Khan Exponent Challenges

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Objective

SWBAT to demonstrate proficiency with properties of exponents on tougher problems.

Big Idea

Online tutorials can be used to challenge students as well as to enable them to develop proficiency.

Assessment Overview

10 minutes

This assessment can be done at home or in class, but from an 8th grade perspective I treat these three modules as challenges. Challenge modules are great opportunities for all students. Some students will take on the challenge at hand and others will use the same time to catch up on past content that they found challenging. I always have something set up for all students to work on. I base it on the weekly feedback I get from students and prepare tasks and suggestions based on the standard based grading system. In other words, students and I are very aware of their current struggles and we use that information to help them pick an appropriate task. 

These exponent modules are challenging because they frequently deal with fractions and negative exponents. The first module tends to keep the exponents as integers, but frequently have fractions for bases. The second module, called "fractional exponents" adds simple fractional exponents into the mix. The third module, called "fractional exponents 2" adds improper fractions and large calculations into the process.

For these modules I encourage students to use calculator when they are stumped with a calculation. However, I ask them to save the calculation that stumped them and share with me during class. I like to list these on the board and share at the end. I ask students if they can find an easy way to think about the arithmetic. 

The challenge of the lesson is to complete any of the modules and if they are feeling brave, to finish more than one module. For anyone feeling like a superhero, I encourage them to finish all three modules. 

I have students start with the intro video and circulate to see how they are doing with the laws of exponents. 

Here is the video for the negative exponents module:

http://youtu.be/oZLsOkJnKM4

 

Here is the video for the fractional exponents video:

http://youtu.be/uS4C_Hb7AbQ 

 

Here is the video for fractional exponents 2:

http://youtu.be/lrPWMjwTCaA

 

This activity can be incredibly helpful if students slow down and take the time needed to think about what they are doing and why it makes sense. The series of exponent activities on Khan Academy can be done all in a row or spread out. They can be placed anywhere in the unit with great success. I wouldn't worry if students are perfectly prepared for any particular exercise at any time. Instead, use these assessments whenever you want. The digital environment is such a different experience that students will approach the problems with excitement and will often not connect the work to their experiences in class (at least not automatically and especially not in the beginning). As a teach, you need to make that connection for them. To do that, I end each class with a discussion about the problems they tried and even review them in the beginning of the next class session. The goal is to constantly spiral over the same concepts, but take every approach possible. 

 

Khan Academy is constantly changing its layout and its scoring system, but for this activity I would ask students to log in (optional) and work until their achieve mastery. This is a topic you need to discuss with students. "Mastery" in Khan Academy might mean something like getting 20 correct, but I want students to complete about 10 questions and only continue if they think they need more practice. I have had many students complain about Khan Academy. They get frustrated, because if they make a single mistake they need to basically start from the beginning. They find this discouraging. They kept working and working even when they understood the topic. They spent hours trying to get "mastery" and would give up if they hit the wrong button or number. Instead, they need to stop and reflect. They need to think, "do I need more practice?" 

The Assessment

40 minutes

There are currently 12 exponent exercise sets on Khan Academy and this is one of the most fundamental in terms of fluency. This is one that I encourage students to even try and get 10 in a row (as opposed to 5 or less on other modules). 

The key to this assessment is to make sure students write out the steps involved with the laws of exponents. For example, if they are subtracting exponents, they might show the terms canceling or explain which laws of exponents help them solve the problem. I like to review the laws of exponents on the board so that students can reference the law in their notes. They might write, "This example is a clear example of how we can add exponents when we are multiplying two or more numbers with the same base, where x^a * x^b = x^a+b

The structure of the site is overwhelming to many students. To simplify the process, I have them log in to Khan Academy and then open a second tab and go straight to this link:

http://www.khanacademy.org/math/arithmetic/exponents-radicals/negative-exponents-tutorial/e/exponents_2

 They could also go to the exercise dashboard and type in "negative exponents."

http://www.khanacademy.org/exercisedashboard

They could also find fractional exponents here:

http://www.khanacademy.org/math/arithmetic/exponents-radicals/negative-exponents-tutorial/e/exponents_3

And fractional exponents 2 here:

http://www.khanacademy.org/math/arithmetic/exponents-radicals/negative-exponents-tutorial/e/exponents_3

 

The key is to ask students a follow up question. The guidelines are as follows:

  1. Finish the module until you reach "mastery." We encourage you at least 10 problems in a row. 
  2. As you work, write the questions and answers in your notebook.
  3. When you are finished, annotate your notes and explain some general observations you made as you worked. Identify the laws you used to solve each problem. 
  4. Create solve and explain a challenge problem that would fit nicely in each module.
  5. What is (a^2 * b^15)^3? Explain how you know.
  6. Does (x+y)^a = (xy)^a?

I usually ask for parts 5 and 6 in email and ask for very detailed explanations. 

Since all students have set me up as a coach I can easily monitor their progress after class. I circulate during class and help students by asking them reflective questions, like "when you move the decimal, what are you doing to the number?"

I collect the notes from at least 1 student who has mastered the topic and 1 who is struggling.

Assessment Review

10 minutes

I finish this assessment by reviewing questions with the class. I log into Khan Academy and project for the whole class to see. I popcorn around the room and ask students to solve and explain. For each question I get at least 2 algorithms, since students love to hear other strategies. I have noticed that many students use one strategy throughout all the problems and are usually so tired of it by the end that they crave a more efficient strategy. I wait until the end to share all strategies because I believe that process of struggling helps them process the importance of a more efficient strategy. If we just shared at the start, I think many students would blindly plug in the more efficient strategy without understanding why or how it is efficient.