This semester I am redeveloping Mathematics for Primary Educators, the (ideally) first maths content course for our pre-service teachers who choose to specialise in maths. I have a notebook full of ‘things I want to do right’ this time, but it mostly boils down to giving students many opportunities to have mathematical conversations, and for me to learn from their responses. (Side note: this means I’m finally digging deeper into Desmos too.)
I began the first lecture with a WODB (Which One Doesn’t Belong?), presented as a Stand and Talk. (I’ll link this in another blog post.) In this post I’ll talk about the second WODB I presented in the lecture, developed with input from Twitter.
In case you aren’t familiar, ‘Which One Doesn’t Belong?’ is a great prompt for looking for points of commonality and difference. The idea is to identify a property three items have in common that are not shared by the fourth item. Ideally, we can find reasons why each item doesn’t belong. WODB helps students to see that there is more than one ‘right way’ to think about maths. It also provides a good launch point for terminology and for more closely examining particular mathematical properties. For more, look at #wodb, wodb.ca, and Christopher Danielson’s books.
A first look: different representations of numbers
To begin, I simply asked ‘Which One Doesn’t Belong?’ and collected their responses in Desmos. I wanted to elicit the different representations of numbers, as well as terminology and misconceptions.
[Note: I’ll write the top left as 0.(3) for convenience.]
- The top left doesn’t belong because the others show 0.3. This was a prevalent response. We talked about the meaning of the vinculum, and recurring decimal numbers.
- The top right doesn’t belong because it’s the only one written as a percentage. G said that it suggests it is out of 100 (and presumably that the others aren’t), which is an interesting idea I didn’t explore in class.
- The bottom left doesn’t belong because it’s the only one written as a fraction. It’s also the only one that involves two separate numbers. (No one picked the bottom left in class.)
- The bottom right doesn’t belong because it’s not displayed as a number. K said that it ‘could mean anything’. S said that it is open to interpretation: it’s either 3/10 or 7/10. I volunteered that it could be the only one with a value more than 1 (that is, 3). We briefly talked about ten frames.
I asked: Are these all numbers?
A second look: the number line
I asked students to put 0.(3) , 30%, 3/10 and 3 on a number line.
I revised the number line I planned to use several times. My first version had tick marks and labels with different colours for positive and negative numbers. Then I removed the labels. Then I removed the zero. Then I removed the tick marks. I settled on a horizontal line with arrows at either end.
We discussed needing to identify the largest and smallest number so that we could make sure all the numbers fit on our number line. This led to the question: ‘Which is bigger: 3 or 30%?‘ Someone helpfully asked ‘30% of what?’ This led to a need to put all our numbers in the same representation.
We placed 3 on the right of our number line, and found a need to put 0 at the left of our number line.
We talked about where to put 3/10 = 0.3 on our number line. H suggested we first mark 1 and 2 on the number line. (We talked about placing them at equal intervals.) Someone then placed 0.5 as a reference point. We then partitioned the interval from 0 to 0.5 so it showed tenths, and placed 3/10.
Finally we discussed where 0.(3) belongs. Should it go to the left or the right of 0.3? We started connecting this to place value.
I asked: What role do different representations play in how we perceive a number?
A third look: the different sets of numbers (N, Z, Q, R)
I asked students to find 0.(3) , 30%, 3/10 and 3 on these posters courtesy of David Butler: www.adelaide.edu.au/mathslearning/handouts. I love these posters because they show numbers in different representations, and because numbers appear in the same place when they are on multiple posters. (I slightly modified them to include our numbers.)
I asked what it means: To be on a poster? To be on more than one poster? To not be on a poster?
From here we progressively discussed our numbers:
- The number 3 prompted discussing the difference between natural numbers and integers.
- Our different representations of 0.3 (as 3/10, 30% and 0.3) prompted a closer look at the definition of rational numbers. The ‘quotient of two integers’ connected to representing 3 as a rational number. The ‘can be expressed as terminating or as repeating decimals’ helped us consider the placement of 0.(3).
We drew the Venn diagram representation and probed what it means to be an irrational number. (Every eye was drawn to pi on the poster!)
We wrapped up the discussion with some ‘give an example of’ statements:
- an integer that is a natural number
- an integer that is a rational number
- a rational number that is not an integer
- an integer that is not a rational number
In all, this was a good elaboration of WODB into some of the key concepts of this first lecture.