Embedded Files
The prompt
The prompt
Mathematical inquiry processes: Make connections; extend patterns; reason. Conceptual field of inquiry: Multiplication and division by powers of 10; and place value.
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 Raffaella Borasi wrote in Learning Mathematics Through Inquiry (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
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. They use place value to convince their peers that multiplying by a tenth is equivalent to dividing by 10. Both operations shift each digit one place to the right. The teacher could use a zoomable number line to facilitate the explanation (see illustration).
Powers of 10 structured inquiry (PowerPoint)
Forms of reasoning
Forms of reasoning
In a year 7 lesson, Zeynab responded to the prompt by extending the calculations in both directions (see picture), She made up her own example using a different multiplicand and then noted some pairs of operations whose product and quotient are the same.
Interestingly, Zeynab struggled to break away from the example 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 an empirical pattern-spotting approach. She reasoned that as “the middle numbers are sort of flipped round” - that is, the multipliers and divisors are reversed - the second list in the prompt could be extended 'upwards' to 4 ÷ 100 = 0.04.
Adam, however, started to use transitional reasoning when he compared the next line of each list. 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.
Extending patterns and reasoning
Extending patterns and reasoning
Students often take an inductive approach at the start of an inquiry. They attempt to spot a pattern that then forms the basis of a conjecture. As students extend the pattern, they create more examples and practise calculations. However, mathematical inquiry requires an explanation of the pattern and even a proof that it will continue in the same way.
Moser's circle problem shows the risk of generalising from a pattern after only a few cases. The problem involves finding the greatest number of regions for different numbers of points on the circumference of a circle. Each point is joined to the others with a straight line. Up to the fifth case, the greatest number of regions for n points follows the geometric progression 2n - 1 - that is, 1, 2, 4, 8, and 16. However, when n = 6 the greatest number of regions is 31, not the 32 that follows the pattern.
Contradiction
Contradiction
Extending patterns can also lead to a contradiction. In her book Learning Mathematics Through Inquiry (1992), Professor Raffaella Borasi, presents two patterns (below) to a pair of students. She writes: "If the heuristic of extending a pattern is applied when we try to evaluate 00, a potential contradiction follows, since two alternative patterns seem both possible and reasonable and yet each suggests a different value for the result."
The patterns can form the basis of an interesting discussion with older students in secondary school. However, the discussion can be difficult to resolve when students take sides on the basis of what feels right. 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.
Ultimately, the inquiry teacher must draw on the convention that 00 = 1, which the wider mathematical community applies in most contexts (although Borasi supports the case for it remaining undefined).
Reasoning
Reasoning
The examples of patterns that break down after a few cases or lead to contradictions are reminders that students should not extend a pattern automatically without trying to understand its mathematical structure.
This applies to the powers of 10 prompt as well. Students use the concept of place value to explain the multiplication and division of decimal numbers by powers of 10.
A novel approach
A novel approach
One year 9 student suggested a novel approach in response to the contradictory patterns for 00. He used an Excel spreadsheet to show that xx tends to one as x approaches zero. However, the formula does not define xx on the spreadsheet when x = 0 (see graph). When others in the class followed the student's example and used a spreadsheet to graph values (up to two decimal places) between zero and one, they noticed that the graph of xx starts to increase between 0.36 and 0.37.
One year 9 student suggested a novel approach in response to the contradictory patterns for 00. He used an Excel spreadsheet to show that xx tends to one as x approaches zero. However, the formula does not define xx on the spreadsheet when x = 0 (see graph). When others in the class followed the student's example and used a spreadsheet to graph values (up to two decimal places) between zero and one, they noticed that the graph of xx starts to increase between 0.36 and 0.37.
Resources
Resources
Page updated
Google Sites
Report abuse