I taught Absolute Value equations and inequalities for the first time in depth this year. There were the usual student struggles, but I kept being reminded of a conversation with Sam Shah that I had a year or more ago about this topic…He was frustrated that his students were more focused on memorizing “the procedure” rather than understanding how/why we solve absolute value problems the way we do. Now, I find myself with the same problem. Here’s my current idea for next year:
Day 1: Teach only problems of the form a|x| + b = c, where a,b,c are in R.
Day 2: Repeat/review this, or move onto inequalities of the form a|x|+b>c (or <c).
I want to make three big points in having only the variable inside the absolute value:
- It is important to isolate the absolute value.
- There is no “inverse” for an absolute value.
- When you get down to |x| = d, you HAVE TO THINK!
My hope is that when students get to the point of |x| = d or |x| > d, they stop to think what the possible answers are. There is no way to magically “cancel out” the absolute value. However, once we isolate that side of the equation, we can reason through the answer. I have been stressing in my class this year the idea that there are moments when you have to “put your hands in your lap and think” for a second. This would be one of those moments.
Only after students are comfortable with this, would I extend to problems of the form a|px+q|+b=c, etc. My hope is that they will get good at isolating the absolute value, and then THINKING about what makes sense next.
We will see how it goes next year…
I’m just going to shout out my former student teacher Matt for not only being an amazing student teacher and now amazing struggling first year teacher, but for jumping into the MTBoS from Day 1 of teaching! He’s much more active on twitter than I, but he mentioned me in a recent tweet about the meaning of radians. So, I wanted to follow it up.
I don’t think I ever learned what a radian really was in school. Somewhere along the way, I figured it out, and I love teaching it!
I usually begin by having students (in PreCalc) put masking tape around the circumference of circular objects so that it makes exactly one circumference. They then lay the masking tape flat on the desk. Next, they measure the radius of their circle. Last, they measure how many radii fit onto their masking tape. “For some reason” everyone keeps getting 6 and a little bit. Quickly, students catch on to 2pi, and they start to relate it to the circumference formula.
But that’s just the first half. Then, students trace the circle onto paper, and put the masking tape back on the circle. By now, they have all these little marks on the tape at the measurements of each radius length. We then draw a central angle that is the width of one of these radius lengths. This is an angle that is one radius wides. A RADius ANgle. A RADIAN!
Tuesday night kicked off the first of four monthly meetings I’m co-facilitating with Steve Viola–a Professional Learning Team focused on debate and discussion in the math/science classroom. We were nervous and excited, and we ended up with 45 awesome teachers!! Way more than expected!!!!
There’s lots to share, and I hope some of them will be sharing their thoughts on it too. Just a quick overview of what we did:
1. Soapbox Debate. Standing up and saying “My Claim is…and my Warrant is…”
2. Research supporting debate in the classroom.
3. Circular Debate. Same as soapbox with summary of the previous speaker added in.
4. How to Start/Examples – Steve did a quick talk about the awesome way he introduces these structures into his classroom: using superheroes!
5. Then we broke out into groups. Groups completed (1) Table Debates and (2) Discussion about how to start including this in their own classrooms (maybe as soon as the next day?!).
6. Hopes & Fears. A final share out, followed by an exit slip.
At then end, we asked them if they were planning and able to come back to part 2 in November. ALL 45 SAID YES!!
For MTBoS Mission #1, I was tasked to blog about either (1) a favorite open ended question or (2) one thing that makes my classroom distinctly mine. I’m going to go double or nothing and answer both questions together.
My classroom has a distinct culture of debate (see a few earlier posts like this one or this one). I try to foster an environment where students are open to discussion, making convincing arguments and critiquing the reasoning of others. This makes my classroom uniquely me (as I’m also a Speech & Debate coach!).
My favorite rich open ended problem is a debate about the music industry. The activity where I use it is the culmination of the many debate structures I introduce throughout the semester, ending in a full-scale, class-long debate. It goes something like this:
A new but promising young artist is trying to decide how to produce his new record. Should he choose a major record label, a legitimate indie label or independent producing? Below are the options.
• Major Record Label wants to sign the new artist. They will give the artist a $200,000 signing bonus, plus the artist will get $0.10 in royalty for each song sold.
• Indie Record Label wants to sign the new artist. They offer a $50,000 signing bonus plus $0.60 in royalty for each song sold.
• Self-Employed artists get to keep all their earnings. It will cost $20,000 for recording time and supplies, but once the record is made, the artist makes $0.80 per song sold.
• Eccentric Billionaire is always interested in new business adventures. He offers the artist $300,000 for full rights to the songs. The billionaire keeps all earnings of the song.