Puzzler

To all my fellow math lovers, today I was posed with a question by my friend Ms. K. Ms. K is a math major like myself who finds puzzling math problems very intriguing, she came across this problem when her mom called and asked her how it worked. This problem came from a grade 7 text.

So, three guys walk into a hotel (sounds like a joke right? keep reading). The receptionist at the desk charges the men $30 to stay at the hotel for the night; each man pays $10. Later the receptionist realizes that he made a mistake and should have only charged the men $25. The receptionist then gives the bellhop the extra $5 to bring back up to the men. On the way up to the room the bellhop realizes that he doesn't know how to split $5 into three equal parts for the men, he decides to give each man back $1 (for a total of $3) and keep the remaining $2 for himself as a tip. If the men originally paid $10 and got back $1 each then that means they really only paid $9 each. $9 times 3 men is only $27. The bellhop keeps $2 for himself, making a total of $29.

What happens here? Why is there a difference in the numbers? If you add up $25 + $3 + $2 = $30 but( 3 X $9) + $2 = $29. What explanations can you come up with? The only explanation I could come up with was that if you write all the dollar amounts on a number line then there are 31 numbers (including 0) and there are 30 spaces (for the cents) in between them. Therefore I think this problem has something to do with integers, but that's open for debate.

In keeping with our problem solving activities of recent classes I thought that this problem was appropriate for us to think about. If anyone else has any suggestions as to the discrepancy in the numbers feel free to comment on this post, I will gladly take any ideas you have. I hope this puzzling question is as intriguing for you as it was for me.

Reflecting on the text

During our classes we have touched on several sections from the text book. There is one in particular section that I found interesting that I didn't really think about before. This section is called Helping Children Master the Basic Facts.

When I learned how to add I don't really remember the exact methods that I was taught but according to the text, students now have a variety of strategies to use for addition. One-More-Than and Two-More-Than facts give all the numbers of 1+n and 2+n (where n is any number between 0 and 9) twice on an addition chart. Facts with Zero covers all the numbers 1 to 9 twice and 0 once. Doubles are all the numbers added to themselves to cover the numbers 0,2,4,6,8,10,12,14,16, and 18. Near Doubles (also known and doubles-plus-one or doubles-less-one) covers the numbers 1,3,5,7,9,11,13,15,17, and 19 twice. The Make-Ten Facts are the numbers of 8 or 9 plus any given number. Generally you can make a set of 10 and then add on the rest. For example 9 +4. 9 is 1 less then 10 so if you take one from 4 and place it on the 9 then your new equation becomes 10+3 which is easier to add. The remaining six facts do not fall under any of these strategies. Under the commutative property these six facts are 5+3=8, 6+3=9, 6+4=10, 7+3=10, 7+4=11, and 7+5=12. Lucky for us that there are four more strategies that help students cover these facts. Doubles-Plus-Two, Make-Ten Extend, Counting On and Ten-Frame Facts.

The same is true for Subtraction. When I learned to subtract I remember having physical objects such as counters, blocks and even my fingers that you could use for help. Now students can Think-Addition when trying to do subtraction, what plus n will give you x (n+_=x). To me this seems logical and I wish that someone had told me that when I was younger. Another method is Subtraction Facts with Sums to 10, make sets of ten and then add the remaining number.

I think these methods are incredibly useful for younger students. Students who may be having difficulty mastering addition and subtraction using one method may try several others before finding the right one to match their learning style.