I was inspired to do this by the neat quadrilateral hierarchy sketch shared this morning on Twitter. But I got wishing they had made the types accurate - that you could only make squares in the square spot. And that their hierarchy used the inclusive definition of trapezoid. (Pet peeve of mine.) Then I thought what if the types lit up when you make the shape? That led to the sketch pictured below, available as a ggb file (EDIT: in GGB 4!) or as a webpage.
When I included it as an applet here, it just didn't work as smoothly as it does over at the geogebra hosting, or by displaying the file directly in a browser.
It was very fun figuring out the conditional tags to make the names show up. I think I've covered most of the corner cases. Figuring out a way to do convex/concave and quadrilateral or not
It was quite handy knowing multiple definitions of each type, which me wonder about a scaled down version of this as a problem to assign to students.
Where do you stand on the trapezoid definition? Is a parallelogram a trapezoid? (I say yes!)
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.
They were able to tap into the arguments that people have contrarily, though. Math is about number-crunching, plug and chug, is boring, about right and wrong. I asked them to discuss at there tables how to counter those arguments, and they said:
Everything can be mathematical - there are numbers everywhere. I think in numbers! Any lesson can tie in.
Numbers can be manipulated in many different and unique ways.
The basic skills are concrete, but the application is creative. Like physics: need math to describe creatively.
The reasoning is creative. People discovered mathematics.
The communication of math is creative. Explaining how things work.
Creative how mathematicians come up with the formulas. Might have just discovered it accidentally.
The different ways to represent: graphs, tables, etc.
Math is the universal language. Everyone can communicate in it.
Creativity is important to me (see these other posts) as a goal for my students, and I think opportunities for creativity add a lot to the likelihood of engagement.
It was very insightful to me how they focused on the communicative aspect of mathematics. If math is a language, it may or may not be creative, much as uses of language may or may not be creative. As we share our own genuine thinking, and the way we perceive the world (or a problem), we create opportunities for creative expression.
The class moved on to look at how they communicated their work on a problem of making quadrilaterals by folding a square (an extension of this nice 3rd grade problem filmed by Annenberg - Teaching Math, Lesson 20). And immediately focused on what the answer was and were they right. After our years of school mathematics, we have definitely been well trained.
In the second workshop, we considered the quadrilateral types, and in particular the idea of nested categories or hierarchical sorting. I show this weird little travelogue:
(I want to update it and maybe make that into an animoto, but the ppt file is corrupted, so it will be more work than I have time for right now.)
The students then made posters of some of the quadrilateral types, striving for a variety of examples, and to be creative in making the posters. Critique and discussion of the posters brought out the discussion points I was hoping for, like which properties are necessary, considering symmetry as an important characteristic, and whether trapezoids should have exactly or at least one pair of parallel sides.
Trying to make space for creativity is not going to be a one lesson effort, but hopefully a theme for the whole semester. I can't wait to see what happens.
EDIT: updated the slideshow to have more visual cues and a couple extension slides. I wanted there to be more to notice.
EDIT2: added student posters for the quadrilaterals. We worked on generating a variety of examples. Some made only the specific types, but some made a variety of types that fit the required properties. I like having both kinds of posters!
Grand Valley State University sponsors a terrific little math conference each year called Math in Action. I think started by Jan Shroyer back in the whereupon. It has 30ish workshops for K-12 math teachers, mostly very practical.
I'm hosting 6 wonderful preservice teachers presenting geometry games for K-8 teachers, so I'm posting the electronic versions here for people to be able to download. If you were a participant and wanted a Word file to edit instead of a pdf, just email me. The address is available on my workpage, linked on the right.
As a bonus, here are the two bonus games from the Quadrilateral Concentration sheet. Both are other uses of the Quadrilateral Cards. Not included here is Quadrilateral Euchre, which is Euchre for a partially ordered card set. Verrrry geeky in a midwest sort of way.
Quadrilateral Go Fish
Materials: Deck of Quadrilateral cards. Best with 3-5 players.
Setup: Deal 5 cards to each player. Put the rest face down in the middle, either in a neat stack, or mixed up in a big pond. Gameplay: Start to the left of the dealer. On a player’s turn they can ask a particular player for a specific property. For example: “Do you have a shape with opposite angles congruent?” You can not ask for a shape by name. (“Do you have a rectangle?”) If the player has a card like that, they have to give it over. If they have more than one, they get to choose which card to give away. If you have a matched pair of the same type, you can play them down.
Winner: First winner is the first player to go out. Second winner is the player with the most pairs.
Variations: • Instead of asking by properties, ask by name. • Start with 7 cards. • Allow players to play cards on other people’s pairs. (If you have a pair of rectangles I can play a rectangle.)
Quadrilateral Guess Who
Materials: Quadrilateral card deck. 2 players.
Setup: Sort the quads by type. Each player puts one quadrilateral of each type face up in front of them, and the others go face down in the middle. Each player draws a card from the middle and keeps it hidden from the other player.
Gameplay: On your turn you can ask one question about the other player’s hidden quadrilateral. That player answers yes or no. Turn face down the quads you have that don’t match.
Winner: first player to guess the other player’s card.
