The new school I’m joining this fall gives something they call “fact quizzes” to all students repeatedly throughout all three of their years at the school. The two topics they “require” students to memorize are the perfect squares up to 30 and fraction to decimal conversions (for some of the major ones especially). The idea (by the well-meaning teachers) I’m sure is that the fact fluency students will have from those numbers being “readily accessible” is useful/helpful. However, I don’t think they’re familiar with the research that Jo Boaler talks about here, https://www.youcubed.org/think-it-up/speed-time-pressure-block-working-memory/ and https://www.youcubed.org/fluency-without-fear/ and https://www.youcubed.org/category/teaching-ideas/depth-not-speed/
Unfortunately, as new man on the totem pole, I don’t feel comfortable yet, speaking up about doing away with these tests. So I’m still going to need to give them – they’re 10% of my student’s grades! However, I’ve been trying to think about ways to give the students experiences with the numbers that will be meaningful so it’s NOT just about memorizing numbers! I also want to deemphasize the timed portion of the test, and give students unlimited amounts of time to recall/derive the numbers, knowing the impact on working memory and anxiety levels (however, I don’t want to get myself into trouble by not being consistent with my coworkers who I believe time the quizzes).
I think for the squares, I can start (perhaps?) with a basic “visual pattern” whose formula would be y = x^2 and get students to talk about ways to see those numbers (especially the idea that each square number is the sum of consecutive odd numbers related to a great diagram with “colored” L’s to indicate each new row). I could also potentially use the “locker problem” (about students switching the open/close position of lockers based on factors, and students realizing that only numbers with odd amounts of factors – and thus perfect squares – would remain open at the end). I don’t like the locker problem as much because I don’t think it’s as visual with regards to the numerical part, and it emphasizes number of factors rather than a way to determine the squares. So again, I’m going to try to find some kind of growth pattern to use to introduce my students to the perfect squares (got any ideas or links or good pictures? Share them with me!).
For fractions and decimals, however, I’m having a harder time envisioning what to do. Obviously, some fraction/decimal conversions will be easier for students to memorize due to previous repeated exposure (for example, many may already be familiar enough with “basics” like 1/2 = 0.50 because of money or even 7/10 = 0.70 because of money). I’m wondering how I can get students familiar enough with the fraction to decimal conversions (and methods for thinking about these numbers in a useful manner) that they are able to convert quickly.
I’m thinking back to my experiences reading the chapters on fractions in Pam Harris’s book, and wondering if there’s a way to incorporate number strings to help students make use of it. I was completely blown away with how much easier operations with thirds and sixths (and tenths, halves, fifteenths, twelfths, fourths, thirtieths, etc.) were when using a clock to do arithmetic (i.e. thinking “A half hour is thirty minutes and a third of an hour is 20 minutes, so together that’s 50 minutes, and 50/60 is 5/6”). Based on the quizzes I’ve seen, it looks like students will be expected to convert the following:
1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6, 5/6, 1/8, 3/8, 5/8, 7/8, all 1/9, 2/9, 4/9, 5/9, 7/9, 8/9, 1/10, 3/10, 7/10, 9/10
When I think about different ways to convert (especially without relying on long division), I know that to convert anything that terminates is pretty easy because of the relationship to money at this stage. In other words:
1/2 is a half dollar, so it’s $0.50. 1/4 is a quarter dollar, so it’s worth $0.25. 3/4 is three quarters, so it’s worth $0.75 (most 6th graders no longer need to multiply this to think about how much three quarters are worth when considering money explicitly). For tenths, we can talk about 1/10 being one dime and thus worth $0.10, and so 3/10, 7/10, and 9/10 is just 3, 7 and 9 times that. I think tenths can also build into fifths, because there are 5 groups of “twenty cents” within a dollar (or we could start with the reverse, of thinking about twentieths as groups of five since there are 20 nickels in a dollar). This seems somewhat visual because most 6th graders can visual money and money conversions with some ease.
So that just leaves thirds, sixths and ninths. I’m not sure if there’s a way to think about converting these to decimals easily. I know that the idea of time can help with operations with these fractions (since 60 is a good common denominator), but I don’t know that 60 helps in converting 20/60 to 0.3 repeating. I wonder if the place to start is ninths, and helping students see that the pattern is 1/9 = 0.1 repeating, 2/9 = 0.2 repeating, etc. and then making correlations between 1/3 and 3/9 and 2/3 and 6/9 – though that still leaves sixths (since it’s not obvious to me that 0.16 with 6 repeating is “half” of 0.3 repeating). I’m also not sure (quite frankly) how to make this more visual – the connect between the repeating part and the fractions.
The fact quizzes don’t include 7ths (“thank god!”) but a 7th grade unit on rational numbers does. I’m wondering how people teach the 7ths (which are easy-ish if you discover the pattern and modular arithmetic but not at all intuitive for young kids – something I didn’t learn much about until I did some adult math investigations one summer with other math teachers). I don’t like the idea of relying on long division, because for most students, even ones who can use it fluently, the algorithm doesn’t actually make SENSE – and my goal is for everything in class to make SENSE.
So, as a reminder, things I’m asking for from you, dear readers:
- Visual patterns to help my students remember the first thirty square numbers
- Visual ways to connect fractions with their decimals that terminate with a denominator 10 or less (i.e. the halves, fourths, fifths, eighths, and tenths)
- Visual ways to connect fractions with decimals that repeat with a denominator of less than 10 (i.e. the thirds, sixths and ninths)
- Something beyond long division to use when converting sevenths, perhaps before or in addition to the pattern.