Instructions
Select a month from the menu above and a calendar will drop down.
School days are in white. Click on any school day to see the challenge problem for that day.
Many of the challenge problems come from the
Math 1 course material
used and developed
by the math department of Phillips Exeter Academy. (Problems from PEA are numbered: page number -
problem number, ex. 1-7 is page 1 problem 7.) These problems are free to downlaod.
Thank you Phillips Exeter Academy math department for
allowing your problems to be used.
Additionally, at the bottom left side of the teaching materials page (Math 1 course material) you
will find a link to 'peanut software'. There are several math programs free to download.
One of them I use is winplot. It is great for graphing which we do a lot of in Algebra and Pre-algebra.
Wednesday August 12, 2009
1-7.
Your class sponsors a benefit concert and prices the tickets at $8 each. Kim sells 12 tickets,
Andy 16, Pat 13, and Morgan 17. Compute the total revenue brought in by these four people.
Notice that there are two ways to do the calculation.
1-8.
Kelly telephoned Brook about a homework problem. Kelly said, "Four plus three times two is
14, isn't it?" Brook replied, "No, it's 10." Did someone make a mistake? Can you explain
where these two answers came from?
Thursday August 13, 2009
1-9.
It is customary in algebra to leave out multiplication symbols whenever possible. For example,
11x means the same thing as 11•x . Which of the following can be condensed by leaving out a
multiplication symbol?
(a) 4 • 1/3 (b) 1.08 • p
(c) 24 • 52 (d) 5 • (2 + x)
2-2.
Pick any number. Add 4 to it and then double your answer. Now subtract 6 from that result
and divide your new answer by 2. Write down your answer. Repeat these steps with another
number. Continue with a few more numbers, comparing your final answer with your original
number. Is there a pattern to your answers?
Friday August 14, 2009
2-3.
Using the four integers 2, 3, 6, and 8 once each - in any order - and three arithmetic
operations selected from among addition, subtraction, multiplication, and division, write
expressions whose values are the target numbers given below. You will probably need to use
parentheses.
For example, to hit the target 90, you could write 90 = (3+ 6)(8 + 2).
(a) 3 (b) 24 (c) 36 (d) 30
Monday August 17, 2009
2-4.
When describing the growth of a population, the passage of time is sometimes described in
generations, a generation being about 30 years. One generation ago, you had two ancestors
(your parents). Two generations ago, you had four ancestors (your grandparents). Ninety
years ago, you had eight ancestors (your great-grandparents). How many ancestors did you
have 300 years ago? 900 years ago? Do your answers make sense?
Tuesday August 18, 2009
A perfect square
is a product formed by the product of two identical factors.
For this problem we are going to be using positive integer factors only.
Generally when you hear the term perfect square it means a number formed
by the product of two identical positive integer factors. In higher level
math classes this definition is expanded to include any two identical factors
regardless of the type of number (rational, irrational, etc.), the sign, or even if
variables are part of the factors.
The first few perfect squares are:
1, 4, and 9.
1 = 1•1 = 12 (one squared)
4 = 2•2 = 22 (two squared)
9 = 3•3 = 32 (three squared)
The concept of a perfect cube
is similar to that of a perfect square but instead of using identical
factors two times identical factors are used three times.
The fisrt few perfect cubes are:
1, 8, and 27.
1 = 1•1•1 = 13(one cubed)
8 = 2•2•2 = 23(two cubed)
27 = 3•3•3 = 33(three cubed)
From the examples you can see that 1 is both a perfect
square and a perfect cube. Is there any other
number besides 1 that is both a perfect square and perfect cube? Is
there more than one number that is both a perfect square and perfect cube? If there are
other numbers that are both perfect squares and perfect cubes, can you find a pattern to determine
what numbers they are?
Wednesday August 19, 2009
An easy way to test if a number is divisible by 6 is to remember that if a number
divides by 2 and it divides by 3 then it divides by 2x3. After learning about this test Bill
went home and made up another divisibility rule. His rule claims that if a number divides by 2 and it divides
by 4 then it divides by 2x4. He tested it for the following numbers: 8, 16, 24, 32, and 40 and it
worked every time. He came to the conclusion that he had discovered a test for divisibility by 8.
His rule is actually called a conjecture
because it is a conclusion based on evidence but
it is not proved mathematically. Do you think his rule always works? If you think so you must
prove it mathematically. If you do not think his rule always works you must prove it wrong
with a counter example.
A counter example is any example that shows a conjecture is not true
for at least one situation. For example if I said that all dogs are black and you showed me
a brown dog that brown dog is a counter example to my conjecture and my conjecture does not hold
true every time so it is a false conjecture.
Is Bill's conjecture true or false?
Thursday August 20, 2009
1-10.
Wes bought some school supplies at an outlet store in Maine, a state that has a 6.5%
sales tax. Including the sales tax, how much did Wes pay for two blazers priced at $49.95
each and 3 pairs of pants priced at $17.50 each?
1-11.
(Continuation) A familiar feature of arithmetic is that multiplication distributes over
addition. Written in algebraic code, this property looks like a(b+c) = ab+ac. Because of
this property, there are two equivalent methods that can be used to compute the answer
in the previous problem. Explain, using words and complete sentences.
Friday August 21, 2009
2-5.
On a recent episode of Who Wants to Be a Billionaire, a contestant was asked to
arrange the following five numbers in increasing order. You try it, too.
(a) 2/3 (b) 0.6666 (c) 3/5
(d) 0.666 (e) 0.67
2-6.
The area of a circle whose radius is r is given by the expression πr2. Find the area of
each of the following circles to the nearest tenth of a square unit of measure:
(a) a circle whose radius is 15 cm
(b) a circle whose radius is 0.3 miles
Monday August 24, 2009
3-3.
You are already familiar with operations involving positive numbers, but much mathematical
work deals with negative numbers. Common uses include temperatures, money,
and games. It is important to understand how these numbers behave in arithmetic calculations.
First, consider addition and subtraction. For each of the following, show how the
answer can be visualized using a number-line diagram:
(a) The air temperature at 2 pm was 12°. What was the air temperature at 8 pm, if it
had dropped 15° by then?
(b) Telescope Peak in the Panamint Mountain Range, which borders Death Valley, is 11045
feet above sea level. At its lowest point, Death Valley is 282 feet below sea level. What is
the vertical distance from the bottom of Death Valley to the top of Telescope Peak?
(c) In a recent game, I had a score of 3. I then proceeded to lose 5 points and 7 points
on my next two turns. On the turn after that, however, I gained 8 points. What was my
score at this moment in the game?
Tuesday August 25, 2009
4-2.
Before you are able to take a bite of your new chocolate bar, a friend comes along and
takes 1/4 of the bar. Then another friend comes along and you give this person 1/3 of
what you have left. Make a diagram that shows the part of the bar left for you to eat.
4-3.
Later you have another chocolate bar. This time, after you give away 1/3 of the bar,
a friend breaks off 3/4 of the remaining piece. What part of the original chocolate bar do
you have left? Answer this question by drawing a diagram.
Wednesday August 26, 2009
4-4.
Profits and losses for the Whirligig Sports
Equipment Company for the six years indicated
are graphed on the chart above. The vertical
scale is in millions of dollars. What was the
change in profit and losses from
(a) 1993 to 1994?
(b) 1994 to 1995?
(c) 1997 to 1998?
For the six years graphed, did the company make
an overall profit or sustain an overall loss? How
much was the net change?
Thursday August 27, 2009
3-1.
What is the value of 3 + (-3)? What is the value of (-10.4) + 10.4? These pairs of
numbers are called opposites. What is the sum of a number and its opposite? Does every
number have an opposite?
For many problems you will find that by drawing a simple diagram, number line, chart, or graph
using the information given in the problem your understanding of the problem and thus your abiltiy to solve
it will be greatly increased. For each of the problems below I have included a number line for you to use should
you want to.
State the opposite of:
(a) -2.341
(b) 1/3
(c) x
(d) x + 2
(e) x - 2
Friday August 28, 2009
3-2.
As shown on the number line below, k represents an unknown number between 2 and
3. Plot each of the following, extending the line if necessary:
(a) k + 3
(b) k - 2
(c) -k
(d) 6 - k
Monday August 31, 2009
6-1
In the balance diagram below, find the number of marbles that balance one cube.
6-9
Draw a balance diagram that is modeled by the equation:
c + m + c + 7m + c = 2c + 2m + 3c.
How many marbles will one cube balance?
Tuesday September 1, 2009
6-4
Using the four integers 1, 2, 3, and 4 once each - in any order - and three arithmetic
operations selected from among addition, subtraction, multiplication, and division, is it possible to write an
expression whose value is 1? Using the same numbers and conditions, how many of the integers from 1 to 10 can you
form? You will need to use parentheses.
Wednesday September 2, 2009
6-3
On each of the following number lines, all of the designated points are evenly spaced. Find
the values for the letters that represent the coordinates of these points.
Thursday September 3, 2009
3-5A
Locate the following numbers relative to each other on a number line:
(a) 3.03
(b) 3.303
(c) 3.033
(d) 3.333
(e) 3.33
3-5B
Locate the following numbers relative to each other on a number line:
(a) 3.03
(b) 3.303
(c) 3.033
(d) 3.333
(e) 3.33
3-5C
Questions 3-5A and 3-5B are acutally the same question but the way they are
displayed actually makes one of them easier to solve than the other one. Which
one was easiest for you to solve? Explain why it was easier. What does this
suggest to you about sorting (arranging from smallest to largest for example)
numbers that are in decimal form? What could you do to make the problem even
easier to solve?
3-7
Mark a random number x between 1 and 2 (at a spot that only you will think of) on
a number line. Plot the opposite of each of the following:
(a) x
(b) x + 5
(c) x - 4
(d) 6 - x
Friday September 4, 2009
6-3
The division problem
12
÷
4
is equivalent to the multiplication problem
12
•
3
.
Write each of the following division problems as equivalent multiplication problems:
(a)
20
÷
5
(b)
20
÷
5
(c)
20
÷
5
(d)
a
÷
c
(e)
c
÷
a
6-4
What is the value of
3
•
2
?
What is the value of
4
•
4
?
These pairs of numbers are called reciprocals. What is the product of a
number and its reciprocal? Does every number have a reciprocal?
State the reciprocal of the following:
(a)
3
(b)
-
2
(c)
2000
(d)
b
(e)
x
Tuesday September 8, 2009
5-8
Consider the sequence of numbers 2, 5, 8, 11, 14, . . ., in which each number is three
more than its predecessor.
(a) Find the next three numbers in the sequence.
(b) Find the 100th number in the sequence.
(c) Using the variable n to represent the position of a number
in the sequence, write an
expression that allows you to calculate the nth number.
The 200th number in the sequence
is 599. Verify that your expression works by evaluating it with n equal to 200.
8-6
Which number is closer to zero, -4/5 or 5/4?
Wednesday September 9, 2009
6-6
Use the distributive property to explain why 3x + 2x can be simplified to 5x.
6-7
(Continuation) Write each of the following as a product of x and another quantity:
(a) 16x + 7x (b) 12x - 6x (c) ax + bx (d) px - qx
6-8
Solve each of the following equations for x:
(a) 16x + 7x = 46 (b) 12x - 6x = 3 (c) ax + bx = 10 (d) px - qx = r
Thursday September 10, 2009
3-6
The area of the surface of a sphere is described by the formula
S = 4πr2, where
r
is the radius of the sphere. The Earth has a radius of 3960 miles and dry land forms
approximately 29.2% of the Earth's surface. What is the area of the dry land on Earth?
What is the surface area of the Earth's water?
Many problems in real life require you to do more work than meets the eye at first. This
problem is no exception. Even though you are not asked to find the surface area of the Earth
you need to calculate it to be able to solve the problem. Start by calculating the suface
area of the Earth. Use 3.14 as the value for π.
Once you have found out what the surface area S of the Earth is you need to calculate what 29.2%
of S is. Are you aware that you can not use
percents in equations? You need to convert the percent
to a decimal or a fraction first. For this problem convert the percent to a decimal. This is usually
the best way to solve problemls involving percents. What is 29.2% converted to a decimal?
Once you have calculated the area of dry land on Earth there are two ways to calculate the
surface area of water on the Earth. One method uses the area of dry land and the area of the
Earth. The other way uses the area of the Earth and the percent of the surface that is covered
by water. Calculate the surface area of water both ways. Should your answers be the same no
matter what method you use? Are they the same? You realize that you are not given the percent
that is covered by water, how can you figure out what it is?
Friday September 11, 2009
7-5
Because 12x2 + 5x2 is equivalent to 17x2, the expressions
12x2 and 5x2 are called like terms.
Explain. Why are 12x2 and 5x called unlike terms?
Are 3ab and 11ab like terms? Explain.
Are 12x2 and 5y2 like terms? Explain.
Are 12x2 and 12x like terms? Explain.
7-6
In each of the following, use appropriate algebraic operations to remove the
parentheses and combine like terms. Leave your answers in a simple form.
(a) x(x+5) + 2(x + 5)
(b) 2x(5x-2) + 3(5x-2)
(c) 5m(3m-2n) + 4n(3m-2n)
Monday September 14, 2009
8-2
Place a common mathematical symbol between the numerals 2 and 3, so as to produce a
number that lies between 2 and 3 on a number line.
8-3
Simplify the expression
k-2(k-(2-k))-2 by writing it without using parentheses.
12-4
Find the smallest positive integer divisible by every positive integer less than or
equal to 10.
Tuesday September 15, 2009
7-8.
Jess has just finished telling Lee about learning a wonderful new algebra trick: 3+5x
can be simplified very neatly to just 8x, because a + bx is the same as (a + b)x. Now Lee
has to break some bad news to Jess. What is it?
7-10.
If m and n stand for integers, then 2m and 2n stand for even integers. Explain. Use
the distributive property to show that the sum of any two even numbers is even.
Wednesday September 16, 2009
17-4.
Chase began a number puzzle with the words
"Pick a number, add 7 to it, and double the result."
Chase meant to say, "Pick a number, double
it, and add 7 to the result." Are these two instructions
equivalent? Explain.
Suppose I tell you that I will give you my CD collection plus I will buy you 7 more CD's.
The
point of reference in this problem is the number of CD's
that I have because it is the number that is being modified
(it is being increased by 7 because I am going to buy you 7 more CD's).
It is very important to translate phrases correctly into algebraic expressions when solving word
problems so that you will set the equations or inequalities up correctly. You always start with
the point of reference and then apply the math operations to that point. For example, say that
I have X CD's. To show that number being increased by 7 it would be written as X + 7. Likewise
to show it being decreased by 10 it would be written as X - 10.
The trick is to determine what number (or variable) is the point of reference and make sure you apply the math
operations to that number (or variable).
11-8.
Write an expression that represents the number that:
(a) is 7 more than x (b) is 7 less than x
(c) is x more than 7 (d) exceedes x by 7
(e) is x less than 7 (f) exceedes 7 by x
Thursday September 17, 2009
4-5.
The temperature outside is dropping 3° per hour. If the temperature at noon was
0°, what was the temperature at 1 pm? 2 pm? 3 pm? 6 pm? What was the temperature
t hours after noon?
2-10.
Simplify
x + 2 + x + 2 + x + 2 + x + 2 + x + 2 + x + 2 +
x + 2 + x + 2 + x + 2 + x + 2 + x + 2
3.
The greatest common factor (GCF) of 12 and 30 is the largest integer that divides evenly into
both numbers. What is the GCF of 12 and 30? Start by making a list of all the numbers that divide
evenly into 12. This list will contain all the factors of 12. Next make a list of all the numbers
that divide
evenly into 30. By comparing the two lists you will find the largest number that is in both lists.
That number is the GCF of 12 and 30.
4.
Repeat the process to find GCF(24,44). The preceedinng expression means to find the GCF of 24 and 44.
Friday September 18, 2009
The absolute value of a number is defined as the distance from 0 to that number on a
number line. On the number lines below you see that 5 and -5 are both the same distance
from 0, 5 units so they both have an absolute value of 5.
You can also see that they are on different sides of 0.
Absolute value is not concerned with which side of 0 a number is on, only how far from 0 the number
is. That is why absolute value can never be negative, it measures distance which can only
be positive not direction which can be positive or negative (up down, right left, backwards forward).
Can absolute value be 0?
How far from 0 is 0?
The symbol for absolute value is two vertical bars with the number or expression
inside.
The absolute value of 5 looks like this: |5|.
The absolute valule of -17 -33 +8 looks like this: |-17 -33 +8|.
Absoulte value is both a math operator and a grouping symbol similar to parenthesis. Here is how
it works. First you do all the math inside the absolute value symbols and then you make the
result positive if it is not already positive. Another way to think about absolute value is
"just make it positive".
1. Evaluate the following:
(a) | -6 |
(b) | -6 +9 |
(c) | -6 +9 -7 |
(d) | -6 +9 -7 | + | -6 +9 -7 |
(e) | -6 +9 -7 | - | -6 +9 -7 |
(f) 2| -6 +9 -7 |
(g) -2| -6 +9 -7 |
(h) | -6 +9 -7 | ÷ |16 ÷ 4 ÷ 2 - 2 |
(i) -2| -5 + 23 | + |(-2)3|
Monday September 21, 2009
A suggestion for the problem below is to put the terms of the expressions in alphabetical order.
For example, the expressions
3z - 2x + y and y + 2x - z
could be rewritten as
-2x + y + 3z and 2x + y - z
This makes the comparison easier because the x terms, y terms, and z terms are all in the
same relative locations.
Simplifying expressions is a very common practice in Algebra. In fact it is a necessary step
when solving equations. It is also a good idea to simplify expressions before you compare them,
after all simpler expressions are easier to compare.
15-7.
Which of the following eight expressions does not belong in the list?
a-b+c
c-b+a
c-(b-a)
-b+a+c
c+a-b
a-(b-c)
b-(c-a)
a+c-b
23-11.
Without using parentheses, write an expression equivalent to
3(4(3x-6)-2(2x+1)) .
Tuesday September 22, 2009
42-4.
How much money do you have, if you have d dimes and n nickels? Express your answer
in (a) cents; (b) dollars.
42-5.
How many nickels have the same combined value as q quarters and d dimes?
Wednesday September 23, 2009
64-2.
Evaluate each of the following expressions by substituting s = 30 and t = -4.
(a) t2 + 5t + s
(b) 2t2s
(c) 3t2 - 6t - 2s
(d) s - 0.5t2
67-5.
Use the distributive property to factor each of the following:
(a) x2 + x3 + x4
(b) πr2 + 2πrh
(c) 25x - 75x2
(d) px + qx2
Thursday September 24, 2009
54-9.
A catering company offers three monthly meal contracts:
Contract A costs a flat fee of $480 per month for 90 meals;
Contract B costs $200 per month plus $4 per meal;
Contract C costs a straight $8 per meal.
If you expect to eat only 56 of the available meals in a month, which contract would be
best for you? When might someone prefer contract A? contract B? contract C?
Friday September 25, 2009
44-6.
Explain how to evaluate 43 without a calculator. The small raised number is called an
exponent, and 43 is
a power of 4. Write 4•4•4•4•4 as a power of 4. Write the product
43•45 as a power of 4.
Hint for the problem below:
Did you know that a clock that is stopped still shows the correct
time twice a day?
46-5.
A slow clock loses 25 minutes a day. At noon on the first of October, it is set to show
the correct time. When will this clock next show the correct time? Tell me the date and what time
it will be when this occurs.
Monday September 28, 2009
You are familiar with the concept of integer exponents. A number raised to the 2nd power
is the square of the number, for example 42 is 4 squared which is 16, 122 is 12 squared
which is 144. Besides integer exponents there are fractional exponents. While an integer exponent
tells you how many times to use the base as a factor, a fractional exponent tells you to
find the root of the base. For example 41/2 is asking for the square root of 4,
641/2 is asking for the square root of 64. Sometimes
the fractional exponent 1/2 will be used instead of the symbol for square roots. That is the
situation for the problems below.
83-5.
Evaluate (x2 + y2)1/2 using x = 24 and y = 10.
Is (x2 + y2)1/2 equivalent to x + y?
Does the square-root operation "distribute" over addition?
83-6.
Evaluate ((x + y)2)1/2 using x = 24 and y = 10.
Is ((x + y)2)1/2 equivalent to x + y?
Explain.
83-7.
Evaluate ((x + y)2)1/2 using x = -24 and y = 10.
Is ((x + y)2)1/2 equivalent to x + y?
Explain.