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### Laws of exponents inquiry

Derek Christensen (Mathematics Lead Teacher at A. Blair MacPherson School, Edmonton, Canada) devised this prompt for his grade 9 maths class. He reports on the development of the inquiry, which he guided with five questions, and reflects on the "rich learning experience" created during the inquiry:

I decided to try to use a prompt to introduce the idea of exponent laws. In the past I always guided students by having them expand 67 x 64 into repeated multiplication and then back into a single power. Then we would draw a conclusion and state the rule. After viewing this website and using a couple of prompts I wondered if using a prompt would be more effective. So I gave my class, seated in groups, the prompt above.

I then had them record their thoughts using the following questions:
1. What do you notice? (With this we asked, "Is it true?")
2. What do you wonder? (This includes questions like "What if ....?" or "What would happen if ....?")
3. What does this make you predict? ("If this is true then would .... also be true?")
4. How can you test your predictions and make new predictions if necessary?
5. Any Conclusions?

Students came into this class knowing what exponents were, what they meant, and how to evaluate powers. They did not know anything about exponent laws. These are the students' responses to the first two questions:

 1 The bases are the same.The exponents are added. The statement is true. The little number is an exponent and means repeated multiplication. 2 Does this work for other powers with the same base? I wonder what happens if the bases aren't the same. I wonder what happens if you divide instead of multiply. What if the exponents are the same but the bases are different? What happens if you add powers with the same base? Why do you add the exponents?

Question 3 on making predictions led to the following questions, wonderings and propositions:

 Questions and propositions Students' responses Does this work for other powers with the same base? Most students felt that yes it would. I wonder what happens if the bases aren't the same. • I think you multiply the bases and add the exponents. • I think maybe you average the bases and add the exponents.• I don’t think you can do anything but evaluate each power. I wonder what happens if you divide instead of multiply. I think that you will subtract the exponents if you are dividing powers with the same base. What if the exponents are the same but the bases are different? • I think you will add the bases and keep the exponents the same? • I think you multiply the bases and leave the exponent the same. • I don’t think you can do anything with these powers. What happens if you add powers with the same base? • I think you might add the exponents. • Nothing? Why do you add the exponents? Many students expanded both sides to show that they are the same.

Question 4 on testing predictions was
a big part of what we did. Students were asked to predict answers to the “I wonder / What if” questions and then find a way to test their predictions. One important part of this is making sure they know it is not bad to make a prediction and prove it to be false. Knowing something does not work is just as powerful and important as knowing something is true. I am still working out how I want to do this part in a more effective way.

At the end of the inquiry, the class returned to its initial responses to the prompt.

 Initial response Conclusions Does this work for other powers with the same base? Yes it does. I wonder what happens if the bases aren't the same. You can’t combine them into a single power, you can only evaluate. I wonder what happens if you divide instead of multiply. You subtract the exponents. (Following on from this, one student wondered if 64 ÷ 67 gives 6-3, what 6-3 means, and whether, in fact, you can have negative exponents.) What if the exponents are the same but the bases are different? You can multiply the bases and keep the exponent the same. What happens if you add powers with the same base? You can only evaluate each power and then add. Why do you add the exponents? Because they each mean repeated multiplication of the same value, so there is just more repeated multiplication.

The inquiry went way better than I ever imagined. The students went further than I thought they would and wondered and made predictions beyond what I expected. This was a very rich learning experience for the students and for me.

Resources
Prompt sheet
PowerPoint

Sharing ideas during inquiry
Kelly Anne Garner posted these pictures on twitter. They show the ideas and questions from her class of middle school students at the Frankfurt International School. The question about whether 3-4 equals 3 ÷ 3 ÷ 3 ÷ 3 shows the students are operating at the edge of their knowledge in attempting to make sense of exponents. They went on to share their inquiry pathways with each other as part of small and large groups.