The indices prompt is suitable for 11- and 12-year-old children, although it can be adapted for older students.
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. The main focus at this early stage, however, should be on explaining why some squares can be linked by the lines. Why, for example, do the squares of numbers ending in four and six always have six in the units column? By extending the pattern 'upwards', students find links for 5 and 10; extending the pattern 'downwards' has required students to multiply two negative numbers.
The prompt can be changed in the following ways:
- Use a different exponent, such as 14, 24, 34, and so on. Students should decide if this activity is better arranged in separate groups or as a whole class. They can then explain the links between the unit digits.
- Keep the base number the same and change the exponent, such as 71, 72, 73, and so on. Again, students can decide on the nature of a collaborative activity, before explaining the links between the unit digits.
- Link numbers in other ways, such as link two numbers to their product (possibly guided by the teacher). So, for example, 22 x 52 = 102. Can the student prove that a2 x b2 = c2 if a x b = c? With the base number held constant, this 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.
- Use negative exponents (such as -2 instead of 2), which can lead to an independent inquiry for individual younger students or, perhaps, a jointly constructed understanding for older classes.
The inquiry ends with students presenting their mathematical findings and evaluating their decisions on the course of the inquiry.