amount of evidence required
• Make available support from resource specialists
• Modify learning objectives to reduce the level of depth and complexity required in final products

• Give the student the freedom to explore mathematical offshoots of the investigations that intrigue the student and to craft evidence in new and original ways.
• During the discussion on the conjectures, encourage the student to explore in detail the similarities and differences among the conjectures and to craft extensive arguments as to which of the methods best represents the context of the polygon problem.

English Language Learner

• Group the student with bilingual students and work with English language instruction specialists.
• Use multiple representations (especially visual and tactile) to help with mathematical access to problems and methods.

## Credits

A teacher contributed this idea for a classroom project. A team of educators expanded the plan into the example you see here.

Examining Change

#### At a Glance

Subject: Mathematics

Topics: Algebra

Higher-Order Thinking Skills: Reasoning, Evaluation, Generalizing

Key Learnings: Conceptualization and Representation of Linear Functions, Making Conjectures, Using Inductive and Deductive Reasoning, Developing and Evaluating Mathematical Arguments

Time Needed: 8-10 hours

Background: California, United States

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