Wednesday, October 14, 2015

WODB: Polynomial Functions

If you haven't experienced the conversation stemming from Which One Doesn't Belong? activities, you are missing out.  
As far as I can decipher (#MTBoS feel free to correct me), this all began with Christopher Danielson's Shape Book centered around this structure.
From there, a crew of tweeps (headed up by Mary Bourassa) established (YES! Canadian) to curate a collection of problems of this format.
My unit on polynomial functions (either in Foundations of Mathematics 30 or Pre-calculus 30) requires students to decipher attributes of polynomial functions from their graph and vice versa. These include end behaviour, sign of the leading coefficient, y-intercept, domain, range, degree, and possible number of x-intercepts.
Each question has four panels. Each panel needs to have an attribute that is unique. (i.e. that could be the reason that it doesn't belong). I sat down to build some polynomial WODB questions, and had a hard time isolating unique attributes with sets of four. I ended up with three examples like the one below.

If you open up space for rich conversation and critique, these three could last upwards of 30 minutes.
(One student insisted that colour was always the answer, which, in hindsight, was my bad). After we were done, I should have had students try and create their own. That activity would show a deep understanding of the parameters. Alas, I succomed to the age old barrier of time.

Download the .pdf with all three questions.

Steal, modify, and adapt. If you make more, link in the comments.


Central Tendency: 10 Burning Questions

My intern just started a unit on statistics with my favourite starter question of all time.

(First blogged near the end of this post in 2011...)

The question is simple: floor is very low, and ceiling is very high.

Create a data set with the following characteristics:
Mean = 3
Mode = 3
Median = 3

During the teacher rotation between groups, I picked up on some lines of reasoning. (Not being directly responsible for the teaching of the lesson, allows me to sit back, be inspired, and follow the lines of inspiration).

Student justifications for their data sets were very interesting. It inspired me to pen some burning questions I may have asked groups if I were "in charge".

  • If the mean of a data set is 3, all data points must be multiples of 3.
  • You cannot have a mode of 3 without at least 2 threes.
  • Adding a data point lower than the mean will always lower the mean.
  • Adding a negative and positive data point will never change the median of a data set.
  • If the mean and mode of a data set match, so must the median.
  • The median of a data set must appear as a data point in the set.
  • The mode of a data set will always be closer to the mean than the median.
  • Adding a data point lower than the median will always change the median.
  • Adding a group of points with the same mean as a set of data will not change the mean.
  • If we keep adding threes to a data set, the mean will eventually become 3.

This type of extreme agree or disagree questions are great for discussion. I believe some teachers have coined them as "talking points". (If you have a link to a great description of "talking point" comment below.)

Asking students to prove or disprove these statements (ranging in difficulty) would allow them to surpass the calculation of the statistics and work directly with the mechanism of central tendency. That is a huge win for me.

If you have more questions, comment them!


Friday, October 2, 2015

Navigating Collectivity: Grade 9 Fractions

"I hate fractions"
- Everyone
Today an amazing thing happened; students put aside the endemic disdain for rational numbers and had a conversation. I'd go further, they weren't discussing their views on fractions, they were collectively conjecturing--the moves of the room enacted each other. I don't think that a written document can capture the movement of the body of learners, but I have to try something. Think of it as less of a remembering and more of a re-membering, a reconstruction of a living learning event from the past.
My intern and I have worked at fostering a spirit of collectivity in our grade nine classrooms. This begins with a starter problem where they are asked to respond and fully explain their thinking in whichever modality they are most comfortable with. After a partner discussion, the teacher anticipates, orders, and debriefs different ideas that appeared. Today, my intern gave a starter on fractions as an introduction to a review day. The thinking was mind blowing. The purpose of this post is hence two-fold: 1) try to capture the spirit of collectivity and 2) an open job application to school divisions looking for a fantastic teacher-to-be.
The *question was as follows:
Place four different digits (2-9) and one operation ( +, -, *, / ) in the boxes to create an expression with the largest result possible. Explain your thinking.
There were three amazing pieces of thinking that came from the collective that would not have occurred if their thinking wouldn't have met perturbations.
1)   She began by collecting answers from the class and placing them on the board. They stood as artefacts of intelligence, but needed to be investigated. As she asked probing questions, the following conversation occurred (paraphrased, of course).
Teacher: How do you know this is the largest?
Student 1: Because I made the bottoms as small as I could.
Student 2: Denominators.
Teacher: Right, denominators. So you chose 2 and 3?
Student 1: Yes, because they were the smallest, the best.
Student 2: Because you said we couldn't use 1.
Student 1: We would have used 1, because it is the best.
Teacher: 1 is the best denominator?
Student 3: Yes, well no, zero would be the best. It is as small as possible.
Many Classmates: No! Can't divide by zero! Zero doesn't work! etc.
Student 3: I  know you can't, but if you could, it would make the largest number.
The interesting (and very mathematical) idea of "best" comes out of the students' method of making a claim and supporting further claims based on their classmates. Student 3 is following a pattern established by the other two. When he takes it past the area where they were comfortable, the collective self-corrected. I'm not sure it gets to this moment with teacher questioning; the teacher constantly deflects back to the collective.
2)    Quickly after this discussion on division by zero subsided, another student made a conjecture and the collective employed rules to decide whether the move was mathematically legal.
Student 1: We could make the fraction bigger by adding a number.
Teacher: How do you mean?
Student 1: Like if we took a digit, 4, and put it out in front of the fraction. (motioning to create a mixed number)
Student 2: But that wasn't the question.
Student 1: I know, but it would always make it bigger.
Teacher: Creating a mixed fraction would always make it bigger?
Student 2: Yes, they are wholes.
Student 3: But that can't work.
Student 1: Why?
Student 3: Because you can't add a whole to an improper fraction? It already has wholes?
Student 1: But this would just add wholes?
Teacher: Adding wholes makes it bigger?
Here, we see the conceptual understanding of fractions (wholes make things bigger) and legalistic understanding of mathematics (can't be improper and mixed) at war. The students do not shy away from making a suggestion even though they are aware that the question forbade it. They are operating on proscriptive barriers and the question doesn't restrict their function. Notice how little the teacher says; the students form conclusions with each other.
3)   At the tail end of the discussion, my intern asked if anyone used addition. One student raised their hand and offered their solution. In a familiar line of questioning, she tried to explain why 2 and 3 would be the ideal denominators.
Student 1: Because they are the smallest.
Student 2: But, wait, they don't create the smallest number.
Student 1: Fewer pieces, larger.
Teacher: What do you mean?
Student 2: 2 and 3 make 6, but 2 and 4 make 4.
Teacher: When we add?
Student 2: Yes. Common denominator.
This initially blew my mind. What an astute comment to make. The chosen numbers would not create the smallest common denominator, so would they still create the largest fraction? Student 2 did not entertain an additive strategy until Student 1 suggested it; it was her explanation that spurred Student 2 to wonder about the effect of common factors in creating the largest possible fractions. While it may seem like the collectivity is one-sided, the thinking formed through communal action recursively challenges the entire class.
What stands out to me is three things:
1)  A simple question that allows for student action is the first step toward collectivity.
2)  The depth of thought the students achieved.
3)  The collective interplay of their thoughts when given the arena to collide.
I hear you:
"But I don't have time"
"My students are too weak"
In the end, it took the provision of a space, curation of conjectures, and presentation of a perturbation to open up collective space.

* Source of Problem: