She chose to shade in 5/100 and 49/1000.
The question for me is whether I try to pin this down in terms of shading in/part whole representations, or try to embed these decimals in a grouping word problem. I think I’m going to go for a bit of a combo approach.
I put “0.1” on the board and asked students what they’d call this. A kid said “one tenth,” but that quickly became controversial.
Question: how do you think these kids are seeing 0.1?
"Find the midpoint of the line segment connecting (2,5) and (2,396)," at the beginning of the unit. Can you predict the top three responses?
— Michael Pershan (@mpershan) April 7, 2014
What would you predict? Here are some twitter responses:
@mpershan Is (0,391) one of them? That seems like a way to get an answer from those two numbers.
— Evelyn Lamb (@evelynjlamb) April 7, 2014
@mpershan (3.5, 199) (2, 200.5) (2, 198)?
— David Wees (@davidwees) April 7, 2014
@mpershan (2,198) is probably one of them. Maybe(1,198). But I am otherwise stumped. Oh! (201.5) maybe?
— Christopher Danielson (@Trianglemancsd) April 7, 2014
@mpershan er…(2, 195.5) as a top answer.
— Christopher Danielson (@Trianglemancsd) April 7, 2014
@mpershan I’ll bet at least one of the top 3 responses includes *two* points.
— Chris Lusto (@Lustomatical) April 7, 2014
—
Here’s your answer key…
First Place:
Second Place:
Third Place
I keep on seeing this in my Geometry classes this year. Tasked with finding the area of a right triangle, kids move toward the hypotenuse even if two of the other sides are given. Then they end up stuck looking for a height that they can’t find.
I’m pretty convinced — based on talking to kids and looking at their work — that this is all about how they see right triangles. These kids must be seeing hypotenuses as bases, and it must feel weird for them to treat the legs as bases. Or maybe instead it’s about the height? Maybe it feels strange to them to use a leg as a height?
Shared by Tracy on twitter, and a great conversation ensued.
@TracyZager @mpershan My first thought on this to see where students are making the mistake. pic.twitter.com/7X7spt0BMI
— Bryan Anderson (@And02B) April 6, 2014
@mpershan @Anderson02B I think I would go concrete right away, and link it to sharing. If this were a brownie, which piece would you want?
— Tracy Johnston Zager (@TracyZager) April 6, 2014
@mpershan @Anderson02B @TracyZager This one is a bit different, tho. Kids seem to cue in on the equal widths here.
— Christopher Danielson (@Trianglemancsd) April 6, 2014
https://twitter.com/BHS_Doyle/status/452804242672476160
@mpershan @TracyZager I'd bet that most 7th grade students have never seen the problem with a shape other than a circle or rectangle
— Bryan Anderson (@And02B) April 6, 2014
@BHS_Doyle @mpershan @Anderson02B @TracyZager So this task needs revision and should only ask about 1/5 or 4/5.
— Christopher Danielson (@Trianglemancsd) April 6, 2014
We’ve been studying graphs of rational functions in Precalculus.
Me: “Take 1 minute with your group: what will the graph of y = x/(x+1) look like?”
One group, during discussion, asserted that it had to be a line, using a sort of process of elimination: it’s not a parabola, it’s not cubic, it’s not a hyperbola.
Interesting, right? Why does this seem like a linear equation? I guess that it sort of looks like one…
What can we say that this student does or does not understand about inverse functions?
One of my little obsessions is teaching complex numbers, but it’s really hard to find genuine instances of complex number mistakes. You start looking around at the most pernicious complex number mistakes, and they’re a lot like this one here: essentially algebraic. These mistakes, to my mind, are indistinguishable from the sorts of mistakes you’d expect from
That observation is probably helpful in itself, though. The mistakes that we see from kids working with complex numbers are essentially algebraic mistakes. That means that kids aren’t really seeing much of a difference between the algebra that they’re usually asked to do and their work with complex numbers. Complex number arithmetic is just algebra with a twist.
Thanks for this, Graham!
What’s interesting about this to me is the mental connection between division and subtraction. I doubt that this kid has anything like an explicit model of division that involves “taking away,” but it makes sense to me that the ideas of subtraction/division would be associated much in the way that addition/multiplication are.
All the more reason to make sure that there’s a robust understanding of multiplication that goes beyond “repeated addition,” no?
Help, if you don’t mind!
I’d like to flesh out our collection of complex numbers mistakes. If you’re teaching this topic in the next few months, could you send me some pictures of mistakes? I’m looking for all sorts of mistakes — either common or uncommon errors.
In the meantime, would you take a second and comment about a complex numbers mistake that you’re used to seeing from your students?
When you’re done with that, check out some of the new mistakes that I just posted…
Thanks for all the great mistakes and comments lately, guys. Keep up the great work!
