- This video shows multiplying by using a parabola. Completely impractical, but I was curious why it worked. I figured it out and then wondered if my pre-calculus students could figure it out too. I wanted a demo instead of a video, so I built something in Desmos. (Hide the equations, and click on the three dots. The middle dot will always multiply the absolute values of the other two.) It's not perfect, but it might be good enough to impress my students.
- I've seen this cute list of functions, with the person's arms illustrating the graph, on a number of blogs lately. I see two that are wrong. Henri sees one wrong, and has quibbles with four of them. What do you see?
- Common Core for math... I keep hearing that the math standards are pretty good. But if the tests ignore the most important standards (the process standards, which describe mathematical thinking), then they're being used badly. This post by Jonathan Katz goes into some detail.
- Nice exercise. One person looks at the board, and describes the graph drawn there. Their partner must draw it from the verbal description.
- Quintic polynomials. There is no formula for the roots. But there is this. I want to learn more!
- Fawn's lesson for proportional thinking.
- Papert on "hard fun."
- I like this diagonal problem, but when I tried it in class my students were not persistent enough to succeed with it. David Cox's post on how he used it with his students makes me want to try it again.
- In whatif?, xkcd's creator, Randall Munroe, takes a silly question and analyzes it with math and physics to come up with an answer. In this episode, he figure how how big a splash you'd get from a tree as big as all trees on earth falling into an ocean with the water of all the ocean's on earth.
- In this post from her calculus for kids series, I like Maria's thoughts on how we help kids learn problem-solving.

## Saturday, April 5, 2014

### Linkfest for Saturday, April 5

## Sunday, March 30, 2014

### Linkfest for Sunday, March 30

- Michael Pershan asks some good questions about how to approach the teaching of complex numbers.
- Fawn Nguyen has another Math Talk with her students. (I'm wondering if students think about which sorts of visual patterns will give a quadratic function and which will give a linear function.)
- OpenMathBook, a blog "to promote, discuss, and develop free and open source mathematics textbooks".
- Ihor describes a high school class recreating the experiment Eratosthenes conducted 2200 years ago. (Some day I want to try it...)
- Maria is conducting calculus math circles with 7 to 11 year olds. I am fascinated. My understanding of calculus is so steeped in algebraic thinking, it was hard for me to imagine at first. I love what she's doing.

## Friday, March 28, 2014

### Guest Post: John Spencer Addresses "Frustrated Parent"

*John's was the first post I saw about the silly complaint going around from "Frustrated Parent". (Now I've seen about three more. They all have good things to say.) John has graciously allowed me to share his whole post here. (But the comments over at his place are an interesting mix, so go on over there too.) Here's John:*

There are many things I hate about the Common Core standards. I hate the way teachers were pushed out of the creation and adoption phase and how we have little voice in the implementation. I hate the fact that the standards will continue to be assessed with standardized, multiple choice tests and that these scores will be used with Value Added Measures in both teacher salary and teacher evaluation. However, I think it's important that in our criticism of bad policy we are careful to avoid blasting good pedagogy.

I'm seeing many of these posts making their rounds on Facebook.

I'm seeing statements like, "What the hell is a number line and why do kids need it?" Or, "just teach them the basics." The notion of using a manipulative, playing with numbers, breaking them them apart and comparing processes is somehow viewed as non-mathematical.

The truth is that number lines are powerful tools for understanding integers. True, when subtraction is something simple that requires no "borrowing" it feels like a joke. However, the goal is to build up number sense. It's to help them understand math conceptually. If you flip the numbers and end with a negative number as an answer, suddenly a number line helps make the negative-positive relationship more powerful.

This parent's snarky answer about "the process used would get you terminated" is based on a faulty assumption that a first grader needs the same approach as an engineer. And yet . . . this "new math" approach that people mock is something we use constantly in real-world, mental math.

Consider it this way: You have fifty-three dollars and you need to give someone twenty-seven dollars. What are you going to do to figure it out? If you find yourself breaking by tens and going backward, chances are you are using a mental number line.

Oh, you could pull out a piece of paper and do that math that way, but chances is are that as an engineer, you'd be fired . . . or at least laughed at.

I remember someone posting an angry rant about doing multiplication by breaking it up into different pieces. "Just teach the algorithm!" the parent posted.

I posted a response. "If the bill is 27.42 and you want to leave a twenty percent tip, what's the answer? How did you find it?"

Some people divided by five. Others multiplied by .2. Still others moved one decimal over and doubled it. Some rounded up to thirty. In other words, there were multiple processes that worked and each of them involved understanding the properties of numbers. In other words, most people used a process mentally that they were openly mocking on Facebook.

* * *

Oddly enough, many of these same people who are mocking "new math" in their posts are also lamenting the fact that Singapore is kicking our butts in math. What they fail to realize is that the places where math is working are the places where they are building number sense.

I've seen what happens when students lack number sense. They learn a lockstep process and think that math is the same as baking a cake. They follow the recipe without understanding why they are doing what they are doing. However, when they get into something as simple as linear equations, they struggle to know what to "do first," when there are often two or three options.

When students lack number sense and they get the wrong answer, they fail to understand why an answer is illogical. You end up with a student who misplaces a decimal number and never finds his or her mistake. Asking students to think conceptually and engage in diagnostic problem-solving isn't superfluous. It's actually part of "the basics."

I know that the "new" math looks different, but instead of criticizing it for being hard or being complicated, try thinking about the theories behind it. There's a reason we're using manipulatives, breaking things apart, using number lines and comparing processes. This is how math works.

## Thursday, March 27, 2014

### Linkfest for Thursday, March 27

- I got my master's degree at Eastern Michigan University. It makes me happy to see them on the web.
- Colorful related rates problems, and colorful descriptions of solution strategies.
- Turning multiplication table practice into a scrabble-like game sounds like a good idea to me.
- Teaching metacognition in math class.
- Fawn is teaching absolute value. (And giving us some great ideas.) And James Cleveland has a great inspiration, using estimation180 to help teach absolute value and analysis of errors.
- Math Munch collects other games like 2048.
- Mr.Honner once again finds a weird graph on a standardized test. It purports to be exponential, but it's not.
- Justin Lanier did a math circle on Celtic knots.
- Interesting puzzle - place the houses on the number line. I know there's some math here, but I haven't figured it out yet.
- Optimization, a calculus problem solved by dogs, ants, and light.
- Henri Picciotto has a nice visual Pythagorean Theorem proof.

## Monday, March 17, 2014

### Linkfest for Monday, March 17

- Looking for pictures of water coming out of a hose in a parabolic arc, I stumbled across this post about using geogebra to come up with an algebraic function that describes the path of the water.
- I have not yet ever taught Calc III. I'm thinking about working through this linear algebra based course, which pointed me to this video on higher dimensional spaces.
- The newest addictive game, 2048.
- Another cool looking site from NCTM, with a wide variety of math challenges.
- Cheesemonkey describes what gave her the courage to persist when learning involved lots of failures.
- How to inscribe a semicircle in a square, using geogebra.
- Gas Station Without Pumps thinks about what keeps women out of engineering.
- Some nice sine and cosine animations.
- 3D printing for hyperbolic honeycombs.
- How would you design a game, using only a fair coin, that you have a 1/3 chance of winning? No answers at the link, only thoughts about the value of hard questions.
- Nice challenge problems with exponents. Can you solve it without logarithms?
- More from This Old Train... on the roller coaster project. And a new project called Parametric Playground.

I still have too many tabs open, but the rest of them are not math...

## Friday, March 14, 2014

### Calculus: e and pi Are Both Transcendental

Yesterday I was introducing the number e, and telling the students that my preferred definition of e is "the number that makes a slope of 1 for the graph of y=a

As I made this introduction, I mentioned that, like pi, e is irrational (not a ratio of whole numbers) and transcendental (not the solution to an algebra equation using whole numbers). I showed them the proof that the square root of 2 is irrational. But it is the solution to x

And then I decided that since pi was in the air, and Pi Day was coming, we'd do an activity I'll be doing today with a different group. I had them all stand up around the outer edges of the room, and hold hands, arms outstretched. The extra people made a diameter along the middle of the room. While I was getting them to stretch, I pulled people out by saying "You're out." They laughed. I told them it was a good kind of out, that we were the circle makers.

The reason I'm posting this silly, super simple activity? When we were done, and they counted off, it turned out that there were 22 around and ... (you know I was holding my breath) ... 7 across the middle! I couldn't believe it. I was glowing, further proof to my students that I am a total nerd for math. ;^)

In case you're wondering why getting 22 around and 7 across was so special, that makes circumference / diameter = 22/7 = 3.1428, almost perfect. And before calculators made people think decimals were cooler than fractions, 22/7 was

(Added on 3/14: Tried it again today in a smaller space. We got 17/4. Bummer.)

^{x}at x=0." I also told them that this definition makes mathematicians unhappy, and wrote out the limit definition. But I like them seeing a concrete meaning for this strange number.As I made this introduction, I mentioned that, like pi, e is irrational (not a ratio of whole numbers) and transcendental (not the solution to an algebra equation using whole numbers). I showed them the proof that the square root of 2 is irrational. But it is the solution to x

^{2}=2, so it's an algebraic number, not a transcendental number. I told them that I have not yet completely understood the proof that pi is irrational. (Updated link: Brent Yorgey, one of my favorite bloggers, posted a good series explaining it at The Math Less Traveled. But I never managed to get all the way through it.) "So it's not really math for me to say pi and e are irrational, when I don't know it by a proof but only by believing what I've been told."And then I decided that since pi was in the air, and Pi Day was coming, we'd do an activity I'll be doing today with a different group. I had them all stand up around the outer edges of the room, and hold hands, arms outstretched. The extra people made a diameter along the middle of the room. While I was getting them to stretch, I pulled people out by saying "You're out." They laughed. I told them it was a good kind of out, that we were the circle makers.

The reason I'm posting this silly, super simple activity? When we were done, and they counted off, it turned out that there were 22 around and ... (you know I was holding my breath) ... 7 across the middle! I couldn't believe it. I was glowing, further proof to my students that I am a total nerd for math. ;^)

In case you're wondering why getting 22 around and 7 across was so special, that makes circumference / diameter = 22/7 = 3.1428, almost perfect. And before calculators made people think decimals were cooler than fractions, 22/7 was

*the*estimate for pi.(Added on 3/14: Tried it again today in a smaller space. We got 17/4. Bummer.)

## Tuesday, March 11, 2014

### Join the Math Circle Institue at Notre Dame, July 7 to 11

**Math Circle Institute**

July 7th – 11th, 2014

The seventh annual Math Circle Institute will be held on the Campus of Notre Dame, in South Bend, Indiana, from July 7th to 11th, 2014. Both novice and experienced Math Circle leaders are welcome. Bob and Ellen Kaplan, Amanda Serenevy, and Nathan Pflueger will demonstrate the Math Circle approach, and participants will prepare and teach their own Math Circle classes with 1st to 12th grade students who attend each afternoon.

Applications can be made by e-mail to kaplan@math.harvard.edu.

The institute is wonderful. If you can go, do it. You'll be so glad you did.

[Past blog posts about my experiences there: year three, year five, year six. (I can't believe I never posted about the first two amazing years.) And a post sharing Rodi Steinig's experiences at the Institute, including her great list of quotes from Bob, which became the lovely list now known as Becoming Invisible.]

Here's a repost of what I wrote about it a few years ago...

Bob & Ellen Kaplan (founders of the Boston area math circles, and authors of

*Out of the Labyrinth: Setting Mathematics Free*and many other intriguing math books), along with Amanda Serenevy (founder of Riverbend Community Math Center), run the fabulous week-long Math Circle Institute, held on the beautiful campus of Notre Dame. This summer is their 7

^{th}annual Institute.

For only $850, you get room and board... (The food is amazingly good for such a huge operation - not quite up to my personal organic, local, sustainable standards, but really yummy, with an incredible variety of choices.) ... and then you get to play with math

*all day*. My friend, Ellen H, and I used to call it the math spa. (We went swimming every morning before breakfast.)

The first year I was amazed that I could do math all day long and never get tired of it. There's lots of freedom to explore in whatever way works best for you.

__Schedule__

Arrive: Sunday, July 6 (or as early as you can on Monday, July 7). Notre Dame is in South Bend, Indiana, 2 hours from Chicago.

Five days of math circle mania:

- participate in math circles in the mornings,
- run one math circle for kids late one afternoon, and watch your colleagues try it out on the other afternoons,
- discuss the ins and outs of math circles and other fun ways of doing math in the early afternoons,
- informal math play in the evenings,
- play with Amanda's collection of math toys and browse her collection of math books, whenever you want,
- plot, plan, and socialize to your heart's content in between all that.

## Sunday, March 9, 2014

### Linkfest for Sunday, March 9

I have to clear out all my open tabs!

I really like Maria Droujkova's response to someone who asked about the Common Core. She (oh so diplomatically) mentioned what she likes. The mathematical practices:

- First off, John Golden has created a great Carnival of Mathematics #108. (John, how can I get any work done if I play with all these posts?!)
- Maria Andersen announced the release of Algeboats.
- I think of the roller coaster projects as being about trig derivatives, but they might work well when I get around to the limits chapter, where we'll talk more about differentiability. (I do it after derivatives and before integrals.)
- Glenn is thinking about how to make some smooth connections from Algebra I thought Calc (and stat). I like what he's doing.

I really like Maria Droujkova's response to someone who asked about the Common Core. She (oh so diplomatically) mentioned what she likes. The mathematical practices:

1. Make sense of problems and persevere in solving them.Although I'm not sure their list is any better than Avery Pickford's Mathematical Habits of Mind. In other words, it's a good list, but not the only good one.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

### Chain Rule: Experiments, Proof, Practice

**Part One: The Plans**

Tomorrow I start our third unit in calculus with an introduction to the chain rule. This deep into the semester, I have usually lost most of my creative energy, and don't find time to think this lesson out carefully. I'm teaching great groups of students this semester, which is helping keep my enthusiasm high. So today I'm planning my chain rule lesson.

**Describing Composition**

First, some functions that we cannot find the derivatives for, because there's more than the variable inside the function. Like y=sin(2x). Here we will review the language of composition.

**Experiment by Graphing**

Then a graph of the function, to see if we can figure out what the derivative ought to be.

I don't usually write blog posts before I teach the class. This is an interesting point in my process. Where I go next with this may depend somewhat on the students' response to the graph exercise. I know right now that I'm not sure of their response, and that their response may inspire me to move a different direction than what I'm planning right now.

**Experiment Algebraically**

Where I think I'll go is this. We next take a look at functions which exhibit composition, but don't really have to (ie, they can be simplified). My three examples for now are:

y = (5x-6)

^{2}
y = (2x)

^{3}
and

y = √(9x)

I'm going to work through finding the derivative without chain rule, and trying to get the derivative in a form that matches the function. I think this will be another way to experimentally figure out the chain rule.

**Prove It**

Then we'll do the proof. I found a much nicer proof here (click on the discussion link) than the one in our textbook (Anton).

*Except that one vital step seems to be missing.***Practice It**

And after that, we'll do lots of practice. The textbook is probably fine for that, if I can't think up enough examples on my feet.

[I found some inspiring posts, but I'm not sure if or when I can use them: Sam's filing cabinet led me to lots of good posts, one of which was this on the monkey and the mathematician. I might suggest my students check this linked wheels applet out on their own.]

**Part Two: How It Went**

I got up to the first algebraic example on Monday. On Tuesday, we did the three algebraic examples. They weren't enthusiastic about walking through the proof, so I asked them to look it over on the handout shown below. We finished up by practicing with lots of examples.

## Tuesday, March 4, 2014

### Linkfest for Tuesday, March 4

So many good posts lately! How will I ever remember all this good stuff?!

- Sam, on having courage, and math as a creative endeavor.
- Parametric Functions, and a rational parametrization of the circle.
- Better polar explorations. (This would even be good in Calc II, I think.)The key is to make it into a puzzle.
- And here's a puzzle approach to inverse functions.
- Determining distance traveled from velocity, from Robert Talbert, for calc I.
- Shireen has made a Where's Waldo themed worksheet to help calc students think about position, velocity, and acceleration.
- I can't watch video right now, my son is sleeping. I have to watch these classroom videos, with students debating in math class, and creating theorems.
- I don't always agree with Curmudgeon, but I'm sure glad I follow this blog. Here's a great post on requiring Algebra II.
- Malke's thinking about spatial reasoning. Here's a post she linked to, with a cute question. If we fold the paper like this, and hole punch it while folded, where will the holes be when we unfold?
- Maria Anderson likes to think about the future. She's made four predictions about technology-enhanced learning.
- Have you heard of Osmos? It's a physics game app. I want to buy it.

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