Eight Hours a Day of Math

At Siobhán's archery lesson a homeschooling mom and I were getting to know each other. I told her I do math for eight hours most days. I realize that is a bit of an exaggeration, but not by much.

The radical as a chuppah

Have I mentioned that I have my own unorthodox ways of describing math processes? I refer to the ubiquitous unwritten one in algebra as sneaky one or as his evil twin negative sneaky one (who often trips up students). I tell my students (in a Catholic school) that the absolute value sign works a lot like going into a confessional. If you have sinned (a negative), you leave forgiven ( a positive). If you had no sin (a positive), you still leave forgiven (a positive). Of course, this follows after the discussion of the true meaning of the absolute value being the distance a number is from zero (and distance is always positive as we don't say, "I drove 140 miles to Chicago and -140 miles back giving me a total distance of zero miles round trip.") Some students need an image to help them to remember what is going on with the math symbols. Sometimes a wacky (to them) story helps.

When studying square roots, the radical is the name of the square root sign which unfortunately looks too much like one of the division signs. The number under the radical is called the radicand. When you want an exact answer (not a decimal approximation) with a simplified radical expression, some students have a difficult time looking for perfect squares to pull out. The students already know how to find the prime factors of numbers using factor trees so this an opportunity to use this method to find the roots. Using a factor tree we get √ 8 = √ 2*2*2.

This is where the chuppah comes in. In the Jewish faith, a couple gets married under a canopy known as a chuppah. In Christianity and Judaism, we both draw from the biblical ideal of marriage being two people coming together to become one. I point out that we have a couple of 2s under the chuppah (the radical looks like a canopy) so one 2 comes out. The other 2 is "left at the altar" or is still waiting for his/her match. Therefore we are left with 2√2. For √72 = √3*2*3*2*2 √we see two couples 3*3 and 2*2 with one extra 2. In this case the 3 and the 2 come out married (as one number each) and the other 2 is left behind with the answer being 3*2√2 = 6√2. One girl in the class was especially taken with the word chuppah announcing it like "Opa" at a Greek restaurant each time she did a problem. Another girl was saying under her breath, "You circle the couples, and one comes out."

Mixed metaphors, stretched analogies, and cross-cultural fusions in a math classroom? Welcome to my world.

GEMDAS

Today while looking at PEMDAS with the 8th graders, we noticed that some students tried to solve parentheses which were being used as multiplication and weren't taking notice of the fraction bar as a grouping symbol. When I pointed out that we needed to take into consideration that PEMDAS is called BEDMAS and BODMAS in other countries because brackets and parentheses are both grouping symbols, one of the students (that's Nate) and I both came up with GEMDAS. The G stands for grouping symbols. Of course we need a new mnemonic device because Please, Excuse My Dear Aunt Sally doesn't work with the G.

Some thoughts: Gee, Excuse My Dear Aunt Sally; Goodness, Excuse My Dear Aunt Sally; Gracious, Excuse My Dear Aunt Sally. I am taking a cue from some former students who came up with Please Excuse My Dog's Awful Slobbering a few years back. My suggestion: Gross, Excuse My Dog's Awful Slobbering. Of course, Great Education Makes Doing Algebra Simple is not a bad second (but not really catchy enough for the middle school crowd).

RIP PEMDAS. Long live GEMDAS!

The delights of a new class

This Wednesday began my sixth year working with sixth graders at Jordan Catholic School doing Enrichment Math. I am fortunate to have 23 students signed up for this 7 am class. What a great bunch of children. We're off to a promising start. Most of the students and I have not met each other before. I asked the children whose siblings have been in any of my classes to raise their hands. That accounted for six. Then I asked which children were sitting at a desk where the name tag on the front corresponded with their name. That gave me another ten or so. The rest I had to work on the old-fashioned way - saying their names over and over to myself while trying to visually tie it to the name I see in my mind. I pride myself in learning my students names within a two or three classes.

After I passed around permanent metallic markers for the students to use to write their names on their brand new calculators (and the face plates), I went over the usual routine of the class. As we are at a Catholic school, we begin every class with a prayer. I ask a student to come forward, ask for prayer requests (intentions), and then lead us in a prayer of his or her choosing. When we asked for prayer requests, the first girl called upon said, "How about the Our Father?" I tried very hard not to laugh. I shook my head and said, "I'm sorry. I didn't mean to ask for your request of the top ten greatest prayers of the Holy Catholic Church. I meant for you to say something like 'My big toe hurts.' or ' My mom is having surgery.' or ' My dog is old.'" Fortunately we all had a good laugh and continued on. Thankfully the girl leading prayer chose to end with the Our Father, so the request was not in vain.

The worksheet I handed out had the numbers 1, 3, 5, 7 , and 9 printed prominently at the top. I wrote the numbers on the board and asked the students to tell me what they knew about these numbers. "They're single digits." "They're odd." "They skip a number in between." I asked if anyone could tell me how many prime numbers were represented. The answers were all over the map. I asked if anyone could define prime. They kept getting closer and closer. One student said, "It's a number that can be multiplied by one." Nice try, but all numbers can be multiplied by one, but I think I understood where the student was headed.
I asked them to draw a little house.
_______
/______\
|______|
|______|

We put the number in the attic and the factors on the floors. Like so.
_______
/__3___\
|__1x3__| Notice that three is a two-story house.
|__3x1__|
All two story houses are primes.

Notice that one is not a two story house.
_______
/__1___\
|__1x1__|

What's up with one? It is unique. Singular. Neither prime nor composite. I asked the students to look at the word unique. How many wheels does a unicycle have? How about a unicorn? We live in the United States of America. Out of many one. We are one. I then point out that zero is another unusual number in this respect.

Then we test out all the other numbers on the board to see how many stories are in each house. Five and seven follow the same pattern as three so we tally up two more primes. Then we look at nine. Uh, oh. Three times three can only be written once. Then we have the other two facts (remember all numbers can be multiplied by one). Then I point out another interesting fact. Houses with an odd number of stories have a special name. Squares. Then I draw a 3x3 square on the board next to a 1x1. Any number times itself makes a square. (At this point I know that they have not studied this concept, but I like to show them this visual point.)

Ah, now we have three primes, two squares, and one composite to add to our list of what we know about these numbers. We haven't even talked about natural numbers, whole numbers, integers, or rational numbers. They will get that later in the year. As I check to see that students are taking notes, I add, "All that and we haven't even begun to look at what the worksheet is asking us to do."

It's time to hold on to your desks as we take math apart. We get to explore the inner workings of numbers and deconstruct the language that we use to describe mathematical realities. We are off on another grand adventure, and I am so privileged to be the tour guide.

Quip

This is my new acronym for what I recommend to parents who ask me how they can best help their children with math.

Qu - Questions: Ask questions if you don't fully understand something. Write down your questions while doing homework. Ask for help. Raise your hand during class if you need clarification. Don't be afraid to ask questions.

I - Integration: Math shows up everywhere and in many other subjects. Parents who integrate math language into everyday life and who point out where and how math is being used daily will help their children to make these connections.

P - Practice: Math is a skill that requires some practice. Just as in music, some people are naturally more talented, but everyone has the potential to improve with practice.

Digits at the Medieval Math Club

33 kiddos showed up at the Medieval Math Club this week. I started by reviewing some of the terms we learned last week. One boy stretched out his arms and announced, "Fathom!" Another child remembered step. Someone held up fingers and said, "Digits." Which brought us to the biggest question from last week - How many digits do I have on my hands? (Last week I counted 11. 10-9-8-7-6 on one hand and 1-2-3-4-5 on the other. 5+6=11 Right?!?)

For the past two days I have had two boys who were at math club the week before trying to sneak a peek at my fingers during face painting at VBS. One of them stopped me in the parking lot. "Do you really have eleven fingers?" he asked. His brother shouted, "I want to count your fingers." When they would come to my face painting station, they would try to count my fingers on my left hand. I sat on my left hand and continued to paint with my right hand. "Not until math club!" I would gently remind them.

"How many digits do you have?" I asked the group. "Ten!" the children shouted. "How many digits do I have?" I asked with a mischievous grin. Some kids said, "Eleven." Some shook their heads and said, "You told us eleven, but everybody has ten." Back and forth it went for a minute, "Ten, eleven, ten, eleven." Then someone said, "We want to count your fingers." I showed the children how I counted 10-9-8-7-6 on one hand and then 1-2-3-4-5 on the other. "You need to count forwards on both hands," someone suggested.

I asked the two brothers to come up. They gleefully counted forward from 1 to 10. Triumphantly they held up my tenth finger and shouted, "She has ten. We knew it!" A boy from the back screamed, "Liar!" (Directed at me for the deception.) I shook my head. "It's a trick. You can do it, too." As a whole group we practiced counting backwards on one hand and forwards on the other.We read Blockhead: The Life of Fibonacci by Joseph D'Agnese. Check out the link below for more info.

http://www.blockheadbook.com/Blockhead/Welcome.html

Next week: Base Ten

Summer Math Club Again

Here's a rundown of what we will be doing.

June 14 - Measurements using How Big is a Foot?
We will measure our digits, hand, hand span, cubit, fathom, foot, step, and pace. Discover what these historical measurements mean and how we still use them today.

June 21 - Fibonacci sequence using Blockhead
We will read Blockhead : the Life of Fibonacci and Growing Patterns : Fibonacci Numbers in Nature. We will explore the Fibonacci sequence in natural objects.

June 28 - Sir Cumference and All the King's Men
Counting by tens.

July 5 -Chess and the powers of two
We will examine the legend of how chess was invented and also learn the basic moves of all the pieces.

July 12 - Platonic Solids
We will explore Platonic solids using gumdrops, marshmallows, and toothpicks.