Task 63 ... Years 2 - 8
SummaryFor some students, the materials in this problem are really an aid to logical thinking rather than a measuring exercise. The challenge is to make particular masses using containers that limit what can be measured. The story shell provides a possible reason why someone might have to do this.
This cameo has a From The Classroom section which corrects the 'lazy' reasoning given in the iceberg notes, illustrates the correct reasoning with a PowerPoint used with students and opens the door to whole family of such problems.
IcebergA task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.
An Unanswered Question
Curses! Just as happened to Andrew Wiles when he published his first solution to Fermat's Theorem, a colleague, Damian Howison in this case, has shown our incomplete reasoning. If this 20g is going to be stored, then it must be in one of the only two containers we are allowed to use.
Have another think about the problem before looking at Damian's contribution below. He offers not one, but two solutions, in a slide show link. But best of all, he places his experiences with this problem in the context of how he worked like a mathematician.
Whole Class InvestigationTasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.
Rather than use this as a whole class investigation with everyone simultaneously, it is better (from the point of view of rice on the floor) to set up the task at a work station and rotate the students through it over a couple of days. Follow this with a class discussion which summarises what has been discovered and raises the 'What happens if...? questions. Students can then follow these up in groups according to the cups chosen, or assigned.
An alternative (which is much easier to clean up) is to do as MacKillop College did and interpret the problem as liquids, then use it as one of the activities of a Maths Fun day.
The stimulus for this was a sequence from the film Die Hard with a Vengeance in which the heroes were set the equivalent of Fried Rice, using water. Either in the whole class investigation, or as an extension when students are working on the task as above, you might include additional challenges of this type using one of the applications available from the Google Play Application Store. Type Water Logic into the Search box and you will (at time of writing) find two applications (Water Logic & WaterCapacity), both of which are free.
At this stage, Fried Rice does not have a matching lesson on Maths300.
Is it in Maths With Attitude?Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.
The Fried Rice task is an integral part of:
From The Classroom
|Damian recognised that a clip from Die Hard with a Vengeance in which the heroes must save the world by solving a problem involving 5 gallon and 3 gallon bottles (and a bit of coarse language) was equivalent to Fried Rice. This stimulated the staff to prepare a Maths Fun day built around the mathematics of life and death. It involved this problem and others. Damian prepared this PowerPoint explanation of the solution for his students and has sent the email below explaining his investigation of the problem. For where we sit it also confirms the process of Working Mathematically.
Doug,Thanks Damian, I have learnt much more about this problem because of your contribution.
This bottles (or rice) problem seems to have a great connection with prime numbers. (It seems) ... that you can do these sorts of problems with any pair of relative prime numbers**. So I started trying it and it appears very true. For example:If I were to choose a 7 unit bottle and an 11 unit bottle, then I should be able to find ways to measure out 8, 9, and 10 units exactly. In fact I think any problem has two solutions, although I haven't really tested this out properly. But during the discussion with kids, it was these ideas that started to engage them more deeply.And there is a definite pattern to the way in which you go about finding the solutions - the solution being the sequence of filling/transferring/emptying required to attain the given amount. I sat down this morning and nutted out the solution for using a 17 unit bottle and a 23 unit bottle to get exactly 22 units. One solution was fairly short but the second was much longer in comparison. Each solution is arrived at in choosing the bottle you first start to fill. The pattern is that you are continually filling one of the bottles but emptying the other with as much transferring as possible in between.
You can find on the Internet that the general rule is: ax + by = k
It's a beautiful little generalisation and helps you see why there seem to always be two solutions.
- x and y being the capacity of the two containers (relatively prime).
- k being the amount you want to measure.
- a and b being the number of times you need to empty/fill x and y respectively. (Positive values for filling and negative values for emptying).
So for the example above:
4(17) - 2(23) = 22would lead you to predict that you will need to fill the 17 container 4 times and empty the 23 container twice, which turns out to be perfectly true.
But it is also true that:
15(23) - 19(17) = 22so you would fill the 23 container 15 times and empty the 17 container 19 times. As you can see this is a much longer sequence, but it works none-the-less and I imagine learners would find it a little bit of fun to try it out - like I did.
Actually I can imagine a piece of software where learners can try this with a visual of the bottles on the screen and them choosing what to do next. Each step you choose to do one of 6 possible actions:
The longer the problem the better you get into the rhythm (pattern) of finding the solution.
- Fill A
- Fill B
- Transfer A to B
- Transfer B to A
- Empty A
- Empty B
You can use a 3 and a 5 to get 4, but you'll never be able to use a 4 and a 6 to get 5. However, you can use a 4 and a 7 to get 5, because 4 and 7 are relatively prime, and you know that 3(7) - 4(4) = 5. You also know already about how long it will take and that it is quicker to start by filling the 7 because 10(4) - 5(7) = 5, which is a much longer sequence if you start by filling the 4.
These are easy to find if you know your multiples/tables, which is a nice little application.
** Relative Prime Numbers are integers whose only common factor is 1.