Welcome Math Commanders!

Battle in the Polygon – The Proof is on the Paper

Round 4: This is it Math Cadets! The final round is here. Hopefully you’re feeling confident about your answer. Unfortunately, in mathematics we can’t just take your word for it. In order to convince someone that your decision is correct, we need to complete a mathematical proof. Alright now let’s see if we can figure the problem out with an algebraic proof. Don’t be worried! I’m going to walk you through every step of the way! For this part of our exploration, the work and the process is MORE important than the answer. This is how mathematicians demonstrate to other mathematicians their new discoveries. While you may only think of math as what’s written in a book, in fact new mathematics is being discovered/invented all the time. The way mathematicians show that their new math is correct is by the process of mathematical proof, where we begin with ideas that are obviously true, use standard mathematical operations, and end up with a new idea at the end. Since we started with something we know to be true and followed the rules of mathematics along the way, the final result must be true. Are you ready to complete your first (or maybe not your first) mathematical proof? OK. Let’s do it together!

Follow along with the instructions below and do all your work on a clean page in your notebook. When you are finished, please take a picture or scan your work and upload it below for full credit.

To start, let’s set the variable (a fancy name for a number we don’t know) X equal to \(.\bar{9}\)

Write that as a mathematical statement (like x=2)

Alright, now we’re going to multiply (times) both sides by 10.  10 times X is just called 10X.  On the other side, we need to do 10 times \(\overline{{.9}}~\). Good news, we just did that problem a few minutes ago. Write your new equation and then keep going to the next step.

Ok, now what would happen if we took an X away from both sides?  We had 10X on one side, so we should have 9X left. What about the other side? What would we have left? Remember what X is equal to. If we subtract an X from one side, we can subtract a number away from the other side as long as that number is equal to X. Write your updated equation and then get ready for the mind-blowing final step.

Alright, last step: Now we need to divide both sides by 9.  9X divided by 9 is going to leave us with just X, the same thing we started with.  When you divide the other side by 9, what do you get? Write your final step down and then get ready to have your mind blown!

Did this proof change your mind? No? Well consider the following questions. There is a major breakthrough here that you may not have noticed.

In addition to showing your work, please answer the following 3 questions in complete sentences.

  1. What does X equal in the final step?
  2. What did X equal at the beginning?
  3. Alright, round 3 just ended.  Time to update your prediction of which number is bigger, \(\overline{{.9}}~\)or 1. Do you want to change your prediction? Be sure to explain your reasoning.


Alright Math Cadets, we’ve battled in every possible way we can, using basic fractions and calculators, using algebra, even with coloring! I think the answer is obvious, so write 1-2 paragraphs about the outcome of this problem, and be sure to discuss if you found this surprising and new! Your challenge for today is to find a friend, brother, sister, parent, or other family member and ask them the same question, “Which number is bigger \(.\bar{9}\) or 1? Or are they equal?”  See if you can prove the correct answer to them!

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