Math 221 – F07
Activities, page 1
Fill-In the Digits
1. Replace each letter with a digit so that the following sum is correct. Repeated letters must
be replaced by the same digit each time. Different letters must be replaced by different digits.
+
S
F
U
U
N
N
S
W
I
M
2. 1–9 Triangle Puzzle
a) Using each of the digits 1 through 9 exactly once, fill in the circles of a triangular
arrangement similar to the one above so that the sums of the four digits along each side
is the same.
b) Find as many different solutions as you can in which the side sum is 20. (How should
you define “different”? Are some solutions more different than others?)
c) Can you find a solution in which the sum on each side is 15? 18? 21? 22?
3. 3-Digit Sums
Using each of the digits 1 through 9 exactly once, fill in the boxes below so that the sum is
correct:
Math 221 – F07
Activities, page 2
+
Math 221 – F07
Activities, page 3
Some Problems
1. Ron’s Recycle Shop was started when Ron bought a used paper-shredding machine. Business
was good, so Ron bought a new shredding machine. The old machine could shred a truckload
of paper in 4 hours. The new machine could shred the same truckload of paper in only 2
hours. How long will it take to shred a truckload of paper if Ron runs both shredders at the
same time?
2. One day my son Jason (at the time a first- or second-grader) made an observation and had a
question.
• The Observation: “If you have two numbers to add together and you take one away from
one number and give it to the other number they still add to the same thing.” (Example:
11 + 8 = 10 + 9.)
• The Question: “Does the same thing happen when you multiply two numbers?”
In my usual manner I answered his question by saying “What do you think?” And he proceeded to think about the problem (and explain to me what he was thinking about the
problem) until he was satisfied that he understood the situation.
What can you learn about this observation and question?
3. At a local neighborhood festival, there was a tug-of-war competition.
• In round 1, one side had four acrobats, each equally strong, and the other side had five
neighborhood grandmas, also each equally strong. The result was dead even.
• In round 2, one side had Ivan, a dog, and the other side had two grandmas and one
acrobat. Again it was a draw.
• In round 3, Ivan and three grandmas went up agains four acrobats.
Which side won round 3?
4. A trapezoid is a four-sided figure with one pair of opposite sides parallel. Draw a trapezoid
that has an area of 36 square units.
5. Here is a pattern exploration exercise called Start and Jump Numbers. It goes like this. Pick
a start number and a jump number. Then make a list of numbers starting with the start
number and increasing by the jump number each time. For example, if the start number is 3
and the jump number is 5, then the list starts 3, 8, 13, . . ..
Your task is to examine this list of numbers (using 3 as the start number and 5 as the jump
number) and to find as many patterns as you possibly can. You will probably want to start
by writing down the list up to numbers of about 130 or so.
Math 221 – F07
Activities, page 4
Some Problems to Solve
Work on the problems on this sheet in two phases:
1. First read through all the problems and make a list of ways you might attempt to solve the
problem. (Keep in mind the heuristics we have seen.) Jot down some notes for each problem,
then move to the next one.
2. Then go back and (attempt to) solve the problems. For each of the problems, explain the
reasoning you use to solve the problem. Whether you solve the problem completely or not, keep
a list of things you tried, patterns you observed, conjectures you made, facts you discovered,
etc. Be sure to note any problem solving strategies you used that did not occur to you when
you first read the problem (before working on it).
You may work the problems in any order.
1. What is the last digit of 399 ?
2. How many rectangles (of any size) are there in the picture below?
3. Jan is having a party, to which she invited 24 people. She plans to serve the meal on card
tables, arranged in a long rectangle, with each table pushed up against the next. If the card
tables are only large enough to seat one person on a side, how many tables will Jan need?
4. How many ways are there to cover the large rectangle below using “dominoes”. (A dominoe
is a rectangle that is 1 square wide and 2 squares tall.)
Dominoes:
[Hints: That large rectangle is pretty large, perhaps you should try a some smaller examples
first. How can you use smaller examples to help you with the original problem?]
5. A pail with 40 marbles in it weighs 175 grams. The same pail with 20 marbles in it weighs
95 grams. How much does the pale weigh alone? How much does one marble weigh alone?
6. How many ways are there to make change for 27 cents using (any number of) pennies, nickels,
dimes and quarters.
Math 221 – F07
Activities, page 5
Some More Problems
1. What is the last digit of 1399 ?
2. A regular pentagon is a figure with five equal (straight) sides and five equal angles. How many
diagonals are there in a regular pentagon? (The diagonals are the line segments that connect
one corner with another but are not sides. All together they look like a star.)
Can you generalize this for other polygons?
3. How many ways are there to make a path connecting adjacent letters in the diagram below
so that the path is labeled with the alphabet (each letter exactly once and in order)?
A
ABA
ABCBA
ABCDCBA
ABCDEDCBA
ABCDEFEDCBA
ABCDEFGFEDCBA
ABCDEFGHGFEDCBA
ABCDEFGHIHGFEDCBA
ABCDEFGHIJIHGFEDCBA
ABCDEFGHIJKJIHGFEDCBA
ABCDEFGHIJKLKJIHGFEDCBA
ABCDEFGHIJKLMLKJIHGFEDCBA
ABCDEFGHIJKLMNMLKJIHGFEDCBA
ABCDEFGHIJKLMNONMLKJIHGFEDCBA
ABCDEFGHIJKLMNOPONMLKJIHGFEDCBA
ABCDEFGHIJKLMNOPQPONMLKJIHGFEDCBA
ABCDEFGHIJKLMNOPQRQPONMLKJIHGFEDCBA
ABCDEFGHIJKLMNOPQRSRQPONMLKJIHGFEDCBA
ABCDEFGHIJKLMNOPQRSTSRQPONMLKJIHGFEDCBA
ABCDEFGHIJKLMNOPQRSTUTSRQPONMLKJIHGFEDCBA
ABCDEFGHIJKLMNOPQRSTUVUTSRQPONMLKJIHGFEDCBA
ABCDEFGHIJKLMNOPQRSTUVWVUTSRQPONMLKJIHGFEDCBA
ABCDEFGHIJKLMNOPQRSTUVWXWVUTSRQPONMLKJIHGFEDCBA
ABCDEFGHIJKLMNOPQRSTUVWXYXWVUTSRQPONMLKJIHGFEDCBA
ABCDEFGHIJKLMNOPQRSTUVWXYZYXWVUTSRQPONMLKJIHGFEDCBA
4. There are 20 people in a room. If each person shakes hands with each other person exactly
once, how many handshakes occur?
5. How many squares (of any size) are there on an 8 × 8 checkerboard?
Math 221 – F07
Activities, page 6
Fairy Problems
Grace Church School’s ABACUS International Math Challenge is open to any child, anywhere
in each of three age groups (see details and rules below). Science and math teacher, Tivadar Divki
posts eight problems a month in each age group. Students are asked not only to solve the problems
but also to present the reasoning in their solutions. If a student submits an incorrect solution or is
simply having trouble with a problem, a teacher will offer hints as to how to go about solving the
problem, giving students multiple chances to succeed.
The Abacus Project, launched in 1997 by Mr. Divki, is based on a printed journal that originated in Hungary over 100 years ago. The original journal focused on gifted students – participants
over the years included Edward Teller, Leo Szilard, John von Neumann and other notables in
physics, computer science and mathematics – and showed that mathematical talent can be stimulated and developed through individual attention, instant feedback, and challenging subject matter
with flexible levels of difficulty. The advent of the World Wide Web has made the same methodology available to a much broader range of students. Current and archived problems are available at
http://www.gcschool.org/pages/program/Abacus.html.
These are the September, 2007, problems for third- and fourth-graders.
A.585 It is Spring time in Fairy Land. In the flower beds in front of the king’s palace the flowers
started blooming one after another. How many flowers started blooming in 10 flower beds in
8 hours if 10 flowers started blooming in 2 flower beds in 4 hours?
A.586 In Fairy School the students received a homework assignment to paint the edges of a cube
either red or blue, so that each face of the cube would have at least one red edge on it. What
is the smallest number of red edges the students can use to complete this assigment?
A.587 The next homework assignment was to find four positive whole numbers so that their sum
and their product are the same. Help the little fairies!
A.588 In Fairy Land one year is 4 months, one month is 6 weeks, and one week is 8 days. Fairy
Starr is one and a half years, 2 weeks and 7 days old. Her sister, Fairy Bella, is one year, 3
months and a half week old. Who is older and by how many days?
A.589 On Fairy Land’s main street the houses are numbered by even numbers on the left hand side
and by odd numbers on the right hand side. The fairies are superstitious and do not use the
digit 7, but they use all possible house numbers that avoid using the digit 7. What is the
street number of the 30th house on the right hand?
A.590 The Fairy King showed his palace to his guest. They stopped at a painting, and the guest
asked the king whose portrait it was. The king said: “The father of the person on this painting
is the only son of the answerer’s father.” Who is on the painting?
A.591 The Fairy King asked his gardener to plant trees in a triangle-shaped part of his garden. He
wanted one tree in the first row, 2 trees in the second row, 3 trees in the 3rd row, and so on
for a total of 30 rows. The gardener planted the first 20 rows, and his helper did the rest.
Who planted more trees, the gardener or his helper?
A.592 In Fairy Preschool the square shape playground was divided into 4 identical smaller squares
for the 4 groups. For fencing it around and to divide the sections completely they used 360
m of chicken wire. How many square meters is the area of each small square?
Math 221 – F07
Activities, page 7
More ABACUS Problems
These problems all come from the ABACUS International Math Challenge for 5th and 6th graders.
These are the September, 2007, problems for 5th and 6th graders. (There is also a set for 7th and
8th graders.)
B.585 How many triangles with different shapes and sizes can you draw so that its vertices are on
the dots of the following diagram? (Two triangles of the same shape but different sizes are
considered to be different.)
B.586 Helga has two candles with different thicknesses and different lengths. One of them, the 10-cm
long candle, burns down in 5 hours, the other in 6 hours. She lights them up at the same
time and in 2 hours they are the same length. How long was the other candle originally?
B.587 Tom bought notebooks for 1/3 of his money, then he bought chocolate for 1/3 of the rest of
his money, and then he bought a present for his brother for 1/3 of the rest of his money. He
has $80 left after all the shopping. How much money did he spend on chocolate?
B.588 Out of the following statements exactly one is false:
• A is older than B.
• C is younger than B.
• The sum of the ages of B and C is twice the age of A.
• C is older than A.
Who is the youngest of them all?
B.589 Alan, Ben, Conrad, Danny and Edward are going horseback riding. Five horses are waiting
for them at the stable: Kansas, Leila, Mirza, Nina and Oliver. Alan can ride only Kansas;
Leila wants to carry only Ben or Conrad; Ben, Conrad, and Danny are the only ones who can
ride Mirza. How many different ways can they horseback ride together?
B.590 If you increase one side of a square by 2/5 of that side, and you reduce the neighboring side
by the same amount, will the area of this new rectangle be different?
B.591 Two positive whole numbers are called competitors if the product of the digits of either one of
them is greater than the sum of the digits of the other number. 6a5 and 31a are competitors.
What is smallest value of a?
B.592 We erected a flag on top of a sand castle by sticking its pole vertically half way into the sand.
The top of the flag pole is 76 cm above the ground. The bottom of the pole is 60 cm from
the ground level. How tall is the castle and how long is the flag pole?
Math 221 – F07
Activities, page 8
A-Blocks
1. What do you have? Sort the A-blocks in some convenient way. What attributes and values
for these attributes enable these blocks to be distinguished? Make sure you have complete set
of materials. (How wil you know it is complete?)
2. Name the blocks. Devise a convenient naming system for the blocks so that you can record
your work in the subsequent activities. For example, you might use R to abbreviate red and
T to abbreviate triangle.
3. Difference loops. (This is like Activity 18.11 in VdW.) Construct a loop of 10 blocks,
each differing from each adjacent piece on exactly one attribute. Record the loop here.
Math 221 – F07
Activities, page 9
4. Two loops. (This is like Activity 18.6 in VdW.) Make two loops of yarn or string in the
pattern below.
A
B
1
2
3
4
a) Label loop A RED and loop B DIAMOND-SHAPED. Place all of the pieces in one of
the four regions 1–4 as appropriate.
i. List the pieces in region 2. We could describe these as the pieces which are RED
and DIAMOND-SHAPED.
ii. List and describe the pieces in region 3.
iii. How many pieces are in region 4?
b) Label loop A YELLOW and loop B NOT-TRIANGULAR.
i. List and describe the pieces in region 1.
ii. List and describe the pieces in region 4.
iii. How would you describe the collection of pieces that are not in region 4?
Math 221 – F07
Activities, page 10
5. Three loops. Lay out three loops as follows.
B
1
8
2
3
4
5
A
6
7
C
a) Label loop A RED, loop B LARGE, and loop C CIRCULAR. Place the pieces in the
appropriate region 1–8.
i. List and describe the pieces in region 5.
ii. List and describe the pieces in region 2.
iii. List and describe the pieces in region 1.
b) Label loop A YELLOW, loop B BLUE-OR-LARGE, and loop C NOT-CIRCULAR.
i. List and describe the pieces in region 5.
ii. List and describe the pieces in region 4.
iii. List the pieces in region 1.
Math 221 – F07
Activities, page 11
6. Puzzles. Your goal in these diagrams is to place an A-block on each of the squares so that
if two squares are connected by one line, the A-blocks on those squares differ on their values
on exactly one attribute and if two squares are connected by two lines, the A-blocks differ on
exactly two attributes. You may want to draw larger versions of the diagrams on your paper.
You can record your results using your abbreviations on the diagrams below.
7. Cooperative Game. On a large sheet of paper, draw a four by five grid. In turn, place a
piece on any square of the board. The rule is that adjacent pieces up and down must differ in
two ways, and left and right must differ one way. You win (as a group) if you fill the board.
Play several times, recording your best attempt here.
Math 221 – F07
Activities, page 12
Find the Pattern
1. For each of the following patterns,
•
•
•
•
(a)
(b)
(c)
(d)
(e)
describe the pattern using words,
decide if it is a repeating pattern or a growing pattern,
see if you can determine the 10th and 50th numbers in the pattern,
see if you can express the nth number in the pattern. (For some this may be very hard,
for others it is much easier.)
2, 4, 6, 8,
,
,...
, 5, 8, 11, 14,
,
,...
5, 10, 20, 40, 80,
,
...
3, 10, 7, 3, 4, 1,
,
,
,...
2, 5, 7, 12, 19,
,
,...
2. Some Sums. Consider each of the following patterned sums.
a) 1 + 2 + 3 + · · · + n. (adding all numbers from 1 to n)
b) 1 + 3 + 5 + · · · + n. (adding consecutive odd numbers)
c) 2 + 4 + 6 + · · · + n. (adding consecutive even numbers)
For each of these sums do the following:
• Determine the sum when n is 1, 2, 3, 4 and 5. (Example: for (a) the first 5 terms are 1,
3, 6, 10, 15.)
• See if you can determine the value of the sums when n = 10 and when n = 50.
• See if you can express the value of the nth sum using a general formula for n.
How sure are you of your formula(s)? What is the basis of your (level of) belief in your
formulas? What would make you more confident?
Do you see any relationships between the various sums? Do the results of one part help you
with any others? Do the methods of one part help you with any others?
3. Find a formula that allows you to quickly compute a + (a + 1) + (a + 2) + · · · + b for any a < b.
4. Consider the following zig-zag number pattern.
1
2 3 4
8 7 6 5
9 10 11 12
16 15 14 13
In which column will the folowing numbers go: 50, 100, 1000, 2007, 12345
5. Diagonals. How many diagonals are there in a regular pentagon? (A regular pentagon is a
figure with five equal (straight) sides and five equal angles.) Can you generalize this for other
polygons?
Math 221 – F07
Activities, page 13
Set Operations
1. Let R be the set of red blocks, let T be the set of triangles, let L be the set of large blocks,
let B be the set of blue blocks and let D be the set of diamond-shaped blocks. For each of
the following sets,
• locate the the set in terms of loops and A-blocks,
• draw a picture (Venn diagram) shading in the loop regions that represent the set,
• list the elements of the set, and
• describe the elements (with words).
(a) R ∩ T
(b) R ∩ T
(c) R ∪ T
(d) R ∩ T
(e) L ∩ B ∩ D
(f) L ∩ (B ∪ D)
(g) (L ∪ B) ∩ (L ∪ D)
(h) (L ∪ D) ∩ B (Hint: something in your previous work might be useful here.)
Find two examples among the sets above of one set being a subset of another. How do you
write this using set notation? Which of the four sets contain the large blue square? How do
you write this?
2. Invent a new set operation. Give it a name, make up symbolism for it, explain what it does,
and give a few examples. Can you express your new operation in terms of the “standard”
operations on sets?
3. Repeat as much of problem 1 as makes sense using R = {1, 2, 3, 4, 5}, T = {0, 3, 6}, L =
{1, 2, 3, 4, 5, 6}, B = {5, 6, 7, 8, 9, 10}, and D is the set of numbers between 0 and 10 (inclusive)
that divide 10 evenly.
Math 221 – F07
Activities, page 14
Classification of Addition and Subtraction Word Problems
1. Van de Walle describes four types of addition/subtraction problems. What are they? Each
of these main types has two or three subtypes. What distinguishes the subtypes?
For problems each problem of 2 – 12 below, determine its type and subtype, and write one or
more number sentences associated with the problem, putting a box around the unknown in your
number sentence. Your number sentences should include the sentence you feel is most “natural” to
the problem as well as a number sentence in computational form. Indicate which number sentences
are in computational form.
2. Al is 49 inches tall. Bea is 52 inches tall. How much taller is Bea than Al?
3. Deb had 7 cookies. She gave some to Cy. Now Deb has three cookies. How many did she give
to Cy?
4. Ed has eight cousins. Three of them are boys. How many girl-cousins does Ed have?
5. After Flo lost five of her new back-to-school pencils, she had 7 pencils left. How many did
Flo start with?
6. Hal had two sweaters. Grandma knitted some more for him. Now Hal has 8 sweaters. How
many did he get from Grandma?
7. Ida has three more brothers than Jose. Ida has five brothers. How many does Jose have?
8. Kendra had 6 hats. Lea gave her 3 more. How many hats does Ken have now?
9. Maria has 5 trucks and 8 cars in her sandbox. How many vehicles does she have in all?
10. Nadia had twelve wine glasses, but Oliver broke four of them. How many wine glasses does
Nadia have left?
11. Pa has five fewer teeth than Ma has. Pa has 8 teeth. How many does Ma have?
12. Rolando had done some of his homework problems, but was stuck on some others. His friend
Syd helped him on four more problems. Now Rolando has 11 problems finished. How many
did he have finished before getting help from Syd?
13. Did all of Van de Walle’s subtypes occur above? If not, write your own problems for any subtypes that did not occur. Identify the type and subtype of each, and write number sentences
for you problems just as you did for problems 2 – 12.
14. Which of the subtypes did you find most natural to express in their computational form?
Math 221 – F07
Activities, page 15
Meaning of Multiplication and Division
Van de Walle identifies four categories of multiplication/division problems, among which two
are especially important: equal groups and multiplicative comparison. Each of these categories has
three subcategories, based on what piece of information is unknown.
1. If there are two categories and each has three subcategories, how many classifications by
category and subcategory are there? Explain how you arrived at your answer.
2. What type of problem is the problem above?
3. For each type of problem (category and subcategory):
a) write a brief story problem of that type, making sure that no number occurs more than
once in the problem or its solution,
b) solve the problem using an array model and one other model (make a sketch of each
model on your paper)
c) write down a number sentence for your problem and put a box around the unknown
information,
d) write down any related number sentences (again with a box around the missing information).
4. Which of the problems above were hardest to invent?
5. Which of the problems above were hardest to model?