Modular Arithmetic Stations
Fellows: Richard Abdelkerim & Yeranuhi Semerdjian
The purpose of this lesson is to expose students to the everyday uses of modular arithmetic and solidify their intuitive understanding of the topic.
We did an introductory lesson the previous day starting with a long division warm-up with various numbers divided by 5. We told them to do it “elementary school style,” meaning that they had to find the integer remainder. (This is a great opportunity to have students come to the board confidently!) Then we labeled the fingers of one hand 1, 2, 3, 4, and 0 and counted each of the warm-up’s dividends as a class to see at which finger the count ends. Also as a class, we compared this information with the remainders obtained in the division problems.
For this lesson, it may be most practical to have students wait outside if possible (supervised, of course) until the setup is complete. Five stations are set up around the classroom in such a way so that students in groups of 5-6 can easily maneuver their way from one station to the next. Each station is entitled “Mod X” where X represents 3, 5, 10, 7, and 12. A pile of handouts and some important signs are placed at each station.
Groups are loosely connected and merely exist for the purposes of understanding the activity at each station (reading directions is an important part of this lesson) and traveling in manageable packs. We had students work in pairs, and students only received one handout per pair, so both names went on the paper for each station. If the number of students comes out odd, have three students work on one paper from each station. Students were good about checking that the work was distributed evenly among their pair, and these were not exactly the best behaved students to begin with!
Mod 3 (This station requires cereal and some monitoring. We used Joe’s O’s, like plain Cheerios. Anything vitamin-fortified and not too sweet would work nicely.)
Students each take a bag/cup/napkin of cereal and eat it three pieces at a time. They keep written track of how many times they eat. They stop eating when they have zero, one, or two pieces left. They should then be able to figure out how many pieces were in the bag and write that number down. Then they answer the following questions:
If there were 31 pieces in the bag and you ate them three at a time, how many would be left at the end (zero, one, or two)?
If there were 72 pieces in the bag and you ate them three at a time, how many would be left at the end (zero, one, or two)?
Mod 5 (This station requires a piano or an electronic keyboard and heavier supervision.)
Students count the number of white keys on the piano. There is a picture of a right hand on the poster with Thumb=1, Index=2, Middle=3, Ring=4, and Pinky=0. They start with their right thumb and play each white key until they get to the end, changing to the next finger for each key (i.e., Thumb, then Index, then Middle, then Ring, then Pinky, then Thumb again for the next key, etc.). Then they answer the following questions:
Which finger ends up on the last key? What number is it? (See diagram.)
Now starting at the green arrow with your right thumb and playing the same way, which finger plays the last key? What number is it? How many white keys did you play this time?
Now starting at the yellow arrow with your right thumb and playing the same way, which finger plays the last key? What number is it? How many white keys did you play this time?
Now starting at the red arrow with your right thumb and playing the same way, which finger plays the last key? What number is it? How many white keys did you play this time?
If you only play the black keys, starting with your right thumb, which finger plays the last black key? What number is that finger? How many black keys are there?
Extra Credit: If you play all the keys (white and black) in order, which finger plays the very last key? What number is it? How many keys are there total?
Mod 10 (This station only requires two hands that have five fingers each and the diagram!)
Students place their two hands in front of them, face down. Then students count each of the following numbers with their fingers, starting with the left pinky and going to the right pinky and back to the left pinky again if necessary (L. Pinky=1, L. Ring=2, L. Middle=3, L. Index=4, L. Thumb=5, R. Thumb=6, R. Index=7, R. Middle=8, R. Ring=9, R. Pinky=0):
11 What finger did you end on? What number is it? (See diagram.)
23 What finger did you end on? What number is it?
32 What finger did you end on? What number is it?
If you were to count to 57, what finger would you end on? What number is it?
If you were to count to 275, what finger would you end on? What number is it?
If you were to count to 1007, what finger would you end on? What number is it?
Mod 7 (This station requires knowledge of the days of the week and the diagram.)
Today is Thursday. What day will it be…
…3 days from today?
…4 days from today?
…7 days from today?
…14 days from today?
…18 days from today?
What day was it…
…5 days ago?
…7 days ago?
…13 days ago?
(Monday=1, Tuesday=2, Wednesday=3, Thursday=4, Friday=5, Saturday=6, Sunday=0)
Today is 3/29/07. It is 278 days until New Year’s Day 2008. What day of the week is the New Year?
Extra Credit: 2008 is a leap year, so it is 366 days long. What day of the week is New Year’s Day 2009?
Mod 12 (This station requires knowledge of the months of the year and the diagram.)
It is now the month of March.
Three months from now, you will graduate from Sepulveda Middle School. What month will it be?
Six months from now, you will begin high school. What month will it be?
Eleven months ago was my birthday. What month was it?
Six months ago you started eighth grade. What month was it?
In June 2011, you will graduate from high school. How many months from now until then?
Extension: We plan to use their familiarity with these moduli to introduce the notation and definitions of modular arithmetic. Upcoming lessons include computer programming (using Excel) and a survey of how CDs generate random numbers to play shuffled lists of songs.