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.
Saturday, November 19, 2011
Friday, October 21, 2011
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!
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!
Friday, September 9, 2011
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.
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.
Thursday, September 1, 2011
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.
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.
Thursday, June 23, 2011
Digits at the Medieval Math Club
howed up at the
"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.htmlSunday, May 22, 2011
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.
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.
Monday, April 18, 2011
Fractions, Decimals, and Percents
My sixth grade enrichment math class just finished a unit on fractions, decimals, and percents. These are probably the most practical lessons these children will learn throughout their years in school. As adults, most of us use these concepts on a daily basis. Our money system uses decimals. This is a revelation to some of my students.
When I present the fraction 1/2, almost everyone understands that the decimal equivalent is 0.5. When I present 1/4, the looks on their faces tend to change. My first question is, "What else do we call this fraction in English?" If I get no response, then I start talking about clock time. "What's another way to talk about 3:15?" Or I bring up money. "If one dollar is one whole, what is one fourth of a dollar?" By now someone has mentioned the keyword, a quarter.
This realization usually helps most older children who have some experience with money. One quarter is $0.25. Then I show them two quarters (2/4 = 1/2) is $0.50 which leads to 3/4 = $0.75. From there we go through all the tenths. 1/10 = $0.10, 2/10 = $0.20, etc. Every tenth represents a dime because a dime is one tenth of a dollar. This seems quite obvious to us as adults, but this sort of connection for many children is not necessarily apparent.
This can be taken through the twentieths (1/20 = $0.05) and the hundredths (1/100 = $0.01). The hundredths lead us to the presentation of fractions and decimals as percents. What does percent mean anyway? Per cent (pronounced as if one is speaking French because cent = 100) means per one hundred. How many cents are there in a dollar? Point out words that include cent: century, centennial, centenary, centipede, centimeter.
Let's go back to the halves. Most children understand that half of something is 0.5 or 50%. (Notice that 0.5 is the same as 0.50 not 0.05 which is another distinction that my students often miss.) See how the decimal point moves two places to the right to name the percent (which means that we are multiplying the decimal by 100).
The next step is tackling percent problems using direct translation. That will be the topic of another post.
When I present the fraction 1/2, almost everyone understands that the decimal equivalent is 0.5. When I present 1/4, the looks on their faces tend to change. My first question is, "What else do we call this fraction in English?" If I get no response, then I start talking about clock time. "What's another way to talk about 3:15?" Or I bring up money. "If one dollar is one whole, what is one fourth of a dollar?" By now someone has mentioned the keyword, a quarter.
This realization usually helps most older children who have some experience with money. One quarter is $0.25. Then I show them two quarters (2/4 = 1/2) is $0.50 which leads to 3/4 = $0.75. From there we go through all the tenths. 1/10 = $0.10, 2/10 = $0.20, etc. Every tenth represents a dime because a dime is one tenth of a dollar. This seems quite obvious to us as adults, but this sort of connection for many children is not necessarily apparent.
This can be taken through the twentieths (1/20 = $0.05) and the hundredths (1/100 = $0.01). The hundredths lead us to the presentation of fractions and decimals as percents. What does percent mean anyway? Per cent (pronounced as if one is speaking French because cent = 100) means per one hundred. How many cents are there in a dollar? Point out words that include cent: century, centennial, centenary, centipede, centimeter.
Let's go back to the halves. Most children understand that half of something is 0.5 or 50%. (Notice that 0.5 is the same as 0.50 not 0.05 which is another distinction that my students often miss.) See how the decimal point moves two places to the right to name the percent (which means that we are multiplying the decimal by 100).
The next step is tackling percent problems using direct translation. That will be the topic of another post.
Monday, January 24, 2011
Teaching Trigonometry to a Fifth Grader
One of my students (I will name him Joy Boy) is in the fifth grade at an elementary school in Pleasant Valley. He and I have been working together for almost a year. His mother contacted Augustana College's math department to locate a tutor for him. Luckily for me, I happen to have a connection with one of the math professors there who knows about my penchant for working with kids doing enrichment for math.
Joy Boy sees patterns quickly. Last year he had a teacher who did not understand how to work with an advanced learner. (I am avoiding the term gifted, but it's probably accurate.) His parents wanted him to move ahead as he was bored, but no acceptable alternative was given to them by the school. This became my great fortune as I have the pleasure of watching Joy Boy's mind at work.
Joy Boy and I have done many things together over the past year including using Calculus By and For Young People by Don Cohen and working through worksheets from MathCounts. We discovered that Joy Boy needed to reinforce his skills when working with ratios and proportions. Ed Zaccaro's Challenge Math has a chapter on this subject, and Joy Boy already had a copy of the book. Guess which chapter follows Ratios and Proportions? Trigonometry.
Last Saturday Joy Boy and his mom met me at our usual time and place. He was not happy about trigonometry. He told me it was his "worst favorite" subject. His mom and I shared a quick glance and a shrug. Given that trig. is based on ratios, his second worst favorite subject, I told him it was no wonder.
Turns out that he didn't mind the ratios. He was just upset that instead of explaining how to figure out the trig. values for yourself, he was told to use a table or his calculator. He felt that he was cheating by doing so. He wanted to know how to figure everything out on his own. Oh, boy!
My first thought was to ask him why he would want to spend the time working out something that other people had already figured out, but I knew better than to go there. I asked him about pi instead.
Pi is a ratio (which is oxymoronic because pi is not a rational number, but that is a whole other discussion). It is the circumference of any circle divided by the diameter of the same circle. This ratio has a wonderful and rich history. I mentioned that pi crops up in the Bible (1 Kings 7:23). This led to some discussion as I had read that some people have used gematria to come up with a more accurate approximation of pi than the 3:1 ratio in the text. I showed Joy Boy how to convert the alphabet to numeric values and showed him how to find the numeric value of his name. We used digit sums and that brought up how we use those in our divisibility rules. I then told him that we were now entering the land of numerology, and I wanted to get back to math. Joy Boy was willing to accept that C/D = pi, which we approximate to 3.14. On to the next step.
Special Case Right Triangles to the rescue! I drew a picture of a 45-45-90 triangle and showed him how to figure out the sides using Pythagorean's Theorem (which he already knew). Then I showed him how to do the same for a 30-60-90. (I would like to point out that when I have helped to teach this concept to 8th graders, it usually sends them into a tailspin.) So far, he was with me. Then I showed him the mnemonic device to remember the trig. ratios: SOH, CAH, TOA. When we put it all together, I realized that neither of us had brought a calculator with trig. functions. Ooops.
Fortunately, his mom remembered that the book had a trig. table at the end of that chapter. Nice job, Mr. Zaccaro. The book is geared for elementary and middle school students who might not have an advanced calculator at their disposal (although one can be bought for just over ten dollars, even at a drugstore). After showing Joy Boy how to derive some of the values, he decided that he could use the table or a calculator to do the work. He just wanted to know that he could figure it out for himself and that he understood where the values were coming from.
Big sigh of relief from both his mom and me. He happily cranked out the answers to the rest of the problems with no signs of distress. Joy Boy, indeed!
Joy Boy sees patterns quickly. Last year he had a teacher who did not understand how to work with an advanced learner. (I am avoiding the term gifted, but it's probably accurate.) His parents wanted him to move ahead as he was bored, but no acceptable alternative was given to them by the school. This became my great fortune as I have the pleasure of watching Joy Boy's mind at work.
Joy Boy and I have done many things together over the past year including using Calculus By and For Young People by Don Cohen and working through worksheets from MathCounts. We discovered that Joy Boy needed to reinforce his skills when working with ratios and proportions. Ed Zaccaro's Challenge Math has a chapter on this subject, and Joy Boy already had a copy of the book. Guess which chapter follows Ratios and Proportions? Trigonometry.
Last Saturday Joy Boy and his mom met me at our usual time and place. He was not happy about trigonometry. He told me it was his "worst favorite" subject. His mom and I shared a quick glance and a shrug. Given that trig. is based on ratios, his second worst favorite subject, I told him it was no wonder.
Turns out that he didn't mind the ratios. He was just upset that instead of explaining how to figure out the trig. values for yourself, he was told to use a table or his calculator. He felt that he was cheating by doing so. He wanted to know how to figure everything out on his own. Oh, boy!
My first thought was to ask him why he would want to spend the time working out something that other people had already figured out, but I knew better than to go there. I asked him about pi instead.
Pi is a ratio (which is oxymoronic because pi is not a rational number, but that is a whole other discussion). It is the circumference of any circle divided by the diameter of the same circle. This ratio has a wonderful and rich history. I mentioned that pi crops up in the Bible (1 Kings 7:23). This led to some discussion as I had read that some people have used gematria to come up with a more accurate approximation of pi than the 3:1 ratio in the text. I showed Joy Boy how to convert the alphabet to numeric values and showed him how to find the numeric value of his name. We used digit sums and that brought up how we use those in our divisibility rules. I then told him that we were now entering the land of numerology, and I wanted to get back to math. Joy Boy was willing to accept that C/D = pi, which we approximate to 3.14. On to the next step.
Special Case Right Triangles to the rescue! I drew a picture of a 45-45-90 triangle and showed him how to figure out the sides using Pythagorean's Theorem (which he already knew). Then I showed him how to do the same for a 30-60-90. (I would like to point out that when I have helped to teach this concept to 8th graders, it usually sends them into a tailspin.) So far, he was with me. Then I showed him the mnemonic device to remember the trig. ratios: SOH, CAH, TOA. When we put it all together, I realized that neither of us had brought a calculator with trig. functions. Ooops.
Fortunately, his mom remembered that the book had a trig. table at the end of that chapter. Nice job, Mr. Zaccaro. The book is geared for elementary and middle school students who might not have an advanced calculator at their disposal (although one can be bought for just over ten dollars, even at a drugstore). After showing Joy Boy how to derive some of the values, he decided that he could use the table or a calculator to do the work. He just wanted to know that he could figure it out for himself and that he understood where the values were coming from.
Big sigh of relief from both his mom and me. He happily cranked out the answers to the rest of the problems with no signs of distress. Joy Boy, indeed!
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