Laws of exponents inquiry
Mathematical inquiry processes: Make connections; test different cases; infer and explain rules. Conceptual field of inquiry: Laws of exponents (indices).
Derek Christensen (Mathematics Lead Teacher at A. Blair MacPherson School, Edmonton, Canada) devised this prompt for his grade 9 mathematics class. Below Derek reports on the development of the inquiry, which he guided with five questions, and reflects on the "rich learning experience" created during the inquiry.
Kelly Anne Garner adapted the prompt, using 24 x 23 = 27 to initiate inquiry (see below). The new prompt can be verified more easily than the original: 24 = 16 and 23 = 8, 16 x 8 = 128 = 27. However, the original prompt, in being harder to verify, might focus students' attention on the law of exponents more than the calculations.
Another alternative sees the exponents and base number having something in common. For example, in the prompt 67 x 65 = 612 the final exponent could be related to the other exponents or the base number. The design could lead to more conjectures about the structure of the equation, which, in turn, could lead to more lines of inquiry.
Rich learning experience
Derek Christensen 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 the Inquiry Maths 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.
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:
The bases are the same.
The exponents are added.
The statement is true.
The little number is an exponent and means repeated multiplication.
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:
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.
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.
Making connections through inquiry
These are the notes Derek Christensen made on the board as his grade 9 class inquired into the prompt. Derek recounts how one student noticed a connection between the laws of exponents and fractions: "This is like adding fractions, you keep the bottom the same and add the tops." Derek thought this was an interesting parallel and a good connection to prior learning. That is why, he explains, you see the equation with fractions on the right of the lower picture.
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.
Adapting the prompt
After exploring the potential of the original prompt, Kelly Anne Garner adapted the prompt for her class in the following year. She posted this picture of her students' questions and observations on twitter, declaring that the inquiry continues to be one of her favourites.