Download Weekday Lesson #2 (NYC) or #4 (SF): Exploring Number Patterns

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)
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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.
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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.
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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
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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”
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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.
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