To begin things, I had every student read a quote I have posted outside my room:
"There is no limit to what [you] can
learn except for the limits [you] create inside [your] own head."
Jan
Guillou
Birth
of the Kingdom
“We are what we repeatedly do. Excellence, therefore,
is not an act but a habit."
Aristotle
Greek philosopher
and
“In any moment of decision the best thing you can do
is the right thing, the next best thing is the wrong thing, and the worst thing
you can do is nothing."
Theodore
Roosevelt
26th president of the United States
These are the only quotes I have posted in my room as they represent my personal philosophy and approach to life.
I then placed the students in a new seating arrangement and told them I was deeply disturbed to hear they had used "magic" and "faith" as reasons on Friday as to why mathematical relationships worked. I explained to them the reason for the quotes and how it had served me well in my career in high tech, enabling me to rise to high-ranking positions running multi-million dollar operations. I then gave the following "quiz" on intellectual laziness:
Intellectual laziness is defined as not utilizing your intellect to its utmost potential to achieve success. This includes creating mental barriers or excuses, inattention to the situation at hand, lack of perseverance and the inclination to stop rather than try different approaches to solve a perplexing problem.
1. Explain three ways you may have exhibited intellectual laziness in this class during the past month.
2. Describe three things you will commit to for the remainder of this semester to overcome intellectual laziness.
Students appeared to take the lesson seriously and seemed reflective about the issue. I will read through their responses and score their responses based upon the honesty of their reflection. I will also make copies so that I can refer back to their commitments if I see habits slipping.
With that, I told students I never wanted to see magic or faith used as a reason again in class and we proceeded on to the lesson.
I revisited the Chinese Remainder Theorem and related it back to the three problems that we worked through. For the last of the three problems, I wrote on the board that we needed to find an unknown quantity, x, such that:
- x/3 has r2
- x/5 has r3
- x/7 has r2
I then introduced the class to congruence notation, showing that two integers a and b are congruent modulo m is written as a ≡ b (mod m). I provided the definition for this notation by stating that a and b are congruent mod m if a and b leave the same remainder when divided by m. I also provided the example that 5 ≡ 17 (mod 3).
This usually takes a little time for students to absorb. I allow them time to consider the statements and example and wait to see if there are any questions. Students need time to understand why 5 and 17 are congruent mod 3. They need to dissect the meaning to see that the remainders are the same under division by 3. I try to reinforce that it is the remainder that is the defining value here and what we are concerned with. It turns out that remainders are an important mathematical concept and not some throw-away value.
I ask students to consider other values under which 5 and 17 will be congruent. I have students restrict their search to values between 2 and 10, non-inclusive. Students find that 5 and 17 are also congruent mod 4 and mod 6.
I revisit the third congruence problem we worked on, the Chinese remainder problem and write the system of linear congruences using congruence statements.
This usually takes a little time for students to absorb. I allow them time to consider the statements and example and wait to see if there are any questions. Students need time to understand why 5 and 17 are congruent mod 3. They need to dissect the meaning to see that the remainders are the same under division by 3. I try to reinforce that it is the remainder that is the defining value here and what we are concerned with. It turns out that remainders are an important mathematical concept and not some throw-away value.
I ask students to consider other values under which 5 and 17 will be congruent. I have students restrict their search to values between 2 and 10, non-inclusive. Students find that 5 and 17 are also congruent mod 4 and mod 6.
I revisit the third congruence problem we worked on, the Chinese remainder problem and write the system of linear congruences using congruence statements.
- x ≡ 2 (mod 3)
- x ≡ 3 (mod 5)
- x ≡ 2 (mod 7)
This is, symbolically, a much cleaner way to represent the problem than what was written before. Again, it takes time for students to absorb this notation. I wait and ask for questions. I then ask if someone can explain how the first statement represents the problem's first condition.
Students struggled with this, which is okay as it is the first time they have seen this notation. I pointed out that we want a remainder of 2 when dividing x by 3. I also point out that 2 divided by 3 leaves a remainder of 2. Since both x and 2 leave the same remainder they are congruent mod 3. After some more thought and question, students felt comfortable enough with this first look that I could move on.
At this point, one student asked if there was a connection between the fact that 3 x 7 = 21 and that both of these congruences contained a remainder of 2. We will be exploring systems of congruences and taking a look at a more formal way to solve these systems shortly. For now I told the student that they do have a connection that we'll be looking at soon.
I presented a more formal representation of the Chinese Remainder Theorem. The point I emphasized was that in order to obtain an unique solution the mi are all relatively prime. As we proceed ahead I will emphasize that the unique solution is modulo the product of the mi.
I asked students to consider where they encounter congruences in their everyday life. I gave them a few minutes and we shared out their thoughts. Several of their ideas were examples of unit rates, such as cost per item. One person thought about shifting in a car and that the interplay of clutch and gas pedal formed a congruence. This gets closer to the mark, as you repeat the process of pedal movement while shifting.
I asked students about a clock. What happens in 12 hours? The clock repeats itself. In 7 days what happens to the day of the week? The day repeats itself. What happens in 12 months? We are in the month of April again. The same can be said for days in a year (excluding leap year), minutes in an hour, or seconds in a minute. These are examples of congruences that we deal with every day.
We looked at a simple example of this using time. Starting on a Monday at noon, the question is what time is it after 43 hours? Students determined that the time was 7 a.m. in the morning on Wednesday. I had students share how they approached this problem. Almost everyone determined that 48 hours was noon on Wednesday. They then subtracted 5 hours to back up to the required 43 hours. Subtracting 5 hours from noon resulted in the equivalent time of 7 a.m.
We'll look at this in terms of a congruence statement next class.
As we only had a minute left in class I had students write down questions they have about congruences and the Chinese Remainder Theorem. I'll start next class by having the class discuss their questions.
As a snow day was just announced for tomorrow, our discussion of congruences will have to wait a couple of days.
Visit the class summary for a student's perspective and to view the lesson slides.
I presented a more formal representation of the Chinese Remainder Theorem. The point I emphasized was that in order to obtain an unique solution the mi are all relatively prime. As we proceed ahead I will emphasize that the unique solution is modulo the product of the mi.
I asked students to consider where they encounter congruences in their everyday life. I gave them a few minutes and we shared out their thoughts. Several of their ideas were examples of unit rates, such as cost per item. One person thought about shifting in a car and that the interplay of clutch and gas pedal formed a congruence. This gets closer to the mark, as you repeat the process of pedal movement while shifting.
I asked students about a clock. What happens in 12 hours? The clock repeats itself. In 7 days what happens to the day of the week? The day repeats itself. What happens in 12 months? We are in the month of April again. The same can be said for days in a year (excluding leap year), minutes in an hour, or seconds in a minute. These are examples of congruences that we deal with every day.
We looked at a simple example of this using time. Starting on a Monday at noon, the question is what time is it after 43 hours? Students determined that the time was 7 a.m. in the morning on Wednesday. I had students share how they approached this problem. Almost everyone determined that 48 hours was noon on Wednesday. They then subtracted 5 hours to back up to the required 43 hours. Subtracting 5 hours from noon resulted in the equivalent time of 7 a.m.
We'll look at this in terms of a congruence statement next class.
As we only had a minute left in class I had students write down questions they have about congruences and the Chinese Remainder Theorem. I'll start next class by having the class discuss their questions.
As a snow day was just announced for tomorrow, our discussion of congruences will have to wait a couple of days.
Visit the class summary for a student's perspective and to view the lesson slides.
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