Unit Plan: Inductive and Deducting Reasoning

Unit Plan

Erika Thompson
Hugh Boyd Secondary, Foundations of Math 11

Inductive and Deductive Reasoning

Textbook: Foundations of Math 11

Pre-planning questions:

(1) Why do we teach this unit to secondary school students? This unit seems very important to teach to secondary students, and I was pleasantly surprised to find out that it is included in the Foundations of Math 11 course (as I never had a unit like this going through high school in Alberta). This unit focuses on some of the most widely applicable skills that we learn through mathematics: logic and reasoning. Teaching students the skills to look at information or data, make conjectures, and then investigate those conjectures to prove or disprove their validity is an important skill not just in math, but in many aspects of life. The skills learned in this unit are the same kinds of skills that help people make sense of politics, economics and world events, as well as the skills that help people make important life decisions.

(2) A mathematics project connected to this unit: Students will work in groups to research a visual (or unorthodox in some other way) proof of a mathematical concept - such as the origami proof of the pythagorean theorem. Students will create a poster or artifact of the proof, and present it to the class. They will explain how the proof works, how it covers all possible cases, and any ways this type of proof (or kind of thinking) could help explain another mathematical concept. They will also reflect on whether the unorthodox proof helped them understand the mathematical concept better than a more standard proof. The aim of this project is to connect the ideas in this unit to mathematics concepts they have learned in the past, and solidify their understanding of both. It also aims to expose the students to different types of proofs then the standard ones they will learn in the unit. Students will be assessed on in two areas. They will be assessed on their mathematical communication skills when they are presenting their project. And they will also be assessed on their knowledge and understanding of the ideas from the unit (such as inductive and deductive reasoning, proof structure, counterexamples, etc).

(3) Assessment and evaluation: Each assessment in the unit will look at slightly different things. There will be weekly assignments as formative assessment, that will be checked for completeness. Students will have class time to complete these, so I will also check in with students orally to see where they’re at. There will also be a short mini-quizzes where students complete 1-2 questions in class. There will be two pieces of summative assessment for the unit. There will be a unit test that focuses more on the student’s mastery of the content in the unit, and a project (described about) that will assess for mathematical communication and research skills. The unit test will include a portion completed in pairs.

Elements of your unit plan:

Lesson	Topic
1*	Introduction
2*	Making Conjectures - Inductive Reasoning
3	Finding Counterexamples to Conjectures
4	Proving Conjectures - Deductive Reasoning
5	Invalid Proofs
6*	Reasoning to Solve Problems
7	Analyzing Puzzles and Games
8	Project Work Period
9	Review & Deductive Reasoning Games (word morphing, 20 Qs)
10	Unit Test

Full lesson plans are included below for lessons marked with a (*).

Lesson Plans

Lesson 1: Introduction

• This lesson includes e (open ended problem solving on vertical surfaces) and a bit of d (arts and mathematics).

Subject: Foundations of Mathematics		Grade: 11	Lesson #: 1	Duration: 75 mins
Lesson Overview		This lesson will be the introduction to the unit. It will introduce the concepts of deductive and inductive reasoning, and give students a chance to practice their reasoning skills by working through puzzles
Class Profile		Block 2-1: 19 students, 3 IEPs, 4 ELLs Block 2-3: 22 students, 2 IEPs, 2 ELLs
Objectives		Introduce the concepts of  inductive and deductive reasoning Explore concrete examples of both inductive and deductive reasoning, so students have a clear sense of what they are, and the difference between them
Curriculum		1.1 Make conjectures by observing patterns and identifying properties, and justify the reasoning. 1.3 Compare, using examples, inductive and deductive reasoning. 1.6 Prove a conjecture, using deductive reasoning (not limited to two column proofs).
Materials & Equipment Needed		White board space for all students, or chart paper, and markers. Four colour theorem colouring sheets and colours.
Lesson Stages		Learning Activities			Time
1	Hook & Introduction	Students will be divided into groups in different corners of the room - each group will be in front of a white board, or have a piece of chart paper. The students will be given the following riddle: Lincoln and Michael get acquainted in a bar. Lincoln offers a drink to Michael and they begin chatting. After a small conversation, Lincoln comes to know that Michael is married and has three children. He asks him, "How old are your children?" Michael thinks and then replies that the product of his children's ages is 72. Lincoln is puzzled and says that the information is not enough. Michael tells him that if he goes outside and looks at the building number posted on the door to the bar, he will get the sum of their ages. Lincoln goes outside and returns. Still puzzled, He tells Michael that the information is still not enough. Michael smiles and tells him that his youngest child loves strawberry ice cream. How old are the children? The students will have about 25 minutes to work through the riddle. Every 5-10 minutes, we will come together as a group to discuss what each group has deduced/where they are stuck. Once a few groups have solved the riddle, we will go through their logical reasoning as a whole class, making sure to break it up into distinct logical steps.			35 mins
2	Presentation	The teacher will explain that in solving the riddle, we were using deductive reasoning. In deductive reasoning, we start with a hypothesis, or conjecture, and make a series of logical steps to reach a conclusion, or prove the conjecture. In this example, the hypothesis, or conjecture, is that Lincoln can figure out the children’s ages from the information given. By working through the logical steps, we have proven that there is only one possible set of ages that satisfy the information given, and we have reached a conclusion. The teacher will then explain that there is another type of reasoning called inductive reasoning. In inductive reasoning we look at concrete information, and try to make a conjecture or hypothesis about the general case. This is the opposite of deductive reasoning.			15 mins
3	Practice & Production	Students will be given different “colouring sheets”, and asked how many colours are needed to colour each section so that no two sections sharing a border are the same colour. Each student will get one sheet, and will be asked to make a conjecture based on their own sheet. To do this they can investigate it in any way they want - colouring it, numbering the sections, etc. Then, students will be asked to discuss their conjecture with the conjectures of 3 of their peers, and make a new conjecture together.			20 mins
4	Closure	The teacher will explain how we were using inductive reasoning to make conjectures, and how our conjectures might have changed when we got more information from our peers. Inductive reasoning can lead to false conjectures, so often in mathematics, and in other disciplines, we use deductive reasoning to prove conjecture we made using inductive reasoning. Student will be asked to keep thinking about their conjectures about how many colours are required, and that we will talk about it more next day.			5 mins
Assessment/Evaluation of Students’ Learning		As this is an introductory lesson, there will be no formal assessment. However, the teacher will ask students questions throughout the lesson to gauge students understanding of inductive and deductive reasoning, and the difference between the two.

Lesson 2: Making Conjectures - Inductive Reasoning

• This lesson includes a (the history of mathematics) and a tiny bit of b (social justice)

Subject: Foundations of Mathematics		Grade: 11	Lesson #: 2	Duration: 75 mins
Lesson Overview		This lesson will focus on inductive reasoning and making conjectures. It will draw on famous historical conjectures in mathematics - proven or unproven, and underscore the important role inductive reasoning plays in mathematics. The lesson will also touch on other areas of life where we make or prove conjectures.
Class Profile		Block 2-1: 19 students, 3 IEPs, 4 ELLs Block 2-3: 22 students, 2 IEPs, 2 ELLs
Objectives		Solidify understanding of inductive reasoning and making conjectures Relate inductive reasoning to the history of mathematics as well as real life situations
Curriculum		1.1 Make conjectures by observing patterns and identifying properties, and justify the reasoning. 1.2 Explain why inductive reasoning may lead to a false conjecture.
Materials & Equipment Needed		Powerpoint about famous historical conjectures/mathematicians http://www.tylervigen.com/spurious-correlations
Lesson Stages		Learning Activities			Time
1	Hook & Introduction	The teacher will ask the students for their final conjectures about colouring the maps from the last day. They will write the different answers on the board. Assuming different groups came up with different conjectures, the teacher will lead a class discussion about why each group made the conjecture that they did.			10 mins
2	Presentation	The teacher will explain that this activity is actually based on the 4-colour map theorem, which was an unsolved conjecture for many many years, until it was solved by a computer. From here, the teacher will go into a presentation about other famous conjectures in the history of mathematics, solved and unsolved. They will start with the pythagorean theorem, which the students should be familiar with, but also touch on Fermat’s last theorem, Legendre’s conjecture, Goldbach’s conjecture, and the twin prime conjecture. The teacher will explain how the heart of the discipline of mathematics is making conjectures based on evidence, and then working to prove those conjectures. Often conjectures rely on other proven conjectures. Science also relies on making conjectures. A hypothesis in the scientific method is a conjecture based on the knowledge we have at the beginning of an experiment, and the scientific method works to prove it. Inductive reasoning does not prove that something is true. In fact, often conjectures are proven to be untrue, both in math, and other areas, as shown by this website: http://www.tylervigen.com/spurious-correlations You could use inductive reasoning to make the conjectures on this page, based on the correlation graphs, but those would clearly be invalid conclusions.			35 mins
3	Practice & Production	Students will have 10 minutes to think about and write down 3 times when they have used inductive reasoning to make a hypothesis or conjecture, and at least one occasion when that hypothesis has been invalid. These examples can be from school, but students will be encouraged to think about when they use inductive reasoning in their daily life. Then they will get into groups and discuss the examples that they came up with. Finally the class will come back together to share each group's favourite examples. The teacher will open up a discussion about how inductive reasoning is the type of reasoning that leads to harmful stereotypes - people make false conclusions about groups of people based on limited information (ie. representations in media, gossip they have heard, etc).			25 mins
4	Closure	The teacher will recap what was talked about (inductive reasoning as a key part of mathematics, other disciplines, and daily life), and explain that next class we will talk about some systematic ways to check the validity of conjectures.			5 mins
Assessment/Evaluation of Students’ Learning		During the group discussion, the teacher will walk around the room and listen to conversations as formative assessment - checking that students have a clear understanding of using inductive reasoning to make conjectures, and times when a conjecture was invalid.

Lesson 3 - Reasoning to Solve Problems

• This lesson includes f (telling only what is arbitrary)

Subject: Foundations of Mathematics		Grade: 11	Lesson #: 6	Duration: 75 mins
Lesson Overview		This lesson will be a problem solving circuit based on proving or disproving conjectures. There will be four tables set up in the room, each with a different conjecture set up on it. Students will have to work together to disprove or prove these conjectures using any deductive reasoning approach they want. The teacher will be there for support, but will not give many hints at the solutions - the aim of this lesson is for students to start to feel comfortable working through difficult logical problems, even when they aren’t sure how to approach them.
Class Profile		Block 2-1: 19 students, 3 IEPs, 4 ELLs Block 2-3: 22 students, 2 IEPs, 2 ELLs
Objectives		Practice problem solving, and using deductive reasoning, Encourage students to take risks in problem-solving, and to try different approaches even when you have no idea how to solve a problem
Curriculum		1.6 Prove a conjecture, using deductive reasoning (not limited to two column proofs). 1.8 Identify errors in a given proof; e.g., a proof that ends with 2 = 1.9 Solve a contextual problem involving inductive or deductive reasoning.
Materials & Equipment Needed		Paper (with triangles drawn on), scissors, activity cards for each of the problem solving activities.
Lesson Stages		Learning Activities			Time
1	Hook & Introduction	The teacher will play this vihart video: https://www.youtube.com/watch?v=D2xYjiL8yyE (that demonstrates the problems with a proof that pi =4) as a recap of last day, and to introduce today’s activity.			5 mins
2	Presentation	The teacher will explain that today we’ll be doing a problem solving rotation about deductive and inductive reasoning. Students will get ~20 minutes at each station to try to prove/disprove the conjecture. The teacher will explain that this kind of mathematics requires you to sit in the discomfort of not knowing the answer, and try different things even if you’re not sure that they’ll work. Student’s will have to submit their work for one of the problems, but they will not be marked on whether or not they managed to prove/disprove the conjectures, but rather on their deductive reasoning approaches.			5 mins
3	Practice & Production	There are 3 different problem solving stations: Station 1: Prove or disprove that it is possible to cut a triangle out of the paper with a single cut. This station will have sheets of paper with a triangle drawn on them, with the center marked, and scissors. Station 2: Prove or disprove that 0.9999… = 1 Station 3: This image shows that 1 + 3 + 5 + 7 + 9 = 5^2 What do you think 1 + 3 + 5 + … + 99 will be? Make a conjecture for the sum of 1 + 3 + 5 + … + n, and try to prove your conjecture. While students will be encouraged to rotate every 20 minutes, if they are engaged in solving a problem, they will be able to stay at that station for as long as they want.			60 mins
4	Closure	The students will fill out a self reflection about how they felt solving these problems. Did they find it exciting? Satisfying? Stressful? How is their experience solving problems like this different from their experience in a more standard math class.			5 mins
Assessment/Evaluation of Students’ Learning		The teacher will collect the students’ solutions (or possible solutions) to one problem, and they will be assessed formatively based on their approach to the problem. They do not have to solve the problem, but rather the teacher is looking to see that each step they take is logically valid, or they identify why the step they took wasn’t valid.