Math 121 Assignments

Due Wednesday, January 16

Read section 1.1 and preread section 1.2

Do activity 1B in the activities manual AND these problems:

1.  If you have 18 identical boxes of Cheerios and 2 identical boxes of Rice Chex, in how many different ways can you line them up on a shelf? (For example, the first two boxes are Rice Chex and the rest are all Cheerios, or the first and last are Rice Chex and the ones in the middle are Cheerios, etc.)

2.  Your office is 18 blocks north and 2 blocks east of your home.  How many different ways (different routes) can you drive to work, if the east-west streets are one way in the east direction and the north-south streets are one way in the north direction?

3.  You have a class of 18 students and they all get As, Bs or Cs. How many different ways could you assign grades?  (For example, they all get As, or the first one on the list gets an A, the next two get Bs and all the rest get Cs, etc.)

(This one, #3, is reassigned for 1/18 with clarifications, see below)

Due Friday, January 18

1. Read section 1.2 and preread sections 2.1 - 2.3

2. Section 1.2, pp. 7 - 8, give a puzzle and three solutions.  The text suggests that the three solutions are different. In what ways are they different?  In what ways (if any) are they the same?

Here is a clarification/correction of  old #3. 

2. You have a class of 18 students and they all get As, Bs or Cs. How many different ways could you assign grades?  (For example, they all get As, or two of them get As and 16 get Bs or two get As, 1 gets a B and the rest get Cs, etc.)

Re think this one. It is a fact that the answer is our favorite number, 190. So for this assignment, try to connect this problem  to clinking glasses, triangular numbers, cereal boxes arrangements, or routes to your office. You might use one or more of the strategies we talked about, e.g., "shrink the problem".

3.  (Down to the river and back again)
You have two jugs, one holding 5 quarts and the other holding 7 quarts.  The jugs do not have any markings on them, nor are you allowed to make any marks.  You walk down to the river and hope to come back with precisely 1 quart of water. 

a) How could you manage this?

b) There is more than one solution to this problem -- in fact there are many.  Find one that is different from your answer in part (a).

4.  (One dimensional Peter Rabbit)  A rabbit who lives on the number line can hop either 7 units to the left or 5 units to the right. (Some strange anatomical deformity prevents him from making any other length of hop.) If he starts at 0, can he get to the head of lettuce planted at 1?  If so, how? What if “left” and “right” are switched? How is this problem related to "down to the river and back again"?

 

Due Tuesday, January 22, 2008

1. Re read 2.1 - 2.3 and pre-read 2.4 - 2.5

2.  Our final in class problem was:

Find a two digit number so that if you append a 7 at the right you get a new number which is exactly 700 more than the one you started with.

Problem:  Generalize this problem in some way-- in other words, make up a problem like this problem. Then solve your invented problem.

3.  Suppose that A and B are two numbers represented in base three, neither of which is divisible by three.

a) What are the possibilities for the units digit for A?
b) What are the possibilities for the units digit for A2 ?
c) Waht are the possibilities for the units digit for 2B2 ?
d) Is it possible that A2  =2B2 ?

4. Consider the set { A, B, C} and the set {X, Y, Z}. We can see that these sets are the same size, because we can match them up, for example A matches with X, B matches with Y and C matches with Z.

Use this idea to show that the set of natural numbers has the same size as the set of whole numbers, that is, find a matching scheme by which every whole number would be matched with a natural number and vice versa.

5. In our text, do problem 2 on p. 30 and 1, 2, and 6 on page 46.

Due Thursday, 1/24/08

1. Re-read 2.4 and 2.5.

2. Complete the worksheet we started in class.

3.  Start numeration systems project:

a) Experiment with the applet that I demonstrated in class
http://www.mhhe.com/math/ltbmath/applets/ch3/

b) Choose one of the numeration systems from the applet, and investigate the system so that you can describe how it works -- what is the base, if any, how are the numerals represented, how is "zero" handled in this system?

Write a short (1 - 2 pages is fine) description of your conclusions -- be sure to include examples and explain how you arrived at your conclusions.

This will be due on Monday 1/28.

4.  Do p. 48 #17 and p. 58 #19

Due Monday, January 28, 2008

1. Pre-read sections 3.1 and 3.2

2.  Complete project described above.

3.  Do Activity 2G on page 19-20 in the workbook and on p. 62 do problems 1, 2, 3.

Due Wednesday, January 30, 2008

1. Read sections 3.1 and 3.2 and pre-read sections 3.3 and 3.4.

2. Do all of class activity 3Athat begins on page 31 of the activity book; problems 6, 7, 12, and 17 on page 75 and 2 and 3 on page 82 of the text.

Suggestion: Before you work on problems for a section, take a close look at the practice problems for that section!

Due Friday, February 1, 2008

1. Read section 3.5. (This won't be on the test, but in addition to reviewing we will need to do some catch up.)

2. Do these problems in the text: p. 93: 9, 13, 21 and 26 and p. 106 4, 11, 20, AND this problem:

Given two fractions A/B and C/D, find another fraction E/F which is strictly in between A/B and C/D.

Due Tuesday, February 5, 2008

Prepare for Exam 1 on Chapters 1, 2, and 3.1 - 3.4

Due Thursday, February 7, 2008

1. Read section 3.5.

2. Do this problem:

Mouse Heaven:  The god of mice has laid out infinitely many chunks of cheese in a rectangular array, as indicated below:

…

…

…

…

 

 

Note that each row extends infinitely to the right and each column extends infinitely down.  The mouse starts in the upper left hand corner and eats that chunk of cheese then moves on to another one and eats that, etc.  What route should she take in order to be able to eat all the cheese, assuming she can keep on eating forever?  (That assumption is OK, since, after all, this is mouse heaven!)

Note: I know several different answers to this question. Maybe you will come up with a new one!

Due Monday, February 11, 2008

1. Read sections 4.1 - 4.2. Pre-read 4.3 - 4.5

2. Do these problems:  p. 120:  4, 6, 10, 21, 22, 23 and p. 137: 1 a,b, 2 a, b, 7, 8

3. Finally, in class we interpreted A - B as A + (-B) in order to use the filled in circle/ open circle model to deal with negative numbers.  But we also interpreted A - B as "take away".  Find a way to use the "take away" idea in the filled circle/open circle model  and use it to illustrate each of the following:

(i) 2 - 5

(ii) -2 - (-5)

(iii) 2 - (-5)

(This has the advantage of the intutiveness of "take away" but steers around the "opposite of the opposite" challenge.)

Due Wednesday, February 13, 2008

1. Problems:  p. 148 8, 9, 15

2. Read section 4.5 and , in Liping Ma, the Forward, Introduction and Chapter 1.

3.  In class we will spend a significant amount of time in small group discussion of the Liping Ma Text.  To help guide and generate discussion, please draft and bring with you answers to the following questions:

a)  Summarize the U.S.  and Chinese approaches to teaching subtraction.

b) What are the fundamental differences?  Are there points of agreement or similarity?

c) In your view, what are the strengths and weaknesses of each approach?

d)  What (if anything) intrigued you about this reading? What would you consider implementing or trying in your own classroom? Are there points with which you strongly agreed or disagreed?

e)  Summarize Ma's point(s) about conceptual and procedural understanding, and give an example from your own experience (not necessarily classroom) to illustrate the difference she describes.

4. Note:  I will ask you to turn your draft, enhanced by your discussion, into a short paper due next Tuesday. 

Due Friday, February 15, 2008

1. Read section 4.6

2. Do exercise 4R on page 92 of the workbook and 4U and 4V beginning on page 98 of the workbook. In 4R, note that vat B is half as tall, half as wide but the same depth as vat A.

3. Solve this cryptarithm:

SIX + SIX + SIX = NINE + NINE

3. Work on the paper due Tuesday:

Using the discussion questions as an outline, turn your draft answers into a coherent response to the reading. (Do not just write answers as numbered paragraphs.) You may assume that the reader is familiar with the reading.  Feel free to add to your answers as prompted by your in class discussion or your own further thinking.  I am expecting about 2 - 3 pages, double spaced and word processed. Criteria for evaluation include quality of writing (coherence, organization, grammar, etc.) as well as the depth of your response. (I do not have a dog in this fight, so I will NOT lower your grade if I disagree with your conclusions. I will lower it, however, if your conclusions are not supported either by the text or your own reasoned argument.)

Due Tuesday, February 19, 2008

1. Paper (described above) due today.

2. Read sections 5.1 - 5.3  (I am going to try to catch up.)

3. Do these problems: in the text, p. 190, 8,9. In the activity book, activity 4X on page 102 (just identify the properties used -- you don't need to do the "describing" part.);  activity 4Y, #1; activity 4BB #3.

Due Monday, February 25, 2008

1. Read sections 5.4 - 5.6

2. Do these problems in the text: p. 204: 9, 10, 13; p. 207: 5; p. 213: 7, 8  AND part c of the end of class problem, that is

How many five digit numbers have exactly one 3?

Due Wednesday, February 27. 2008

1. Read section 5.7 and pp. 28 - 54 of Liping Ma. Be prepared to discuss how the different groups of teachers handled the problems described in the text.

2. Do activity 5V on page 149 of the activities manual.

3. In our text, do problems 13, 14 on page 225 (For #13, you may find it useful to consult www.moneyfactory.com), problems 9, 21, 23 on page 236 and problem 14 on page 248.

Due Friday, February 29, 2008

1. Be prepared for review on section 3.5, plus chapters 4 and 5.

2. Do these problems:  in activity book, p. 161 5 BB; in tht text, p. 257: 1, 2, 12, 13.  (Note: For 13a) what is 21 in base 2?)

Due Tuesday, March 4, 2008.

Prepare for Exam 2.

Due Thursday, March 6, 2008

Pre-read sections 6.1 and 6.2.

Due Monday, March 10, 2008

Reread 6.2 and pre-read 6.3.

Do problems 1, 2, 3, 13 and 17 on page 269 in the text, and activity 6E in the activity book.

Due Wednesday, March 12, 2008

Pre-read 7.1 and 7.2.

In the text, do problem 10 on page 278, problems 2 and 9 on pate 286, activity 6Q in the activity book, and this problem:

Here is an newspaper clipping from the Rochester (New Hampshire) newspaper. Read the clipping and answer the questions that follow:

"Subtraction error to cost Rochester schools $3000"

An arithmetic error may end up costing the  School Department almost $3000 next year.

School Board Chairman Roland Roberge said Thursday night the subtraction error during a comparison of milk bids led the board to accept a bid it thought was only $298 higher than a second. It was actually $2986 higher.

“Well, the decimal point was put in the wrong place,” Roberge told the board.

He said the error was in turning a half-cent difference in milk prices into a .05 cent difference, which was then multiplied out over the more than 650,000 cartons of milk used in a year by the School Department.

 

1.        1.What is the decimal for one-half cent?

2.        2. If a bid for 650000 cartons of milk is .05 cent per carton higher than another bid, how many dollars greater is it?

3.        3.  If a bid for 650,000 cartons of milk is one half cent per carton higher than another bid, how many dollars greater is it?

4.       4.  How much money did the schools lose by misplacing the decimal point?

5.    5.  Was this really a subtraction error? Explain.

D Due Friday, March 14, 2008

  1 1. Re-read 7.2 and pre-read 7.3.

     2.  Do these problems: in the text, p. 299: 3, 5, 11a and b, , 16 and 21

      Due Wednesday, March 26, 2008

     1.  Read sections 7.3 and 7.4. 

     2.  In a little bit, we will be doing another reading and possilbe a paper from Liping Ma, so if you want to get started on the reading over break, it's pp. 55 - 83 and is closely related to the text reading for this day.

     3. Do problems 24 and 25 on page 302 of the text, and problems 5, 8, 9, 10, and 14 on page 312 of the text.

      Due  Friday, March 28, 2008

     1. In preparation for next class, read section 7.5

     2.  Do problems 6 and 10 no page 325 and problems 3, 4, 12, 24 and 28 on page 339.

     Due Tuesday, April 1, 2008

     1. In prep for next class, re-read section 7.5 and read section 7.6

     2.  Do all the problems on the worksheet 7U in the activity book.

  Due Thursday, April 3, 2008

     1.  Read the already assigned Liping Ma section. Bring draft responses to the discussion questions to class.

     2. For practice with ratios and proportions, do all the parts of activity 7EE in the activity book. THen do these problems on page 373:  17, 23 a,d, 26, 28, 30, 33.

     Due Wednesday, April 9, 2008

  Paper on article from Teaching Children Mathematics due today.

  Due Friday, April 11, 2008 Tuesday, April 15, 2008

     1.     Read sections 12.1, 12.2 and skim 12.3.

     2.  Do problems 5, and 9 on page 653. Hint for #9:  Try this first for 30 lockers instead of 1000 to get a feeling for the problem.

     3.  Do these additional problems. use the defintion of "divides" to give an explanation of each of the facts below:

     A.  Suppose that A is a divisor of N and A is a divisor of M. Explain why A is a divisor of N+M.

     B.  Suppose that A is a divisor of N and K is any integer. Explain why A is a divisor of K times N.

     C.  Suppose that A times B is a divisor of N. Explain why A and B are both divisors of N.

     D.   Suppose that  A and B are both multiples of N. Explain why A times B is a multiple of N.

     4.  Suppose that P is a divisor of A times B.  Give an example that shows that it is possible that P is either a divisor of A or B.  Give an example that shows that it is possible that P is a divisor of neither A nor B.

     Due 4/17/08

     1. Read sections 12.3 and 12.4

     2. Do these problems:  p. 660:  15, 17, 18 and 19 and activity 12G in the activity book.

     3.  Explain why the following conjecture is true:  If A is a divisor of B, then gcf(A,B) = A and lcm (A,B) = B.

Due 4/21/08

1. Re-read sections 12.3 and 12.4.

2. Do these problems:

a)  In the text, p. 667: 3, 4, 5.  You may find it useful to review the practice problems before hand.

b) Use Euclid's algorithm to find the GCF of the pairs of numbers in 5a) and 5b).

c) Using the data sheet on numbers of factors that we did in class last week, choose four numbers greater than 20 which are NOT prime, and do the following:

(i) Write the prime factorization of the number using exponent notation

(ii) Write the prime factorization of each factor of the number using exponent notation.

(iii) With this information in hand, look for a formula that gives the number of factors of N in terms of the exponents you find in the prime factorization of N. (The data in (ii) may help explain the formula.)

Due 4/24/08

1. Read section 12.5 and, for next time, 12.6.

2. Do these problems: p. 668:6, 7; p. 671: 8, 10, 12; p. 676: 4, 5, 11, 15 AND

Here are two tests for divisibility by 7. Choose either one and complete the exercise for the one you chose.

a) A number is divisible by 7 if 7 divides the number you get when, ignoring the last digit, you multiply by 3 and then add on the last digit. For example, 7|42 because 7|4 ∙3 + 2 (7|14)

(i) Try this test on a number you know is divisible by 7 and one you know is NOT divisible by 7.

(ii) Try to explain why this test works.

b) A number is divisible by 7 if 7 divides the number you get when, you take the number you get by ignoring the last digit and then subtract two time the last digit. For example, 7|42 because 7|4 - 2 ∙ 2 (7|0). Or, 7|49, since 7|4 - 2∙9  (7|-14.)

(i) Try this test on a number you know is divisible by 7 and one you know is NOT divisible by 7.

(ii) Try to explain why this test works.

Due Monday, May 28,2008

1.  Read sections 13.1 and 13.5.

2. Do problems 3, 4, 6, 10, 11, and 19a on page 688 in the text AND

A.  Suppose that 1/B has a terminating decimal expansion. Explain why A/B is also terminating.

B.  Suppose that 1/B has a repeating decimal expansion where the length of the period is P.  Can you predict the length of the period of A/B?  If so, what is it and why? If not, give some examples that show that the length is unpredictable.

C. Suppose that 1/B has a repeating decimal expansion where the length of the period is P.  Can you predict the length of the period of 1/kB where k is any integer?  If so, what is it? If not, give some examples that show that the length is unpredictable.

Due Wednesday, April 30, 2008

Read sections 13.5 and 13.6

On page 714, do problems 30, 32, 33, 34 AND

A.  a) Do #2 on the handout from today.  (Hint: draw a picture.)

b) The answers to #3 that students gave in class were NOT correct.  Solve #3 correctly -- test your answer with some actual values to see if it makes sense.  Also, change the problem so that 1<x < 2. What happens to your formula if 0<x<1?

 

Due Friday, May 2, 2008

1. Final paper on Liping Ma reading  (here)

2. On page 774, problems 3, 4, 12, 15 AND

A.  Consider a graph that looks like a circle centered at (0,0).  Could this be the graph of some function?  Explain carefully

B.  Do the last problem on the worksheet -- the flea market problem.