Currently reading Mathematical Mindsets by Jo Boaler and continuously stewing over some geometry problems I 'taught' last year.
I need your help.
I did numerous examples with students and students did numerous examples with each other. But I don't know that I actually taught them how to problem solve. To me, these type of problems are very intuitive. How do I make students feel that way?
I didn't give students anything to figure out. I did an example for them, then asked students to do an identical example with different numbers. I didn't ask students to notice or wonder anything. I didn't ask students to think about any patterns.
This has literally bothered me for over two months now. When I would 'help' students and ask them if they had checked their notebook, more than one told me "My notebook doesn't help me." Looking at worked examples was not helpful because I didn't give them anything to make meaning out of. There was nothing to help them do the problems.
Problem Type 1: The Rectangle
It's easy enough for students to figure out and remember that the opposite sides are equal. It's also pretty normal for them to know the diagonals are equal to each other and you can even cut them in half. Then it's pretty easy for them to agree that alternate interior angles are equal. It's a step up for them to know intuitively that each vertex is a 90 degree angle and actually be able to find the measures of the alternate interior angles. It's a bigger step up to realize this diagram contains 4 right triangles and that you need to use the Pythagorean Theorem to find the side lengths or diagonal length. And then piling more and more steps....Angle EDA is the same as angle EAD because there's also an isosceles triangle in here..Then we can use the triangle sum theorem to find angle AED.
And then there's the leap to giving them 3 values and asking them to find 8 other values while also switching back and forth between sides and angles.
Problem Type 2: The Parallelogram
The parallelogram jacks up everything. The diagonals aren't congruent anymore. The opposite vertex angles are congruent but no longer equal to 90. There's no more right triangles, Pythagorean Theorem, or isosceles triangles. But we can still find all the same values.
It just seems like so much. So many different values to find using different strategies. The two problems almost undo each other. "Hey, you know all those things you just figured out? Yeah, they no longer apply and will mess up everything if you try to use them."
How do I give students an experience or notes to look at that actually explain to them how to find these values?
All I have come up with so far is giving them a variety of diagrams with some values already given and ask them to look for patterns. Then they could write down and color code relationships. I also thought about creating a parallelogram and rectangle that they could cut apart or lift up flaps to write the relationships between them.
How do I make these problems make sense instead of like I am just pulling numbers and strategies and out of thin air? {BTW, why do we say out of thin air like air can be thin or fat? Hmm...}
How do I make this into a puzzle where students can figure out meaningful relationships between angles, sides, and missing values?
I made this flip-book that you can use if you like... http://newellssecondarymath.blogspot.com/2016/02/quadrilaterals.html
ReplyDeleteMy students do very well with Quadrilaterals (probably best unit that they perform on). As we write out the properties, students mark it on their diagram and we discuss it further. Also, as we go through the example problems, students will solve it but I ask why? What property did you use? They get annoyed at first, but quickly catch on and can easily justify why they are setting up a problem a certain way.
Thanks for sharing those resources. I liked reading your reflection about the lesson as well.
ReplyDeleteHow many class periods did this take? Did the students add the notes from their whiteboard to paper anywhere? I would definitely want to include that. After this lesson did students spend more time solving problems for each individual quadrilateral?
BTW, your resources are lovely! =)
It takes about 2 weeks for our Quadrilaterals unit... I'm hoping to make more time for it next year because I want them to make a comprehensive quadrilaterals project. (know any? :))
ReplyDeleteNo my students didn't add their notes to their paper anywhere... I really want to add that for next year in their notebook because they referred back to it (more than their flip-book) or took pictures of it for reference.
Yes, they did... I am thinking of making a "challenge problem of the week" on my whiteboard and have difficult problems where they have to solve for many missing variables.
Thank you so much! That means a lot coming from you :)
I'm already planning to use the flip book on one page and the whiteboard lists on the other in our INBs.
Delete;)
With this sort of thing, I get the best results by circulating and prompting (what do you know about these two lines? What makes you say that? etc.). Also, making posters of geometry theorems is a good activity for those days when half the class is gone for a sports game or some such.
ReplyDeleteAlso, your students may enjoy this: http://map.mathshell.org/lessons.php?unit=7325&collection=8.
ReplyDelete(I haven't used it, but I've used other MAP resources. In my experience, they engage the kids at first but then they get sidetracked socializing when they're meant to be comparing strategies.)
I definitely agree with the prompting. I just felt like I had to do that so much for each student that no one could work without talking to me first. Katrina's resources look like a much better reference to get students started on their own.
DeleteThanks for the link, I am a fan of MAP resources but rarely use them. This looks like a great follow up and way to emphasize Katrina's vocab and properties.
Hi - I have a few geogebra lessons that my students work through individually or in pairs. They need work, but my intention is that students figure out the pattern themselves and write the "rule" in their own words. (All of our kids have MacBooks - geogebra is something I use A LOT in geometry.)
ReplyDeletePS - Blogspot thinks that I am my husband....I can't figure out how to change this. If you would like to follow up, my email is khicks@madison-schools.com.
Have you ever used Geogebra on ipads?
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