Emile's Math

Negative numbers

2008-09-07T13:35:00.001-07:00

Hi Emile,
So far your work in class can be successfully described using negative numbers. For instance, last week out of 8 assigned geometry exercises, you solved: at home 4, and in class -4 more than at home; so it remains to solve ?? exercises.
Your next algebra assignment is, after reading Section 13 "Addition of negative numbers," to compose 5 word problems whose solution would be given by:
(1) addition of a positive and a negative number,
(2) addition of two negative numbers,
(3) subtraction a positive number from a negative one,
(4) subtraction of a negative number from a positive one, and
(5) subtraction of a negative number from a negative one.
In the book, there are examples of such problems involving temperatures and annihilation of elementary particles. I'd like you to compose problems about years of life of ancient Greek geometers, as written on p. viii of Kiselev's Geometry / Book I.
After finishing this assignment, you are allowed to torture your teacher and classmates by asking them to solve your problems.

Summary of recent homework

2008-09-03T09:24:00.000-07:00

Hi Emile!
I sum up here your recent work at home, and post next assignments.

In Algebra, it took you only a few minutes to do section 11,
and we've finally completed Problem 31 from section 10 about use of parentheses. I'll write down the answer so that we don't forget.
The problem is about the number of ways to put parentheses into a product of several factors. Your answer was
for # of factors 1, 2, 3, 4, 5, 6, 7, ...
# of the ways is 1, 1, 2, 5,14,42,???, ...
(these are called Catalan numbers )
For instance, abc can be computed as either (ab)c or a(bc),
so for 3 factors the number of ways is 2. Likewise, the product abcd of 4 factors can be computed in 5 different ways. Namely, designating the last multiplication to be: a x bcd or ab x cd or abc x d , we find 5=1x2+1x1+2x1. Here 1, 1 and 2 are the numbers of ways to compute the product of 1, 2, and 3 factors. Similarly, for 5 factors abcde, the number of ways is: 1x5+1x2+2x1+5x1=14, and so on.
Can you compute (correctly!) the 7th Catalan number?

Your next assignment in Algebra is to work on section 12 "Letters in Algebra." Plan to do this in class on Fri, Sept. 5.

In Geometry, you've finished the previous assignment, but there is a "research" question left: Out of 3 altitudes of a triangle, how many can lie outside the triangle, and for which triangles this can happen?
The next assignment, to work in class on Th, Sept. 4, will be Exercises 60, 56, 55, 54 (in this order).

Looking forward to your solutions!

Polygons

2008-09-01T11:17:00.000-07:00

Hi Emile!
Here is the geometry assignment to do in class on Tue, Sept. 2.
Read paragraphs 32, 33, 34.
Write down the definition of a polygon's diagonal .
Exercises: let us start with 57, 59, 61, 58 (in this order!), and deal with the rest of them later.
Write your solutions in your notepad or as comments to this post.
Good luck!

Outstanding classwork

2008-08-29T23:04:00.000-07:00

Hi Emile,
Here is what we discussed last week:

Geometry: Aug 25 (due by Wed, Aug 27). Read paragraph 22. Solve exercise 51 and write down your solution.
For the proposition: "If a whole number is divisible by 3 then the sum of its digits is divisible by 9," write down the converse proposition.
For each of the direct and converse proposition, if it is true, prove it, and if its is false, give an example where it is false. Write down the proofs and examples.

Added on Tue: Give an example of two angles whose halves add up to 90 degrees, but whose bisectors are not perpendicular.
In your solution to the 2nd question in exercise 51, find where you use the hypothesis that the angles are supplementary.

Algebra: (due by Th, Aug. 28). Section 10. Write down solutions of Problems 31--33. Hint for 31: Start with designating a multiplication to be performed last.
If done with 31--33, try the variant of Problem 31 with 6 numbers to multiply.

Geometry: (due by Mon, Sept 1).
Read paragraph 31. Write down definitions:
- of a broken line, its sides, and its vertices;
- of a convex broken line;
- of a closed broken line.
Draw an example that satisfies the definition, and another example that does not (for instance, draw an example of a closed broken line and of a broken line that is not closed). Solve exercise 53.

Since Monday, Sept. 1 is Labor Day, may I suggest that you spend enough time on Sat, Sun, and Mon to finish all this before school resumes on Tue?