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Saturday, April 28, 2018

3 ACT Tasks

As I am browsing through my previous blog posts, it appears as though I have never blogged about 3-ACT tasks.  I've been using these types of activities for the past five or six years.  I first found out about them in Dan Meyer's blog back in 2011. 

A large majority of the 3 ACT tasks I use come from Dan or Andrew Stadel (follow the links for a list of their activities).   There are quite a few really good ones to use this time of year in geometry, including Andrew's File Cabinet activity that I used in Applied Geometry class on Friday.

I don't have time to get into the fine details, but let's just say that the activity was a hit for my class of student who usually don't get too excited about doing math each day.  I wasn't fully prepared to record a video, but as we started watching Act 3, I could sense the excitement building.  Here is what I was able to record.



Needless to say, it was a fun day in class.  Thanks, Mr. Stadel, for the awesome activity!

(Last summer, I imported this 3 ACT activity into Desmos.  If you'd like the Desmos activity, here you go: 3-ACT File Cabinet Desmos.)

Tuesday, April 10, 2018

When Conceptual Understanding Fails... My Dilemma with Special Right Triangles

I started writing this blog post over a week ago and have realized that I can't give it enough attention until this summer.  So, I am going to post what I have thus far and resume this summer after a little more thought and with a little more time.

**WRITTEN OVER A WEEK AGO**
I want my students to leave geometry with a strong understanding of the two special right triangles (45-45-90 & 30-60-90).  I know that the two triangles are foundational when it comes to the unit circle and trigonometric functions & their graphs.  I also strive to connect as many concepts together in my lessons as I can; I believe those connections are critical and lead to true conceptual understanding.

This past year, here is how I approached teaching the section on special right triangles.
For the record, my learning target for these lessons are:
  • I can derive and apply the properties of special right triangles.


On day 1, students worked in pairs with their elbow partner to complete this task, which was handed out on paper:

Task: Find the length of the diagonal. 
Express answer as a decimal rounded to the nearest hundredth and a radical in simplest form.

Each group had a different side length on their square.  Groups had sides ranging from 1 to 12.  No two groups had the same square.

At the same time, I handed each student the following table on a full sheet of paper:


Students have no problems identifying the shape as a square.  Groups had very little problem finding the diagonal length.  We had reviewed how to simplify basic radical expressions the previous week, so very few groups struggled with that.  As I roamed around the room, I made sure each group had the correct answer and had found correct place to fill their answer into the table. After I confirmed their answer was correct, I invited one person from the pair to go to the Smart Board and fill in their row of the table.

As students filled in the table on the Smart Board, I asked that each student fill in the table on their paper.  After about three rows were filled in, I could hear rumblings around the room about a pattern students were seeing.  When all pairs had shared their answer, the table looked something like this:


None of my classes had more than 22 students this day, so the last row was blank each hour.  At this point, I lead a little discussion about what we're trying to derive.

I start by having students share with their partner what they think should go in the last row our table.  Again, this patterns is not hard to see, so I hear a lot of "twelve root 2" being whispered.  I call on a volunteer to share and ask to explain their thinking.

I then go to the wipe board that hangs adjacent to my Smart Board and talk briefly about what it is exactly that we're doing today.  I will draw a right isosceles triangle on the board and ask students to tell me what they know about the angles.  I then introduce the term "45-45-90 triangle".  I remind them that because each group's triangle was a 45-45-90 triangle, all of the triangles we are working with are similar via Angle-Angle.  I will put a random, much larger side length on the leg and again ask them to share with their partner what they think the length of the hypotenuse is.
Tell your partner what you think the value of x is.
Again, I hear a lot of "eighty root two" whispered between partners.

At this point, I ask for someone to summarize the relationship between the lengths of the leg and the hypotenuse.  The response I hear is "the hypotenuse is the leg times the square root of two".   I write the relationship on the board and ask students to write it on the sheet with their tables.


I will ask the group that had the leg of length 1 to share what they found for the decimal value of the hypotenuse.  They respond with "1.41".  I will mention that the rule we derived is saying that the hypotenuse is always about 1.41 times as long as the leg (on a 45-45-90 triangle).  I remind students that each group used the Pythagorean Theorem to find the value of the hypotenuse, and that they can always fall back on that process if they would happen to forget the rule we just derived.


That whole process takes about 15 minutes, which leaves about 30 minutes remaining for the next part of the lesson.

I ask groups to turn the paper with their square over and draw an isosceles right triangle on the back.  Then they are to take the value that was their square side and label the hypotenuse of their triangle the same value.  (So the group that had the square with side length of 4 from above would now have this triangle.)
Find the value of x.
Express your answer as a radical in simplest form.

I also have this table included on the back of the paper that has the previous table:


At this point, student thinking typically takes one of two paths.  Some groups will use the rule that we had previously derived and come up with this answer:


Or, they will use the Pythagorean Theorem and get something along these lines (with sometimes a little guidance):


Once again, I floated around to make sure groups are getting one of the two answers above and are filling in their table correctly.  After I confirmed their answer was correct, I invite one person from the pair to go to the Smart Board and fill in their row of the table.

Each hour, I saw a mixture of solution methods and our class table looked something like this:

Prior to this lesson, we hadn't reviewed how to rationalize the denominator.  I know students see the process briefly in algebra 1, but rarely does anyone remember it.  I took the next five minutes and led a discussion about why "simplest form" doesn't include a radical in the denominator and how we can manipulate these radical expressions and rationalize the denominator.  (There is a good explanation on coolmath.com if you're curious.)  

At the conclusion of the discussion, each hour had a table that now looked like this:

With the expressions in this form, students now had no problem detecting the pattern to our expressions.  I then ask students to generalize the patterns they see, and we arrive at the following two equations:


(Side mini-rant: When will mathematicians decide that it's okay to leave a radical in the denominator?  I vote now.  Each year I have students look at me funny when I try to sell them the idea that three root two divided by two is simpler than three divided by root 2.  The simplification rules are extremely outdated... we have calculators now if we should happen to want to find a decimal approximation of these expressions.)

This concludes the lesson.  Students have a homework assignment that I post online that isn't due for a few days later since the assignment also includes 30-60-90 triangles.

**END OF MY ORIGINAL POST**

The next day, we do a similar exercise to derive the rules for a 30-60-90 triangle.  Students get some practice working with these rules when they complete the homework, and we assess with a quiz a few days later.


Here is my dilemma and questions for the audience:


One of the quiz questions is to find the missing variables in the diagram below.  Many students did well on this question, but I had a handful of students who claimed y = 8 and x = 4 root 3.  The error the students made was they applied the wrong rule.  But what I struggle with is the idea that these same students would have done just fine with this question prior to us having our lessons on special right triangles.  We had previously learned about isosceles triangles and the Base Angle Theorem Converse.  If they were to analyze the triangle and notice that the two acute angle were congruent, they would have concluded that the legs were also congruent.  Then, with two legs known, the Pythagorean Theorem would have led them home.




It's also like the lesson clouded their thinking.  

My questions for those still reading...
1.  How would you improve the first day of this lesson?  What could I change to do better?
2.  Are there any other resources you'd recommend to use with this lesson?
3.  What other insights do you have about my dilemma?


Thanks for reading!