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Exploring Indices Visually (Investigation)



  • To visualize exponents.
  • To use standard form to estimate size and distance.


  • 5 Pieces of construction paper
  • Scissors
  • Markers
  • Measuring tape
  • Internet
  • 2 Post-it Notes
  • Computer Paper
  • Pencil


Creating the Accordions

  1. First take a piece of computer paper and cut several small squares about a half an inch big each.
  2. On the back of each piece of paper place a loop of tape.
  3. Make at least 8 of these then place them to the side.
  4. Hold your piece of construction paper horizontally and fold it into thirds.
  5. Cut along the folds you just made. You should now have three equal rectangles.
  6. To create the exponent accordion first fold the edge paper so that there is about an two inches of space. 
  7. Fold the rest of the strip of paper back and forth, but make the folds smaller than the original 2 inch fold. The last fold should be larger than all of the others so it sticks out behind the original. When you’re done the paper should look like this:
  8. On the front flap write your exponential expression in exponential form using your marker. For example  2^{5} would look like this:
  9. Create exponent accordions for the following exponentials: 
    • 3^7
    • 3^5
    • 5^3 (make two for this one)
    • 4^3
    • 4^{ - 6 }
    • 10^3
    • 10^6
    • 10^{ - 5 }
    • 5^6
    • 9^{ - 4 }
    • 9^{ - 6 }
  10. Unfold your paper. Then write the distributed notation form that corresponds to your exponential expression.  Hint: Write out the distributed notation form for negative exponents as fractions ex.   3^{-2}=\frac{1}{3}\times \frac{1}{3} etc.
  11. Lastly, fold your paper back up as it was. Draw an equal sign on the front and write the answer to your exponential on the flap sticking out from the back. Fold this flap behind so that the answer cannot be seen while you are working.


  12. On each of the Post-it Notes write either a multiplication or a division symbol. Make two Post-it Notes for each.

How to use the exponential accordions

  1. On construction paper place the accordions that you need to solve the problem.
  2. Between the accordions place the necessary Post-it Note sign. Let’s say I’m trying to solve the problem \left ( 2^{5}\times 2^{8} \right )\div 2^{4} , I would set up my problem as follows:


  3. Notice that I used my pencil to write in parenthesis. I will erase these before starting the next problem. You may need to write in parenthesis or a missing number depending on the problem.
  4. Unfold your accordions so you can see the expanded form of the problem.
  5. Look for terms you can cancel out. When two terms cancel place one of the small squares with tape over each.


  6. Perform the indicated operation on the remaining terms. In your notebook write down the problem you did as well as your answer both in simplified exponential form and the final numerical result.    


  1. Use the appropriate exponent accordions to solve: 3^7 \times 3^5  
  2. Use the appropriate exponent accordions to solve: (5^3)^2 \div  5^6  
  3. Use the appropriate exponent accordions to solve: 4^{3}\times 4^{-6} 
  4. Use the appropriate exponent accordions to solve: 9\times 9^{-4}\times 9^{-6} 
  5. Use the appropriate exponent accordions to solve:4^{3}\div 4^{-6}
  6. Use the appropriate exponent accordions to solve:9^{-4}\div 9^{-6}  
  7. Use the appropriate exponent accordions to solve:\left ( 7\times 10^{6}\times 10^{3} \right ) \div 10^{-5}  
  8. Look back in your notebook at all of your results. Can you connect each of your answers to an exponential law (multiplication, division, power of power)?
  9. How do you think you could write your answer to number 2 in exponential form? Why?
  10. Combine one or more of the numbers 10^{6},10^{3}, 10^{ - 5 } with any single digit 1-9 to create an expression that could roughly represent the distance from New York to Florida, or any two major cities in a country of your choice. You can use any operations you would like to create the expression.
  11. Simplify the expression you created.
  12. What kind of notation is your expression in?
  13. What units did you use for your measurement of distance?

Work with a friend!

  1. Did they come up with a similar expression? Compare and contrast your expressions and the methods you used to create them.
  2. Use standard form to express your guess for how large the following distances are:
  3. the distance between the desks in your classroom
  4. the distance from your school to your house
  5. the distance from the east coast of the USA to the west coast
  6. the distance from the earth to the moon
  7. Explain your reasoning behind each guess.
  8. What units did you use for each guess? Why?
  9. Check your guesses either by measuring them with your measuring tape or looking them up on the internet.
  10. Were your guesses close?

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