Park City Mathematics Institute
High School

[CalcReady] [Geometry] [ModelingAlg] [Technology] [TransformationGeo] [StatisticsH] [StatisticsL]
[AssessingMPs] [Algebra] [ModelingGeo] [Parents] [GroupTasks] [LowThreshold] [Quadratics]

Grade Level: 8-12
Subject: Algebra (or other HS courses focused on functions)
Topic: Rate of Change
Authors: Nathan Goza, Gabriel Rosenberg, Liem Tran
This PD is motivated by the fact that Calculus students often struggle with the concept of derivatives because they lack understanding of rates of change. Student exposure to this idea starts as early as 6th grade, and their conceptual understanding should continue to grow throughout high school according to the CCSS. In this PD participants will attempt to solve a Calculus problem using their knowledge of rates of change and slope. Participants will then explore an Algebra lesson that helps develop conceptual understanding about rates while looking at specific standards that connect to the lesson. Participants will receive a guide to the progression of rate from 6th grade to Calculus. Finally they will discuss the connections between prior, present, and future learning goals as they think about how they teach the concept of rates of change. Participants will develop their ability to plan vertically for all the standards they teach.
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Grade Level: High School
Subject: Geometry
Topic: Recognizing mathematical practice in student work
Authors: Marla Mattenson, Peter Sell, David Newhouse
Recognizing mathematical practice in student work is the primary focus of this professional development. Teachers will engage in a geometry task from MARS ( individually, in pairs, and small groups to solve the task in multiple ways. Teachers will experience the task from a student perspective as well as through the lens of pedagogy. After teachers work on the task and share out, they complete a reflection slip. Next, teachers analyze four different student solutions for use of the 8 Mathematical Practices with emphasis on what feedback they would give a student to guide them in developing the practices further. Facilitator then leads a discussion to have teachers think deeper about the mathematical practices. Finally, the PD ends with possible mathematical extensions to the original math task as well as a recap of math practices used during the PD and teacher insights.
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Grade Level: HS Mathematics
Subject: Algebra I
Topic: Modeling Algebra
Authors: Titin Suryati Sukmadewi, Andrew Richardson, Jake Leibold
This workshop is designed to identify what mathematical modeling is and how it is different from problem solving. The workshop will first ask you to reflect on any prior knowledge you have on mathematical modeling and what makes a good modeling task. You will use your ideas on what modeling is as you work through an Algebra Task titled, "Cell Division", that is aligned to the CCSS Standards MP4 and F-LE. We will then look at examples of what is and what is not mathematical modeling and highlight how modeling is different from problem solving.
download zipped folder [username/password required]
Grade Level: 9 - 12
Subject: HS Mathematics
Topic: Technology in the Classroom
Authors: Debbie Seidell and Will Stafford
Technology can be used to enrich classroom learning by giving students a chance to explore and experience mathematical situations. In this workshop, participants will play a classroom-style game involving graphs of polynomial functions, using the free online graphing program Desmos. They will then reflect on how the experience allowed them to experiment and try out ideas that develop an understanding of how parameters affect the shape of a graph. For students, this sort of concrete experience can lead to deeper understanding of abstract concepts as the unit progresses. Next, participants will graph a function by hand with some help from data points that they gather on the geometry program Geogebra. They then reflect on how students might draw connections between the geometry of a circle and the shape of a sine wave through this lesson. We discuss CCSS Mathematical Practice 1, making sense of problems, and Mathematical Practice 5, using technology to explore and deepen student understanding of concepts. Our examples come from Algebra II and Precalculus, but the ideas can be used in any subject of high school math.
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Grade Level: High School
Subject: Geometry course
Topic: An introduction to Transformational Geometry and how to show congruence using Transformations, both by hand and with software.
Authors: Clint Chan, Wendy Jennings, and Teri Hulbert
This is a 60-90 minute PD intended to demonstrate the differences between traditional geometric approach to congruence and CCSS geometric approach using transformations. We explore the CCSSM for High School Geometry, namely HSG-CO. A.2 and HSG-CO.B.8 which deal with rigid motions of the plane and geometric congruence. The participants begin by using technology to perform transformations, and map one congruent shape to another using rigid motions. The participants will then use rigid motions to demonstrate that any mapping of one congruent plane figure to another can be accomplished by using a sequence of rigid motions. The next activity is an exploration of congruence with specific respect to the SAS theorem of triangle congruence done by hand. For workshops longer than 60 minutes, there are two additional congruence activities involving SSS and ASA and transformations. The three triangle congruence activities are fully explained in the Solution Guide.
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Grade Level: Upper High School
Subject: Algebra2/IntegratedMath3
Topic: Using Probability to Make Informed Decisions
Authors: Brian Shay and Eleanor Terry
This session will have participants simulate activities, calculate probability, and make decisions based on these probabilities. Participants will experience two simulations where they will gain firsthand experience in sampling data, calculating probabilities, and discussing if their sample is consistent with given assumptions. Participants will also be solving problems using the Empirical Rule for Normal Distributions, as well as learning how to use the margin of error to approximate the mean of a normally distributed population from a random sample of that population.
download zipped folder [username/password required]
Grade Level: 9-12
Subject: HS Mathematics, Geometry, and Common Core Math 2
Topic: Exploring Conditional Probability
Authors: Vicki Lyons, Phylicia Lockhart, Paul Winston
This workshop is organized to help teachers be more knowledgeable of what organizing and interpreting data could look like in a classroom setting under the Common Core standards. Two-way tables are emphasized in the CCSSM. This workshop focuses on the construction and interpretation of two-way tables. One activity will allow teachers to make connections between various representations of data. Teachers can expect to use their representations to answer interesting probability questions. Marginal, joint, and conditional probabilities will be defined. Another activity develops the concept of independence by first looking at dependent relationships among data. The workshop focuses on the following Common Core State Standards: S-CP.4, S-CP.5, S-CP.6, and S-MD.7
download zipped folder [username/password required]
Grade Level: High School
Subject: HS Mathematics
Topic: Identifying Evidence of Mathematical Practices in Student Work
Authors: Katie Waddle, Miriam Cukier, Kieran Flahive, Jason Lang
The inclusion of the eight Mathematical Practices in the Common Core standards provides teachers with a mandate to teach and assess problem solving strategies. In this workshop, we will focus on three of the Mathematical Practices. Teachers will become "experts" on these Practices by doing a close reading of their descriptions. Through a series of problems, participants will learn to look for evidence of these Practices in 1) their own problem solving 2) written student work, and 3) student discussion. By the end of the workshop, teachers will have identified characteristics of tasks that lead students to use the Mathematical Practices.
download zipped folder [username/password required]
Grade Level: 8th and 9th grade Algebra
Subject: Algebra 1
Topic: Systems of Equations
Authors: Anne Springfield & Jonathan LaManna
Building Intuition through Multiple Representations with Systems of Equations
This workshop is designed to present participants with a method to approach systems of equations intuitively with students and provide multiple access points to the topic. It focuses heavily on using visual models to reinforce mathematical concepts such as equivalence, and to provide opportunities for students to contextualize and decontextualize the mathematics. Participants will also learn to modify existing tasks to promote discussion, critical thinking and justification of methods in the classroom.
download zipped folder [username/password required]
Grade Level: HS Mathematics
Subject: Geometry
Topic: Modeling Geometry
Authors: Elmer Calvelo, Andrew Richardson, Caitlin McCaffrey
This workshop is designed to identify what mathematical modeling is and how it is different from problem solving. The workshop will first ask you to reflect on any prior knowledge you have on mathematical modeling and what makes a good modeling task. You will use your ideas on what modeling is as you work through a Geometry Task titled, "One for the Road", that is alligned to the first CCSS Modeling with Geometry Standard. We will then look at examples of what is and what is not mathematical modeling and highlight how modeling is different from problem solving.
download zipped folder [username/password required]
Grade Level: All
Subject: Mathematics
Topic: Parents Overview of the Common Core
Authors: Moe Burkhart, Heather Brown, Marty Schnepp
The twofold purpose of this webinar is to give parents an overview of the Common Core State Standards for Mathematics (CCSSM), and help parents develop positive communication strategies with their child and teacher. The overview focuses on the content and practice components of the CCSSM, how they evolved and why they are important. Parents solve a problem presented in context and later abstractly. They will have access to a list of strategies and questions they can use at home with their child, and another list they can use to help them communicate successfully and positively with their child's teacher, including a video demonstration.
download zipped folder [username/password required]
Grade Level: 9-12
Subject: Group Task
Topic: : Implementing Group Tasks in the Classroom
Authors: David Herron & Esther Song
Group tasks provide a perfect environment for students to practice the Mathematical Practices outlined in the Common Core State Standards. Participants at this professional development will explore their own answers to the following questions:
  • What makes a task well-suited for group work?
  • How can I implement a group task effectively?
During this ninety-minute session, there will be two major components. The first half will focus on working on a task in groups to understand what group work is like from a student perspective. Participants will then decide on the components of a "group worthy" task. The second half of the session will focus on implementation strategies for effective group tasks. Participants will analyze video footage of group tasks, critique the efficacy of different strategies, share their own best practices, and sift through resources provided by the facilitators to find those that are most helpful for their own school context.
download zipped folder [username/password required]
Grade Level: 9-12 (High School, potentially upper Middle School as well)
Subject: HS Mathematics
Topic: Identifying and Using Low Threshold and High Ceiling Tasks
Authors: Constance Bowen, Usha Narra, Donna Young
How can teachers engage students in learning mathematics so that they all achieve the high expectations put forth by the Common Core State Standards of Mathematics? One way would be to use low threshold/high ceiling tasks to entice students to participate in learning and to strive to learn more. This presentation gives definitions of low threshold and high ceiling tasks and allows teachers to experience tasks meant to have many access points in order to pave the way to higher mathematics. There are opportunities for teachers to reflect personally and to discuss the uses of these tasks in their classroom. Through transparency about how the tasks presented were adapted and through practice during the workshop, it is hoped that teachers will leave the workshop prepared to adapt tasks on their own.
download zipped folder [username/password required]
Grade Level: Middle or High School
Subject: Algebra 1
Topic: Introducing Quadratics
Authors: Sarah Burns and Jennifer Outzs
Introducing Quadratics:
This activity is designed to assist departments or other groups of math teachers to understand how an emphasis on the standards for mathematical practice MP4 (model with mathematics) will enhance the way students learn about quadratic functions (specifically standard F.IF.4). We will investigate the value of using an "anchor" problem to ground student understanding of quadratics in a real-world scenario throughout the development of the unit. We include a PowerPoint and facilitator guide for a 1 1/2-hour professional development session. The presentation focuses on how to approach the content in such a way that promotes meaning and context. We conclude by looking at student and teacher mindsets in the common core.
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© 2001 - 2018 Park City Mathematics Institute
IAS/Park City Mathematics Institute is an outreach program of the Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540
Send questions or comments to: Suzanne Alejandre and Jim King

With program support provided by Math for America

This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.