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Posts from the ‘algebra’ Category

6174

June 6th, 2011

alecmce

Task

Choose any 4 digit number except for 1111, 2222, 3333, 4444. I’ll choose 1502.

Rearrange the digits to give you the biggest and the smallest numbers you can. 5210 and 0125.

Find the difference of these two numbers. 5210 – 0125 = 4995.

Repeat for the new number you get (using zeros to supplement any missing digits if necessary; you must always have 4). 9954 – 4599 = 5335.

Keep repeating until you have a good reason to stop.

Example

5210 – 0125 = 4995
5533 – 3355 = 2178
8712 – 1278 = 7434
7443-3447 = 3996
9963 – 3699 = 6264
6642 – 2466 = 4176
7641 – 1467 = 6174
.

I’m not going to repeat any more because 6174 => 7641 – 1467 which is the calculation I just did!

Discussion

  • So every number we’ve tested came to 6174. Do you think all numbers will come to 6174? Why? In maths we can’t say we’re sure about something unless we have a good reason, so unless you want to go through every number and check it, we can’t say that we know that every number sequence we choose will go to 6174! (This is how mathematics differs from science, we don’t just make hypotheses and wait until the next time they break, we find ways to be certain!)
  • Would we have to check every different number to be certain or are there shortcuts we can take? (should we check both 1234 and 1243 separately?)
  • My rules stipulated that you couldn’t choose all the same number: 1111 or 2222. What would happen if you did use those numbers?
  • What about two digit numbers or three digit numbers?
  • Could we frame the problem as (1000a + 100b + 10c + d) – (1000d + 100c + 10b + a)?
  • What about five or six digit numbers (or more…)? You may need a computer to help!

Thoughts

I just came across this interesting bit of mathematics via Twitter: http://plus.maths.org/content/os/issue38/features/nishiyama/index. It offers the opportunity for a beautiful open-ended task for secondary maths classes of all abilities. The initial task is easy and produces a startling result that feels like a trick; that hook can lead to discussion and further work about probability, permutations, algebra and programming.

Bracket Expansion

August 29th, 2006

alecmce

This activity is for a first introduction to expansion of brackets or for a review activity.

Bracket Expansion Excel Spreadsheet

Imagine telling a class that 4(x+3) = 4x+12 and them asking you “why?”. What if instead you presented the information to them in a way that they generated the idea that they were equal and the explanations for their ideas themselves?

This sheet generates pairs of equations (for just expressions, read on)*, one with the bracket in place, one with it expanded. It is a very simple spreadsheet. It contains simple macros that allow either equation to be hidden. Though you can use it however you like, my suggestion follows…

I would use the sheet with both equations or expressions* visible and change the values a few times. I would ask students to try to explain the relationship between the two expressions.

Most likely the first thing that they notice is the methodological relationship the product of the two numbers in the first expression is the last number in the second expression. A brief glance at some other variations will demonstrate that this is the case.

However, they are unlikely to have grasped the numerical relationship; that they each have the same value, irrespective of the value of x. This will probably require some prompting. You could ask each student to choose a value of x and substitute it into the two values. What do they notice. Choose random numbers a few more times. Repeat the activity. Once they have established that they are always getting the same numbers then you can move on to the big one: the conceptual relationship.

The concept we’re aiming at is surely for the students to grasp that the link between the two expressions they first noticed is necessary for them to have the same value. It will be difficult for this activity alone to convince them of that, but I would at this point urge them strongly to consider whether the two things they’ve noticed are related. Since they’re expecting a relationship, they may trust that linkage. I haven’t linked it for them however, at least not in their minds.

I would then remove the expanded expression and ask them to construct it themselves, making sure that for whatever number they make x the two expressions match. This is a crucial check that should be encouraged early.

* The ‘y=’ on both the equations can be easily removed to just give the expressions. To be honest, that’s my preferred usage, because of the problems students have understanding when we’re using equality as a question signifier and when we’re using it as a binary relation between the two expressions on either side. To get rid of the ‘y=’ click on the ‘control’ tab and delete G4 and G5, where the ‘y=’ text is seen. Go back to the ‘display’ tab and they will have gone.

Also note that you must have macro security set to medium and allow macros for the buttons to function. Please disable macros and view the macro code if you are unsure whether this is safe, or if you do not understand code, ask someone who does.