Assignments

August 31 Class Activity

Patterns with Tiles

Here is the beginning of a pattern using tiles:

A. Your group should try to come up with at least 6 different ways to continue the pattern.  For each pattern, you should

i.                     Draw the first four examples of the pattern on your paper.

ii.                    Record the number of tiles in each pattern in a chart.

iii.                  Predict how many tiles will be used in the 10th and the 99th pattern.

iv.                  Find a formula (expressed in words and also expressed in symbols) for predicting the nth pattern.

v.                   Note whether the pattern is linear, quadratic, etc.

B. Make as many observations as you can about what patterns you could find, how alike or different your 6 patterns were, and anything else you notice.

Here is the beginning of a second pattern:

 

Repeat steps A(i-v) and B above.

 

Due September 7

 

Journal I:

C. As you solve the problems below, keep notes on your thought processes.  Do not erase any mistakes you make!

Write an equation for each of the following English descriptions:

i.                     At Mindy’s restaurant, for every 4 people who ordered cheesecake, 5 people ordered mousse.

ii.                    There are six times as many students as professors in the master’s program.

iii.                  Juanita has four more pens than Karen.

iv.                  Linda has ten fewer eggs than Nuno and Ophelia combined.

v.                   Pablo has four times as many pens as Quincy.

D. Write two different possible English sentences that translate to each of the following equations

i.                     k = s – 4

ii.                    4d = e

iii.                  f = ¼  g

iv.                  g = 4f

v.                   3 + 2y = p

Read the article “Teaching Algebraic Expressions in a Meaningful Way,” by Chalouh and Herscovics, and reflect on both the article and your experience with problems C and D above.

 

Journal II. Read the introduction and Chapters 1 and 2 of Driscoll’s Fostering Algebraic Thinking, including the Euclidean Algorithm.  Write a journal entry reflecting on the questions:  How does having a framework of algebraic thinking as doing-undoing, building rules to represent functions, and abstracting from computation help you see algebraic thinking in new ways or places?  How does your personal classroom questioning technique compare with Driscoll’s list of teacher questions?

 

 

Homework (collected September 7)

1. Here is the beginning of a third pattern:

 

a. Instead of coming up with any patterns you like, you should now try to come up with

i.                     One pattern which has linear growth

ii.                    One pattern which has a formula which is quadratic

iii.                  One pattern which has a formula which is cubic

iv.                  One pattern which is not linear, quadratic, or cubic

b. Describe your process in doing (a).  In particular, what did you know immediately about constructing a pattern?  Were you able to systematically build a pattern that had the prescribed formula, or was there guess-and-check involved?  Did it get easier or harder as you proceeded to build successive patterns?  Describe what you learned from the activity and make any generalizations you can about the process of building patterns.

 

2.        Equations, Number Sentences, and Linear Functions

2.1.      Alejandro and Barbara are twins.  By their 10th birthdays, Alejandro has read 20 books, while Barbara has read 36 books.  Each month after their 10th birthday, Alejandro reads 6 books, while Barbara reads 2 books.

2.1.1            How long will it take until Alejandro has read as many books as Barbara?  Show as many solution methods as you can.

2.1.2            Pose at least two variations on this problem which ask questions that are mathematically different, i.e. don’t just change the numbers and/or the names and ask the same question.

2.1.3            Generalize 2.1.1 as much as possible by considering first the possibility that Barbara reads a different number of books each month, then by varying the other numbers in the problem, until your solution is completely general and includes a discussion of when there is no solution.

2.2      Three districts offer different pay scales to a credentialed teacher with less than 20 additional units of credit (all three districts refer to monthly salary, and pay salaries over 10 months):  Beach District pays $3000 plus $175 for each year of experience, City District pays $3000 plus $150 for each year of experience, Dismal District pays $2800 plus $175 for each year of experience.

2.2.1            Express each district’s pay scale in a formula, in a table, and with a graph.

2.2.2            Find out when two different teachers (with the same number of units) in different districts can make the same salary.

2.2.3            How is (2.2.2) similar to or different from (2.1.1)?

2.2.4            Create another district which has a pay scale so that its teachers will never make exactly the same amount of money as teachers in City District.

2.2.5            Generalize the situation of finding when teachers in different districts make the same salary.

2.3      Chandra has some quarters and dimes in her piggy bank.  The total value of her coins is $2.00, and there are a total of 14 coins.

2.3.1            Solve for the number of quarters and dimes that Chandra has in as many ways as you can find.

2.3.2            Compare (2.3.1) to (2.2.2) and (2.1.1).

2.3.3            Generalize the situation…

 

 

September 7 Class work

The Jug Problem

You are given a 5 gallon and a 3 gallon jug.  Each jug is made of clay, so that you cannot tell how much water is in the jug except when it is full. 

  1. You are next to a stream and need to put exactly 1 gallon into one of the jugs.  Find a way to do this.
  2. Find another pair of jugs with different capacities and see whether you can accomplish the same feat of getting exactly 1 gallon of water into one of the jugs.
  3. Look at more examples as necessary, in order to answer the following questions:
    1. Is it always possible to use two jugs of any given capacities to get 1 gallon of water?  Explain your reasoning.  Is there a smallest nonzero amount of water you can produce in a given pair of jugs?  Can you predict this amount?
    2. Can you connect this process with the Euclidean algorithm from your reading in Driscoll?

 

Journal III.  Discuss your experience in dealing with the vaguely worded problems on homework 1, and how your ideas about them did or did not change after the class discussion.  Also include any observations, mathematical or otherwise, that you wish to share from the homework or class activity.

 

Homework due September 14

Journal IV. Read Chapters 3 and 4 of Driscoll.  Discuss anything you got from the readings, or any related thoughts.  Also discuss homework problems (3.3) and (4.2).  Compare working visually or with manipulatives to working with the usual algebraic symbols in (3.3); In (4.2), what questions did you leave unanswered?  Are there other questions or remarks you wish to share concerning these problems?

 

3          More Equations and Inequalities

3.1      A car travels from A to B at an average speed of 60 miles per hour.  The car returns from B to A (along the same route) at 40 miles per hour.

3.1.1            Explain whether the average speed of the car for the round trip from A to B to A will be more or less than 50 miles per hour.

3.1.2            Is it possible to find the car’s average speed for the entire trip?  If so, what is it?  If not, what additional information is needed?

3.1.3            Pose some related problems:  either generalizations, similar problems, or some other sort of variations.  You should give me at least two related problems.  Do not restrict yourself to problems you can solve.

3.1.4            Solve whichever problems you can from (3.1.3).  Report any partial solutions to problems you did not solve.

3.1.5            Report on your observations.  What can you predict based on the initial speeds?  What other relationships can you find?

3.2      Let x satisfy 3 / (x - 1) < 2 / (x + 1).

3.2.1            Find x graphically, using the graphing calculator.  Report on your process.

3.2.2            Verify your work in part (3.2.1) by solving the inequality symbolically.

3.2.3            Pose some related problems, with the same guidelines as in (3.1.3).

3.2.4            Solve the problems in (3.2.3).  Report any partial solutions to problems you did not solve.

3.2.5            Report your observations.  When will an inequality not be solvable?  In solving the inequality symbolically, what can you say about so-called “extraneous solutions”?

3.3      The Haybaler problem, p. 81

3.3.1            First solve the problem using only reasoning or manipulatives or diagrams.

3.3.2            Solve the problem by translating it into an equivalent algebraic problem.

4          Number Theory and Modular Arithmetic

4.1      Warm-up

4.1.1            Use the Euclidean Algorithm to find integers a and b so that 511a + 37b = 1.

4.1.2            Use the Euclidean Algorithm to find integers c and d so that 1024c + 1622d = 2.

4.2      Postage Stamp problem, p. 24 of Driscoll—Go as far as you can to report on the behavior for any pair m, n.

 

September 14 Class Activity

Journal V.  Complete the following problems and identify the algebraic thinking involved in solving the problem.  Pay particular attention to the questions you ask yourself as you work through each problem.

i.                     Take any three digit number.  Subtract 5.  Multiply the result by 7.  Add 33 to that number.  Multiply the result by 11.  Add 22.  Multiply the result by 13.  You should have your original three digits repeated twice in a six digit number.  Explain why this happens.

ii.                    Pick any single digit number.  Multiply it by 12,345,679.  Then multiply the result by 9.  What happens?  Explain why this works.

iii.                  The number 9 can be written as a fraction using all ten digits 0 through 9, such as 9 = 95,823/10,647.  Find more ways of writing 9 as a fraction using all ten digits.

iv.                  Find the next two lines in the pattern below, and then write a general rule/formula and explain why it works:

1×2×3×4 + 1 = 5×5

2×3×4×5 + 1 = 11×11

3×4×5×6 + 1 = 19×19

Due September 21

Journal VI.  Reminder:  Journals will be collected September 21.  Read Chapter 5 of Driscoll.  Pick one problem previously assigned from Driscoll (any chapter).  Write a sample lesson plan for your students (if you don’t have any, imagine some) using this problem in which you describe

  • How you would set-up the problem with your students (would you change the context or any other details in giving the problem, would you have the students read the problem from an overhead and make sense of it  together, or would they be asked to read and interpret it individually or in small groups)
  • How you would divide up the time among individual, small group, and whole class work
  • How you will address students questions or “stuck points”—you may want to make a list of potential sticking points or questions that students might ask and how you would respond to each of these, perhaps using the question framework from Chapter 1 of Driscoll
  • Discuss how you would get students to reach beyond the initial problem to generalize, pose similar questions, or pose related but different questions
  • Discuss any follow-up activities or problems you might give to reinforce the lesson

5          More Number Theory and Modular Arithmetic

5.1      Given a divisor d (a natural number), and integers m and n, with rm and rn the remainders when m and n (respectively) are divided by d, find the remainder when the following expressions are divided by d:

5.1.1            mn

5.1.2            m + n

5.1.3            m – n

5.1.4            If e is another divisor, r and s are natural numbers with  0≤r<d and 0≤s<e and, using the Euclidean algorithm, there is an equation ad+be=1, then ads has remainder s when divided by e, while ber has remainder r when divided by d.  [If this one seems confusing, use some examples in which you have a, b, d, e from the jug problem and make up and r and s and try it out.]

5.2      Show that the square of any odd number is always one more than a multiple of 8.  Discuss the algebraic thinking involved in solving this problem.

 

September 21 Class work

H. Constructing Parabolas from Lines—Graphic Viewpoint

i. Graph the lines y = x and y = 1 - x.  Then graph the parabola y = x - x2.  Describe any information you can gain about the parabola by looking at the graphs of its linear factors.

ii. Repeat this process for at least two other pairs of lines (and the parabola generated from the product of those linear factors).

iii. Summarize your findings.

iv. Are there any parabolas which cannot be obtained from two lines in this way?  Explain.

 

Journal VII.  Reflect on this new perspective of parabolas and how it fits with your previous knowledge of parabolas.  In particular, discuss this activity with reference to Driscoll’s chapters 4 and 5 on Expressing Generalizations.

 

Due September 28

6          Euclidean algorithm application?

6.1      Here are four similar problems.  Look for ways to generalize beyond the specific numbers used in the problems.

6.1.1            When Mr. Jones arranged the desks in the classroom into groups of 3, he had one extra desk left over.  When he arranged the same set of desks into groups of 4, he had 3 desks left over.  When he arranged the desks into groups of 5, he had one desk left over.  How many desks could be in the class?

6.1.2            The age problem, p. 33

6.1.3            A number of eggs were brought to market.  When put into groups of 2, 3, 4, 5, 6, and 7, there were remainders of 1, 2, 3, 4, 5, and 0, respectively.  How many eggs could there have been?

6.1.4            On Halloween, I divided all the candy I bought into groups of 2, 3, 4, 5, 6, but each time I had 1 candy left over.  Finally, when I divided the candy into groups of 7, there was no candy left over.  How much candy might I have had?

 

Journal VIII.  Read Chapter 6 of Driscoll.  Discuss 2 problems from this chapter or in any previous chapter with regard to “Fostering Symbol Sense.”

 

September 28 Class Activity on Inequalities

Journal IX.  Discuss the levels of solutions possible for problem (6) from the homework.  Discuss how algebraic thinking is a theme in these problems and what questions (a la Driscoll’s list) are relevant to solving the problem at different levels.  Discuss how to get students “into” the problem, i.e. how to help them get started.

 

 

 

  1. On a previous homework, you solved the inequality (*)  3/(x – 1) < 2/(x+1).  Some of you found that the solution set contained two regions:  {x | x < – 5} and {x | -1 < x < 1}. 
    1. Based on this information, predict the solution set to the inequality (**)  (x+1)/(x – 1) < 2/3.
    2. Which of the inequalities (*) or (**) above is equivalent to: (***)  (x+5)/[(x – 1)(x+1)] < 0 ? 
    3. How is the other inequality (*) or (**) NOT used in (b) related to (***)?
    4. How does the form of the inequality (***) help you find the solution?
  1. Graph the left and right sides of the inequality (**).  Does your algebraic work coincide with the graph?  Go to the graphing calculator’s table mode and set up an appropriate table to compare the values of the two sides of the inequality.  Does this table fit with your earlier work?

 

 

Due October 5

Journal X.  Read Chapter 7 of Driscoll.  Be sure you have “Algebraic Thinking” by Moses, because we will start it next week.  Develop a series of 3 accessible but deep problems around the following problem to build up your students’ habits of mind:

                Jade rides her bike to and from school every day, Monday through Friday, this week.  By the end of the week, she rode a total of 35 miles.  How far is it to school for Jade?

The problems can be at any level, as long as they deal in fostering algebraic thinking (use Driscoll’s questions as a guide for whether they foster algebraic thinking).  You don’t absolutely have to make up your own problems (as opposed to finding them in a resource somewhere), but it’s a good exercise in deep mathematical thinking if you do make them up.  I am hoping to have most or all of us share at least one problem in front of the class.

 

 

 

7          Quadratic Functions

7.1      A quadratic function f(x)= x2 + bx + c has roots m and n.  For instance, f(x) = x2 – 3x + 2 has roots 1 and 2.

7.1.1            Find b and c as functions of m and n, i.e. b = b (m, n) and c = c (m, n).  This is sort of the inverse to the quadratic formula.  You should first make a table with columns labeled “b,” “c,” “m,” and “n,” find some parabolas and fill in the values (from my example, b=-3, c=2, m=1, n=2), and try to find the functions from this table.  Then, try to show that your formulas for b and c work for any parabola with roots m and n.

7.1.2            From (5.1.1), if you are given values for b and c, you have a system of two equations in two unknowns.  In my example, you are given b=-3 and c=2.  Graph on your calculator this system for several different values of b and c.  Based on 5.1.1 and your graphs, how does changing b change the values of the roots of the quadratic?  How does changing c change the values of the roots of the quadratic?  When (in terms of the graphs) does the system not have a solution with real values for m and n?  Report on any other observations you made from your exploration of the graphs.  (You do not have to turn in the graphs, but you should note what values of b and c you graphed.) 

7.2      This is similar to 7.1 but put into a “real-life” context.  A ball is launched vertically into the air from a height of h meters and with an initial upward velocity of v meters/second.  The ball’s height above ground is given by the equation H(t)=-4.9t2 + vt + h.

7.2.1            What is the contextual meaning of the roots of the equation H(t)?

7.2.2            What is the effect of changing v on the roots of the quadratic H(t)?

7.2.3            What is the effect of changing h on the roots of the quadratic H(t)?

7.2.4            Is there any way for H(t) to have no real roots?

 

October 5 Class Activity

Share Journal X.

 

Journal XI.  Discuss two techniques, algebraic thinking discussions, or student learning discussions that were of value to you and how they have affected/will affect your teaching.

 

 

 

Due October 12

Journal XII.  Read “Doing Algebra in Grades K-4,” “Conceptions of School Algebra and Uses of Variables,” and “Communicating the Importance of Algebra to Students.”  Reflect on Usiskin’s uses of variables and any other points in the reading you found interesting.

 

 

8          More Inequalities

8.1      Here is another inequality: (+)  6/x > 2/(x – 2).

8.1.1            Solve the inequality (+).

8.1.2            What is the relationship between (+) and this inequality:  (++) 6(x – 2) > 2x ?  Can you relate their solution sets?

8.1.3            Are either of the inequalities above equivalent to 4(x – 3)/[x(x – 2)] > 0 ?

8.1.4            How is the form (+++) above helpful in solving the inequality?

8.2      Using the graphing calculator:

8.2.1            Repeat part (J) for (***).

8.2.2            Repeat part (J) for (+).

8.2.3            Repeat part (J) for (++).

8.2.4            Repeat part (J) for (+++).

 

October 12 Class Activity

 

Journal XIII.  Read “Does Everybody Need to Study Algebra?” by Steen, pp. 49-51 of Algebraic Thinking.  Write your reaction.

 

K. Gasoline in the US is sold in three grades:  87 (regular), 89, and 92.  Typically, pricing of the three grades is done so that the difference in cost between consecutive grades is ten cents. 

a. Is pricing proportional to the grades?  If regular gas sells for $1.999 per gallon, how much should the other grades cost if pricing were proportionally dependent on the grade? 

b. Better grades of gasoline are supposed to yield better gas mileage.  If Matt gets 24 miles per gallon on 89 gas at $2.099 per gallon, what mileage should he expect to get from the other grades in order to get his money’s worth?

c. Pose and investigate other problems of your choosing around this topic, such as looking at the situation for your car’s mileage…

 

 

 

Due October 19

Reminder:  There is an exam October 19 at the beginning of class.

 

Journal XIV.  Everyone must read “Why Elementary Algebra Can, Should, and Must…” by Usiskin.  In addition, you will be assigned one of the following three articles:

i.                     “Why is Algebra Important to Learn?” by Usiskin

ii.                    “Algebra:  What Should We Teach and How Should We Teach It?” by Thorpe

iii.                  “The Transition from Arithmetic to Algebra” by Lodholz

These articles will be discussed in small groups on October 19.  In order for the discussion to succeed, you must read the article assigned to you!  Write a one paragraph summary of the article assigned to you, and then write your reaction to your reading.

 

 

9          Linear problems—Revisited

9.1      Alejandro and Barbara are twins.  By their 10th birthdays, Alejandro has read 20 books, while Barbara has read 36 books.  Each month after their 10th birthday, Alejandro reads 6 books, while Barbara reads 2 books.  In solving these problems, work graphically, work from an equation, and, if helpful, also use a table.  You may find more than one graphing and equation approach is possible.

9.1.1            How long will it take until Alejandro has read twice as many books as Barbara?

9.1.2            How long will it take until Alejandro has read three times as many books as Barbara?  Explain your answer.

9.1.3            Change the numbers in the book problem so that Alejandro will eventually have read 10 times as many books as Barbara, but will never have read 11 times as many books.  Explain your process.

9.2      I buy CDs from several different sources, let’s call them Amazin’ Discs, Venezuela House, and Big Music Group.  One particular month, Amazin’ had CDs for $11 each, Venezuela offered the first CD for $24 plus $6 each additional CD, and BMG offered the first CD for $18 plus $6 for each additional CD.  At the end of the month, having charged all purchases to my credit card, my wife was checking the bill and said I spent $120 on CDs.

9.2.1            If I ordered all my CDs from one company, how many CDs might I have purchased?

9.2.2            Is it possible that I ordered 2 CDs from Amazin’ Discs and the rest from the other two companies?  Explain.

9.2.3            The next month, I spent $66 on CDs.  Find all possibilities for what I ordered.  Explain your method.

9.2.4            In another month, I spent $D on CDs, all from one company.  For what values of D is it possible that I ordered from

9.2.4.1        Amazin’ Discs?

9.2.4.2        Venezuela House?

9.2.4.3        BMG?

 

Due October 26

Journal XV (no journal will be assigned in class on October 19 because of the exam).  Read “Prealgebra:  The Transition from Arithmetic to Algebra,” by Kieran and Chalouh.  Write your reaction.

 

  1. Sums of consecutive numbers, p. 37 of Driscoll

 

October 26 Class Activity

  1. Find positive integers n and a1, a2,  … , an so that a1+a2+…+an=1000 and a1×…×an is as large as possible.  Note:  Although the special case when a1+a2=1000 and a1×a2 is as large as possible might be remembered as a calculus problem, calculus is not needed to solve this problem, since we are dealing with integers.

 

Journal XVI.  Read “What Should Not Be in the Algebra Curriculum…” by Usiskin, pp.76-81.  Write your reaction to the following questions:  Do you agree with Usiskin’s suggested deletions?  Make your own list of 3 topics (which may or may not overlap with Usiskin) and explain why they could or should be deleted from the Algebra I curriculum.

Due November 2

Journals will be collected November 2—it will include Journals up through XVII.

Problem 11 is due November 2.

 

Journal XVII.  Read “Procedures for, and Experiences in, Introducing Algebra in New South Wales,” by Pegg and Redden, pp. 71-75.

 

 

November 2 Class Activity

 

        Journal XVIII.  Each group will be assigned an article from the Patterns section of the book.  Your job will be to summarize the activity in the article, explain for what level of students it is designed, and offer a critical review of the activity.

 

Below are two challenging problems.  We will work on them in class on November 2, but they will be collected as homework due November 9.  I expect that everyone make an effort to solve both problems, but to receive full credit on the homework, it is only necessary that you record your attempts and explain where you got stuck (or how you solved the problems).

 

  1. Prove that for any whole number n, among a set of n + 1 positive integers, none of them larger than 2n, that at least one member of the set must divide another member of the set (divide means divide evenly with no remainder).
  2. Given that a, b, and c are odd numbers, show that ax2 + bx + c = 0 cannot have a rational root.  I have some hints ready for this one; ask me if you need help.

 

Journal XIX due November 9 

Read “A Technology-Intensive Approach to Algebra,” by Heid and Zbiek, pp. 82-89.  Write your reaction.

 

November 9 Class Activity

Given any nine integers, show that there exists a subset {a,b} consisting of two of these nine integers, with the difference of a and b a multiple of 8.

 

        Journal XX.  Each group will be assigned an article from the Variables, Expressions, and Equations section of the book.  Your job will be to summarize the activity in the article, explain for what level of students it is designed, and offer a critical review of the activity.

 

Due November 16

Journal XXI.  Read “Revitalizing…with HALP,” by Dougherty and Matsumoto, pp. 90-96.  Write your reaction.

  1. For each type of function below, identify two real world situations in which that type of functional relationship exists.  I have offered a couple of examples.

a.        Y=kx, with k nonzero:  One example is distance traveled as a function of the number of turns of the pedals of a bicycle (assuming this is a track bicycle in which coasting is not possible).

14.2  Y=mx+b, with m and b both nonzero.

14.3  Y=ax2, with a nonzero.

14.4  Y=ax2+bx+c, with a and one or both of b and c nonzero:  As we have seen, one example is the height of a ball launched into the air as a function of time.

14.5  Y=k/x, k nonzero.

14.6  Y=kxn, with k nonzero and n a whole number greater than 2.

14.7  Y=Cax, with C nonzero and a > 0.

 

November 16 Class Activity

There is an algorithm known as the Russian peasant algorithm for multiplication, which I will illustrate on 25 × 28.  First, make a column of numbers in which you double the first number and halve the second number, dropping any fractional parts.

25            28

50            14

100          7

200          3

400          1

Take all the rows in which the right-hand column number is odd (I’ve indicated these by underlining them), and add the left-hand column numbers in the underlined rows.  Thus the product is 25 × 28= 400 + 200 + 100 = 700. 

  1. Verify that the algorithm gives you the same answer for 28 × 25.