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Class Time Estimate:

Core + Formative Activities: 50 min
Longer Introduction + Core + Formative Activities: 65-75 min

Key Ideas for Instructors

Problems are under-determined on purpose. Students must make and state assumptions. There are different correct interpretations of each problem, as well as different correct final answers; the important thing is justification and reasoning.
Emphasize teamwork, process and reasoning over “the final answer” or “the right method”.

Student Prerequisites

None! This lesson can be used on the first day of class; it is in fact designed for that purpose.

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Binomial/Number Triangle

PURPOSE & HIGHLIGHTS

LESSON OVERVIEW

Plan for the lesson

Introduction and Teamwork Preparation (7 min): Introduce the binomial triangle. Make sure students understand the pattern used to construct the triangle and establish some way of discussing locations within the triangle. Assign groups and roles.

Core Activity, Thinking about subsets with puppies (24 min): Students count possible combinations of puppies based on the number of leashes. They use their reasoning to understand and explain the standard recurrence for binomial coefficients. Students share.

Extension Activity, Doing the problem two ways

(34 min): Students explore the relationships between binomial coefficients and the binomial triangle. They use the values from the Core Activity to link and compare the notations from both concepts. Students share.

Formative Activity (5 min): Students discuss the pros and cons of using the binomial triangle or recurrence as opposed to an algebraic formula (which your students may or may not already know). Students create examples and support/challenge arguments. Students share.

Wrap-up (5 min):

Conclude with student survey or further reflection.

This activity is designed to elicit students’ ideas about subsets, binomial coefficients, and counting; prompt alternative ways to think about a problem; and layer binomial coefficient vocabulary and techniques over intuitive experiences.

Highlights: Students see the value of approaching a problem in different ways. Discovering the connection between the visual triangle and the algebraic approach to binomial coefficients creates joy.

Class Time Estimate:

Core + Formative Activities: 50 min
Core + Extension + Formative Act.: 75 min

Key Ideas for Instructors

Some students may do the Core Activity with the algebraic formula; encourage them to think in terms of sets as well.
The goal is to make the connections between visual/diagrammatic and formula-based methods and be open to alternate approaches.

�Student Prerequisites

The activity assumes the students have seen combinations, in particular:� (n choose k) = the ways to pick k unordered objects from n objects. The activity does NOT assume students have used the factorial formula for binomial coefficients: � (n choose k) = n!/(k! (n-k)!)�though it is okay if they have seen or used it.

BINOMIAL RESOURCES

7 of 12

Handshake

PURPOSE & HIGHLIGHTS

This activity is designed to review some concepts of graph theory and the meaning and practice of mathematical conjecture as the students explore properties of graphs.

Highlights: Students enjoy discovering the handshake lemma instead of being told it as a law. They also get to experience math as a living practice that they can discover, discuss, and explain.

LESSON OVERVIEW

Plan for the lesson

Introduction and Teamwork Preparation (5 min): State the lesson goals: Notice interesting properties of graphs, discover new ideas, and develop arguments in teams. Assign groups and roles.

Core Activity 1 (18 min): Student groups make a graph with at least 6 vertices and count degrees. They draw more graphs with variations in vertices and edges. For these graphs, they tabulate various quantities pertaining to graphs. They do whole-class share-out, review for any discrepancies, discuss, and adjust their conclusions accordingly.

Core Activity 2 (25 min): Groups develop a conjecture that relates the sum of degrees and number of edges, validate the conjecture with the quantities in the tables from Activity 1, and attempt to work on a proof. They do whole-class share-out, and work on feedback.

Formative Extension Activity (25 min): Groups make two more new conjectures, determine if a conjecture is “reasonable” and “interesting.” Groups pair up, exchange conjectures and validate them with their own group. Pairs of groups identify the ‘most reasonable and interesting conjecture’ of the total four from the pair of groups.

Wrap-up (2 min):

Conclude with student survey or further reflection.

Class Time Estimate

Core Activities: 50 minutes
Core + Extension (Formative) Activities: 75 minutes

Key Ideas for Instructors

Core Activity 1 is the prep work of creating examples of graphs and tabulating quantities. Core Activity 2 is essential for students to learn what a conjecture is and how to validate a conjecture using known examples.
If time permits, incorporating the formative Extension Activity will give students more practice in building, testing, proving, and validating conjectures.

Student Prerequisites

The lesson assumes students know what a graph is and reminds/introduces the definitions of the terms vertex, edge, loop, adjacent, and degree.

HANDSHAKE RESOURCES

Team Roles Handout

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Ramsey/Dot Game

PURPOSE & HIGHLIGHTS

Students will explore properties of graphs; formulate mathematical conjectures using graphs; and share and compare mathematical conjectures within and across teams.

Highlights: Learning with games is fun! With luck, students may also learn that the 5-dot game can end in a draw and the 6-dot game must end with a win, i.e., the Ramsey number R(3,3)=6.

LESSON OVERVIEW

Student Prerequisites

We do not assume that the students have any previous knowledge beyond high school mathematics. In fact, this activity is an excellent way to introduce the topic of graph theory. It might even be used on the first day of class, if combined with the teamwork-building portions (beginning of the Introduction) of Team-Worthy Tasks.

Plan for the lesson

Introduction and Teamwork Preparation (5 min): Introduce lesson and assign groups and roles.

Game procedure (3 min): Explain the rules of the game: given a set of a few dots, players alternate drawing edges (no loops, no multi-edges). Players use two different colors. Game ends when a single color triangle is created (creator of the triangle loses); or there is a complete graph (tie or draw).

Sample Game (2 min): Give an example of an entire game being played. Either one student against another at the board or one student against the instructor. Highlight the winner and edge counts.

Core Activity 1 (25 min): Within each team, the game is played with 5 dots and 6 dots several times. Teams document observations, organized in a table drawn in public space stating whether there was a draw or a winner for each of these game trials. Whole-class review and discussion

Core Activity 2 (15 min): The teams observe the organized data to build conjectures. Students share.

Formative Activity 3 (20 min): Each team pairs up with another team (or two) to share and discuss interesting takeaways from observations made while they built their conjectures. Whole class discussion

Wrap-up (5-10 min):

Conclude with student survey or further reflection.

Class Time Estimate

Core: 50 min
Core + Formative Activity: 75 min

Key Ideas for Instructors

Students may have difficulties with conjecture formulation, and maybe even with the definition of conjectures; the instructor guide has information to help with this.
Tables with data should be clear and visible/accessible to all students.

RAMSEY RESOURCES

Team Roles Handout

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Logic & Inference

PURPOSE & HIGHLIGHTS

This activity is designed to apply logic to solve a puzzle delivered in “word problem” form; and analyze logical fallacies.

Highlights: Many students greatly enjoy solving the “Who is the Thief?” puzzle. Examining mistakes by (fictional) ChatGPT responses underlines the importance of valid arguments.

LESSON OVERVIEW

Student Prerequisites

The activity assumes the students have seen some variation of rules of inference or valid argument forms (see page 3 of the instructor roadmap for an example of a table). Works well after covering truth tables and the contrapositive.

Plan for the lesson

Introduction and Teamwork Preparation (10 min):

Refresh familiarity with the rules of inference. Assign groups and roles.

Core Activity: Looking for the Thief (25-30 min). With the theme of statements about finding the thief, students work on shared surfaces (e.g., standing at whiteboards) to: (1) Translate statements into symbolic form; (2) Arrange the statements in logically valid sequences; (3) Draw a conclusion. In the final 10 minutes, selected team Reporters share solutions.

Extension Activity: ChatGPT Fallacies (15 min). Students examine computer generated logical arguments as answers to the core activity. They turn the ChatGPT narratives into symbolic forms, identify potential errors, and then answer: Which steps were justified? Which ones were not? Teams share their answers with the class.

Formative Activity (10-15 min): Students discuss the pros and cons of using symbolic logic and/or natural language to construct and validate arguments.

Wrap-up (5 min):

Conclude with student survey or further reflection.

Class Time Estimate

Core + Formative Activities: 50 min
Core + Extension + Formative Activities: 70 to 75 min

Key Ideas for Instructors

Core Activity has some built-in logical blind alleys/dead ends. It’s OK if student answers are not maximally efficient, as finding the Thief efficiently is NOT a goal. Getting there eventually is a goal.

LOGIC RESOURCES

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INDUCTION RESOURCES

Animated GIF, Claims Sheet, Team Roles Handout

Induction

PURPOSE & HIGHLIGHTS

This activity is designed to encourage students to validate and build on the reasoning of others, prompt precision in communication, and explore mathematical induction as a proof technique.

Highlights: The “proof by assembly line” structure of the activity is exciting and fun. Students must understand other students’ work in order to build their own, leading to reflection about their own work.

LESSON OVERVIEW

Student Prerequisites

The activity assumes the students have worked with the Principle of Mathematical Induction at an introductory level and have seen at least three examples of proof by induction.
The activity does not assume students have independent experience creating proofs by induction.

Class Time Estimate

Introduction + Core Activity : 70 min
Introduction + Core Activity + Formative Activity: 85 min

Key Ideas for Instructors

Suggested Timing: Use on day 2 of proof by induction (e.g., as day 2 in a 3-day lesson).

Plan for the lesson

Introduction and Teamwork Preparation (15 min): Review with students the proof of the Claim: 11ⁿ – 6 is divisible by 5 for integers n > 1. Assign groups and roles.

Core Activity (55 min):

After an activity overview, set up for each group to begin work on vertical surfaces (e.g., whiteboards) and give each team a Claim to prove using induction.

Stage 1: Each team starts the proof of their claim in their vertical work space, writing Claim and Basis Step, and stating the Induction Hypothesis.

Stage 2: Teams physically move from their proof-start to relocate and work on the induction step for a neighboring group’s proof-start. Teams review what is on the board for this new (to them) claim and pick up where the neighboring group left off. Their job is to do the induction step and finish the proof.

Stage 3: Teams physically move again to a third Claim. They examine the completed proof and validate it. Students share their validation work.

Presentation & Discussion: Have selected teams present the proofs they validated.

Wrap-up (5 min):

Conclude with student survey or further reflection.

Formative Extension (15 min):

Students create and prove new claims based on the Introduction claim.

Students work in teams on a given but incorrect Linear Search Algorithm to:

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Algorithms & Recursive Functions

LESSON OVERVIEW

Plan for the lesson

Introduction to Tag ID# Search & Find (6 min):

Introduce the search and find tag ID# problem and the Robot and its command set. Illustrate with an Animation.

Teamwork Preparation (4 min): Assign groups and roles.

Linear Activity/Linear Search Algorithm (30 min):

Debug the algorithm, and compare corrections with another team.
Find and share the number of elementary steps required for a robot to perform the search and find task for particular targeted tag positions.
Find the number of steps as a function of the position of the tag (closed and recursive forms).
Formative Assessment: Teams check each other's’ work on time complexity.

Binary Activity/Binary Search Algorithm (30 min):

Teams explore the Binary Search Algorithm using the same mechanisms as the core activity, with X=32 (a power of 2) tags.

Wrap-up (5 min):

Conclude with student survey or further reflection.

PURPOSE & HIGHLIGHTS

This lesson is designed for students to explore linear and binary search algorithms, to learn about recursive and time computing functions, and to make sense of (and debug) an algorithm.

Highlights: Discovery of key mechanisms for efficient algorithm through the engagement on applied and fun activities for basic robots to perform tasks, with team reflection and supporting feedback.

Student Prerequisites

Students need some basic knowledge of arguments occurring in a particular order (as they might typically see when studying logic), recurrence relations, and functions in discrete math. Experience with induction is not necessary, but is helpful. Lesson can be done after introducing the concept of recurrence relations.

Key Ideas for Instructors

Start with introduction to the problem and the Robot and then form teams and assign roles.
For a 50 min class, choose one of the linear or binary activities, which are basically independent; for a 75 min class, do both.
It is critical to start with the Intro Activity. The students become engaged in close examination and building of algorithm via finding errors and counting steps.

LESSON OVERVIEW

RESOURCES

Animation used in-class

Team Roles Handout