Whispered Words of Wisdom

The students were not quite seated before the whispering began.

“I’ll ask if you don’t want to!”
“No, Ben should be the one to ask.  He’s the one who brought it up to begin with.”

As you can imagine, my interest was piqued.  I looked at Ben whose cheeks were bright red.  “Ben? Do you have a question for me?”

I hope you can picture just how big my smile was at that moment!  This was the first orthographic question of the year that was inspired by something happening outside of our classroom!  I was delighted, and I hoped my smile conveyed that!  “What a great question!  Tell me more.  What were you talking about when this question came up?”

Ben began by explaining that in math  class they were discussing polygons.  Specifically they were talking about shape families.  When they got to quadrilaterals, the teacher asked if students knew any other words with <quad>.  As students named words, it was the consensus that words with <quad> have something to do with “four.”  When Ben asked whether or not <squad> was related to <quad>, the teacher suggested they bring that question to me.  Perfect!

The first thing we did was to recreate the list of words the students had thought of earlier in math.  They included:

Then I asked, what is the spelling they have in common?  What specific string of letters do you see in each and every word?

The first response was <quad> (no doubt because that was what they had been talking about earlier).  I asked them to look again and more carefully.  That was when several hands shot up at once.  “I see q-u-a-d-r!”

Great!  Now I underlined the <quadr> in each word so we could look at the rest of each word.

Before I could even ask a question about this word, a student raised their hand to say, “The <i> could be a connecting vowel!”  Awesome!  I didn’t expect that, but it is true!  It could be!  Next I asked if anyone recognized any suffixes.  Someone called out <er> and <al>.  Great!  Those might indeed be suffixes.  They often are.  (Notice that instead of saying, “You’re right,” or “Sorry, you’re wrong,” I’m using words like “might” and “could.”  At this point we are doing some out-loud thinking about this word.  We will consult a resource when we have had a chance to think through our observations.)

At this point I asked if anyone knew what <lateral> meant.  No one did.  So I said, “What if I told you that a fish has lateral fins?  Does that help?”
There was a moment of hesitation as students mulled over this idea.  Then someone said, “Side fins?”
“Yes!  Do those of you who love to play football know what a lateral throw is?”
“Yes. It’s when you throw the ball in a backwards or sideways direction.”
“Right.  So we’re seeing a sense of “side” in both when we refer to a lateral fin and a lateral football throw.  So now tell me what a quadrilateral is.”

Several students at once responded with, “Four sides.”

Right away I wanted someone to tell me what quadruplets were.  Everyone seemed to know that it was when four babies were born in a single birth.  None of us knew much about the <uplet> part, but had heard it as part of <triplet>, <quintuplet>, <sextuplet>, <septuplet>, and <octuplet>.

Having identified the base as <quadr> made the rest of this word recognizable.  I could just ask, “What is a quadrangle?”  And several students replied that it was a shape with four angles.  Instead of quickly moving on, I wondered aloud whether a quadrangle and a quadrilateral could refer to the same shape.  Hmmm.  After a bit of thought, the students agreed that a shape with four angles would also have four sides, and a shape with four sides would also have four angles.

The students quickly named <million>, <billion> and <trillion> when thinking of the second part of this word.  I went on to name <quintillion>, <sextillion>, <septillion>, <octillion>, <nonillion>, <decillion>, <undecillion>, and <dodecillion>.  (I love knowing this list because I can see the same <sept> in <septillion> as I do in <September>, the same <oct> as in <October>, and the same <dec> as in <December>.)

The students weren’t as familiar with the use of this word.  I explained that if an area were to be split into four areas, one of the areas would be called a quadrant.

At this point a boy raised his hand and stated, “I don’t think <squad> fits with these.  None of these words begins with an <s>.”
I loved knowing that the original question sat in his head as we were discussing all the words with <quadr>.  I replied by saying, “You might be right, Sam.  But then again, we can often be surprised by what we find.  I don’t know the answer, but it’s almost time to look.”

But there was still something the students were wondering about.  “Isn’t quad a word all by itself?”
“Yes.  I think you’re right.  I wonder if it isn’t a clip of one of the words we’ve looked at.”

All in all, this glorious discussion took about 25 minutes.  I enjoyed identifying what we knew already, and what things we could relate to other things without running immediately to a resource.  There is such value in recognizing the connections one already knows.  This is how the students will strengthen their confidence in their ability to connect one word to another.

What an opportunity to point out that both<quadr> and <squad> began in Latin, but had different journeys into Modern English.  Both were used in French, but <squad> was also used in either Spanish or Italian and that different journey has been reflected in their spellings.  It turns out that they ARE related!  They are related etymologically, but because they do not share spelling, they are not morphologically related.

Now isn’t that something worth whispering about?

Pi, Pi, Mathematical Pi

Every day this week we took a look at Pi.  On Monday and again later in the week we listened to the Mathematical Pi Song (to the tune of American Pie). I gave each student a paper plate and assigned them a number.  They were to write that number on the plate and decorate.  We hung the plates in a “Pi Number Line” down the hall.  That way we could think about this irrational number all week!  I gave each student a copy of the Greek Alphabet and we practiced reciting it.  I wanted them to see the letter Π and know where it came from.

On Tuesday we reviewed the words <radius>, <circumference>, and <diameter>.  Earlier in the year we had discovered that the word sum for <circumference> is  <circum> + <fer> + <ence>.  The prefix <circum> means around and the base <fer> means to carry.  The circumference is the distance carried around the outside of the circle.  The word sum for <diameter> is <dia> + <meter>.  The prefix <dia> means through and the base <meter> means to measure.  You measure the distance through the circle to find the diameter.    The word sum for <radius> is <radi> + <us>.  The base <radi> means rod, spoke of wheel, and ray of light.  I like to picture the radius of a circle as a spoke of a wheel.  I brought miniature donuts.  We multiplied the diameter of a donut by Pi, and in doing so calculated the circumference of the donut.  We watched another video having to do with Pi.  This one was the  ‘The Dance of the Sugar Pi Fairy’!  How fun to find creative ways to memorize digits of Pi.  (Certainly not a necessary thing to do, but certainly a tempting thing to do)

On Wednesday I brought cookies.  We measured the circumference and divided that by Pi to calculate the diameter of the cookie.  We also listened to some very creative math/musicians who put digits of Pi to music!  The first one is called Song from Π.  It is an enchanting piano piece with interesting facts about the number Pi.  The digits of Pi float by as they are being played on the piano.  Quite honestly, we loved the music, but found it difficult to follow the digits.  The second one was fascinating for other reasons.  The artist repeats the first 31 digits of Pi.  He uses different instruments and different tempos.  It’s called What Pi Sounds Like.

On Thursday I brought peanut butter cups and we measured both the circumference and the diameter.  We then found Pi by dividing the circumference by the diameter.  Several students came up with 3.166.  That’s pretty close!  We watched a video about art created from the digits of Pi.  It is called Pi is Beautiful.   We were inspired to see what kind of art we could create using the digits of Pi.  I found a circular graph.  We numbered around the outside using the digits of Pi.  If the number was a 3, then 3 squares toward the center were colored in.  Some students assigned a specific color to each number 1-9 to see what that would look like.  Some students stuck to using 1, 2, or 3 colors.  We were very pleased with the results and hung them in the hall above our Pi number line.

Then came Friday – March 14!  Students brought in pies – 17 in all!  We ate pie several times throughout the day and shared with adults around the school.  We held a contest to see who had been able to memorize the most digits of Pi throughout the week.  Our winner memorized 100 digits of Pi!  We were amazed!  Our second place winner memorized 61 digits and our third place winner memorized 24 digits!  Wow!  What fun.  Enjoy the following video.  It’s kind of a medley of our day!

Celebrating Our Thousandth Day of School!

If our school year is approximately 180 days long, and we multiply that by the five years my students have been in school (kindergarten, first grade, second grade, third grade, and fourth grade), at the end of fourth grade they’ve been in school for 900 days.  So that means that on the 100th day of fifth grade they are celebrating their 1000th day of school!

I decided to center our celebration around the number 1000.  Mid-morning I mixed up a snack stew.  In the stew were 100 each of 10 different ingredients.  There were things like animal crackers, marshmallows, cheese crackers, pretzels, and conversation hearts.  Yum.  Throughout the day we jumped for joy in ten sets of 100 jumps.  Whew!  Students answered questions such as, “How old will you be in 1000 days?  1000 months?  What are the factors of 1000?”

Take a peek and just see how much fun we had!

It’s Time for a Math Break!

Today was such a fun Friday.  My math students have been improving all week when it comes to  staying focused during work time.  Staying focused means finishing work in class and making less errors.   Making less errors means less fix ups and a better chance of building a deep understanding of the skills being practiced.

Today we decided to take a break from learning math and instead focused on teaching math.  We are currently learning how to find the fraction of a number.  For example, do you know what 5/9 of 27 is?  If the answer doesn’t come quickly, watch this video and then see if you can figure it out.

I know.  Entertaining, wasn’t it?  But besides all that, do you think you could figure out what 5/9 of 27 is now?   Leave a comment, and we’ll let you know if you’re right.

Celebrating Pi Day with Pizazz!

Pi.  Pi.  Mathematical Pi!   We had so much fun today!  We sang songs.  Some were based on familiar Christmas tunes, one was a rap, and one went to the tune of American Pie.  We learned about William Jones who first recorded the symbol for Pi.  We  read about Gaurav Raja who at one time held the record for reciting 10,000 digits of Pi.  We learned about Ludoph van Ceulen who had the first 35 digits of Pi engraved on his tombstone (until his wife swapped it out for something more proper)!

We felt that in order to really understand Pi, we needed to really understand circumference, diameter, and radius.  The class divided into three groups and the word investigations began!  With limited time, no group quite finished, but they made great progress.  I find it interesting that many students still fall back on old habits of dividing words by syllables instead of beginning with their lists of tried and true affixes.  Patience and practice.  They must discover the logic of that for themselves.   In one part of the video, the group investigating the word <diameter> started laughing.  I had just asked if the words they found that begin with <dia> had to do with the definition of <dia> which is through.  You see, diarrhea was on the list and they definitely saw the connection!

Next we held a Pi Digit Contest.  We were looking to see who could memorize the most digits of Pi with only two days of preparation.  Our first place winner recited 65 Pi digits!  In second and third place, students recited 46 and 42 places respectively.  The fourth, fifth, and sixth place winners recited 36, 28, and 27 places.  What an amazing accomplishment for all who gave it a try!

Then there were the pies!  Yum!  We had cherry, lemon meringue, apple, turtle, peanut butter, chocolate, toasted coconut creme, pecan, banana creme, and brownie pies.  Heavenly!

During math, we actually measured circles of all kinds and calculated Pi for ourselves by dividing the circumference by the radius.

Lastly, we sang our favorite Pi song just one more time.  Love Pi Day!

Divisibility Rules!

Simplifying fractions becomes easier when you can recognize what both the numerator and denominator are divisible by. Hope you enjoy our video on the divisibility rules for 2, 3, 4, 5, 6, 9, and 10.

Frog Races … predicting, graphing, fractions to percents

Frog races have been a part of my math teaching for a while now.  It’s fun!  It’s exciting!  It’s unpredictable!  But we try to predict it anyway!  We don’t use live frogs (just in case you were wondering).  We use stuffed frogs.  I’ve made a giant race track out of a huge piece of felt by drawing a 6 x 12 grid on it.  We race six frogs and they are lined up at one end of the track.  Before we begin I ask each student to predict who will win that day’s race.  The frogs are numbered one through six.  They all have names, of course, and are more often referred to by their names.  Just in case you are curious, frog 1 is Mr. Polyester, frog 2 is Lightning, frog 3 is Fatty, frog 4 is Smoochee, frog 5 is Leaper, and frog 6 is Bobby Bo.  The students take turns racing the frogs.  The rest of the students are our spectators.

Once predictions have been made, I walk around to the spectators and ask them to shake three dice.  What they roll determines which frogs move.  Let’s say the first person shakes two 4’s and one 2.  That would mean that frog 4 (Leaper) would move two squares, and frog 2 (Lightning) would move one square.  It’s as simple as that.  We keep going until a frog has crossed the finish line.  It gets pretty exciting when there are two, three, or four frogs in a tie for the lead!

Once we have established a winner, then the students head back to their desks to fill out their data sheet.  I also send someone to update the bar graph we have on the wall in our room.  The bar graph has a picture of each frog at the bottom of the column they represent.  Their frog number is at the top of the graph.

The students keep track of each week’s results in a notebook.  They record the date, the frog they predicted would win, the frog who actually won, the fraction of times they have predicted correctly, and the percentage of times they have predicted correctly.

Next I list on the board all of the possible fractions for this race.  On this particular day it was our 8th race so a person might have guessed right either  0/8,  1/8,  2/8,  3/8,  4/8,  5/8,  6/8,  7/8, or  8/8 of the time.  I asked the students to tell me the equivalent percent for each fraction listed.  I asked them to start with the obvious ones … such as 0/8 = 0% and 8/8 = 100%.   Then someone recognized that 4/8 = 50%.   Well, if there was a 50%, I wondered out loud if there would be a 25% or 75%.  Immediately hands shot up as students recognized that 2/8 = 1/4 = 25%  and that 6/8 = 3/4 = 75%.   Once we had that much figured out, I asked if we could figure out the rest of the percentages without a calculator.  Again, some students recognized that 1/8 was half way between 0/8 and 2/8, so 1/8 would be half of 25% .  Then there was a scramble of hands up because once the pattern had been established, figuring out the rest of the percentages became fun!

The very last step is to ask the students to come up to the board and place a tally mark below the percentage that represents their personal data on predicting frog races.  Back when we had only raced twice, we had some students who had been right 50% of the time.  Currently, the best percentage is 38%.  Two students are still at 0% correct predictions.  It is good practice to talk about what observations one can make by looking at data.

In the weeks to come, we will continue to enjoy the excitement of the race, but then also become very familiar with writing fractions as percents.  The students will also begin to recognize that some fractions are easy to write as a percentage if you simplify them first.  Other fractions (with a prime number denominator) require you to get out your calculator!

Besides the fraction to percent work, it is also interesting to figure out what the chances are of shaking three of any number.  I could tell you how, but instead I challenge you to find out and submit your answer as a comment to this post.  The first person to post with a correct answer wins \$50 froggy buck!  Go for it!

Note for teachers:  This activity takes about 20-25 minutes.