Khan Cube Roots

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Objective

SWBAT to find perfect and non-perfect cubic roots

Big Idea

Students need as many opportunities to practice and reflect as we can provide to them.

Assessment Overview

10 minutes

This assessment can be done at home or in class. These assessments help students recognize perfect cube roots and practice using prime factorization to break down non perfect cubic roots. It is important to tie this specifically to the common core standard and ask them related questions with variables: "if we know that x^3 = 343, what is x?" Students will also have a difficult time with some of the basic cubic roots (they tend to range between the third powers of 1 through 13). I would encourage them to use a calculator as they tinker with these numbers. 

I give them a cubic root handout and ask students to record some of the basic patterns they notice that could help them remember these number facts.

Here is a link to the handout: Khan Academy Cube Roots

Students aren't required to memorize these numbers, but I find that it helps them if they can recognize a few. For example, if I remember that 5^3 = 125, then I know a number that is slightly higher is probably 6 or 7 to the third power. I ask them to write down anything they notice and then watch the intro video. I want them to compare the patterns I mention to the ones that they record. 

http://youtu.be/xLj9-Q6Boog

 

 

The second module is much tougher and I certainly encourage students to support their mental arithmetic with a calculator. As always, I encourage them to write everything in their notes, because it is easy to lose track of these numbers without taking the proper space and time to work. 

http://youtu.be/2yuJY3dSaFI

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. 

 

Notes about Khan Academy: 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. They don't all apply to the 8th grade standards, but these two tend to help students work through many other complex problems. If you are only teaching the 8th grade standards, I would only stress the first cubic roots set and offer the second as a bonus or challenge. 

The key is to circulate and ask questions that deepen their thinking about the mathematics in the modules. Ask them "how do they know they are right?"

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 for today's first activity work:

http://www.khanacademy.org/math/arithmetic/exponents-radicals/cube-root-tutorial/e/cube_roots

They could also go to the exercise dashboard and type in "cube roots."

The guidelines for today's work are as follows:

  1. Finish each module until you reach "mastery."
  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. 
  4. Create solve and explain a challenge problem that would fit nicely in each module.
  5. Why does the cubed root of x^3 = x? 

I usually ask for the part 5 via email.

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.