I’m pursuing a National Board Certification and am in the midst of preparing for the content knowledge portion. An area in which I know I struggle: Geometry.
I gave this problem to my students on Thursday, with the hopes that a student would be able to figure out (and show me) how to solve it. When that plan failed, I knew I needed to start preparing more formally for my impending exam.
So I spent my Saturday evening working on geometry. (Math teachers lead such exciting lives, I know.) I started with some problems from A+ Click Math. Wrong answer after wrong answer lead me to seek out a more formal review. I know there are other websites out there for geometry review (and if you can PLEASE point me to them, I’d appreciate it), but I sank to a new low and turned to Uncle Sal. The 6 minute video was excruciating to watch as he repeated himself as he drew , but to be fair, I understood the triangle inequality theorem much better at the end of the seemingly endless 6 minutes.
Today, I tried Brilliant, which is geared toward problem solving. I suppose I should be pleased with my progress and my willingness to try again in the face of defeat.
- Failing sucks. We need to remember that when we ask our students to be okay with failure and mistakes.
- It’s hard to admit you aren’t good at something. We need to acknowledge that when we expect students to approach us with questions about their learning.
- To some, being wrong means admitting you’ve failed. We can’t automatically expect students to transition into that mindset.
- Tell your students that you struggle as well. Give them specific examples of when you’ve failed and let them know that they can persevere.
Oh boy, do I hear you. I am taking an online Calc class for a final 4 credits so I can apply for Math for America’s Master Teacher fellowship. I read the textbook out loud, pore through lecture notes from a number of generous professors’ websites, message a buddy at work, email Sam Shah, and each assignment takes HOURS (sprinkled with some tears). I do finally make some progress, but it is a HUGE eye-opener for what many of our students go through. I salute you for undertaking the NB Cert process.
One of the best things about you sending problems to my geometry class is that I don’t work them before I hand them over. This means that they get to watch my struggle, my process and my discovery alongside their own.
I think it’s a world of difference between helping kids struggle and struggling with them.
Keep up the great work and let your students see that you’re working.
And learning together is so much better than going alone
I would guess the frustration of Geometry comes from probably not teaching it and thus being removed from the subject matter for a long period of time. I believe the education gurus call that “extinction”. It make one feel like a dinosaur. I always thought it was interesting that our colleagues in tha Science Department were always specialists in a particular areas (biology, chemistry, or physical science) where as math teachers had to be generalists knowing all of algebra, geometry, and statistics. Our students are even forced upon this with state testing. How do you suppose they would do if the state science test had all three facets in it??? I guess the rationale is that it makes our students more proficient at mathematics. My suggestion is to get a HS geometry book and review the topics in the second half of the book. Specially 2D and 3D objects, similar figures, proportionality, and circles (with properties involving tangents and secants). I am willing to bet your trig is still fresh so no need to spend a lot of time there. It is good that you know your weaknesses so you can focus on them. Good Luck and Good Skill.
Thank you so much for all of your support, guys! It means so much to know I have such a fantastic PLN.
The answer is 3. I know it’s 3. I verified it on GeoGebra. But I can’t, for the life of me, prove why it’s 3. I know it’s half of the short leg, but that doesn’t work for other right triangles, just ones similar to 3, 4, 5…
I got 3 as well! I drew in an auxiliary line from the radius to the hypotenuse. I knew it would make a right angle because the hypotenuse is tangent to the circle at that point. Now, I can set up similar triangles and do a proportion to solve for x. Its a fun problem and totally stealing it for my geometry students!
Did you ever want to hit yourself over the head for missing the (in hindsight) blindingly obvious? I’m looking at the right triangle created by the radius and the tangent and know the proportion I can set up with x and 10-x and all that, and I’m looking at things that *look* to be true and figure if I could prove those assumptions … and then realize that the hypotenuse has to be divided into 6 and 4 b/c the 2 tangents from the highest point of the triangle have to be 6.
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Similar triangles works nicely to determine solution. Nice problem!
The picture shows half an inscribed circle. I would first draw the other half of both the circle and the triangle. Then use the formula for inscribed circles:
r = 2a / p where a is the area of the circle and p is the perimeter of the triangle. Doing that gives r = 3, which is the answer several other people gave. Hope that helps.
Let theta be the angle on the rhs,
draw auxiliary line from origin of semicircle at right angles to line of length 10. This line is of length ‘r’
Observe that theta also equals sin (r/(8-r))
Just realised that this is a roundabout way of using similar triangles!
Drawing the radius which is perpendicular to the 10 side forms a smaller triangle similar to the whole, in which one side is r, and the hypotenuse is 8-r. Comparing similar sides gives r/6 =(8-r)/10, which gives r=3. Nice problem.