Out of bread this morning so ... tortillas for the kids' sandwiches. (Making tight little appetizer style rolls.) The first piece of salami for my Ysabela's funroll (marketing considerations) instantly prompted my math curiosity.
Zero
Any questions?
But then I wondered about what photo would be best for #anyqs?
One
Two
Three
And then just because...
Mmmm, ellipses. Did you see that eccentricity comic recently? (Xavier won't eat salami.)
I think the original photo (Zero) is the best for getting at the question I like here - how many pieces of salami to cover the tortilla? One gets at diameter comparison, Two does that even more literally, and Three might help create some dissonance. Which would you use?
I neglected to take pictures of the fun rolls (TM) despite my recentinterest in spirals. May have to make a jelly roll.
Yeah! Back in 5th grade today. Been too long. The fabulous Mr. Schiller invited me back in, and we hit the ground running with a nice new area and perimeter game: Area Battle. It's close to Area War with a few new wrinkles that make it a better learning game.
I launched the explanation that it would be sort of like War, but comparing area instead. I showed them these two cards:
And asked which had the larger area. Unanimously the class agreed on the one on the right. What is the area? And someone explained how they got 3. "Do you agree or disagree?" Loud agreement. On to:
Students disagreed about which was bigger here, so we counted carefully. Finally all agreed that the cross was bigger.
The last set took a little longer to count, and then had a tie. The students knew that in War you play more cards and then compare. I described how we have a small deck, so for our ties we'd just play one more card, and then highest perimeter wins the tie breaker.
We talked about how in this game you make your own deck, but it had to have one card each with areas 1 square up to 9 squares, and then they could make two free cards however they wanted.
We then explained the full rules of the game. Each team draws two cards - always keep two cards in your hand. Then the team who's turn it is picks a card and declares 'high' or 'low' - whether the highest or lowest area will win this turn. The other team picks a card to play and the cards are revealed. The winner takes both cards. If it's a tie, each tied team plays another card, and highest perimeter wins. The next team takes a turn going first. Play until at least one team has played all its cards. The team that won the most cards wins.
Students did a great job working in teams of two to creae a variety of interesting shapes. We chose the restriction that shapes had to be made of squares and half squares, or half rectangles. Once they had a deck check (1 to 9 + 2), they were free to find a team to battle. there were some rule clarifications to clear up, and a couple people got too competitive, but they really gave it a go. One group played a three team game and that went well, too. After playing they came back together to discuss strategy. They liked having a variety of cards, that hit both high and low. They talked about recognizing a neat idea from another team and then using it. One team started out with a 1/2 square area card but several had them by the end. Universal thumbs up as to whether they would recommend it to other teachers. They were requesting time to play in class later, which is surely a good sign.
Below are some images of student cards, and then a link to the handout if you want it.
From twitter I found out about the K12 Online Conference. In particular about a Scratch session by Chris Betcher (@betchaboy on Twitter). The session has a nice 22 min video about using the Scratch programming language with Year 5 students. Very worthwhile info in a sitcom sized bite. Chris also has a Scratch wiki devoted to getting students going with Scratch, and all the resources you need to get going. Scratch is perpetually one of those things I'm going to investigate when I have obtained some sparemomentium.
I got a chance to work with a local 8th grade teacher who's looking into Geogebra. He sent me some of the state standards he was interested in exploring, and I made up a couple sketches to play around. The first is just a demonstration of the area formula for a parallelogram. It seems so unreasonable that all parallelograms with the same base and height have the same area. That connects with one of Geogebra's strengths to me - providing essentially infinite examples so that students can notice.
The next is probably pointless. I was considering how to make a dynamic area measuring sketch. I thought the advantage would be being able to change the figure, and easily check your answer regardless of how you've changed the shape. I used sliders to build the shapes so that the distances would be nice. Webpage and geogebra file
The third sketch is for similarity - I'll post that later this week with the world's easiest activity.
I'm trying to shift towards standards based grading (SBG) this semester for the content grade in my geometry class, and so I'm paying a lot of attention to the kinds of problems I write. My tendency is to write big, open, sprawling problems that cross many standards, but I haven't yet worked out how to do that and SBG. I know that I want problems where students can show understanding without necessarily getting to a correct answer. Mathematicians are wrong a lot. What distinguishes us, I think, is that we often know whether we are wrong or right.
My students asked to have the time before the test to practice, instead of the book group I wanted to do. (110 min class and a 1 hour exam.) Very reasonable. But that meant that I had to come up with practice problems! I'll post those now, then the test when all students have taken it.
Photo by scrappy annie @ Flickr
Do you give students practice? What's the relationship between practice and the assessment problems? This was a big topic in my student teacher observation this morning also.
322 Midterm Practice
The Standards
A. Analysis of characteristics and properties of 2-D geometric objects: number of edges; side length; angle size; parallel; perpendicular; convexity, etc.
1. concept: definition and recognition 2. application: use to sort and characterize, build or draw 3. combination: consider multiple properties in a single object 4. familiarity with examples: triangles, quadrilaterals, polygons
D. Distance, area, angle
1. concept: definition and key properties 2. formulas: use and derivation 3. connections among formulas
Try your choice of the following problems. Look for problems that allow you to problem solve and share your thinking.
Which standards could you demonstrate on which problems?
1) Connect each side property to an angle property and draw a different polygon to match each pair.
(at least) 3 congruent sides
no adjacent congruent angles
pair of perpendicular sides
(at least) 1 angle > 180 degrees
3 parallel sides
pair of adjacent congruent angles
Draw 3 different connections and try again!
2) Sometimes quadrilaterals are defined by their diagonals rather than by sides and angles. Determine which quadrilateral goes with which definition below, and make your argument. If no quadrilateral goes with a definition, state why. a. Diagonals both bisect each other. b. At least one diagonal bisects the other. c. Diagonals are equal length. d. Diagonals are perpendicular. e. Diagonals do not intersect. f. Diagonals are perpendicular bisectors.
3) On our Area on a Grid class workshop, find the areas of the shapes using formulas.
4) On graph paper, divide a square up into exactly 7 triangles with as many different triangle types as possible. Can you get all 7 types?
5) Area a) Make an area formula for a trapezoid, or prove the one you know, using the formulas for rectangles and triangles. b) Make an area formula for a kite. c) Make an area formula for a chevron, using the diagonal lengths.
6) On graph paper, find squares with areas listed or argue why you can’t: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. a. Note: 5 is possible. b. Find the edge length for each square you found using the Pythagorean Theorem. What do you notice?
7) Draw a 2 circle Venn diagram with the labels only in your mind (eg. a line of symmetry and at least one pair of parallel sides). Write in the quadrilateral types where they go, and give to a tablemate to figure out the labels.
Made a simple hexagon dilation sketch in geogebra for my geometry class. Let's you vary the objects, measures area and perimeter, and control scale factor. Algebra view let's you see individual side lengths also.
It's part of a set of four similarity problems for stations. Here's the pdf.
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.
