Intro to Math Teaching

THEO2 1) First, as a conclusion to our discussion about fraction division, please visit the website What Works Clearinghouse (Links to an external site.)Links to an external site. - an excellent resource for finding information about classroom approaches. This is a government sponsored site that reviews math ed research to provide summaries of what can actually be shown/concluded - comparing results from small, potentially inconclusive studies with results from multiple well-researched studies. Professor Jon Star, at the Graduate School of Education here at Harvard, has helped review material for this website (Jon often gives guest lectures in our Capstone course). For this question please go to the What Works Practice Guide on Teaching Fractions (Links to an external site.)Links to an external site. page - this lists five recommendations for teachers. Please read the list of recommendations on page 1, and then a summary of the recommendations starting on page 8. After that please skim through the discussions about each recommendation (for instance, Recommendation One is discussed on pages 12 through 18 (there's a fair amount written, again, please feel free to skim these discussions (pages 12 through 46) that discuss the actual recommendations). Now, please write up your reactions to this list of recommendations in a short response that you post on our class discussion board.

2) Next, time for some division - historical style! Start off by showing how Egyptian scribes would have worked out the following two problems. Remember to begin by rewriting the problems in "Egyptian" (i.e. start with "reckon with ...") You can just use Hindu-Arabic numbers instead of writing things out in Egyptian symbols.

(a) 90 divided by 16

(b) 94 divided by 24

3) Reversing this somewhat, try to come up with an example of an Egyptian division problem (i.e. write out the work a scribe would do to solve the problem) whose answer is 7 + 1/2 + 1/4 + 1/8 + 1/16.

4) Next, revisit some of the work we did with Egyptian Fractions. Recall that a fraction X/Y is said to be written in Egyptian fraction form if it is written down as

X/Y = 1/A + 1/B + ... + 1/N

where the denominators A, B, ..., N are all distinct (positive) integers (and there can be any number of terms on the right hand side). For example we found that 5/6 = 1/2 + 1/3, but then it's also the case that 5/6 = 1/2 + 1/4 + 1/12, showing that the same fraction can be written in Egyptian fraction form in more than one way.

Remember the algebraic identity 1/N = 1/(N+1) +1/N(N+1) that we checked out in class.

You can use this to work out that 1/3 = 1/4 + 1/12. Now, multiply both sides of this equation by 2 to yield 2/3 = 2/4 + 2/12 or 2/3 = 1/2 + 1/6 (one of the results we found in class). Use this same approach to work out Egyptian fraction forms for 2/X where X runs through all the odd integers from 5 up to 27 (i.e. work out Egyptian fraction forms for 2/5, 2/7, 2/9 etc. up through 2/27)

5) Now compare your results to the ones that actually show up on the Rhind Papyrus (Links to an external site.)Links to an external site.. Notice that your result for 2/9 differs from the one found on the Rhind papyrus (i.e. that 2/9 = 1/6 + 1/18). (a) How do you think the Egyptian scribes ended up with 2/9 = 1/6 + 1/18?

(b) Next, figure out what they must have done to get the results for all the other 2/n fraction resolutions up to n = 27 that differ from the ones you calculated. Just work out what they did for the ones that have just two terms, you can skip the ones for 2/13, 2/17 and 2/19 (for those interested, you can read more on the Rhind Papyrus link to find out how it's thought that Egyptian mathematicians created the rest of the results).

6) Do a bit more internet sleuthing to find out when the Rhind papyrus was discovered, its rough overall size, why it's so named, and what approximation to Pi can also be found on the papyrus.

7) (a) Write out 3/11 in Egyptian fraction form using the fraction splitting method (i.e. using the 1/N = 1/(N+1) +1/N(N+1) identity multiple times as we did towards the end of class for the fraction 3/7).

(b) Can you find a nicer/shorter Egyptian fraction form for 3/11 than the one that you just found with the splitting method?

8) Next, show that if an integer M divides N + 1, for another integer N, then the fraction M/N can be written in Egyptian fraction form specifically as the sum of two unit fractions (hint, try a few examples using the 1/N = 1/(N+1) +1/N(N+1) identity to check out what's happening first). M dividing N + 1 means that M is a factor of N + 1, i.e. that it goes evenly into N + 1. For example if M is 4 and N is 11, then M divides N + 1 as 4 goes evenly into 11 + 1 = 12.

Bonus - finish off the counterfeit coin problem that we left hanging during the class - there are seven stacks of coins, each stack is either all counterfeit coins (weighing 11 grams each) or genuine coins (weighing 10 grams each). You don't know how many stacks are counterfeit. Can you, with one weighing on the scale, figure out which stacks are counterfeit and which are genuine - good luck! Big hint - think binary!

Bonus Two - continuing on in the Car Talk Puzzler tradition (and please check out the Car Talk Puzzler website (Links to an external site.)Links to an external site.!) here are two more counterfeit coin problems - the first one is a classic that many of you might have seen already, but the second one is less common - good luck!

(i) Suppose you have nine coins (that all look the same) and you're told that exactly one coin is counterfeit and that it's lighter than a genuine coin. Suppose you have a balance scale (a scale with two pans such that when you put items in the balance scale's two pans, the scale shows you whether the two sides weigh the same, or, if not, which side is lighter). What is the minimum number of weighings needed to figure out which coin is the fake one?

(ii) A different scenario... now you have a stack of five coins - you know the first coin is genuine, but you're told that one of the other four is a counterfeit coin that weighs a different amount from a real coin (but you're not sure whether the counterfeit coin is heavier or lighter than a real coin). Can you find the counterfeit coin with just two weighings using, again, a balance pan scale (such as the one used in the last problem) (harder aspect to this puzzle - can you also determine whether the counterfeit coin is heavier or lighter in the process?)