Math Monday Blog Hop: Multiplication and Division

Cindy, over at love2learn2day, runs a weekly blog hop on elementary level math. This week's topic is multiplication and division. I'm writing this post to gather a few resources I like.

Most of the posts that usually gather in the blog hop are activities. Before we get to activities, we might want to ask ourselves some questions, like "What is multiplication?" Multiplication can be modeled in many ways, and it's important to give young kids a chance to think about it through lots of different frameworks. Your own conceptions of multiplication may grow as you think about it.

One resource that will help you broaden your perspective on multiplication (along with a number of other math topics) is the lovely book Moebius Noodles. (It's $15 for the paperback version, and whatever you decide to pay for the pdf version.) It is full of activities to share with young kids.

I also love the art connection made by Waldorf schools, with their multiplication stars. Beauty helps us love what we're doing, which helps us learn.


As we teach, the more we can see the child's understanding, and help that grow, instead of telling them what is so, the better off we are. I can think of two ways we might just tell children our adult understanding, without giving them enough opportunities for exploration. One is in the commutative nature of multiplication, noticing that 3x5 = 5x3. Kids don't see that at first. If we tell them, we take away their chance to discover it. I never knew that until I read ‘Third Graders Explore Multiplication’, a chapter by Virginia Brown in the book What's Happening In Math Class, Volume 1, edited by Deborah Schifter. [You might be interested to know that mathematicians study versions of multiplication that are not commutative. Matrices are a tool for solving systems of equations, if A and B are matrices, AB may not equal BA.]

The other example I'd like to share of this is our knowledge that to multiply by ten we just "add a zero". (Why does that work?) Here's an excerpt from the chapter 'Trust, Montessori Style', by Pilar Bewley in the soon-to-be-published book Playing With Math (edited by me):

Answer Versus Process 
Five-and-a-half-year-old Roland came to ask if he could multiply 8,696 times 10 using the stamp game. I was thrilled to see his interest in math taking off again after an unfortunate temper tantrum with the addition blank chart. I suggested he borrow a stamp game box from another classroom to supplement the one we have in ours, and he got started.

It took him a while to figure out where he would do the work, and then he painstakingly began to make 8,696 with the stamps… Six units, nine tens, six hundreds, and eight thousands… Leave a space and repeat… Six units, nine tens, six hundreds, and eight thousands…

He made the amount five times before it was time to go home, and he left his work out so he could return to it the next day. (Can you imagine what that looks like, once you’ve made 8,696 with little colored tiles ten times? How cool!)

“I’m going to get right back to work as soon as I change my shoes,” he declared before leaving that afternoon. “I won’t even talk to anybody!”

I waited eagerly for him to arrive the next morning, looking forward to the moment when he would put AAAAAALLLLLL those tiles together in neat rows by category, and he would have to exchange several times (not to mention his surprise at seeing all the units disappear when multiplying by ten).

Instead Roland came in, shook my hand, and said: “My dad told me that all I have to do is add a zero to 8,696 and I’ll have my answer, because when you multiply by ten you just add a zero.”

My heart sank. Oh no, Dad! You robbed your son of such a cool experience! He was getting ready to see what ten times 8,696 looks like, and would have discovered the process that takes place during multiplication. He won’t be doing multiplication tables for at least two more years in public school - the answer doesn’t matter yet, but the process could have really taught him something valuable.

Although I firmly believe that math is about understanding, multiplication facts are one of the few things that need to be memorized. Many students come to college not knowing those basic facts. When I was teaching beginning algebra, I gave a quick quiz on all the basic multiplication facts, and required students to keep trying until they could get 85% right. Here are some of the suggestions I offered them:

• Try some of the games at Math Playground.
• Use these flash cards, which help you visualize the numbers. 
• Play the Product Game.

Those suggestions were for adults who were likely to have emotional baggage about not yet knowing something basic. For kids, just have fun. Make up games (like John Golden does) using dice or cards, go on scavenger hunts, or tell stories. (Perhaps you could tell stories about the large family, who have to buy in bulk because they have so many children. Each child needs 3 pair of shoes, for dress-up, play, and beach. How many shoes are in that house?)

Hmm, all of that and I haven't even mentioned division. I guess that will have to wait for its own post.

More Links: Good article, video, tool, problems, game, and activity

What I've stumbled upon in the past two days:

• Can't keep up with the good articles I'd like to read. The De Morgan Forum pointed me to an article by Liping Ma, whose writing always gets me thinking. 
• Malke's love for the math found in stars primed me to take notice of this video by Toomai on stellations. He notices something intriguing that I'd like to explore when I have more time...
•  I think some day I'll want to explore the tools available at SageMath.

And some older ones (as I begin to slowly clear out my backlog...):

• Megan's thoughts on ranking tasks as a way to get students to grapple with definitions. Hmm, I wonder if I could design a good activity for my calculus or linear algebra courses.
• The link takes you to an illustration of adding cubes, but the page that's on is a huge collection of good problems.
• Kate's logic game, whose name has not been decided, looks great. I'd use it in Discrete Math. Reminds me of an old game called WFF 'n Proof.
• On inventing number systems (with third graders).

Links versus Real Writing

I used to share links more often. I used to write substantive blog posts more often, too. Since I've been writing less, I haven't been comfortable sharing lots of links. Didn't want this blog to descend into just a link-share. But it would be helpful to me to have them here. So maybe I'll start sharing my almost daily finds, even if it's not exciting for y'all.

• What is a radian? Here's Sam's colleague's applet (on geogebratube).
• Trig war for review (thanks again, Sam!)
• Measuring the moon's size through geometry (Anna Weltman)
• With an Eye on the Mathematical Horizon, by Deborah Ball (good article, can only be read online)
• Fermat's Little Theorem (the ad before this video is bizarre - if the ad stays the same...)

These were the tabs I've kept open - some for weeks - hoping to figure out how to use some, how to find time to really process others.

Online Conversations: Math Communication, and Understanding Computer Graphing

I am enjoying two online conversations right now.

Michael Pershan asked:

Students don't like to write about their reasoning. They don't present their work in a way that allows anyone else to comprehend their path to a solution. But we want kids to write about their reasoning. Conflict! Drama!

Why do kids hate writing about math?

I am currently trying to grade my students arguments (as prosecutors) for the murder mystery. Some of them really got into it. Most still didn't explain the math well. My take on this is that students will write (maybe even well) if we give them a good enough context.


In the other conversation, Mr. Honner blogged about what happens when you zoom in super far on Desmos, looking for the hole in a rational function. It gets a bit crazy. The conversation got more interesting for me when Alan Eliasen started explaining "interval arithmetic", which I had never heard of.

Starting Circle Trig in Pre-Calc

I'm teaching four classes this semester, which is a lot for me. That's embarrassing to admit - I know most math bloggers are high school teachers, and teach way more hours a week than I do, with more responsibilities for their students. But for me it's a heavy load. So I'm not prepping as much as usual. I've taught calc and pre-calc dozens of times, so I can usually get by with winging it. And, once in a while, I'm able to conduct a better class by improvising than I ever could have with a tight plan.

That's what happened yesterday in pre-calc. The day before that I had worked hard to get their tests graded, so in the morning I printed out the new unit sheet, and walked into class not particularly sure how I wanted to get us started. I had grabbed a problem from my computer, and asked them to start thinking about it while I handed back tests.


The problem:

Consider three circles, all tangent (externally). Their radii are 4 in, 5in, and 6in. What is the area between them? 

I had asked the students to draw a picture. After they had had plenty of time, I drew my picture on the board. Then I asked them how we might start thinking about the problem. A student suggested finding the area of the triangle formed by connecting the centers. I asked if that triangle's sides actually went through the points of tangency. No one answered. Unlike in a math circle, I rescued them be showing a picture of one circle with a tangent line, and reminding them that they likely proved in geometry that the tangent is perpendicular to the radius (the one that ends at the point of tangency). I don't know what that proof would look like. To me, it seems obvious because of the symmetry. (In the afternoon class, they didn't think it needed proving. It already looked necessary to them.)

To find the area of the triangle, one student suggested drawing in the height. We drew it in, but couldn't yet see how to find its length. One of the students suggested that we could find the measures of the angles. They first suggested using law of sines. That didn't work, so we used law of cosines. Sine of that angle gave the height over a triangle side, so we got the height, which gives us area of the triangle. Then we got the other angles and found the sector areas. The afternoon class did it without the height, so they got to use law of sines.

It was a lot of steps for them, but it was a great review of the triangle trig we'd done earlier in the semester. And maybe they got a small taste of what problem-solving looks like.

When we were done, I had just enough time to explain radians to the morning class. The afternoon class had more time, so we worked out the new circle-based definitions of the trig functions.

What are our intuitions about temperature?

I'm teaching exponential functions and logarithms in pre-calc right now. That means it's time to pull out my murder mystery, in which they will use logarithms to solve an important problem - which of their classmates killed John Doe? Since the murder mystery uses temperature to find the killer, I want to lead in with some thinking about how temperature changes over time.

On Wednesday and Thursday, I told my classes a story, and asked them to draw a graph. I said I was mixing some cake batter up to make a Halloween cake. I asked what temperature it should cook at. We decided to set our oven at 350 degrees. (In one class, I talked about how silly the Fahrenheit temperature scale is, but how, even with Centigrade, zero is just attached to water freezing. It's not the same as zero length, volume, or weight. Temperature is different...)

I also asked what temperature the batter was now. They told me room temperature, and we decided that was about 70 degrees. Then I drew axes on the board, labelled them, and asked the students to graph the temperature of the batter over time. Only one person (out of over 50 in the two classes) came close to the right shape. No one seems to have much intuition about how temperature changes. I did this once before, with the cooling coffee we always think about, and got slightly better results.

Here are my approximations of what students thought:

The green one may have been influenced by our attention in the past week to exponential growth, while the purple one seems to have taken the exponential growth we were studying and limited it by the temperature of the oven. I have often seen students give a linear graph like the blue one, and a logistic-like graph like the orange one. No one wants stuff to heat up fast at first, and then slower.

What makes their intuition bad here? Is there a physical experiment / demonstration we could do to improve their intuition? What would make exponential decay feel like the natural choice to them? Maybe cake is the wrong object to be heating?

Please help me think about this.

What is Calculus? Part Two

In our first installment, we saw a bit of history, and began to think about the idea of rate of change (which can be visualized as the slope of a graph). Slopes of straight lines, central to algebra, are defined by rise over run, or the ratio of change in y over change in x. In calculus, we extend our definitions so that we can consider slopes of curvy lines. Since the slope is different for different parts of the graph, we can think of it as a new function. For each x value, there is a y value, and there is a slope. We call this new slope function the derivative of our original function. (This is the part of calculus called 'differential calculus' - differentiating means finding the derivative.)

We need an example for this to make sense. Let's go back to that ball. From part one:

Suppose we throw a small ball straight up, and are able to measure its position perfectly every tenth of a second. We can make a graph for that - [its] horizontal axis will measure time ... in fractions of a second... The vertical axis will simply measure the ball's height. It starts out in my hand, about 4 feet off the ground. I think I can throw it about 20 feet high. We'll imagine together that I do. Working out what gravity does to that ball is one of the things we can do with calculus. Here's the graph (new version):

We see from the graph that the slope is positive until we reach the highest point, and after that it's negative. In fact, finding the highest point turns into finding out when the derivative (slope function) equals zero. This is a powerful tool for finding the most or least of anything that's defined as a function.

So how do we find these slopes? If I asked you to draw a tangent to this parabola at t=0.4 seconds, you would probably know what to do, even though the only definition of tangent line you may have seen before this is the tangent to a circle. Can you draw that tangent line? (I've drawn a picture below, but I'll discuss our next step now, so the picture won't show up right away. Please turn away from your computer, and draw a sketch of this graph with a tangent line added in.)

If we want a more precise description of what's happening to the ball, we need an equation. In this case, height in feet is determined by time in seconds, by h(t) = -16t²+32t+4. Notice that height at time 0 would be 4, as given in the story above. (The -16 is determined by the force of gravity on earth, and would be different under different gravitational conditions. This part can be understood more deeply by the end of a calculus course. The 32 is determined by how hard I throw the ball upward.)

The tangent line touches the graph at just one point. Using our equation for the curve, we can easily find the y-coordinate (height) at that point, but how would we find the slope? Slope requires finding the change in y and change in x, which takes two points. But all we have is one point. We're stuck.

One strategy mathematicians use might be called wishful thinking. Since we need two points to find the slope, let's pretend we have them - let's cheat! If we use a second point on our graph very near the first, and make a line that goes through both points, our line, called a secant, is pretty close to the tangent line we're seeking. What if we try to get closer and closer this way?

You may want to experiment here. Our one point is (0.4,14.24). If we use t=0.5 for our second point, we get (0.5,16). Now we can figure the slope of our tangent secant line. Slope is the ratio of change in y over change in x. We find those changes by subtracting, so we get:

That would be 17.6 feet per second. But that secant line is just a little bit less steep than the tangent line. We could get a better estimate by trying a second point at t=0.41. But we'll never get a perfect answer. Hmm ...

Ok, this is where it starts getting a little weird. We want to imagine the second point getting infinitely close to the first. The use of infinity is what separates calculus from algebra. Newton called his infinitely small quantities 'fluxions', and another mathematician of the time (George Berkeley) complained bitterly:

“What are these Fluxions? The Velocities of evanescent Increments. And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities?”

It was more than a hundred years later before mathematicians were able to develop a solid logical basis for calculus!  But if we're willing to trust our intuition, like Newton did, this method of taking points "infinitely close" to one another works amazingly well. If we call the distance between the time values h, and imagine h getting smaller and smaller, we would say we're "taking the limit as h goes to 0".

The algebra gets very ugly here, and it turns out it's easier to see for a generic time t than for a specific time (like t=0.4). We're imagining that t represents one time, and t+h represents another time very close to it. The height at time t+h would be

Still ugly. And it's going to get a bit worse before it gets better. Now we find the slope:

Here's where the logic gets tricky. We can't have zero on the bottom of the fraction (dividing by zero just doesn't work), but if h doesn't equal zero, no matter how close to zero it is, we can cancel it. What we're left with includes -16h, but h is so close to zero that this term doesn't really matter.  So we get that the slope is simply -32t+32 or 32(1-t). So the slope of the tangent line at time 0.4 is 21.6. We could also say that the ball is moving upward at that moment at 21.6 feet per second.

Finding an average speed (I traveled 122 miles in 2 hours, for an average of 61 miles an hour) is straightforward. It's a ratio of distance traveled over the time it took. But finding an instantaneous speed - how fast I was going at a particular moment - that takes calculus.

I think this is incredibly beautiful, and every semester I relish helping students understand this concept. But this is the first time I've written it all out without feedback from the 'students'. It's time to ask: "Any questions?"

What is Calculus? Part One

The New York Times has an online column called Numberplay, written each Monday by Gary Antonick. It usually features mathematical puzzles, but this week he was asked to explain 'differential calculus', and in his article he wrote:

I’ll lay out a few of my own thoughts but I’m especially looking forward to learning from others on this. Let’s make differential calculus our puzzle for the week.

He shares a delightful story of working in some very eccentric bakeries. Delightful though it was, I didn't feel like it would help anyone who didn't already get something about calculus. I want to offer up just a tiny bit of history, and then see if I can explain this for people who have at least a rudimentary understanding of algebra.

Isaac Newton and Gottfried Leibniz both invented calculus at about the same time. That sort of thing happens often in math. I think if neither of them had been around, someone else would have invented the methods of calculus - the problems being investigated by scientists of the time were just crying out it for it. Planetary orbits, gravity, the paths of projectiles, the shapes of lenses for microscopes and telescopes - understanding any of these requires calculus. (See the first six pages of this book.) The time was ripe.

Calculus mainly addresses two sorts of questions: How fast something is changing, and finding areas of irregular shapes. (The methods of calculus can also find volumes and the lengths of curves, among other things.)  Addressing these questions requires dealing carefully with the notion of infinity, mostly what it might mean to be infinitely close and what an infinitely thin slice might mean.  The Greeks (and probably others I don't know enough about) did some good work with these ideas. Even though they lacked the super-versatile tool of algebra, they made a good start on the area problem. Perhaps through lack of interest, they don't seem to have addressed the rate of change problem. Other cultures - and other individuals before Newton and Leibniz - made progress on both questions. What Newton and Leibniz did that got them famous was to find a simple link between the two, which is now called The Fundamental Theorem of Calculus.

The rate of change questions are addressed by what's called 'differential calculus' and the area questions are addressed by what's called 'integral calculus'. Since the question asked in Numberplay was about differential calculus, we'll start there. Even though the ancients did more with areas, rates of change are where we start in calculus courses these days.

If I made 3 loaves of bread a day (Are Gary and I making you hungry?) and saved them all, I could show how many loaves would be in my pantry with a simple graph, something like this ...

In algebra, we talk about the slopes of straight lines like this. For every one unit you go over, you need to go up three units to stay on the line, so we say the slope is 3. (You might remember "rise over run" - that's part of the definition of the concept of slope.) Well, slope and rate of change are pretty similar ideas. Rate of change just gets a bit more precise about the story that goes with the graph. In this case, we're adding 3 loaves of bread per day to the pantry.

The problem is, this notion of slope or rate of change only works with straight lines and constant rates of change, but most of the interesting things scientists want to study have changing rates of change. Like when we toss something into the air, and gravity pulls it back down. The methods of calculus helped Newton to understand how gravity affects objects, and it's a good place for us to start too.

Suppose we throw a small ball straight up, and are able to measure its position perfectly every tenth of a second. We can make a graph for that, too. Like our bread graph, the horizontal axis will measure time, but now it will be in fractions of a second instead of in days. The vertical axis will simply measure the ball's height. It starts out in my hand, about 4 feet off the ground. I think I can throw it about 20 feet high. We'll imagine together that I do. Working out what gravity does to that ball is one of the things we can do with calculus. Here's the graph. (I don't like that I showed the curve for negative time values. I wonder if I can fix that.)  I'll explain this one in part two.

A math circle...

I mentioned Pythagorean triples to my pre-calc students last month, and told them if enough of them were interested, I would run a math circle on this topic. Ten of them signed up, six came the first week, and three came to the second and third sessions. It's small, but it's going very well, and I may do something bigger next semester.

This is my first time doing an ongoing math circle with many sessions devoted to one topic. It's also my first time getting my own students to come to a math circle. I am really happy that they keep coming. I originally said it would be five sessions, but I can see that we could easily go for six to eight sessions on the questions raised here. (I may let them talk me into extending it.)

I love it when the way they approach a problem is different than the way I would have done it. X saw a pattern I had not seen before, and we explored her pattern at length in the second session. I haven't had time to write up the details, and have probably forgotten much of it.


Week One
What examples can we come up with?  (3-4-5, 5-12-13, ...)
6-8-10 leads us to define primitive Pythagorean triples (in which gcf(a,b,c)=1; 6-8-10 isn't primitive)
Maybe writing a list of all the perfect squares up to 400 will help us find more.
What patterns do we see?

• Odd + Even = Odd
• Middle number is a multiple of 4
• c = b+1 (after which I added 8-15-17 to our list)

Week Two
One person was new, so we reviewed our first week's work for him.

We explored the "family" of triples with c = b+1. a² + b² = (b+1)² becomes a² = (b+1)² - b²
= b² + 2b + 1 - b² = 2b-1. If  a² = 2b-1, then b = (a²+1)/2. This will be a whole number whenever a is an odd number. So we got lots more: 7-24-25, 9-40-41, ...

X noticed that in the triples

3-4-5
5-12-13
7-24-25
9-40-41

the second number is 4*1, 4*3, 4*6, 4*10, ... For the nth one, we use 4 times a number n more than the previous one. I showed them why these (1, 3, 6, 10, ...) are called triangle numbers, and asked them to add 1 to 100. They each came up with their own way of thinking about it. We came back to X's pattern and wrote:

a=2n+1
b=4*n(n+1)/2=2n(n+1)
c=b+1

Week Three
Another new person came, so we summarized for her. Then we explored triples where c = b+2.

I love seeing their creativity and persistence. At the same time, I am blown away by the holes in their understanding of algebra moves. Y was considering (4n)², and thought he might have to distribute.

We verified that we get all of the primitive Pythagorean triples with c=b+2 using:

a=4n
b=4n²-1
c=4²+1

Not sure where we'll take it in Week Four, but eager to find out. I am still struggling to lead less, become less visible, and listen more.

Playing With Geometry

The Math Monday Blog Hop, at the love2learn2day blog, has a theme each week. This week, it's Geometry. At community college, we don't teach geometry as a course - not surprisingly, I feel it's my weakest area in math. So I love playing with games, puzzles, and problems that stretch me geometrically.

A Game
My favorite geometry puzzle/game is Katamino, which makes a two-player game out of pentominoes (I blogged about it here). I love the beautiful wood pieces. (I can't tell if the newer versions are wood or plastic. I recommend trying to find the wood.) Playing with this will work your visualization skills.

A Puzzle
Online, I recently stumbled on a site which offers 40 challenge problems in compass and straightedge geometry, implemented in an online puzzle.

Some Problems
I've recently come to love the Five Triangles blog, where mostly geometric problems are posted a few times a week. Their problems often challenge me to think in slightly new directions.


What's your favorite way to play with geometry?