marykadams
Registered: 03/02/09
Posts: 10
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Reply with quote | #1 |
Hi All,
I have been doing some work at home and going over my algebra curriculum for next year in terms of what students struggle with most. Having looked at the different standards and lessons, I think I would like to focus on the laws of exponents. The students always get the laws mixed-up and confused. I believe it even begins before the formal laws in which students in pre-algebra and before get confused when adding like terms... is 3x + 2x equal to 5x or 5x squared? This may be where their troubles begin. So I am hoping to use this as an area of study and refinement for the summer. Let me know what you guys think and where we should post this... on the middle school or high school page. It fits in both categories. |
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childs_r
Registered: 01/21/09
Posts: 21
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Reply with quote | #2 |
Hi,
Marna proposed to work on the Law of Exponents for the HS prior to the meeting at Lane ESD.
I find this a worthwhile unit to work on. My experience has been that students struggle with that topic in Advanced Algebra. In the past I have only reviewed those laws very briefly in AA. My wishful thinking was students have had and mastered those in Algebra.
The other two reasons I think it is worthwhile 1. the topic of exponents integrates greatly with science and 2. feeds into Logarithms.
Furthermore as Mary mentioned this topic works well to for the High School algebra curriculum and Middle school algebra curriculum and as mentioned for Advanced Algebra/Algebra 2.
Therefore I propose that we work together with the MS on this topic.
Mary and I have been talking about that possibility and thought that this would work great.
Regine |
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aflack
Registered: 11/16/08
Posts: 13
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Reply with quote | #3 |
Hi all, hope you are having a great summer. I think doing a unit on the las of exponents is a great idea. A lot of my low end Algebra students were definitely getting the laws confused.
Aaron
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marykadams
Registered: 03/02/09
Posts: 10
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Reply with quote | #4 |
Regine,
I know we have talked about this in class and looked at the first lesson in your textbook: Zero and Negative Exponents. Would you please post the titles of the rest of the book's lessons on exponents? Based on those we can start collaborating on specific parts of the laws of exponents. Thanks.
Mary
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childs_r
Registered: 01/21/09
Posts: 21
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Reply with quote | #5 |
Hi Mary and hi Aaron ( glad to see you are part of this)
the book we are using is Prentice Hall Mathematics, Algebra 1.
Here is the way our book covers the unit on Exponents and Exponential Functions
1. Zero and Negative Exponents
2. Scientific Notation
3. Multiplication Properties of Exponents
4. More Multiplication Properties of Exponents
5. Division Properties of Exponents
6. Geometric Sequence
7. Exponential Functions
8. Exponential Growth and Decay.
For the purpose of our lesson we would focus mainly on 3./4. and 5.. These three lessons focus on the laws of exponents.
Would be good to see how other books cover the unit.
Regine |
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childs_r
Registered: 01/21/09
Posts: 21
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Reply with quote | #6 |
Hi all,it appears that we need to provide some evidence of the problem from classroom data. I did not teach Algebra last semester but I taught Algebra 2. As mentioned before I have experienced students having difficulties with the Law of Exponents in that class as well. Especially as we move on to rational exponents which would be another topic to investigate. All the data for this is at school but out of memory I recall that students first do well when the Laws are introduced separately but as soon as they are all mixed up they start having problems. Furthermore, when rational exponents are introduced on top of it they not only have difficulties with the laws of exponents but also with adding, subtracting, multiplying and dividing fractions.
I have noticed that students many times understand simple problems such as x*x=x^2 they understand that it is not 2x. But the students struggle when the base is a number and not a variable. Mistakes I have encountered are that students add the exponents and also add the base. Example: 3^2*3^5=3^7 but students go 6^7 they add the bases. Generally, they understand the fact that you only can add exponents when the base is the same but then they wonder what happen to the "extra" 3 at the example. They see the exponents and know how to add but there are also two 3's which need to be added. I let them write out all the factors of the 3s in that case and tell count all the 3s- how many do you count? Is there any 6? and they do see it for that moment. Overall, students just become confused when trying to remember the rules. Maybe from here we could discuss some questions about the content and pedagogy.
I outlined in my previous thread how the Algebra book we use introduces Exponents. It would be interesting to see how other books introduce those Laws. Looking at ours I would mainly focus on 3.-5. bullet. At 4j we might also need to have a look at the Algebra Conceptualized material. Regnine |
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aflack
Registered: 11/16/08
Posts: 13
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Reply with quote | #7 |
I don't have my textbook with me , it is at school. But like you said, the students seem to do ok when the properties of exponents are shown individually. When you start combining the properties, that is when they get confused. I am not sure how our textbook presents the material off hand but I know when coefficients are introduced in the problems such as 2x^2 * 4x^4, students get confused whether to add the coefficients or multiply them. I hope your summer is going great.
Aaron
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childs_r
Registered: 01/21/09
Posts: 21
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Reply with quote | #8 |
Hi Aaron,summer is going good. Glad the heat wave is over. I am wondering if we teach those Laws we should take a longer period of time spread the different laws out. Maybe, presenting the laws so close together causes the confusion. For instance, students learn earlier on multiplying 2x*3x without specifically introducing the laws we teach them that this would be 6x^2. There seems no major struggle with this. We could already introduce the addition law here and expand it out to more complex problems by changing the exponents to different values. Way later we go into the division and so on. Then at one point we put all the laws together. Regine
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mknoer
Registered: 06/30/06
Posts: 33
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Reply with quote | #9 |
Hey Regine and Aaron and glad to add MaryK,I've been pretty much out of it this summer. My apologies. With out district common Algebra assessments, CT 11,12 & 13 were the ones that caused the most problems. I did a summer session make up session for students missing 3 or fewer CTs. I thought it would be lots of kids who didn't understand quadratics but these earlier ones with the exponents were the ones holding students up. It seems to me that while students have no problem writing expressions in expanded form, as soon as we ask them to evaluate the expressions or simplify expressions, they revert to thinking multiplication. I also end up needing to reteach the distributive property. The concept that distribution only occurs at the next lower step in the order of operation seems to be a big part of the problem. I also recall that students had more difficulty than I would think with simplifying fractional expressions using order of operation, even without throwing exponents into the situation. I like the idea of separating the different laws so they don't become so confused in the students' minds. How would you suggest separating them, more specifically what content would you put between the various laws when you separate them?
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Marna |
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mknoer
Registered: 06/30/06
Posts: 33
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Reply with quote | #10 |
I've spent the day thinking about how to make an investigative approach work in developing mathematical concepts and am trying to see how this could apply to teaching concepts around exponents. Perhaps an activity where students take an expression like x^(-2) and evaluate for several values of x (prime values and fractions with relatively prime values, mayhap) and develop their own way of seeing that it is equivalent to 1/x^2. A deeper understanding of the equivalence of these expressions may be better than separating the concepts in time to avoid the confusion people are noticing.
Has anyone found any research into why these ideas are confusing?
I have found one research paper on laws of exponents, its from Greece: Secondary school students’ levels of understanding in computing exponents Demetra Pitta-Pantazi,,Constantinos Christou and Theodossios Zachariades
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Marna |
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