If you teach algebra, you've most likely seen the error $(x5)^2=x^2+25.$ Previously, in explaining this error, I would resort to discussions about what "squaring" means and the value of the distributive property. This year has been different, though. Since the beginning of the year, I've been teaching students that they can check their work simplifying expressions by plugging in a number into the original and the simplified versions and seeing if they get the same result. We emphasized this approach throughout our review of linear expressions. Today in class, we had exactly the problem of simplifying the expression $(x5)^2$. Some students gave the incorrect response of $x^2+25$, but as I listened to their group conversations, I could hear their discussions "I plugged in 2 to both expressions. $(x5)^2$ gave me 9, but $x^2+25$ gave me 29." This was music to my ears. Rather than simplification being a set of rules to remember and apply, simplification becomes attached to the real experience of ensuring that I get the same answer each time.
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If you give a student a platform, they're probably going to use it. This year, in addition to my weekly Problem of the Week challenge problems, I've been giving students the chance to reflect on their past work to see how they can improve in the future. Specifically, I ask students to write a few sentences about a time they had an insight into a problem and what helped them have this insight. My students shared some of their secrets to success:
If you give a student a platform, they may exceed your expectations.
I had more trouble with a recent inclass activity than I had anticipated. Students were given cards with simple inequalities and asked to form groups of 3 around a joining conjunction "AND" or "OR" and graph the compounded, resulting inequality. Students were quite comfortable graphing the compound inequality above. The two arms point outwards from 1 and 4, respectively. However, when asked to graph an inequality where both arms pointed in the same direction, the students had much greater difficulty. For example, students had trouble viewing the inequality $x >0 \text{ AND } x>3$ as equivalent to $x>3.$ To get to the bottom of the misunderstanding, I decided to ask students their thoughts about how AND and OR graphs work. Students responded that an OR graph has "two parts" and a "gap in between," while an AND graph has "one part" and "no gap." I realized that students' experience of AND and OR graphs had only been in the context of absolute value inequalities, where these descriptions are (almost) always true. However, if given a compound inequality that does not come from an absolute value, these descriptions may not apply. I needed to find a way to show my students a more inclusive description of compound inequalities. At this point, as if prompted, another student, gave another description of an AND graph as "two graphs overlap" and an OR graph as "any two lines — doesn't matter if they intersect." While the wording could use some work, the idea is there that AND means to take the two graphs' intersection and OR to take their union. We wrote both studentgiven descriptions of AND and OR graphs on the board in an attempt to judge between them. When the bell rang I had to let students go. It was worth it to spend class time in discussion, even if I didn't get to everything I had planned. We'll just have to pick up on these ideas next week.

About Me
I started this blog to share my transformation from math nerd to math nerd who loves to share math with young people. I teach high school in Philadelphia, Pennsylvania. Your comments are always welcome. Archives
June 2017
