An Ad?

While Math Hombre is traditionally ad free, I do owe a link to my daughter Ysabela. In exchange for her review of Babymouse: Dragonslayer, the excellent math/fantasy graphic novel by Jennifer Holm, I promised to plug her market craft.

Yzzy's Sticky Silverware has recycled silverware affixed with two powerful magnets and artistic embellishments to make a beautiful and practical refrigerator decoration. A steal at $1.25 decorated ($1 plain), you might pay up to $5 for a similar product at a craft show. Check it out! Mailing is at cost.

Some of the math involved was: figuring out the unit cost from the total materials cost, and also balances, profit margin, the length of the total wire based on the wire per silverware, (which was a pretty good upper el measurement problem). What is her break even point? That's the next calculation!

Area Block

New game! Blokus meets Nim, this game works on area and strategy. Tested with fourth and fifth graders, but flexible upwards with sophistication in area computation techniques. Get it as a pdf here: AreaBlock.pdf

Materials: Game Board, 2 different color pens, pencils, crayons or markers.

How to play: Players take turns making a single shape on the board that has an area of 10 or less. The game is done when the board is filled, and the player with the most squares covered wins. The table is for recording how many squares you cover each turn.

The first player to go can only color up to 8 squares on their first shape. (Otherwise it would always be best to go first.) After that the limit is always 10.

The squares colored in to make your shape need to share a side, not just touch at a corner. (The shape you make has to be a polygon.) A player can even use slanted lines – as long as they can figure out the area for their shape! When you are making a new shape, it does not have to touch your old – you can put it anywhere there’s room.

After you make or shade in your shape, record the area of your block – you don’t want to miss any points. The next player then colors a polygon of up to 10 squares of area in a different color.

When the board is completely filled, the game is done. Total up your squares and see who won!

Notice you can check if all the squares are counted by adding both scores – you should get 100.

Whoever lost gets to choose the next time if they want to go first or second.

Variations:
• The board can be changed to have holes or blocks or be a different shape. As long as there are 100 squares. After trying these boards, make your own.
• The game can be played with dice. Roll 2 dice, and you can fill in the total rolled.
• Advanced players could try where their score is not the area – but the perimeter. (You might want to try that with no slanted sides.)

Board 0:

Board 1:

Board 2:

Board 3:

Board 4:

It's pretty fun! I haven't been able to spot a degenerate strategy yet. Give it a try, and let me know what you think.

Group Matematika SMA

Mungkin sebagian besar kita sudah memiliki akun di salah satu jejaring sosial paling banyak anggotanya di seluruh dunia, Facebook. Ya, sebagian besar kita sepertiya lebih betah untuk berlama - lama duduk di depan PC kita hanya untuk sekedar mengupdate status ataupun membalas di wall teman kita. Nah ngga ada salahnya kan kalau sebagian waktu anda ketika mengutak - atik fb anda mampir ke group

GeoGebra: Triangle Tuning

What can we deduce from the side lengths of a triangle?

This is a preservice teacher activity, easily adaptable to middle or high school use. GeoGebra is a free dynamic geometry program available as an online applet or downloadable program, from geogebra.org.

Objective: TLW explore triangle properties relating type and side length.

Schema Activation: What are the 7 triangle types? Fill them in in the ‘Type’ column below. You'll fill in the other columns as you go through the activity.

.....Type ....................... Examples [Like (3,5,5) ] .......................What do you notice?
1.

2.

3.

4.

5.

6.

7.

Focus: One of the reasons dynamic geometry is so powerful is the support it allows the teacher (or curriculum designer) to give students for finding examples. Lots of examples. As we’ve discussed in class, the natural way we reason is to go from lots of specific examples to the general.

Activity:
1) Open the sketch TriangleBySide.ggb. (Online at faculty.gvsu.edu/goldenj/TriangleBySide.html)

a. Collect at least 2 examples of each type of triangle. Where possible, try not to have the triangles be similar, where all the sides are multiples of another triangle.
b. Which were hardest to find? Was it something to do with the type or how you were looking?

2) Open the sketch PythagoreanData.ggb. (Online at faculty.gvsu.edu/goldenj/PythagoreanData.html)

a. Look at your right triangle examples on this triangle. How do you think ancient mathematicians noticed this cool pattern with the squares

b. Check your other examples on this sketch. For each, record whether the sum of the areas of the smaller squares is <, =, or > the area of the large square.

c. What pattern(s) do you notice?

d. Why do you think your pattern(s) might be true?

3) Go to http://teachers.henrico.k12.va.us/math/GeoGebra_Site/pythagoras/pythaPrblm1.html (link on Blackboard) for the Ladder sketch. Answer the questions there.
a. How high will he get if he places the ladder 3 m off the wall? Drag the ladder point and find a solution. Sketch the solution with all its lengths (s, r, h) on paper.

b. Now, calculate the solution of task (a) on paper. Which lengths are given, which are sought? Do you get the same solution?

c. At what distance of the wall should Pythagoras place his ladder in order to reach the window 4.50 m above the ground? Sketch the solution with all its lengths (s, r, h) on paper.

d. Now, calculate the solution of task (c) on paper. Compare your solution to your sketch?

e. As a teacher, what do you notice about the difference between doing the tasks in sketch and on the paper?

4) Open the sketch AdjustableLadder.ggb (Online at faculty.gvsu.edu/goldenj/AdjustableLadder.html)

a. What young Pythagoras might not have known is that a safe ladder ratio is 4:1 for the height to distance from the wall. What length ladder does he need to reach the window safely? Is there one solution or more?

b. How would you solve this problem on paper?


Reflection:
a. What confirmed, new or deepened understanding did you develop regarding triangles through these activities?

b. How did the dynamic environment affect your understanding?

EDIT:
Be sure to look at the comments. Scott Farrar has some notes on using this with 9th grade geometry, and has added a complimentary activity.

• http://scottfarrar.googlepages.com/triangleineq.html
• The worksheet http://scottfarrar.googlepages.com/TriangleCategories.pdf

Math Resources

Moodle
Moodle Math documentation
http://docs.moodle.org/en/Mathematics
Pretty technical information, but accurate and detailed.

Book about teaching math with Moodle
http://www.packtpub.com/moodle-1-9-math/book

Content
http://illuminations.nctm.org/
NCTM’s free lesson and resource center, with many online applets. Note the Calculation Nation link leads to some excellent 1 and 2 player games, but require a free login account. (Guest access to try it.)

http://ohiorc.org/for/math/
An Ohio Resource Center with links to many topics by grade and content. Some high quality resources.

http://nrich.maths.org
From the British equivalent of the NCTM. Once you get past how they say ‘maths,’ there’s an excellent supply of interesting problems, some clever student solutions, and open challenges. Also features some interactive applets.

Internet Resources for in class use
http://math.hippocampus.org/
An extensive collection of online exposition of topics from algebra and calculus. (As well as Bio, Chemistry, Physics, etc. Some support for Spanish.) Includes a voice delivering reading of the text (with a skip forward feature), and assessment along the way. Not clear what the purchase of a license adds, but everything I’ve seen was free.

http://www.imagination3.com/
GE’s interactive whiteboard, open to collaboration, optional graph paper background. Colors and some shapes are possible. You can save files.

http://www.wolframalpha.com/
Wolfram|Alpha, a free computational engine online. Amazing and worth exploring. Also some encyclopedia function. Supercedes some of the older online utilities.

http://geogebra.org
GeoGebra, a free dynamic geometry program that can be used as a web app or by downloading and installing. Some powerful algebraic applications, too.

http://www.plu.edu/~heathdj/java/
A collection of java applets, from many different mathematical content areas.

http://www.univie.ac.at/future.media/moe/onlinewerkzeuge.html
Powerful collection of online tools. One of the best online function plotters (graphing calculator) is here.


Games
http://www.funbrain.com/brain/MathBrain/MathBrain.html
K-8 math games. Not very challenging, but there are students for wom this will be just right. Connected with Poptropica, which my kids love, and features some decent problem solving.

http://gprime.net/game/
Huge collection of flash games – not all kid appropriate. Some excellent math and reasoning games, though. Try 3D logic, a maze game on 3 cube faces (http://gprime.net/game.php/3dlogic), gridlock, the classic game (http://gprime.net/game.php/gridlock) or grow island, a deductive reasoning game (http://gprime.net/game.php/growisland), and one of a family of grow games (http://www.eyezmaze.com/grow/v3/index.html).

http://www.logicmazes.com/
Very challenging puzzles by a very interesting man, Robert Abbott. Also links to some games of his creation.

Utility
http://incompetech.com/graphpaper/lined/
Free graph paper pdf generator. Very handy.

Teacher
http://www.learner.org/resources/browse.html?discipline=5
Annenberg Teacher Resource Center. Extensive collection of lessons, videos of students and teachers, and professional development materials. All free. Sortable by grade level.

Blog guru

Sebelumnya saya pernah memberikan daftar guru - guru yang ikut ngeblog membahas matematika pada artikel blog matematika dan blog matematika part 2. Tentu saja masih banyak lagi guru yang ngeblog tetapi belum saya masukkan ke dalam daftar tersebut, sebab ternyata semakin hari semakin banya guru yang menyeburkan dirinya ke dunia maya untuk berbagi ilmunya dengan anda semua.BTW, setahun yang lalu

Money Problems

Excellent variation on the "how many ways to make change problem" at the New York Times today. The Freakonomics column is reporting the work of Patrick DeJarnette. I can see giving this problem from 4th grade to linear algebra!

Click on the money tag to see my previous money games. Click on the cartoon to go to the excellent Non Sequitur website.

Math Teachers at Play #16

The new carnival is up.

A clock face activity for learning how to tell time supports students with a nice simple representation. The biographies reviewed here look like must buys. Very nice math teaching memoir from Math Mama. We should all write more of these. Denise, carnival originator, starts a good division discussion. Some interesting online math games are highlighted at the Innovative Educator.

The order of operations conversation reminded me of a student teacher I just saw who has picked up the nickname Aunt Sally. ("It's Not Funny," she said in response to my giggle.) I'm hugely in favor of a four step Order of Operations. It's also a great opportunity to discuss with students the difference between a result or theorem and a convention.

Anchor Charts

While we did this as a teacher prep activity, I think it would be interesting with K-12 students as well. I first heard about anchor charts in Mosaic of Thought (I think), and like how they serve as both assessment and culture building. I've had students make charts about a particular concept, about what to do when you're stuck, and for what it means to do, learn and teach mathematics.

For this most recent lesson I had the students read "Mind mapping As a Tool in Mathematics Education", by Astrid Brinkman from Mathematics Teacher, Feb 2003. (They had previously expressed curiosity about and a lack of experience with concept maps.) One reason I love these is that they are frequently surprising. I expect one like this, that echoes the NCTM process standards we emphasize in the first unit: (Remember you can click images to see them full size.)

But then they go and make these completely original things like:
with different processes emphasized

and

Highlighting the difference and connection between
school mathematics and real mathematics.

Instructions to the students:

Activity: Anchor Charts
The following example comes from Ellin Keene’s (2008) To Understand: "Teachers generate anchor charts to capture and celebrate increasing sophistication in oral language use." (p. 278)
If you substitute ‘understanding learning in mathematics’ for ‘oral language use,’ you have the purpose of this activity. (Follow a link for a free pdf of the first chapter, under samples.)

Create an Anchor Chart for Learning in Mathematics
1. Identify the concepts and ideas that you want to remember as they relate to doing mathematics.
2. Develop an anchor chart that captures and celebrates your increasing sophistication in understanding “Doing Mathematics.” This might be a list, or a mind map, or a representation of your own creation. You decide – just be prepared to share your chart during our next class.
3. Be sure you leave 10 minutes to reflect.

Reflection: How well does your group’s anchor chart capture what you want your future students to think of hen you ask them “what does it mean to do mathematics?”


Home Extension: You might want to check out how math teachers use anchor charts at books.google.com - Integrating Literacy and Math. by Ellen Fogelberg, Carole Skalinder, Patti Satz, Barbara Hiller, and Lisa Bernstein.

Other interesting examples:

(OK, the last one's mine.
Fair's fair if I'm putting up my students.
No, I don't know what's up with that guy's hair.)

Send me yours or your students' and I would be happy to post that!

Geogebra

I know I've gone post crazy. I will stop soon, really.

I'm trying Geogebra this semester with my Geometry K-8 class, first computer session tomorrow. It's a freeware java implementation of dynamic geometry quite similar to Geometer's Sketchpad, with an even stronger algebraic interface. In addition to playing with it to get a sense of what's going on, I've got two sketches to look at:

Square Not Square Where you adjust quadrilaterals by vertices
to check the 'real' type, as all 9 look like squares.

Quad By Angle Where you determine a quadrilateral

by the angles rather than by edges or diagonals.

Links lead to webpages with the java sketches, as geogebra has a great export to webpage feature. They are java, so some browsers don't handle them as well as Firefox. If you prefer the actual geogebra sketches, it's SquareNotSquare.ggb and QuadByAngle.ggb. Most of the sketches I've seen so far have been for HS or university, so if you're interested in K-8 uses, please let me know!

EDIT: Followup - the students were mostly positive, and some extremely positive. Definitely a better reaction than to the old computer labs with sketchpad. There was more interest in the program because it is so accessible; many more of them saw themselves as potential users of this time. It also connected with people's desire for visual representation. Makes me wonder, have I been underusing manipulatives this semester?

Book Review: Babymouse

Read a very cute book recently, Babymouse: Dragonslayer, by Jennifer Holm and her brother Matthew. I had never read Babymouse before, but it was recommended by Jen Robinson (who herself was recommended by Kathy Coffey, my guru in all things about teaching reading). Jen Robinson's blog is a great resource for anyone interested in kid's lit as a parent, teacher or reader.

Babymouse is a math-hater, because she is bad at it. Her teacher, shockingly, starts her on her adventure by sending her to be a mathlete. While there is no math per se in the book, it is all about immersion and expectations by the Mathlete coach, and employment and approximation on Babymouse's part. And all with excellent connections and fantasies of our great fantasies, Narnia, Lord of the Rings and Potter. A homerun all around. My only unrequited wish - more actual math. All that for only $6 at Amazon.

The italicized terms here are from one of the best frameworks for teaching and learning I've ever seen. It's really the lens through which I evaluate and work on my teaching. Brian Cambourne's article "Toward an educationally relevant theory of literacy learning: Twenty years of inquiry" is available online here.


My daughter Ysabela's review is here. She says the title misspelling is intentional...

Math Teachers at Play 15?!

The new carnival is up at mathfuture. One of the most interesting things about it is the host site, which looks at Web 2.0 math applications. For example, their GameGroup, which I may be seeking to join.

Other than that the 2 posts that were most interesting to me are a description of a role-playing site for math-centric careers and 25 uses of Wolfram Alpha (which I love).

Denise has a good starter on Mental Math, but I don't see these as techniques you teach as much as ways of thinking that you demonstrate and can grow out of kids' number sense. I was saying to Xavier (my 4th grader) last night, knowing 6x9 is not as important as being able to find it efficiently, and knowing things like it's connected to 6x10 - 6 or 6x5+6x4 is much more important.

Perubahan Tampilan Blog

Blog in memang memiliki beberapa perubahan tata letak sejak kemarin, bukan karena perubahan itu disengaja tetapi lantaran karena terpaksa harus dirubah. Awalnya hanya ingin mencoba fitur baru yang ada di blogger ( fasilitas read more ), padahal sebelumnya blog ini sudah menambahkan beberapa kode agar tampilan read more berjalan. Saya sendiri tidak paham dengan yang terjadi, ketika setingan edit

Angles

What do you see as the big ideas with respect to teaching angles?

To me:

• filling around a point, no gaps or overlaps between two boundaries
  
• connection with a circle (filling all the way around a point) - important for units
• size of the angle corresponds to how open

So I love to begin teaching angle with pattern blocks. The activity I start with is adapted from one taught at GVSU by Jan Shroyer, don't know where she got it or whether she wrote it. The activity as a Word .doc is here. If printing from the web, try to size the pattern blocks so they are life size. (Doesn't affect the angles, of course, but makes it much more natural.)

A Very Special Blossom

A blossom is a special pattern in mathematics when copies of the same shape are arranged to fit together all the way around a point. Try to blossom the following shapes. Record how many fit around a point. Sketch either the shapes or the edges that meet at the point.

What do you notice?

Would the blue or red pattern blocks blossom?

Teaching notes: the white rhombus is chosen especially, since the wide angle doesn't blossom. The narrow angle will provoke a little discussion, 11 or 12 to blossom, because if they're tracing one block, the thickness of the drawn lines add up. The wide angle will draw responses of 1, 2, 3, 4, and 8. 4 will usually be two wide and two narrow angles (tessellating the rhombus) and the 8 is from filling the rest with the narrow angle. The red trapezoid and blue rhombus will sometimes have the students seeing blossoms with the narrow angle but not the wide.

After the connection of 360 degrees with filling all the way around, these blossoms can be used to deduce the measures of the pattern block angles. This is nice in conjunction with measuring practice with a protractor or angle ruler.



Filling Time

Use pattern blocks to measure the following clockwise angles. (Start at 12, and then measure in clockwise direction to the other edge.) Use all of the same unit for each angle. Measure each angle twice using different units, if possible.

Teaching Notes: You will see students make a lot of connections with congruent angles doing this. Also, there will probably develop an appreciation for the smaller angles as units. There is a natural tendency to measure the smallest direction, so that will bring up the clockwise/counter-clockwise thing, which is a nice connection with rotations, which will be a great way to teach angle to kinesthetic learners. The middle left angle brings up the idea of partial units, as it is not a whole number of pattern blocks for any of the shapes. The scientific standard is to measure to half of the smallest unit, so a good answer is 1 1/2 white rhombus (small angle) or 1/2 square. How many green triangles is a nice discussion.

The next activity I'm including the way I work it for preservice or inservice teachers. Easy to adapt for 5-12th grade students, though. The Word .doc version is here.

Telling Angles

Objective: TLW expand their understanding of angles, connect with the angles on a clock face, and use reasoning to find angle measures by comparison with known angles.

Schema Activation: What do you know about angles, measuring them and degrees? List your top 3 facts or bits of relevant knowledge.







Activity:
1) Forget the time, what angle is it? Record the angles made by the clock hands below. Add one sentence of justification for how you know.

2) Teacher question: why might I have sequenced the clocks the way I did?




3) Record the angles made by the clock hands below. Add one sentence of justification for how you know. Notice the hour hands are no longer pointing directly at a number.

4) 11:50, 1:10, 3:20, 7:40. For each time, draw in the hands precisely, and then determine the angle between the hands. Describe your process for each time. Start with the one you think would be easiest.

Reflection: What 3 ideas do you most want your students to understand about angles and angle measure?







Extensions: Challenge questions:
a. Is there a time for any angle?
b. Is the clock more likely to have an acute, right, or obtuse angle when you look at it?
c. How many degrees does the angle change in 1 minute? 5 minutes? 10 minutes? Does it depend on what time it is at the start?

Mohon Maaf

Mohon maaf atas ketidak nyamanannya,saat ini blog sedang dalam perbaikan, anda bisa kembali dalam 24 jam kedepan.