Mind Reader: An intro lesson into Order of Operations

I was lucky enough to have a grandfather who loved Math.  Heck, he was elbow deep in it every day of his career.  He was a plumbing instructor at McKinley High in Buffalo, NY, and even published an instruction book along with his uncle.  He loved to check over my homework and ask how I figured out the problems, and if I didn’t show my work, he made me do the whole thing over!

Any time I would spend the night, it would be filled with games like “shut the box”, “War”, “boxed in”, or something he liked to call “Math Riddles”.

This was one of my favorites!  The basis of it was that he would give me a series of clues and I would have to guess the number he had in his head.  We would go back and forth until one of us got stumped.

It was along this line of thinking that helped me plan an introductory lesson for a 5th grade class that was about to begin order of operations.  The teacher was going to be absent and asked if I could take her class for Math.  We met and discussed where her students were, and we bounced some ideas off each other trying to find the best way to introduce why order matters when different operations are being used in the same expression.  We finally found a task that we believed would help in gaining some conceptual understanding with her students and give her a platform on which to help develop her students thinking.  The only issue was that we couldn’t think of a hook.  I let her know that I would continue to ponder and do my best to find one….I was lucky enough to remember my grandpa…

On Tuesday, the class came to the Math Lab at their regular time, but instead of Journal Work, or an estimation station, I started with a number talk.  Just a quick review of basic operations to see what strategies the students relied upon.  I then told them that I was going to read their minds.  They laughed but played along.

I had the students find a seat at the tables, get comfortable, and then I began:

Me: “I bet that I can guess what number you will end up with after I give you a series of steps, no matter what number you decide to pick”

They laughed, as I explained that my new haircut gave me the ability to allow their brain waves to mix with mine more easily.

Me:“In order to proceed, please make sure that you clear your minds.  That you do not share your thoughts or work with your neighbor as it will interfere with my abilities”

They fidgeted, trying to make sure their neighbors weren’t looking…

Me:“Please pick a number, any number will do”

Me:“Add to that number the number that would come after it on a number line.  For example if you picked 2, then you would add 3 to your original number for a total of 5”

Me:“Now add 9 to your current sum”

Me:“Take your new total and divide by 2”

Me:“Lastly, please subtract your original number from the quotient you got on the last step”

As they anxiously watched me, I played it up some.  I rolled up my sleeves, ran my hands over my head a few times, pinched the bridge of my nose, placed my fingers on either side of my temple, and closed my eyes.

Me:“Alex, the number you have on your paper as of this moment is….5!”

The class exploded because everyone got the same answer in the end.

Student:“How did this work?”

Student: “What did you do?”

Student: “Can you really read our thoughts?”

I turned their attention to the flip chart, which read “How did he do it!?!?!?!?“.

Me: “I will give you 10 minutes max to see if you can figure out what I did and why it worked”

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After 10 minutes we came back together as a class and I wrote a few of the things they asked me to.  The writing in blue was a tie-in to how they learned to solve missing addends in 1st grade.  One student told us how he used 0 to try and see what was happening.  In the end, they came to the realization that the middle part of the order ( +1+9 divided by 2) will always yield 5.  The first and last part of the order “cancel” the original amounts out.  It began a big discussion on what would happen if the order was changed and why we have the order of operations.

*Note: You can extended this into algebraic thinking as in {x + (x + 1) + 9 / 2 – x} and discuss parenthesis and the effect it’ll have on the division!

My favorite part was what came next:  The student asked if they could take some time before the next part of the lesson to see if the riddle would work with any number.  Here are a few:

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As you can see, I had students trying fractions, decimals, and NEGATIVES!  They needed some help, but it was wonderful!  I brought them back together and then let them play around with the following…

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We had some amazing discussions because the students attacked this problem so many different ways, yet they all got the same answer.  It lead to “I wonders” such as: I wonder if instead of twice the amount, Mr. D had half the amount.  Can we move division around like we moved the times 2 and still get the same answers?

This lesson gives the teacher such a wide platform to jump from as she moves through this standard!

My “take away’s”:

  • Riddles are a great way to give students the opportunity to use the strategies they’ve learned
  • When given the freedom to explore, students never cease to amaze me
  • My grandpa was a genius
  • Tasks such as this one, are wonderful lessons to employ when introducing a new concept or standard
  • The students “ate it up” and had so much fun, it was truly enjoyable for me
  • “Playing it up” gave a wow factor, plus I had fun trying not to laugh
  • How can I get more of my teachers to use 3 act tasks, exemplars, games, etc. to create a fun learning environment?

Thanks for tuning in!  Remember, together we achieve more!

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