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 10^{th} and the 99^{th} 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

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

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 10^{th}
birthdays, Alejandro has read 20 books, while Barbara has read 36 books. Each month after their 10^{th} 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.

- You are next to a stream and need to put exactly
1 gallon into one of the jugs. Find
a way to do this.
- 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.
- Look at more examples as necessary, in order to
answer the following questions:
- 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?
- 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 r_{m}
and r_{n} 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 - x^{2}.
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.

- 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}.

- Based on this information, predict the solution
set to the inequality (**) (x+1)/(x – 1) < 2/3.
- Which of the inequalities (*) or (**) above is
equivalent to: (***)
(x+5)/[(x – 1)(x+1)] < 0 ?
- How is the other inequality (*) or (**) NOT used
in (b) related to (***)?
- How does the form of the inequality (***) help
you find the solution?

- 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)= x^{2} + bx
+ c has roots m and n. For instance,
f(x) = x^{2} – 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)=-4.9t^{2} + 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

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 10^{th}
birthdays, Alejandro has read 20 books, while Barbara has read 36 books. Each month after their 10^{th}
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,

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

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.

- Sums of consecutive numbers, p. 37 of Driscoll

**October 26 Class Activity**

- 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

**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).

- 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).
- Given that a, b, and c are odd numbers, show that
ax
^{2}+ 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.

- 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=ax^{2}, with a nonzero.

14.4 Y=ax^{2}+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=kx^{n}, with k
nonzero and n a whole number greater than 2.

14.7 Y=Ca^{x}, 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.

- Verify that the algorithm gives you the same
answer for 28 × 25.
- Perform the algorithm on two other multiplication
problems.
- Explain why it works.
- Express all the steps in base 2 notation.

Journal
XXII. Each group will be assigned an article. You should summarize the article, explain for
what level of students the lesson is designed, and offer a critical review of
the activity.

**Due November 23**

Journal
XXIII. Read “Children’s Difficulties in Beginning
Algebra,” by Booth, pp. 299-307. Write
your reaction.

- Some of you may know the divisibility rule for
3: A number is divisible by 3 if
the sum of its digits is divisible by 3.

15.1 Show that this is true for any three digit natural
number, and then extend to any natural number.
Discuss the algebraic thinking involved in this problem.

15.2 Restate the divisibility rule for 3 as a statement
about mod 3.

- How many zeroes are at the end of 1000000! (one million factorial)?

**Class Activity November 23**

Journal
XXIV. Each group will be assigned an article. You should summarize the article, explain for
what level of students the lesson is designed, and offer a critical review of
the activity.

Show that,
if 2n+1 and 3n+1 are perfect squares, then n is divisible by 40. [You might want to use modular arithmetic.]

**Due November 30**

Journal
XXV. Read “Ideas About Symbolism…” by Stacey and MacGregor, “Some Misconceptions…” by Rosnick,
and “What Are These Things Called Variables?” by Wagner, pp. 308-320.

- Recall that composite numbers are those numbers
which are not prime. A number is square-free if it is not
divisible by any perfect square.
Thus 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19 are the first dozen
square-free natural numbers.
Numbers which are not square-free are called non-square-free.

a.
What is the
longest possible sequence of square-free natural numbers?

17.2 CHALLENGE 1:
Show that it is possible to create a sequence of K consecutive natural
numbers, all of which are composite, where K is any natural number.

17.3 CHALLENGE 2:
Can you have K consecutive natural numbers, all of which are non-square
free, for any given natural number K? If
not, what is the largest possible value of K?

- You have a balance scale and 8 weights. The weights are 1g, 2g, 2g, 5g, 10g,
10g, 10g, and 10g.

18.1 With these weights, show how it is possible to weigh
any object of up to 49 grams, to the nearest gram.

18.2 Find a more efficient set of weights by changing the
gram values of any or all of the original set.

18.3 Is your solution the only one? Is it the most efficient one possible? Explain your thinking.

18.4 If you think you have found the most efficient set of
weights, let me point out that you can put weights on either side of the
balance. If your solution did not
account for this, can you find a new answer?
If you did account for this, what if you are restricted to putting your
known weights on just one side of the balance?

**Class Activity November 30**

Journal
XXVI. Each group will be assigned an article. You should summarize the article, explain for
what level of students the lesson is designed, and offer a critical review of
the activity.

Previously (on September 14), we showed that the product of
4 consecutive integers plus 1 is always a perfect square. Generalize this to show that the product of 4
consecutive terms in any arithmetic sequence of integers plus the fourth power
of the common difference is always a perfect square. Symbolically, a_{k}a_{k+1}a_{k+2}a_{k+3}
+ d^{4} is a perfect square, where k is any natural number, the {a_{k}| k = 1, 2, …} are
the terms of the sequence, and there is a common difference d between
consecutive terms, a_{k+1}-a_{k} = d.

Is 2^{2x+1} a
multiple of 3, where x is a natural number?

**Due December 7**

Journal
XXVII. This is the last journal. Journals will be collected in class on
December 7. Read “From Words to Algebra: Mending Misconceptions,” by Lochhead and Mestre. Write your reaction.

- Look back over the math problems we have done
since October 26. Write 4 problems
which are closely related, that is, are either equivalent or are
extensions. To give you an example
of what I mean, showing that the square of an odd number is 1 mod 8
(problem 5.2) and finding which multiples of 8 are one less than the
square of an odd number (exam question 1) are two related problems.
- Find or write one problem appropriate for your
students (as before, imagine some if necessary) which is not in Driscoll
and which was not done in this class. Discuss the algebraic thinking
involved and the goal(s) you would have for students in terms of both
performance (what solutions or partial solutions would you expect from
them) and in terms of learning (what the students should know as a result
of working on this problem).

**December 7 Class Activity**

Manipulating both sides of an
equation

Describe the manipulations
allowed in solving equations functionally.
If we think of any equation (in one variable) as g(x) = h(x), when is f(g(x)) = f(h(x))?
. Aim to come up with a general
statement of which functions can be applied to both sides of an equation, and
which functions may alter the solutions.
As an example, in the equation x -5 = 11, if we apply f(x) = x + 5 to
both sides then we find x = 16.
CHALLENGE: Extend to inequalities
of the form g(x) > h(x).