I feel like I always have to make the disclaimer that I teach for an online school. That in and of itself poses different challenges than a brick school.
However, not only is my school online, we are also asynchronous. That means we have students starting classes every day of the year that the office is open. Students can pace as quickly or slowly as they want, as long as they are finished with their course after 10 months.
Because of this dynamic, it is not uncommon for a teacher to have a student submitting their first assessment in a course on the exact same day a different student is submitting their final exam. It would be impossible to make presentations to all of our students on the topic they are at every day (or even every week). As a result, we use a static curriculum that the students can move through on their own.
Of course, sometimes students will need help on a topic they are learning. This fact is true for all subjects we offer, but is especially true for our math courses. To meet this student need, we have developed a Math Lab that we run weekdays, usually 4 times a day, for roughly 37 hours per week. Students can get math help outside of the Math Lab hours if a teacher is available, but these 37 hours, we promise a math teacher will be available.
So when I am talking about Teaching My Lesson, it is in the context of helping a student in one of these Math Lab sessions. The drawback to the Math Lab, since it is drop-in first-come-first-served, we never know what subjects or what questions will show up during Math Lab. I have helped a Pre-Algebra student and a AP Calculus BC student at the same time. All that to say, we cannot do much pre-planning of a lesson as they are all relatively spontaneous.
Wow! I’m glad you got through that. It seems like I need a separate blog post just to introduce this blog post!
So the other day, I was working with a student on subtracting polynomials (Algebra 1). She came in specifically to find out why, when subtracting polynomials, do you multiply the second polynomial by −1 (negative one).
Rather than directly answer, though, I thought it would be helpful to start with elementary school arithmetic and move to integer arithmetic and the distributive property in general before looking at the distributive property with algebraic terms.
We started with the simple 8−5. Anyone can do this problem from 1st or 2nd grade. Then we bumped up the difficulty by reminder her that “subtraction” was the same as “adding a negative.” 8+(−5).
Then back to 8−5, but instead of writing 5, I wrote the equivalent phrase (2+3). And then we explored that 8−(2+3) was the same as taking 8−2−3. (Take the 8, then subtract 2, then subtract 3.) All of these methods yielded 3 as a result.
From here, we went back to the 8−(2+3) line. I rewrote it, and then explained that we can multiply a 1 in front of the (2+3) since 1 times anything is just that anything. Multiplying by 1 doesn’t change the value. From here, I played the “subtracting” is the same as “adding a negative” game we looked at earlier to get 8+ −1(2+3).
From here, we used the distributive property to get 8+ −2+ −3. And then finally “adding a negative” is the same as “subtracting” to get 8−2−3. But we had already seen this line earlier in the lesson! I love it when things work full circle.
Now that we played with numbers (the concrete), I thought it was appropriate to bring in the abstract, really only now answering the question she came here for initially.
I will admit, I was disappointed when she asked that last question, “So what would your final answer be?” I loved the initial question she had. I appreciate students not wanting just “the rule,” but also wanting to know why the rule works. This question is especially fun because it isn’t covered in the course. The student doesn’t have to ask to meet the objectives of the course.
But at this point in Algebra 1, she has practiced combining like terms over the last several lessons. Initially just two terms. Then with adding polynomials the lesson before this. At this point in the course, she should be comfortable with combining like terms (especially before even attempting to subtract polynomials). In hindsight, I should have flipped a clean whiteboard out, and reviewed with her the few lessons leading up to this one.
I will say, this lesson wasn’t atypical, but also wasn’t typical. I get more questions akin to “how do I do this?” With those inquiries, I usually make the student do more writing than me. Math is not a spectator sport. In order to learn the math, you have to do the math.
If I get a student looking for a how, I will see if I can do more screenshots and post again (maybe this week, maybe next). Stay tuned…