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Multiplication and division inquiry


The prompt encourages students to compare and contrast two lists of calculations. It aims to 
make students aware of the consequences of dividing by numbers between zero and one and the link between, for example, multiplying by 10 and dividing by a tenth. As Borasi wrote (see below), patterns provide a "valuable heuristic to guide the extension of a known operation to a wider domain." In the initial phase of the inquiry, students have posed questions and made observations: 
  • The numbers you multiply by and divide by have been turned upside down.
  • I know that 4 x 100 = 400 and 4 x 1000 = 4000 and you can carry on.
  • How can you get the same answer when you multiply by 10 and divide by 0.1?
  • How can multiplying make the number smaller?
  • If you divide a whole number, how can the answer get bigger? It doesn't make sense because dividing makes the number smaller.
Lines of inquiry
The use of ascending and descending powers of 10 in the prompt invites students to extend the lists above and below the lines that appear. They might also make up their own examples with another whole number (including two- and three-digit numbers) before moving on to more complex examples using decimal numbers (for example, 2.9 or 8.71). The teacher might challenge the highest attaining students by requiring them to select a 

number between zero and one (0.47 x 0.1 = ..... and 0.47 ÷ 10 = .....). After manipulating the 
equations, the inquiry ends with students explaining why pairs of solutions are the same. The teacher could use a zoomable number line to facilitate the explanation (see illustration).

Resources 

  
A contradiction encountered by extending patterns
Professor Raffaella Borasi, in her book Learning Mathematics Through Inquiry (1992), introduces the students she is working with to the two patterns in the illustration. She writes: "If the heuristic of extending a pattern is applied when we try to evaluate 0
0
, a potential contradiction follows, since two alternative patterns seem both possible and reasonable and yet each suggests a different value for the result." For Borasi, this is a fruitful contradiction because she is running a series of inquiries with two students into what constitutes a mathematical definition.
  
The patterns can form the basis of an interesting starter discussion with older students in secondary school. However, the discussion can become a bit messy with students taking sides on the basis of what 'feels right' to them. So, typically, they argue 00 must equal zero because 'there's nothing there' or 00 is one because the law of indices would not work if it was anything else. While there might be merit to both arguments, this is an opportunity for the inquiry teacher to remind students of the wider mathematical culture in which 00 is normally defined as one (although I note that Borasi supports the case for it remaining undefined).
  
One of the most novel responses I have had to these patterns was when a student suggested using a spreadsheet to see what xx equalled as x approached zero. The answer is one, although the spreadsheet, we decided, does not amount to a proof. When students graphed values (to two decimal places) between zero and one, they noticed that values for xx start to rise between 0.36 and 0.37. More broadly, the activity reminds students that they cannot extend patterns unquestioningly.
Classroom inquiry
In a year 7 lesson, Zeynab responded to the prompt by extending the calculations in both directions, making up her own example using a different multiplicand and, finally, noting some pairs of operations whose product and quotient are the same. Interestingly, Zeynab has struggled to break away from the teacher's model in the prompt. As she develops her own examples, she uses '4' rather than her chosen digit in three equations. This suggests that she is completing the task by direct comparison, rather than thinking about each calculation separately.

In the class discussion about the prompt, students demonstrated different levels of mathematical thinking. Amelia used empirical pattern-spotting to suggest that the second list in the prompt could be extended upwards to 4 ÷ 100 = 0.04 “because the middle numbers are sort of flipped round” - that is, the multipliers and divisors are reversed. Adam, however, started to use transitional reasoning when he compared the form of the next line of the lists. He claimed that 4 x 100 = 400 and 4 ÷ 0.01 = 400 are linked: "I’m just saying that if you look at it, the decimal ones are practically exactly the same only the other way round. Zero point zero one, if you turn that around you get one hundred, so the answer – four times one hundred." Adam went on to say that "I can almost explain it, but I can't" as he wrestled with a structural explanation of why multiplying by one hundred and dividing by one hundredth will give you the same answer.