Annotated Lessons
Kendyll Herauf
April 12, 2020
Activity 1
Outcome: SS8.1 Demonstrate understanding of the Pythagorean Theorem concretely or pictorially and symbolically and by solving problems.
Geometric concept: Pythagorean theorem – distance, measurement
Grade level: 8
Activity name: Eggs in a Basket
Source of activity: My Phys. Ed background knowledge
Description of activity: This activity would be done as an introductory lesson to the Pythagorean Theorem. The intent to this activity is to engage learners, especially those who enjoy kinesthetic movement and learning.
The game “Eggs in a Basket” has been done in Phys. Ed classes for years. There are four hoola-hoops laid out on the ground, and for the purpose for this activity we will say they are laid out to make a perfect square. In each hoola-hoop there is a bunch of bean bags. Students line up in teams behind the hoola-hoop (they are represented by the “x”s).
One student can leave their line at a time, and they can travel to another team’s hoola-hoop to steal a bean bag. That same student then returns to their hoola-hoop to drop their bean bag off. Once that member of their team is back they would go to the back of the line and the next person in line would go. They can only take one bean bag at a time, and only one runner from each team can be running at a time (so no more that four total runners in the game at once). Once the music turns off, or the time is up, each team counts up how many bean bags they have to see who wins that round.
Because I am gearing this to an introductory lesson on Pythagorean Theorem, I would add in a new rule. I would say that students can run to a neighbouring hoola-hoop and then to the next neighbouring hoola-hoop, in order to pick up a bean bag from each, and returning to their hoola-hoop with two bean bags instead of the usual one. If they choose this route, they HAVE to go to another neighbouring hoop, they cannot simply go to one neighbouring hoop and return back (so they must go from hoop A to B to C then back, as an example).
Another rule is that they are able to travel to the opposite hoop from them, but they are only able to take back one bean bag at a time. (A to C or B to D).
After posing these new rules, I would have the students huddle into their teams to strategize. Some would note that it takes more time to travel to two different hoops and it may not pay off to do the extra distance for two bean bags, when in the same amount of time, maybe they could afford to send two people across the floor to receive one each. Some would (hopefully) argue that they don’t think that distance is “double” the opposing distance, and that’s the discussion I would be hoping to create. So students come up with the idea that running across is less distance than running in an “L” shape, but perhaps not half as far.
Tools/supplies required: Hoola-hoops, bean bags, gym attire, gym space or outdoors, music or stopwatch.
Assessment: Because this is an engagement activity I would mostly be doing anecdotal records of the conversations happening during their strategy planning. I would likely get to witness one team’s conversation only, then as I make my way across the gym I would have to ask the rest of the teams to summarize their discussions and to justify what they chose and why. I could also do a quick exit slip for the students (Phys. Ed class lends itself nicely to exit slips as students finish them as they wait for others to change out, get a drink of water, etc.)
The exit slip wouldn’t address the Pythagorean Theorem concretely, rather, just address the topic of distances in relations to right triangles, as was practiced in the class. An example would be:
Activity Analysis: When I designed this lesson I had in mind the article “Lacing Together Mathematics and Culture” by Denise L. Mirich and Laurie O. Cavey. In this article students were designing patterns for moccasins using methods of solving area. In this article one student said “We didn’t do math today.”, and that really resonated with me. I wanted to find a way to create an activity that introduced a mathematic topic, but without seeming like a math class. I figured a pretty easy way to do that, would be in Phys. Ed! I kept in mind a lot of my students who are extremely active and hands on but really seem to burn out come their after-lunch-period of math class. My students have Phys. Ed sometimes in the morning so this could easily be done as an introduction activity for a math class later on in the day. In my discussion director articles I talked a lot about the “driving force” behind inquiry-based learning. What it is that drives students to want to engage and want to dig further to exploring a topic. In this case, it was a game. Now of course this doesn’t always engage students who aren’t as active and aren’t as passionately competitive come Phys. Ed time, however I am hoping that the collaboration between students was still a time for all students to shine in contributing their ideas based on their own perspectives and understanding of distance. This was something else I discussed in my discussion director review. Inquiry based learning, especially in group settings, really is enhanced when you have students who approach problems with different perspectives.
Additionally, once I got thinking about some of my “sporty” students, I found this video below that talks about the Pythagorean Theorem and how it plays an active role in the NFL in terms of defensive football plays. This is another “hook” or “driving force” for some students for sure. Both this activity, and the video below, also show positive relation between what students are learning in school, and real world application of the concept as well.
https://www.youtube.com/watch?time_continue=134&v=Grzy-ZAotB0&feature=emb_title
Activity 2
Outcome: SS8.1 Demonstrate understanding of the Pythagorean Theorem concretely or pictorially and symbolically and by solving problems.
Geometric concept: Pythagorean theorem – measurement, area
Grade level: 8
Activity name: Sticky Note Triangle
Source of activity: My own, with background knowledge from Math Makes Sense Textbook – Pearson
Description of activity: Group students into groups of 4-6, so that they are able to work around a common work space. Hand out three different colours of square sticky note pads. Display the following model on the board with your own sticky notes and have the students recreate it.
Ask students what they note. Conversations I am hoping arise from this would include that they notice the sticky notes form an interior triangle, that there are three red stickies, four pink stickies, and five yellow stickies, etc.
Next step would be challenging students to make a square out of the coloured sticky notes, without removing the original ones put down. With some direction and prompting I am hoping they would be able to create something that looks like this:
From here I would again, ask students what they note. Students could potentially note which square is biggest, side lengths of particular squares, etc. We could then discuss how the area of the red square is 9 units, the area of the pink square is 16 units, and the area of the yellow is 25 units. I would label these on my model on the board. I would then allow conversations for any relationship they see between the areas we’ve discovered. Once we’ve discussed how the area of red + the area of pink = the area of yellow, I would challenge students to try another trio of square lengths that would hold this equation true. We could then discuss side lengths, what we know about square side lengths, and then eventually come to and understanding of the Pythagorean Theorem.
Tools/supplies required: Three different colours of sticky notes (about 4 pads of each colour)
Assessment: Much of the assessing would again come from anecdotal notes done during observations of the student’s communication and group work. Again, because this is more of an introductory activity I would recommend a quick exit slip to help attain knowledge of student understanding. A sample exit slip would be:
Activity Analysis: This activity reflected knowledge attained from the article “Geometric and Spatial Thinking in Early Childhood Education” by Douglas H. Clements. I remember summarizing in one of my reading responses that there was overwhelming evidence that young learners have the innate ability to understand and categorize geometric and spatial concepts. I remember being pretty blown away as there was such an emphasis being put on spatial thinking over numeracy, even though numeracy always seems to be our focus within our school division. This reading reaffirmed teaching practices mentioned in our Math Makes Sense textbook, where students are taught the basis of what the Pythagorean Theorem really is, before simply introducing the formula. Although the formula is simply an expression of a spatial concept, I think understanding where that concept derives from speaks more to our innate ability to spatially reason. This idea inspired this activity, where students explore square area of side lengths of a right triangle, rather than simply being given a formula and expecting to have a holistic understanding of what it represents.
Things to consider about this activity are the use of sticky notes. If students are using time to try and recreate another Pythagorean triple, they might end up going through quite a bit of sticky notes. Another alternative would be providing large chart sized paper to students to simply draw their squares, and likely wasting less paper. Overall, as long as students have an understanding of area of squares and addition, there are not too many constraints to this activity.
Activity 3
Outcome: SS8.1 Demonstrate understanding of the Pythagorean Theorem concretely or pictorially and symbolically and by solving problems.
Geometric concept: Pythagorean theorem – measurement, area
Grade level: 8
Activity name: Pythagorean Origami
Source of activity: https://www.youtube.com/watch?v=z6lL83wl31E; pictures below sourced fromhttps://www.mathsisfun.com/geometry/pythagorean-theorem-proof.html and http://jwilson.coe.uga.edu/EMT668/EMT668.Student.Folders/HeadAngela/essay1/Pythagorean.html
Description of activity: Reflect on last day’s lesson where we analyzed an interior triangle, surrounded by the appropriate sized squares on the outside, outlining the triangle. Pose the following picture on the SMART Board:
Have the students discuss differences between what they see here, and what they saw last day with the sticky notes. Then pose the question; “Does this picture still represent the Pythagorean Theorem? Why or why not?” Multiple points of view could be argued, where some students disagree because the interior shape is a square and not a rectangle, whereas others will notice that the square is just created by the squared hypotenuse of one of the blue triangles. Once students see this, you can begin this origami video. It goes fast, so certainly pausing will be needed, and I would recommend doing your own example at the front of the classroom to help students along.
https://www.youtube.com/watch?v=z6lL83wl31E
This video shows multiple ways of identifying triangles outside the exterior of the square. Again, I believe this video goes by quite quickly so I think it would be very beneficial to pause. Especially once you have your square created, you could then have students count how many triangles, rectangles, and squares they see outside of their interior square. This video also allows the opportunity for students to practice multiple area measurements of different shapes. The overall goal of this video would be to reconfirm what students recognized last day, that corresponding side lengths between right triangles and squares are both established when dealing with the Pythagorean Theorem.
Tools/supplies required: Square origami paper, SMART Board, pencils
Assessment: Assessment after this activity could be a little more formalized, especially after you review labeling the sides of the right triangle appropriately and discuss formulas and square roots. If that has not yet taken place, here is another exit slip for the meantime.
Activity Analysis: This activity reminded me of a takeaway from the article “Habits of Mind: An Organizing Principle for Mathematics Curricula” by Cuoco, Goldenberg and Mark. In this article we hear that experience tells us that students are capable of thinking like mathematicians. In my response I mentioned how I always used to giggle when I would hear someone say “Let’s put our mathematician hats on!” and perhaps roll my eyes at the elementary way of promoting false engagement. However, after reading the article it was clear to me that this was indeed, accurate. We do want our students to be mathematicians because they are capable of being it! We need to promote their ability to think and assess like mathematicians, no matter how complex the topic may be. That’s certainly what motivated me in choosing this third and final activity. This activity has students proving the Pythagorean Theorem. Proving it, not simply just typing numbers into a formula, but genuinely proving that beyond all doubt, this theorem works, and understanding why! I genuinely enjoyed the fact that students, in this activity, are following through steps that mathematicians do, proving theories. The assessment for this activity also reminded me of the reading “Lacing Together Mathematics and Culture” by Mirich and Cavey. In this article a student expresses “I enjoyed this lesson; it was fun seeing how many different ways you can find area and still be right.”. The model in the assessment provides multiple ways for students to show what they now know about the Pythagorean Theorem. It also encourages students to expand their thinking, beyond what they know, and again follow a mathematician process of making predictions and having to prove them. For example, a student could write that the triangle that extends upwards of the green box would be the same as the blue triangles, and they would have to justify why that is true. Another student could explain that similar shapes could be made around the green square to make a larger, symmetrical image, and again have to defend why. To me, this allows for a far more in depth view into what a student understands about the Pythagorean Theorem than if a student can plug things into an equation and solve mindlessly.
This activity does have some constraints for students who struggle with fine motor skills or step by step instructions. There is no reason why this activity couldn’t be done in partners. Really the biggest piece of learning happens once the paper is done being folded, so that certainly opens up the opportunity for students to still explore the concept, even if they were not the one directly folding the paper.
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