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Base and index inquiry

Helen Hindle (an advanced skills maths teacher) devised the prompt to generate short inquiries with her lower attaining classes in years 7 and 10. The year 7 class used the question and comment stems to express their uncertainty about the "little number". These are their questions, observations and comments:
    • Why is the 2 on top of the numbers?
    • I know the little 2 means 'squared', but I don't know how to square.
    • This is a sum. I think it is true because 2 x 5 = 10. At the moment I don't know what 22 or 52 mean.
    • 2 x 5 = 10. With the little 2, it still equals 10. I don't think it will make a difference to the numbers.
    • I think it is false because 22 = 4, 52 = 10 and 102 = 20.
    • I think the real answer is 14.
    • I think it is false because 2 x 2 = 4 and 5 x 2 = 10. So it should be 40.
    • 52 = 25
Contrast these responses to those from the year 10 students (see picture top right) who spontaneously suggested changes to the prompt and also attempted to generalise by considering other cases. In the top right-hand corner of the board we can see three students speculating about the law of indices: one claims 22 x 33 = 66 because you multiply the indices; a second student summed the indices and got 65; and the third went for 64 because the indices increase in consecutive numbers. Note, none thinks that the base number is important in their considerations because 2 x 3 = 6, just as 2 x 5 = 10. This is a rich set of questions and comments that encompasses the meaning of indices, how to manipulate the equation, the laws of indices and numbers written in standard form. 
After one hour, Helen asked the year 10 class to reflect on how the inquiry had contributed to a growth mindset (see the second picture). She reported that the inquiry "really engaged some of my most challenging students and generated the best group work and discussion of the year."

Prompt sheet
Promethean flipchart    download
Smartboard notebook

The picture above shows the initial questions and comments from the year 10 class. Below we see the students' reflections on the inquiry process. The advantages of inquiry include asking questions, using mistakes to learn, the development of fluency and thinking about what has been learned.
Helen Hindle
teaches at Longhill High School in Brighton (UK). She has an excellent website on developing a growth mindset in maths. You can follow Helen on twitter @HelenHindle1.