A blog for sharing my math interests on the web, to post new materials for elementary, secondary and teacher ed, and vent mathematical steam when needed.
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Design
The call: a game for 5th graders just starting with fraction multiplication.
I look at my games. Fraction version of the Product Game... great fun, but more for practice than introduction. The crazy Ant Man game ... fun, good for calculator use, but also dividing fractions, so probably not time for that. Hmph.
Answer the question
(this was the first one)
Get it right to get a chance to
shoot past the goalie.
I look around on the web. Googled fraction multiplication game and got a lot of really awful drill "games." Glgkh - they left an awful taste. Some are obviously just quick flash mass production, but there are a couple that people really put time into looks and animation. For a quiz set to 8-bit music.
So, I'm on my own. Often with introduction time I try to think about representation. One of the things to love about fractions are all the many representations. I think the discrete models are underused, so I thought about about students claiming fractions of a common pot (similar to the GeoGebra percent game I posted recently) - but it was difficult to figure out how to keep to intuitive numbers and overcome the disproportionate effect of going first. Also, I had trouble thinking of a game context that would get students to see it as a fraction of a fraction instead of a fraction of a whole number.
Then I thought about the area model. I imagined carving up a rectangle, having kids carve up rectangles. Scoring a total... connecting two points... then I had a connection. Cutting down bit by bit, it felt like searching for something. I tried a 12x12 grid, and my first pass at a mechanic worked pretty well: rolling a die to get halves, thirds, fourths. I thought of a context - searching for a lost hiker. Too scary if you've been lost? Finding a lost pet... maybe. It was a little too direct. Is it a competition? It was starting to feel like Battleship (a fine game), and that was good. I tried finding multiple objects; 2, 3, 4... and 4 was right. Oh! They could come up with the context - and that would give them the opportunity to add rules of their own. That's worth a try!
They managed to find all three, before... well before nothing. I was pleasantly surprised by how engaged they were just trying to find the rings. Like spontaneous applause when someone found one. (Playing with the whole class, I have them pass the die to someone who's ready of the other gender. Usually works.) Afterwards, I shared how maybe I needed more rules. Or the Mandarin's searching also. Or if you roll two 5's the Mandarin finds a ring. Or...
It was clear this was going to work because there was immediately a crowd of students trying to tell me their context, Minecraft, aliens, how it fit into the story she's writing about two wolves who turn into humans. It was exciting. They experimented with more than 3 objects and asked me why I had chosen three.
The wolfgirls.
Quite complex. This was played on two boards, with interaction between the heroes and villains.
The minecraft game. This had hazards as well as the goal.
The zombie game, which also had a hazard. You had three lives, and had to find the zombie solution before you lost all the people in your party.
The playing went well also. I was impressed by students ability to
divide regions equally, and the many ways they found to do it. They
started inventing their own terminology for how they were doing it, like
the strips or plus method for dividing into four. They used horizontal
and vertical divides, and one group experimented with non rectangular
regions. One group played like Battleship, competing to find all three before the other team did.
In feedback, everyone gave the game a thumbs up (mostly) or so-so. (Rare to have one that no one dislikes.) They liked the Battleship connection, the feeling of searching and the multiple objects to find. They were very excited to tell about their context and rules variations.
Game Evaluation
Goal(s) - good - experience with representation, dividing up rectangular pieces into equal parts. Plus a context for future questions and rephrasing.
Structure - works well.
Strategy - puzzle like. Choice in which region to divide up with which fraction. Choices for where you hide the objects. Not the strongest element of the game, though.
Interaction - good and so-so. One person/team being the mechanic for revealing spots and checking the other team's work on dividing was good mathematically. But Battleship isn't strong on player interaction.
Surprise - die roll, so okay.
Catch-Up ... depends on the variation. It's a bit methodical doing the search, but there's no time element in the basic version. The chance to get lucky with a search or a roll will help.
Inertia - works for this. Students were anxious to play more.
Rules - toughest element is the dividing up equally. Once you've got that idea, rest is simple.
Context - here's the winner. Students being able to set their own context was very engaging for a vast majority.
In yesterday's post on a percent game, I shared two great GeoGebra sketches that students found. I remixed each of them a little, so I thought I'd share them here.
dhabecker's neat rational number arranging sketch lets students place arrows to try to put fractions, decimals, and percents in order. He has a very clever way to check if the randomly generated numbers are in the right spots. It notifies students when they've got all 4 right, and I wanted to them to be able to check their answer along the way. So - thinking of Mastermind - I thought about what if it can give the number correctly placed? Since I was doing that I added a reset button and a bit of color. Next I would add a fifth number, as that makes so many more permutations possible.
On GeoGebraTube: download or applet. (Unfortunately doesn't seem to work in HTML5, because the polygons won't move.)
The other sketch I modified was David Cox's great percent estimation sketch. Almost immediately on trying it on the Smartboard, the students turned it into a game.
So I turned it into a game with turns and scoring. There's 6 rounds. I thought about forcing players to take turns going first but ultimately just decided to ask them to.
Next I would be interested in seeing it go from a percent number line to another quantity. So the game asks you to find 38%, but the number line goes from 0 to 630. The percent and the whole would change each time. Worth a go? Probably that's in my head because of David's nice double numberline percent sketch.
As always, I'd be interested in feedback on either one of these.
But I'd also be interested in what could help develop a remix culture in GeoGebra. I learned it (am learning it) mostly on my own by experimentation, from suggestions on Twitter, and googling stuff from the online help. But in the Learning Creative Learning class they put a big emphasis on remixing as a way to learn that gives a lot of support to learners. With my middle school GeoGebraists, they are struggling to do work of value all on their own.
Are you a remixer by nature? What would it take to get you trying it in GeoGebra?
But as I looked, the idea for a game came to me, just to give percent experiences. There's no context, really, it's just a race game. No strategy, just rolling random percentages. But the mechanic of smaller roll goes first creates some nice percentage situations, and a lot of games wind up surprisingly tight.
I debated having the students find the percentages to subtract, but decided to make it optional. There's a short video of how the game works below.
The test game came out extremely close - most games will be shorter than that. It can be surprisingly suspenseful, though. All students found it pretty playable, and some got very into it. I think the best benefit might be from playing and then using as a context for problems.
Searching through GeoGebraTube, students found two sketches particularly of interest.
Arrange Fractions, Decimals and Percents by dhabecker (who has quite a few rational number sketches), which lets students arrange form numbers of different form from least to greatest. A few students got quite engrossed in this sketch, and the feel of the sketch is terrific - very much like pieces snapping in place.
Estimating Percents by David Cox, which lets students make a first estimate and then a second estimate with the tens showing. Students were happy to see their % error improve from 1st to second guess, and got better quickly through playing. They also made an impromptu game out of it, and I think that would be fun.
Do you know a GeoGebra Percents sketch that you think supports students' understanding?
Quasiperiodic tessellations are my favorite bit of recreational math. They came up in my thesis non-recreationally, and the best bit of Mathematica I ever wrote was to generate them for various different rotational symmetries.
Recently the Penrose magnets came from the successful kickstarter. (I can't tell from their website if they are available anywhere now.) It was fun to see tweeps sharing getting theirs. @MrHonner posted a picture that started the jealousy. This is a picture of my son's attempt to tile with them.
A recent GeoGebra project got me using the RigidPolygon[] tool for the first time, and later I realized that this would enable me to make them in GeoGebra. The construction of the kites and darts is easy with the regular polygon tool. I couldn't make a tool to make copies, because it wanted too much information. I remembered the advice Kathryn Peake was giving to David Wees on Twitter for a sketch: make a button.
So the first sketch was the tiles without matching rules.
Purists, like Edmund Harriss (@gelada) in that Twitter conversation, will correctly point out that these are not Penrose tiles but they can be used to make a Penrose tiling.
Without the matching rules it's really easy to get yourself in trouble.
Here's the GeoGebraTube page. Unfortunately the applet doesn't work well in the HTML5 version, since you can make new tiles, but they can't be moved. Fine in the Java applet, though.
It took me a bit to figure out how to construct the tile alterations to provide the matching rules, but the upside is that it is customizable, so that now you can make the tiles into shapes that are pleasing to you. I can see how Dr. Penrose wound up with chickens, though.
I did get to meet Dr. Penrose when he was working with my advisor - very fun and charming man. As well as obviously brilliant; nice when things work out that way.
Researching 61 as a number tells us it's not just prime, but is a twin prime, a cuban prime, the 9th Mersenne prime exponent, and a Pillai prime. That's primo prime pedigree. It's a Keith number and thrice Fortunate. (Fortunate numbers are pretty interesting and the subject of an open conjecture.) It's a centered square number (1+4+8+12+16+20), a centered hexagonal number (1+6+12+18+24) and a centered decagonal number (1+10+20+30); mostly because it's neighbor 60 is so nice. (Did you see the 60th Math Teachers at Play?) Is there a reason that \( \sum 4n \), \( \sum 6n \) and \( \sum 10n \) overlap at 60? But the fact that really caught my eye was that 61 is an Euler Zig Zag number. How cool can a number get?
Early Learners Preschool Math Ideas from Lilac at Learners in Bloom. "Preschool math can be so creative and fun. This post is a round-up of some of my favorite math activities from the past year (twins age 2.5-3.5)." Lots of fun kinesthetic number experiences shared here.
Make Me Dinner
from Jennifer Bardsley at Teaching My Baby to Read. "Here's a
deceptively simple math game to play with little ones. It works on loads
of math skills through pretend play." Working on cardinality and comparison.
mag3737
Elementary Center Idea For Building Addition Fluency
at Zoom Zoom Classroom. "This blog post highlights a product I
created. The idea is to provide children with extra practice with
solving addition facts in both vertical and horizontal form."
CCSS.Math.Content.1.OA.C.6 There's velcro, but I'm not sure I
understand the point of this center.
Skip-bo Addition Game
(or The Funnest-est Bestest Math Game Ever) from Amy at Ita Vita. "I
wanted a fun way to practice addition with my son and came up with this
one on the spur of the moment. It was such a hit. It can be modified to
make it harder or easier, and variations could be invented for
subtraction as well, or possibly even multiplication or division."
EDITOR'S SUBMISSION: Peeps Math with Tabitha from Christopher Danielson at Overthinking My Teaching. A slice of number thinking, representation and a teaching demonstration all rolled up in one.
Eva the Weaver
Algebra A Visual Approach to Simplying Radicals
from Chris Hunter at Reflections in the Why. "I'm submitting this blog
post because I think the method discussed is bloody brilliant. I can
get away with saying that because my colleague created it, not me. When
you see it, you'll think 'Why haven't I seen this before?' " Perfectly
fits his blog title!
Flippable Fun with Equations from Jennifer Smith-Sloane at 4mula Fun. Jennifer shares one of her Interactive Notebook (INB) on multistep equations.
If you like the INB approach, be sure to check out the resources at the Julie Reulbach's Math Wiki, too. (Among the many, many resources.)
Understanding Slope from Jamie Riggs at Miss Math Dork shares an early slope activity she did with her students in the resource room that seemed to help the idea develop for them.
Half-your-age-plus-seven Rule
from John Chase at Random Walks. "I saw this graph illustrating the
half-your-age-plus-seven rule for dating ages and I immediately thought
about how this is both (1) a wonderful application of systems of
inequalities, and (2) a wonderful application of inverse functions. It
turns out that that 80-year-old has a *range* of datable ages from 47 at
the lowest to 146 at the highest (good luck with that!). I had never
thought of it that way."
EDITOR'S SUBMISSION: More Imbalance Problems from Paul Salomon at Lost in Recursion. I love these elegant and elemental problems for getting at some really powerful algebraic thinking. Plus he's hosting a contest. Check out the whole series.
Domain and Range
from Shaun at Math Concepts Explained. Shaun tries to explain these
ideas in student friendly language. (But be careful!) In a similar vein
he steps through the procedure of Understanding Point Slope Form.
Geometry Squares
at Five Triangles Mathematics. "This is a middle school level problem
that requires some visual and elementary math skills, but does not
require the Pythagorean theorem." Good rearrangement problem with some
room for generalization. What determines the size of the center square?
This actually inspired me to make a GeoGebra sketch to generalize this with a dynamic representation. Download or mobile-ready applet. Thought this dissection was the essential idea in several proofs of the Pythagorean Theorem.
Speaking of inspiring me to do geometry, have you checked out the amazing art at Geometry Daily?
Probability Probability Tree Diagrams as Puzzles! from Bon Crowder at Math Four. "I teach probabiity trees in Finite Math as well as Stats. And students either love them or hate them. Treating probability trees as puzzles helps them to practice and then see the value of them." Two free downloads with 11 probability tree puzzles.
Puzzle Break!
Quento is a fun sum and difference puzzle app for iOS or Android. Free trial version, 99¢ full.
Here's an example. The 2 number and 3 number paths are shown. No reusing a number. How would you make 8? I just discovered there's a 5 number option!
I made one for the carnival at right. My previous turns hosting were 47, 36, 26 and 22, so...
Thomas Hawk
Teaching Spot the mistake
from Colleen Young at Mathematics, Leanring and Web 2.0. "A
collection of resources which allow students to mark work and correct
errors. This is something they enjoy doing and can lead to some great
discussions." Colleen highlights good TES resources for this idea, but I
can't help linking Michael Pershan's Math Mistakes here also.
Start with One Straight Line
at Moebius Noodles. A post in which Noodles contributor, Malke
Rosenfeld, talks about using blue painters' tape to create an
environment that fosters mathematical exploration.
EDITOR'S SUBMISSION: Speaking of Malke, do not miss her post at The Map Is Not the Territory on Adventurous Professional Learning, with a great video about what success looks like in her classroom.
What Goes Here? from David Coffey at DeltaScape shares an outstanding algebra lesson and teacher-thinking from one of our novice teachers, Alyssa Boike.
Mathematics: The Science of Patterns from Guillermo at Mathematics and Multimedia. "A post celebrating the beauty of mathematics through patterns."
This is where I'll throw in a plug for Fawn Nguyen's single serving Visual Patterns site. Many great problems.
On the theme 'Mathematicians Ask Questions', Denise Gaskins at Let's Play Math, the founder of this here carnival here, looks at questions for teachers or students to ask.
Subtraction with Borrowing: The Video
from Julie at Craft Knife. "Faced with a kid unenthusiastic about
learning subtraction with borrowing, I asked her to film a video
tutorial to teach other kids how to do it. With that, I immediately
found myself with an enthusiastic, highly engaged kiddo..." Julie then
goes on to use the video for some analysis of the teacher's
understanding.
Let's Get Talking
from Oluwasanya Awe at MatheMazier. "I'm a student and I have worked
as a tutor for about 3 years and I just thought I should 'pour out' some
of the cries of people I have taught. I want this post to bring to
mind, the problem children face in Mathematics and also to question
teachers." A dozen short student complaints and misconceptions. He's
also wondering about Points, Shapes and Space.
Morris Motorcycles
Recreational Math & Logic Puzzles
from Sue VanHattum at Math Mama Writes. "I went to a Pi Day Puzzle
Party and had a blast. Here are the puzzles. (No pi involved.)" Very fun
story from Sue.
10 Fun Parent-Tested Math Board Games
from Caroline Mukisa at Maths Insider. "Board games are a great way to
make math practice painless. For those of you fellow board games fans,
here are 10 recommended math board games, along with what parents have
to say about them."
Encouraging a Love of Math
from Tricia Stohr-Hunt at Bookish Ways. "I teach preservice teachers
and spend a lot of time thinking about ways to get kids to love (or at
least like) math. This post is about some of the things I suggest with
links to additional resources." Puzzles, art, books and more, with loads
of links.
Math Art - Geometry from Julie at Highhill Homeschool: Many kids are
very artistic and find learning geometry through art very rewarding. Geometric math art is always close to my heart. Julie is using activities from Mathematics in Space and Time by John Blackwood. Julie also shared Secret Codes
"Secret codes are really complicated
patterns. They are fun to figure out and good for building creative math
skills."
Practice Basic operations worksheets generator UPDATED! from Maria at Homeschool Math Blog. She's added more customizable features to their free random problem generator. (Please find a creative way to use such problems.)
Soggy Math from Ritsumei at Baby Steps. "Wherein we have an impromptu math lesson in the swimming pool."
61?
That brings MTAP61 to a close. I think we're still looking for a host for 62, and I highly recommend the opportunity. Just want to submit a post? Here's the link!
But I can't close without acknowledging that we all know the number 61 can really only mean one thing in the spring...
See Mid TN
Roger Maris hit the unimaginable 61st home run in 1961 to break Babe Ruth's single season record. Some people still consider it to be the record as those who have broken it were allegedly using chemical enhancements. I promise you a clean blog carnival with no unnatural math enhancing compounds. Of course, there's math in those stats: what portion of his hits were HR, how to compare his season of 61 to Ruth's season of 60 (in only 151 games), how do they compare to modern hitters, how hard is it to hit a home run... it could go on as long as the whole season. Maris even has some advice that's just as good in math as in baseball: "You hit home runs not by chance but by preparation."
Image credits: Flickr users noted on their photographs. I think I made all the other ones!
Then maybe you'd be kind enough to try the adding and subtracting integers game I made in Scratch? And give me feedback?
I've been wanting to get better with the Sequence command in GeoGebra. It's a powerful tool for repetitive computation or construction, and math, of course, is full of the patterns. And then I saw...
From the perpetually fascinating Geometry Daily. Perfect opportunity for Sequencing.
First I dug into the geometry a bit. It seemed to me like the interesting bits were the 90º turns and the constant (looking) increase in scale. That means you could do them as a series of dilations - but it was complicated to think about the centers of dilation. Probably better to just figure out the radius and center of each circle. It seems important that the circles osculate - the kissing is a big part of the visual effect.
I was also thinking about what you could generalize, as that is the point of making it dynamic.It's also easier to build those things in as sliders at the beginning of a sketch than editing them in later. (Though it's not that much harder.) The angle between circles, the number of circles, the dilation ratio... pretty good start.
To build in the angle I just rotated the the circle defining point by the slider and the opposite of that angle. And then I made vectors in those two directions from the center of the circle. I was thinking I needed those directions to build the new circles. The radii of the new circles would just be a, a^2, a^3 if a is the scale.
Then came the messy thinking. The center of the first circle is a* the original radius in the direction of one of the points. Then the 2nd circle is.. the 1st radius plus a times that radius in the 2nd direction. I was working it out symbolically, but now that we've got a picture...
So thinking about the centers, I had to organize my data. Usually I scribble on an envelope, but didn't have one handy, so I used Word.
Pretty neat once I sorted it out. I decided to separate the circle centers by the two directions since it was really massy trying to come up with a single sequence to describe the pattern. The key here was the Sequence command. (Quotes are from the GeoGebra Wiki.)
Sequence[ <Expression>, <Variable i>, <Number a>, <Number b>]
Yields a list of objects created using the given expression and the index i that ranges from number a to number b.
Example: L = Sequence[(2, i), i, 1, 5] creates a list of points whose y-coordinates range from 1 to 5: L = {(2, 1), (2, 2), (2, 3), (2, 4), (2, 5)}.
A list is what you would expect:
Lists
Using curly braces you can create a list of several objects (e. g. points, segments, circles).
Example:
L = {A, B, C} gives you a list consisting of three prior defined points A, B, and C.
L = {(0, 0), (1, 1), (2, 2)} produces a list that consists of the entered points, as well as these nameless points.
Note: By default, the elements of this list are not shown in the Graphics View.
But I couldn't get it to work. I often try to do too much at once, so
I decided to make a list of the powers of a, and then the coefficients
of u and v. The relevant commands in GeoGebra are:
It
took a little monkeying around to get these next commands to work. I'm
not sure if copying from Word was introducing extra characters or what.
Sequence[B'_1 + Element[cofs, 2n] u + Element[cofs, 2n - 1] v, n, 1, 10] (list1)
Sequence[B'_1 + Element[cofs, 2n + 1] v + Element[cofs, 2n] u, n, 1, 10] (list2)
Then the circles are easy!
CirclesList1 = Sequence[Circle[Element[list1, n], r Element[ays, 2n - 1]], n, 1, N]
CirclesList2 = Sequence[Circle[Element[list2, n], r Element[ays, 2n]], n, 1, N]
I'm pretty happy with the results. If you'd like to play with the result or remix it for yourself, it's on GeoGebraTube for download or as a mobile-ready applet.
Geometry Daily is a good source for GeoGebra inspiration, as well as good geometry, and just beautiful art. (Usually I put them on my Tumblr.)
I've long been curious about Dan Ariely's work. Predictable irrationality? That's students! (The place to start is probably his good TED talk.)
So I'm auditingish his Coursera course. His three books were bundled for Kindle at $20, and there's a number of resources available at the course site. I promise not to just blurt my notes all over the blog like I have been doing for the creative learning course, but only share bits of interest to teachers.
The idea of Behavioral Economics is that, though people are irrational, they are predictably irrational. Ariely is a big believer in the experimental method, which is extremely refreshing compared to the highly theoretical usual economics which relies heartbreakingly on weak correlation coefficients. (That's pretty pure mathematics snobbery, sorry.) His interest in it has a sad start with an early burn tragedy in his own life; his treatment plan got him wondering if there was a better way and how could you find out. The first application to teaching is this mindset. Teaching is so hard and complex, getting an idea and figuring out how you can test it is crucial.
What follows is super-abbreviated and concatenated.
Week 1 key concepts that apply to teaching:
Choice Architecture: Inspiration: two nearly identical Scandanavian countries have vastly different organ donation rates. One is an opt-in system and one has an opt-out system.
Experiment: People are given a choice between two people and asked which is more attractive. 50-50 results. An irrelevant third choice is added that is a less attractive version of Person A. Person A now gets selected as more attractive at a 2 to 1 rate. (This is called the Decoy Effect.)
Principle: how we offer choices has a great deal to do with what people do with them. In general, make the default something you want to happen and keep the number of choices down. I violate this all the time, making choices in class too open, too complex, or too unclear. Typically people vastly underestimate the effect of choice architecture on their decisions.
Participation: a sub-area under choice architecture. More participation: forced choice, color-coded choices, fun registration, simple straightforward information, few steps, reward participation, more complexity Less participation: stress the importance or urgency, require multiple steps,
Imprinting: Inspiration: A savvy pearl producer found no demand for black pearls and created a huge market for them by setting the price point at exorbitant and only offering them from one exclusive Manhattan jewelry store.
Experiment: Participants were asked to write down a random number by a range of items. Later their bids for those items correlated strongly with the random number written down, while they maintained that it had absolutely no effect.
Principle: Our first impression of the worth or value for something has a huge effect on future valuation. This makes me think about how teachers describe assignments or homework. We should be more intentional about the imprinting we can control over students future valuation of the tasks we want them to do. If we want to break an imprinting, we need to do everything possible to distinguish our idea from related ideas. Starbucks made their experience as different as possible from Dunkin Donuts, and people didn't evaluate their prices by DD prices.
Self-herding: Inspiration: having broken down and paid Starbucks' prices for coffee once, we are much more likely to do so again.
Principle: Once we have made a decision, we are likely to do the same thing again, because we remember our actions better than the emotional states that led to those actions. Good and bad news for teachers, in terms of habits formed and repetitive behaviors. The positive spin on this for me is that it is worth a significant teacher investment to start something new because it will have momentum for the students by this principle. We won't have to sell it so heavily the 2nd and 3rd time.
Pluralistic Ignorance: Inspiration: in this video clip of an Ariely lecture, he speaks from a randomly generated script for 3 minutes with no student questions. Finally he stops and poses the question, "why did nobody stop me and ask what I was saying?"
Principle: we make inferences from the behavior of people around us that we value more than our own perception. This is deadly for us, I think. How many times have I thought, or have I heard a teacher say, "but no one had any questions!"
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