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W E E K DA Y M A T H N A O N YC L E S S O N 2 / S F L E S S O N 4 Weekday Lesson #2 (NYC) or #4 (SF): Exploring Number Patterns on a Calendar How can we use algebra to formalize numerical observations? SITUATING THE LESSON: LAST WEEK (Saturday Academy) Lesson 03: Factoring and Divisibility Students learned how to factor sums and differences through identifying a common factor, and explored how factoring and algebraic representation can be used to create mathematical argument (proof) about divisibility properties of even and odd integers. TODAY (Weekday Lesson 2) See Summary. NEXT WEEK (Saturday Academy) Lesson 04: Expressions, Equations, and Identities Students will compare two common ways of interpreting the variable, which is the main building block of algebra: 1. as an unknown but fixed value (as in an equation); and 2. as a value that can vary without changing underlying properties (as in an expression or identity). Summary: In this lesson, students will engage in the paradigmatic mathematical activities of pattern noticing and generalization. They will look for and identify numerical patterns on a Calendar (pattern noticing). Then, they will use algebraic representation to show that the pattern they noticed is always true (generalizing). This lesson is student-centered and the task is open-ended. In other words, different students may come up with different patterns and different ways of showing algebraically that their pattern holds on any calendar. This means that the role of the instructor is to give students the space to explore, help students understand what it means to justify, and manage group discussions when they present their work. Preparation Note: Before students arrive, copy the March 2016 Calendar on the board so that you can refer to it during the Mini-Lesson. Preparation Before Class: Work through Classwork problems in advance. Read through and annotate the Lesson Plan in a way that will be useful to you. There is no Instructor Answer Key this week, see Lesson Plan. Materials: Weekday Lesson #2: Classwork (1 per student and instructor) Weekday Lesson #2: Lesson Plan (instructor only) © Sponsors for Educational Opportunity 1 1. Mini Lesson (15 min) A) REVIEW CONCEPTS FROM SATURDAY ACADEMY Ask students what they learned in Saturday Academy. If they don’t bring it up, ask them what they learned about even and odd integers and review (quickly) on the board with student input: o If a number is “even,” it is “divisible by 2,” which is the same as “having a factor of 2.” Algebraically, all even integers can be written in the form 2𝑿, where 𝑿 is any integer. For example, 12 is even because 12 = 2(𝟔). In this case 𝑿 = 𝟔. Use different color markers strategically, for example by writing 𝑿 and 𝟔 in the same color. o Algebraically, all odd integers can be written in the form 2𝑋 + 1, where 𝑋 is any integer. For example, 15 is odd because 15 = 2(7) + 1. In this case 𝑋 = 7. o Briefly, ask students for examples of even and odd integers, and how they would rewrite these integers in the appropriate form. Remind students that with just this small bit of information, they can prove statements like The sum of any two odd integers is always even Ask students: is it true? Have them briefly give a numerical example to make sure everyone understands what the statement is saying. This is a BIG claim. Ask students “if you tried to calculate all of the possible combinations of sums of odd integers, how long would it take you?” Forever! There are an infinite number of odd numbers, they do not end. And yet, algebra gives us a tool to prove without any doubt that no matter which two odd numbers you pick, their sum is always even. Briefly show this on the board: Let 2𝐴 + 1 and 2𝐵 + 1 be two odd integers. Let’s add them: (2𝐴 + 1) + (2𝐵 + 1) = 2𝐴 + 2𝐵 + 2 Now, we can factor out a 2: 2𝐴 + 2𝐵 + 2 = 2(𝑨 + 𝑩 + 𝟏) This expression is of the form 2𝑿 (in this case 𝑿 = 𝑨 + 𝑩 + 𝟏, but that part doesn’t really matter). So the sum must be even. With two little algebraic manipulations and in like two minutes, we’ve just done something that couldn’t be done by a million people in a million years if they were trying to use brute force to calculate the sums of all possible combinations of odd numbers. Ah, the power of math! Hype this up with your students. 2 © Sponsors for Educational Opportunity THINK/PAIR/SHARE: Ask students “how could you use algebra to represent numbers that are divisible by 3? For example, numbers like 6, 9, 12, 30, etc?” Give them a minute, then share out. Answer: write it in the form 3𝑿. For example, 6 = 3(𝟐); 9 = 3(𝟑); 12 = 3(𝟒); 30 = 3(𝟏𝟎), and so on. What about divisible by 4 ? Yup, all you need to do is write it in the form 4𝑋. B) LAUNCH THE TASK Hand out the Classwork and explain the task. Students have this month’s calendar, and their task is to do the following: 1. Write down a pattern that they notice with these numbers 2. Use algebra to represent this pattern. 3. Create an argument to show that your pattern is always true in this calendar. 4. Repeat! If students do not understand this task or what to do, give them the following example: 1. I noticed the pattern that if you add any two numbers on this calendar vertically, the result is always odd. For example, 2 + 9 = 11 and 11 is odd. Also, 14 + 21 = 35 and 35 is odd. 2. I can use algebra to represent this problem. Let’s say that I start at some date 𝑥 (see Calendar on the right). Every week is 7 days long, so it must be true that the number directly below it can be written as 𝑥 + 7. © Sponsors for Educational Opportunity 3 3. I can create an argument to show that if I add these two numbers, the result is always odd: (𝑥) + (𝑥 + 7) = 2𝑥 + 7 We want to show that this expression is actually in the form 2𝑨 + 1, since that’s what an odd number looks like. This is where smart factoring comes in: 2𝑥 + 7 = (2𝑥 + 6) + 1 = 2(𝒙 + 𝟑) + 1 Yup, that’s an odd number! TEACHER’S NOTE: There are many different patterns in this Calendar, as well as different arguments. Here are two more to get a sense of the possible landscape of this task: The sum of any three numbers in a row is divisible by 3 (can be written in the form 3𝑨) Argument: (𝑥) + (𝑥 + 1) + (𝑥 + 2) = 3𝑥 + 3 = 3(𝒙 + 𝟏) The difference between two diagonal numbers (from bottom left to top right) is 6: Argument: (𝑥) − (𝑥 − 7 + 1) = (𝑥) − (𝑥 − 6) = 6 4 © Sponsors for Educational Opportunity The important thing is for students to be able to track where their algebra came from. For example, in this last pattern they should be able to say “if I start with 𝑥 and move up, I end up at 𝑥 − 7. Then if I move to the right, that is +1. So 𝑥 − 7 + 1 = 𝑥 − 6.” 2. Paired Work and Presentations/Discussion (30 min) Allow the students to work in pairs or small groups. While students are working, circulate to provide hints, encouragement, and praise for strong effort. As much as possible, ask students questions to help lead them to the hit that you want to give them. Try not to give too much away, and encourage students to view struggle and false starts as an entirely normal and valuable part of doing math. When students have a nice pattern and explanation, have them write it up on the board while other students continue working After about 20 minutes, invite students to present their problems on the board to the entire class. Encourage students to ask clarifying questions (just like they do during Saturday Academy presentations), and clear up any misconceptions. 3. Final Activity/Wrap-Up (10 min) Direct students to consider the Final Activity. This activity requires students to assess other hypothetical students’ responses. Have students read each of the three student responses out loud, give them one minute to think about which one(s) make the most sense, then lead a group discussion. Some points to hit on during this (student-centered) discussion: o Wyatt’s reasoning is entirely correct even though he did not use algebra. Many students will say “Wyatt’s answer is wrong because he didn’t use symbols.” Stress to students that making a mathematical does not necessarily have to involve symbols, it just has to make sense! Often it is more concise to use symbols, but at the end of the day logic and reasoning are the foundation of making an argument in math. o Xiao gave two examples, which is not an argument that it is true for all possible examples. There is no mathematical thinking involved here, it’s just (empirical) observation. Using example to see and explore a pattern is great! But once you see the pattern, you need to use logic (or algebra) to show that the pattern is always true. o Gabriel’s answer is nonsense! Many students will say “Gabriel’s answer is the best because he used algebra.” When they do, ask them to explain how his “calculations” © Sponsors for Educational Opportunity 5 relate to the square in the calendar (not at all). Again, reiterate that just because symbols are used does not mean that he argument is correct. At the end of class, take a few moments to summarize what was covered today and point the way forward. Next week during Saturday Academy, students will delve deep into expressions, equations, and identities. 6 © Sponsors for Educational Opportunity