An Edible Method for Teaching Numerical Bases
One of today’s terrible tragedies is students being taught “intellectual” Mathematics. I mean that we ask our students to stop thinking about numbers as anything other than a thought in their head. This is completely contrary to the vast majority of human existence. Our Mathematical roots are intimately associated with our everyday living world. I feel that it helps to use supplemental material to link a concept back to the real world.
Today’s topic will do just this with popcorn. I am using popcorn because it is edible and inexpensive. Eating is a time of low stress. It is a time of camaraderie. It is a time when groups form to share in some event. This is an introduction to how a placement value base system works. Are history taught us to group items together to help quickly evaluate large numbers. Students will remember the process because of the link with eating. And, the system is simple enough that when you assign homework the student will be able to go home and recreate the process . . . especially if this means hooking up the air popper and including parents for a demonstration.
Supplies and Directions
You will need a large clean paper bag full of air popped popcorn. This will be the “bank”. Assign 1 student to be the banker. Be sure to rotate this position frequently so everyone can have this important social political position. This position will have several important parts to play in the game. The most important will be that no one gets to eat popcorn until the banker “pays out” the correct measure at the end of each game.
You will also need several pieces of clean paper (or bowls) sheets. Each sheet will be marked and placed in an orderly manner. Let’s say we are going to first look at Base 3 numbers. Number your sheets from RIGHT to LEFT just as we do in normal Base 10 counting. The first sheet will be marked “1’s”. The next sheet to the left will be marked “3 x 1 = 3’s”. The next sheet to the left will be marked “3 x 3 x 1 = 9’s”. Then do one more sheet and mark it “3 x 3 x 3 x 1 = 27’s” to be the sheet furthest left.
This should give us enough place values without having huge numbers to work with. You will notice that this is analogous with our normal Base 10 placement system. In Base 10 our sheets would be numbered from RIGHT to LEFT this ways: sheet 1 is 1’s, sheet 2 is 10 x 1 = 10, sheet 3 is 10 x 10 x 1 = 100 and so on.
The first game is this. Place 1 piece of popcorn on the furthest left paper, 0 on the next to the right, 2 on the next on to the right and 2 on the paper furthest right. This is to represent the number 1022 in Base 3. (Note: I used Pennies in my images to show up better.)
Now, it is time for the team to turn this Base 3 number in to a Base 10 number. You can of course just let them count them out. Or, you can set up your sheets for Base 10 numerals. Place all of the kernels in the right hand single 1’s sheet. Then count out 10. Put 1 on the next sheet to the left and return the other 9 to the bank. If you do this 3 times you will have 3 kernels on the sheet just to the left of the right hand single 1’s sheet and you will have 5 kernels on the single 1’s sheet.
Further Homework Problems
Continue practicing with some other Base 3 numbers like:
These are just some examples. You can easily make up more if these are not enough for the student to get the idea. Each new number practiced gives another the chance to have a new important banker.
If these students are old enough you can show them the “short cut” to determine a Base 10 count for a Base 3 number without counting out popcorn kernels from the left most paper moving right. I will set up a table so you can see how to do this. I will put the numbers and place values with the smallest at the bottom and the highest place value at the top. I will replace the following place values with a Base 10 value.
1 = 1
1 x 3 = 3
1 x 3 x 3 = 9
1 x 3 x 3 x 3 = 27
The table works by multiplying across so that the number in the right hand column. To get the answer you will want to “sum up” the values in the right hand column. So, here is the table for translating a Base 3 number 1022 into the Base 10 number 35.
Changing from Base 3 to Base 10
What’s interesting about this system is that it works for any two Base place value systems. So you can have your students change numbers from Base 4 to Base 8. You may wish to change a Base 7 number into a Base 3 number. For this article let’s turn a Base 10 number of 53 into its equivalent Base 3 number.
The process will be the same. Count out 53 popcorn kernels and place them in the single number box for Base 10. Now, transfer all of them to the single number box on the Base 3 right most paper. Then pick out 3 kernels. 1 gets moved to the immediate left box and 2 go back to the bank. Keep pulling out 3 kernels, move 1 left and give the bank 2. When they get done get 1222 should be the display they end up finding in Base 3.
Now the question of how the banker rewards the student should be addressed. I’m sure you will agree that each should get 53. Of course, some of your wiser students will soon learn they get a reward without the work. My suggestion is that they do not get the reward without the proper Base 3 representation. Some of them may want to claim 1222 kernels each. This too will require a response as to why that will not happen because they are confusing 1222 in Base 3 with 1222 in Base 10.
This should be enough for one lesson. There is still the need to show how to add two numbers together where they are in the same base or different bases. And, we have to learn how to multiply and divide with different bases. Look for these in the near future.