Cube Root Solutions

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Objective

SWBAT solve simple equations using cube roots.

Big Idea

To avoid rote mechanical solving of equations involving cube roots, good understanding of the concept is a must.

Launch

15 minutes

Before beginning the lesson:

  • Hand each student a CUBE TABLE WORKSHEET.docx and ask them to answer question 1 above the App table. (Acceptable answers are: equal edges, 3 dimension, volume is found by multiplying 3 dimensions)
  • Ask students to download the app called Think 3D Free.

Alternative: Project the app on the board for all to see and work along with the class.

The Think 3D Free application can be downloaded here.


Using the Application: (See Video)

  • The app begins with 1 cube. 
  • Tapping the screen places another cube on the exact spot tapped
  • Ask students to construct the next 3 perfect cubes (the next perfect cube will contain 8 cubic units, followed by 27 cubic units and so on.  Students should complete the white un-shaded boxes in the worksheet as they construct the cubes. 
  • Ask students to analyze the pattern shown in the table and complete the grey shaded region of the Table

After the App

Ask students to answer questions 2 and 3, below the App Table.

2. What can you conclude when analyzing the first two rows?

Students should conclude that the volume of a cube equals the number of cubes that make it up. A common answer is "the number of cubes that fit inside."

3. Write an equation representing the relation between rows two and three. 

Students should conclude with V = e3  : e = edge, or something similar.

Finalize by writing the volume equation V =s3 on the board and tell students that s = side of the cube

 

 

 

 

 

 

New Info

20 minutes

Once finished, project the table on the whiteboard and call on students to come up and fill in the boxes on the whiteboard. Ask the students to explain how they got their answers.

I would ask for thumbs up, sideways, or down to gauge how well students have grasped the completion of the table. Take a few minutes to publicly help clear any confusion among students on the completion of the table. 

 Refer to the equation  V = s3    

Add that in a manner like that of a square, studied in the previous lesson, if the volume of a cube is V, then an edge of the cube is called a cube root of V or  3√v

 and   3√v ∙ 3√v ∙ 3√v  =  3√v3 = v

 Example: 3√3= 3√27 = 3 

Calculator work

Ask students to find the cube root button on their calculator. TI calculators and many others have it under the x3. Have them check some cube roots in their App Table to practice using this key. 

 

Application

15 minutes

Ask students to simplify each expression in question 4 and state what they can conclude about cubes and cube roots. Students should be able to explain that cubes and cube roots are inverse operations, like squares and square roots, division and multiplication, or addition and subtraction.

Now ask students to solve the equation in question 5 and complete question 6.

Call on a volunteer to go to the board and write their work. Students usually are quick to see that taking the cube root of either side solves the equation for x.  

Inform students that up to now we have been seeing cube roots of perfect cubes and that there are perfect cubes that are common and should be learned. I like to have the entire class say the first 6 perfect cubes together: "1, 8, 27...216"

Application Problems: Ask the students to solve for x in each case. Allow students to discuss their work and use their calculators. 

1) 8x3 = 24   (Round answer to nearest hundredth)

This problem can be solved two ways. Make sure all students see both routes and that one route may be easier than the other depending on the equation.

2)  x3  =  - 0.027 

After this is done, tell students to note that the cube root of a negative value exists.

3)  (8/27) = x3

 If students get stuck here, ask to recall the root of a quotient property.

         

 

 

Closure

5 minutes

To close the lesson, ask each student to write two things on the back of their Cube Table Worksheet.

1. What part of the lesson or what specific problem would you want the teacher to go over again? 

2. Where do you think the lesson will go from here? Can you figure what comes next??