Mathematical inquiry processes: Interpret; make connections; identify and extend patterns; reason and explain. Conceptual field of inquiry: Squares and indices; the relationship between integers and their product.
The inquiry starts with students trying to understand the diagram. It is better to delay any discussion about the meaning of the exponent until students have had the opportunity to raise questions about the diagram. In this way, the teacher can evaluate the existing level of knowledge in the class and consider the detail and duration of an explanation (if one is required). Moreover, the types of questions and comments will allow the teacher to decide whether any students can be asked to give that explanation.
Typical questions and statements that arise during the orientation phase are:
Why are numbers linked by lines?
Do the lines carry on in the same pattern?
What does the 'little 2' mean?
Why are there no lines going to 25 and 100?
Why are there four lines? Could there be more?
If the last digit (in the units column) is the same, the numbers are connected.
What happens if you extend the pattern 'downwards' into negative numbers?
Can you extend the pattern 'upwards'?
As the inquiry gets under way, the teacher should dispel any thoughts that 152 means 15 x 2 and not 15 x 15, thereby ensuring that the exponent is applied correctly. By extending the pattern 'upwards', students find links for 5 and 10; extending the pattern 'downwards' has required students to multiply two negative numbers.
One popular line of inquiry is the construction of chains, such as 1 - 81 - 121 - 361 - 441 or 4 - 64 - 144 - 324 - 484. Why can you construct only six chains (for squares ending in the digits 0, 1, 4, 5, 6, 9)? Why aren't there chains for 2, 3, 7 or 8? The answers to these questions requires students to reason about the structure of multiplication. For example, the squares of numbers ending in one and nine will always end in one. Similarly, the squares of numbers ending in two and eight will always end in four.
The inquiry can develop into different lines of inquiry, In a structured inquiry, the teacher directs students (see below). However, often students will come up with the same ideas when asked to change the prompt. The inquiry ends with students presenting their mathematical findings and evaluating their decisions on the course of the inquiry.
Structured lines of inquiry
1. Use a different exponent
Can you connect cubes based on the digit in the units column? What about numbers to the power 4, 5, 6 and so on? How many chains are there in each case?
2. Visual representation
How would the squares or cubes look on a number line or graph? Can you describe the pattern? How would you represent them on a 10-point circle (see below)?
3. Use the same base and change the exponent
What do you notice about, for example, 71, 72, 73, etc. Do they end in the same digit? Why not? Is it possible to find a base number that always ends in the same digit when raised to consecutive powers?
4. Use negative exponents or a negative base
What do you notice about (-2)1, (-2)2, (-2)3, etc? What about 2-1, 2-2, 2-3, etc? How are the results in each case different to 21, 22, 23?
5. Link numbers in other ways
What do you notice about 22 x 52 = 102 ? Can you prove that a2 x b2 = c2 if a x b = c? With the same base number, this line of inquiry could lead to an appreciation of the laws of indices. For example, if the base number is 3, 9 x 27 is linked to 243, which can be recorded as 32 x 33 = 35. (See the base and index inquiry for more information.)
Creating visual representations
In a year 7 mixed attainment class, Shqipdon noticed that the units for the first 10 cube numbers include all 10 digits. When he shared his results with the class, Lily decided that the the observation could be represented on a 10-point circle and sketched a diagram. In the next lesson Lily presented her idea and the students drew the diagram on a template (see the PowerPoint in Resources).
She explained why the pattern repeats: "123 equals 1728. The 8 in the units column comes from two cubed." The class then went on to see if digits of other powers also created patterns that could be represented on a 10-point circle.
Kirsten McGarrie, a teacher of mathematics in London (UK), encouraged her year 7 mixed attainment class to represent cube numbers in different ways.
Starting with the top left diagram below and going clockwise, Kirsten describes the students' designs:
(1) The student noticed that if she wrote the number to be cubed in ascending order around the 'clock' and continued until 19 cubed, then the units of each pair of cube numbers matched. She then doubled the unit of each cube and labelled the clock with that number. Noticing that the units of opposite numbers matched, she joined them with straight lines on the diagram.
(2) The student noticed the commonality of the units of the cube numbers and joined them in ascending order, although I'm not sure what the rectangle in the middle represents.
(3) The student noticed that the units of the cube numbers at either end of the lines she drew were number bonds to 10, except the two numbers left over (zero and five).
(4) The student noticed the commonality of the units of the cube numbers and joined them in ascending order, completing the loop from nine to zero to one.
Differentiation through inquiry
As part of a professional development project about differentiating more effectively in her year 6 classroom, Claire Lee, a PYP teacher in Geneva, used the indices prompt. She presented the results to her colleagues and other teachers. Claire found that using Inquiry Maths gave students the opportunity to challenge themselves at different levels. See Claire's presentation here.
Developing the inquiry
These are the questions and observations about the prompt from a year 7 mixed attainment class. When one pair of students noticed that 9, 49 and 169 were linked because of their last digit, other pairs began to group the squares on the same basis. Initially the students decided that they would all extend the prompt up to 252 and, in so doing, realised that 25 and 100 could be linked to the squares of numbers ending in five and zero respectively. After the teacher had orchestrated a discussion about why the squares of 3, 7 and 13 end in the same digit, the class extended the reasoning to explain the last digit of other groups. In the second lesson of the inquiry, students followed different lines of inquiry:
Change the power (as suggested in one of the responses to the prompt). What happens if we cube the counting numbers?
Use the same base and change the power. Are there any patterns between the final digits?
Extend the prompt in the negative direction. What happens when you square a negative number?
Explore negative indices. What happens if you use the same base and extend the powers in the negative direction?
The inquiry ended with students giving short presentations after which the teacher probed their understanding of the findings.
Questions and observations
These are the responses of Caitriona Martin's year 8 class to the indices prompt. Caitriona reports that the students noticed that the connected number differences go up in a sequence, which is a novel way of approaching the prompt. The observation that the squares of 5 and 10 have no lines attached to them can lead into a discussion about possible digits in the units column when you square numbers
A different representation
The indices prompt was changed to this horizontal presentation to fit in conceptually with an exploration of the number line being carried out by year 7 mixed attainment classes at Longhill High School, Brighton (UK). The students had studied how to represent decimals and fractions on the number line and convert between them. Their learning then developed onto different types of numbers.
The indices prompt was the students' first experience of inquiry learning in Year 7. The first lesson focused on asking questions and making observations. Students went on in the second lesson to explore the patterns they had noticed. They represented the gap between the square numbers (consecutive odd numbers) by drawing squares in the first quadrant of an x-y graph starting from (0, 0). Alternatively, students chose to answer the questions below about their observations at the start of the inquiry.
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 00, 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.