Teachers Across Borders South Africa 2012
James Metz
metz@hawaii.edu
Maths Senior Phase (Grades 7 - 9)Workshop
Mathematics in the Senior Phase covers five main content areas (or domains).
1. Numbers, Operations and Relationships
2. Patterns, Functions and Algebra
3. Space and Shape (Geometry)
4. Measurement
5. Data Handling
Each content area contributes towards the acquisition of specific skills.
Patterns, Functions and Algebra
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describing patterns and relationships through the use of symbolic expressions,
graphs and tables; and
identifying and analysing regularities and changes in patterns and relationships
that enable learners to make predictions and solve problems.
investigating numerical and geometric patterns to establish the relationship
between variables; and
expressing rules governing patterns in algebraic language or symbols.
developing algebraic manipulative skills that recognize the equivalence between
different representations of the same relationship.
analyzing situations in a variety of contexts in order to make sense of them
representing and describing situations in algebraic language, formulae,
expressions, equations and graphs
Curriculum and Assessment Policy Statement (CAPS), Senior Phase (Grades 7 – 9),
Mathematics, page 6.
1
Patterns, Functions and Algebra
Problem Solving Strategies
1.
2.
3.
4.
5.
6.
7.
8.
9.
Act it out
Draw a diagram
Make a table
Make a graph
Work backwards
Systematize the counting process
Look for a pattern
Find a rule
Make the problem simpler
Investigations and Problems to Explore
With special thanks to Dr. Neil Pateman, and Dr. Joe Zilliox, University of Hawai’i
As you investigate the following problems, keep in mind the rubric: IDEAL (Identify the
problem, Define and represent the problem, Explore the possible solution strategies, Act
on the strategies, Look back and evaluate). To expand the problems, look at the elements,
ingredients, or parameters of the question and change one; then change another.
Equal sums
Place the numbers 1 through 6 in the circles using each number exactly once so that the
sum of the three numbers on each side of the triangle is always the same. How many
solutions can you find? What do you notice about the solutions?
Now repeat but this time use the numbers 1 through 8 and place them so that the sum of
the three numbers on each of the four sides of the square are the same.
2
Area and Perimeter
Take several 6cm x 7cm rectangles with centimeter grid on them. Measure the area and
the perimeter of the rectangles. Divide and cut the rectangles into three connected parts of
equal areas (same number of boxes). The division must follow the grid and squares must
be connected by sides. Do it in a different way each time and record the perimeter of each
part. What is the range of their perimeters?
McNuggets
McNuggets are sold in boxes of 6, 9, or 20 pieces. If you wanted to buy exactly 21
pieces you could buy a box of 9 and two boxes of 6. In a similar manner you could buy
30 pieces exactly, but there is no way to buy exactly 10 pieces or 22 pieces.
a. What combinations of 6, 9 and 20 allow you to buy exactly 38, 39 and 40
pieces?
b. Are there number of pieces larger than 22 that you cannot buy
exactly? If yes, which ones?
c. Is there a largest number of pieces you cannot buy; i.e., can you buy all
numbers after one particular one?
If you said yes, find the number and explain how you know it is the largest such
number.
If you said no, explain why there is no largest number you cannot buy.
Extension: Consider 3 other numbers, such as 1, 2 and 6
Coins
What are the best denominations for coins to have? In Hawaii we have 1, 5, 10, and 25
cents. We also have 50 cents and $1 coins, but they are not in a general use. In Europe
the new coinage consists of 1, 2, 5, 20 and 50 cents and also 1 and 2 euros. In the past
and in different countries we have had a huge variety of denominations. Australia
originally had 1- , 2- 5- 10- , -20, and 50 cent coins (but no 1 or 2 cent coins now).
Remember the Spanish pieces-of-eight, and the English sixpence. What should be the
denominations for three coins, such that any transaction up to 10¢ would require at most
two coins? This means that either you pay the exact amount using one or two coins, or
you pay with just one coin and get one coin as change.
Squares on a geoboard
Consider a 6 x 6 geoboard
List the dimensions of the different-sized squares that can be made (dots must be at
corners)
3
Dot to dot
Use a 5 by 6 grid of dots like the one shown below
Rules.
1. You may connect dots only horizontally and vertically.
2. You draw just one line that starts and ends at the same point.
3. The line cannot cross itself.
4. The line must pass through all 30 points.
How many different shapes can you find? What are their perimeters? What are their
areas? We have been making figures from a 5 by 6 grid. Keeping exactly the same rules,
from what other sizes of rectangular grids can we make pictures?
Math Tricks
1. A man died with 17 horses, but his will said the eldest child gets ½ of his property, the
middle child gets 1/3 and the youngest 1/9. The lawyer rode in on his horse and said, "I'll
loan you mine, then we have 18. The eldest got 9, the middle one got 6 and the youngest
got 2, for a total of 17, and the lawyer took back his horse. What happened? Find other
numbers for which this works.
2. Pick a number. Add 5. Double the result. Subtract 4. Divide by 2. Subtract the
number you started with. Prove that the answer is always 3.
3. Pick a number. Double the number. Add 9. Add your original number.
Divide by 3.Add 4. Subtract your original number. Prove that you get 7.
Invent other problems like this using your skills in algebra.
4. Go to one of these websites: http://www.milaadesign.com/wizardy.html or
http://www.regiftable.com/RegiftingRobinPopUp.html
Do it several times and notice the pattern of the numbers in the answer. Then do the
algebra to explain why it works.
Experiment
Take a cup of boiling water and a thermometer. Record the temperature at the beginning
and every 5 minutes. Make a table and a graph to compare time and temperature.
(Investigators use such a graph to determine the time of death.)
Frogs
Ten frogs sit on a log - 5 green frogs on one side and 5 brown frogs on the other with an
empty seat separating them. They decide to switch places. The only moves permitted are
to jump over one frog of a different color into an empty space or to jump into an adjacent
space. What is the minimum number of moves?
4
Trains
1. A square table in a restaurant seats 4 people. A group of 18 comes to the restaurant so
they want one long table. How many square tables are needed? Draw a picture and make
a data table. Find the expression for the number of tables needed for a group of n people.
2.
a. To make a square requires 4 toothpicks. If we place 2 squares adjacent to each
other we need 7 toothpicks; 3 squares require 10 toothpicks. How many toothpicks are
needed for 4 squares? How many toothpicks are needed for n squares? Make a graph
showing the number of squares on the x-axis and the number of toothpicks on the y-axis.
Discuss the graph.
b. Use other shapes (triangles, pentagons, hexagons) instead of squares and
repeat exercise a.
Knotted Rope
How does the length of a piece of rope change as knots are tied in the rope?
Collect, organize, and represent data as you do the following:
1. Begin with a piece of rope about a metre in length. Measure and record the length of
the rope to the nearest cm.
2. Tie one simple overhand knot in the rope (not too tight as you will need to undo all the
knots at the end of the experiment). Measure and record the new length of the rope.
3. Tie a second, third, fourth and fifth knot in the rope, and measure and record the new
length after each knot is tied. You may want to tie even more knots.
4. Graph the data. Be sure to label the axes.
5. What conclusions can you make from the graph?
6. What is the rate of change in the length of the rope?
7. Write an equation that represents the graph.
5
Rectangle
If 24 tiles are used to form a rectangular array, what are the possible dimensions for the
rectangle? What are the possible dimensions if 36 tiles are used? How about for 45
tiles? Generalize to n tiles, where n is a counting number.
Points and Segments
With 2 points I can make one segment, with 3 points I can make 3 segments. With 4
points I can make 5 segments. Find the number of points made by joining n points.
If everyone in our class of 60 learners shook hands with everyone, how many handshakes
would be made made?
Figures and intersections
1. Two lines can intersect in at most 1 points, 3 lines in 3 points, and so on. Find the
relation between the number of lines and the maximum number of intersections.
2. Two circles can intersect in at most 2 points, 3 lines in 6 places, and so on. Find the
relation between the number of circles and the maximum number of intersections.
3. Use m lines and n circles and find the total number of intersections.
Figures and regions
1. Draw a rectangle on a piece of paper. When 2 intersecting lines are drawn, we get 4
regions. When 3 lines intersect we can get at most 6 regions. Find the relation between
the number of lines and the number of regions.
2. Draw a circle. Take 2 points on the circle and connect them with a segment to create 2
regions. With 3 points we get 4 regions and with 4 points we get 8 regions. Find a
relation between the number of points and the number of regions. Test your
generalization for 6 points. Watch out! Surprise!
The wrong way
1. If I write the fraction 19/95 and I cross off the 9s, I get the correct answer, 1/5. If I
start with 16/64 and cross of the 6's, I get the correct answer, 1/4. Find others and then
explain. Find 3-digit numbers.
2. If I write 26/39 and then add the digits in the numerator and the digits in the
denominator, I get 8/12, which is another name for 2/3, the correct answer. Find others
and then explain. Find 3-digit numbers.
3. I can divide 7 into 28 and get 13. Find other examples of this foolishness.
Integers
2
2
2
1. If 3,4 and 5 are the sides of a right triangle, then 3 + 4 = 5 . 3, 4 and 5 are integers.
Find other triplets of integers that satisfy the Pythagorean Theorem and see if you can
find any patterns. Find as many triangles as you can with integer sides and a simple linear
relation between the angles.
2. The product of 3 and 6 divided by the sum of 3 and 6 is 2, an integer. Find other
triplets of integers for which this is true and then see if you can find any patterns.
3. The following 4 polynomials are all factorable over the integers:
x2 + 5 x + 6 , x2 + 5 x − 6 , x2 − 5 x + 6 , x2 − 5 x − 6
Find other examples.
4. What integers can be represented as the difference of two squares?
Rectangular Floor
The rectangular floor in this figure is 6 tiles long and 4 tiles
high. A diagonal of the floor crosses 8 tiles (that is, it divides 8
tiles into two parts; the diagonal does not cross squares A and
A
B). Explore this problem for other size floors and see what
B
insights you can generate regarding the number of tiles that are
crossed by a diagonal.
6
Locker Problem
A school has 1500 lockers numbered sequentially from 1 to 1500. As the students
entered school one morning they did the following:
The first student opened every locker.
The second student closed every even-numbered locker.
The third student changed every locker that was a multiple of 3. That is, if the locker was
open she closed it and if the locker was closed she opened it.
The fourth student changed every locker that was a multiple of 4, the fifth student
changed every locker that was a multiple of 5, and so forth, until 1500 students had
entered the building.
After all 1500 students had entered, which doors were open and which were closed?
Cutting
1. What is the maximum number of different pieces, not necessarily the same size, into
which a pizza can be cut with 5 straight cuts?
2. What is the minimum number of pieces, not necessarily the same size, into which 10
identical candy bars must be cut so that the 10 bars can be equally divided among 6
people?
3. Draw the letter "S". If we draw 1 vertical line through it we get 4 pieces. If we draw 2
vertical lines through it we get 7 pieces. Find a relation between the number of vertical
lines and the number of pieces.
4. L-Shape
1. If a square is removed from the corner of another square, the resulting figure is called
an L-shape. If 24 tiles are arranged into an L-shape, describe the possible L-shapes (if
any) that could be formed. Describe the possible L-shapes that can be formed with 36
tiles as well as with 45 tiles.
2. Use grid paper for the following:
a. Cut out an L-shape, dissect it and reassemble it into a rectangle. What is the
minimum number of dissection cuts necessary? Describe a method of dissection, using a
minimum number of cuts that will work to change any L-shape into a rectangle.
b. Cut out a rectangle, dissect it and reassemble it into an L-shape. What is the
minimum number of dissection cuts necessary? Describe a method of dissection, using a
minimum number of cuts, that will work to change any rectangle into an L-shape.
c. If the edges of the L-shapes and rectangles coincide with grid lines, under what
circumstances is it possible to make the dissections so that the cuts also coincide with
grid lines?
Polyominoes
Polyominoes are shapes made by connecting certain numbers of equal-sized squares
together. How many different ones can be made from 2 squares? from 3, from 4, from
5(Pentominoes)? Draw on grid paper all the possible Polyominoes that can be made
from 1, 2, 3, 4, and 6 squares. Which pentominoes can be folded along the grid lines to
make an open box? Investigate the shapes that polynominoes can make.
Monty Hall
Behind one of three doors there is a prize. You pick door #1, he shows you that the prize
wasn't behind door #2 and then gives you the choice of switching to door #3 or staying
with #1, what should you do? Why should you switch? Make an exhibit and run trials to
"show'' this is so. Find the mathematical reason for the switch.
Figurate numbers
Investigate figurate numbers. What is the 100th triangular number? What is the 100th
square number? What is the 100th pentagonal number? Describe how you arrived at your
answers. Is 200 a pentagonal number? Generalize as you see fit.
7
Printing Books
Suppose you have 3 options for printing a 325-page book and you need 2250 books:
1. 9.50 rand per book plus initial typesetting cost of 5000 rand.
2. The Copy Place: .05 rand per page plus 2 rand per book for binding.
3. Copy Rent: .035 rand per page plus 3000 rand for rental of machine.
If your budget is 35 000 rand, what is the best option? Make tables and graphs and write
an argument supporting your choice of the 3 options.
8
Number Patterns and Patterns from Geometry
Double stairs
STEP 1
STEP 2
STEP 3
STEP 4
Add a block
STEP 1
STEP 2
STEP 3
STEP 4
Arches
STEP 1
STEP 2
STEP 3
“W”
STEP 2
STEP 1
STEP 3
Toothpicks
STEP 1
STEP 2
STEP 3
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Note: Learners may only be able to get a recursive formula t n = tn−1 + n , not an explicit
n ( n + 1)
formula, t n =
.
2
Squares and Triangles
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14
8
4
1
2
5
6
7
9
10
3
10
11
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Counting Lattice Points
The following problems challenge students to organize their work, recognize patterns,
make generalizations, and apply a variety of algebraic concepts and formulas, all within
the context of lattice points on a grid. This activity was inspired by the classic “painted
cube” problem in which a painted cube is cut into smaller identical cubes and the
problem is to determine the number of cubes with 0, 1, 2 and 3 faces painted. When
students have completed these sheets, they should be invited to try the “painted cube”
problem.
The questions on SHEET 1 are designed to lead students to generalize about the lattice
points on a square and a rectangle and to observe the location of the lattice points with a
particular property (the intersection of 2, 3, or 4 segments). Some students may need
more examples before generalizing, so dot paper should be available.
Fig. 1
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SHEET 1
Consider the 3 × 3 grid shown in figure 1. The points of intersection of segments are
called lattice points.
1.
2.
3.
4.
5.
A. How many lattice points are on the 3 × 3 grid?
B. How many lattice points are formed by the intersection of:
a.) 2 segments?
b.) 3 segments?
c.) 4 segments?
C. Show that the sum of your answers to B matches your answer to A.
Consider an n × n grid.
A. How many lattice points are on the n × n grid?
B. How many lattice points are formed by the intersection of:
a.) 2 segments?
b.) 3 segments?
c.) 4 segments?
C. Show that the sum of your answers to B matches your answer to A.
Consider the 4 × 6 grid shown in figure 2.
A. How many lattice points are on the 4 × 6 grid?
B. How many lattice points are formed by the intersection of:
a.) 2 segments?
b.) 3 segments?
c.) 4 segments?
C. Show that the sum of your answers to B matches your answer to A.
Consider an m × n grid.
A. How many lattice points are on the m × n grid?
B. How many lattice points are formed by the intersection of:
a.) 2 segments?
b.) 3 segments?
c.) 4 segments?
C. Show that the sum of your answers to B matches your answer to A.
Where do you find the lattice points that are the intersection of:
a.) 2 segments?
b.) 3 segments?
c.) 4 segments?
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Answers
1. A. 16; B. a.) 4, b.) 8, c.) 4; C. 4 + 8 + 4 = 16
2
2
2
2. A. ( n + 1) ; B. a.) 4, b.) 4(n – 1), c.) ( n −1) ; C. 4 + 4(n-1) + ( n −1)
2
= 4 + 4n – 4 + n - 2n + 1
2
= n + 2n + 1
2
= ( n + 1)
3. A. 35; B. a.) 4, b.) 2(3) + 2(5) = 16, c.) 15; C. 4 + 16 + 15 = 35
4. A.
C.
=
=
=
( m + 1) ( n + 1) ; B. a.) 4, b.) 2(m – 1) + 2(n – 1), c.) (m – 1)(n – 1);
4 + 2(m – 1) + 2(n – 1) + (m – 1)(n – 1)
2
4 + 2m – 2 + 2n – 2 + m – m – n + 1
m2 + m + n + 1
( m + 1) ( n + 1)
5. a.) on each of the 4 corners; b.) on the edges; c.) on the inside
Painted Cubes (Like the Lattice Points problems)
1. The outside of a 3x3x3 cube is painted and then cut into 27 unit cubes. How many
unit cubes have paint on only:
a.) 3 faces
b.) 2 faces
c.) 1 face
d.) 0 faces
2. Consider painting a 4x4x4 cube and then cutting it into 64 unit cubes. Answer
questions a-d above.
3. Make a table and determine the answers to a-d above for a painted and cut-up cube
nxnxn.
13
Hundreds chart
A hundreds chart is an array of the numbers from 1 to 100 set out in rows of 10. What
patterns do you see in the rows and columns? What causes these? What is different if
you go from 0 to 99?
Look along the diagonals. What patterns do you see? What causes these patterns?
Choose a square of 9 numbers anywhere in the table. Calculate the sum of any three
numbers in a line that also contains the center number. What do you notice? Why does
this happen? What is the sum of this set of 9 numbers? Can you find a rule for finding
the sum quickly? Why does this happen?
Pick any three numbers in a row. Multiply the middle one times itself; multiply the two
end numbers together. What do you notice? Check this for a few more rows triples.
Write a general rule.
Pick three numbers in a column. Repeat the process from #4.
What is the sum of the consecutive integers from 1 to 5? Draw a picture of this. From 1 to
10? From 51 to 100? From 1 to 100? Is there a simple rule for finding these sums?
What other questions can you find and answer?
Color the prime numbers. Make a "six array", rows of 6, not 10 and go to 102. Color the
primes. What do you notice?
Use a calendar or other charts instead of a hundreds chart and explore the same questions
as above.
Powers of 2
Make a list of the powers of two. Then make a list of the last (units) digit of each
number. What pattern do you notice? Explain.
Tiling Squares
(from Mathematics Teaching in the Middle School, Nov. 2005)
Observe the 3 patterns shown.
Pattern 1
Pattern 2
Pattern 3
1. Find a way to count the number of white tiles. Use your method to generalize a
pattern for the number of white tiles in the nth pattern.
2. Now find another way to count the number of white tiles. Use this new method to
generalize a pattern for the number of white tiles in the nth pattern.
Are the two expressions equivalent? Explain.
3. Find a third way and a fourth way to count and show that the expressions are
equivalent.
14
Tiles around a garden
Observe the tiled walk around the square gardens.
Garden 1
Garden 2
Garden 3
1. Find a way to count the number of white tiles. Use your method to generalize a
pattern for the number of white tiles in the nth garden.
2. Now find another way to count the number of white tiles. Use this new method to
generalize a pattern for the number of white tiles in the nth pattern.
Are the two expressions equivalent? Explain.
Hawaiian Vintage Chocolates
Big Island Vintage Hawaiian Chocolates are arranged in boxes so that a square caramel is
placed in an array of four round chocolate creams as shown below. The dimensions
above the box tells you how many rows and how many columns of chocolate creams
come in a box.
Develop a method to find the number of caramels in a box if you know its dimensions.
Use words, diagrams and expressions to justify your method.
2x 2
2x 4
3x 5
15
Mirrored tiles
The tile store makes square, mirrored tiles, 10 cm on a side. Three different kinds of tiles
are available: tiles with no bevels, tiles with one beveled side or tiles with two adjacent
sides beveled, as pictured:
Tiles are put together to make a rectangular or square mirror with a beveled edge around
the perimeter. Here are two examples:
1. You want to make a mirror that is 40 cm on each side. How many of each kind should
you buy?
2. What if you wanted a mirror 50 cm on each side or 60 cm on each side? How many
tiles of each kind should you buy?
3. Develop a general formula for the number of each kind of tile needed to make a square
mirror if you know the length of a side.
4. How many of each kind of tile are needed for a rectangle with width m and length n?
Checkerboard numbers
The first four checkerboard numbers are 1, 5, 13, and 25. Tile representations are shown
below. What is the 50th checkerboard number? What is the nth checkerboard number?
16
Placemat
Here is a placemat made with 16 white circles in the center and 20 black circles around
the edge. Investigate how many black and white circles are needed to make different
squares and rectangles. Can you find placemats made with equal numbers of black and
white circles? Can you find placemats that need twice as many whites as black? Three
times as many?
Dots
Describe in words how the pattern is changing. Write a rule for describing the nth step.
step 1
step 2
step 3
Toothpicks
The shapes shown below are made with toothpicks. Find rules that relate the shape
number to the number of toothpicks, to the area, and to the perimeter.
Shape 1
Shape 2
Shape 3
Shape 4
Fractals
Step 1
Step 2
Step 3
If the area of the triangle in step 1 is 1, at each step, what is the area of the shaded part?
If the perimeter of the triangle in step 1 is 3, at each step, what is the perimeter of the
shaded part? Generalize for n steps and explain how the answers change as you move
from one step to the next.
17
Additional Extend and Predict Patterns:
18
POWERS
How High?
Take a piece of paper and cut it in half. Place the two halves atop one another. Cut this
stack of 2 papers and cut it in half. Make one stack from the two parts. You should have
4 pieces of paper. Cut this stack in half and again make one stack. Continue this process,
recording the height of the new stack after each cut. Now consider doing this until you
have made 50 cuts (Do not actually do it). Estimate how high the stack would be. Then
do the math and determine how high the stack would be.
Money
You have a choice: Collect 1 million rand at the end of June, or be paid 1 cent on June 1,
2 cents on June 2, 4 cents on June 3, 8 cents on June 4, and so on, doubling the previous
days pay each day until the final payment is made on June 30. Under which plan will
you make more money? Explain using tables, finding a formula and making a graph to
show both plans.
Folding
Take a piece of paper and fold it in half as many times as possible. After one fold there
will be two regions. How many regions will there be after three folds? Four folds? If we
could continue, how many regions will n folds produce? If the area if the original sheet is
one square unit, then after one fold there will be 2 regions, each with an area of 1/2
square units. Make a table to compare the number of fold, the number of regions and the
area of the smallest region. Make a graph to compare the number of folds with the
number of regions. Make another graph to compare the number of folds with the area of
the smallest region.
Chessboard.
The inventor of chess reportedly asked for a reward from his king for his invention. He
asked that the king put one grain of wheat on the first square, 2 grains on the second
square, 4 on the third, and so on, doubling the previous amount each time. The
chessboard has 64 unit squares. How much wheat would the king need?
Tower of Hanoi
Do you know the game called the Tower of Hanoi? Model the game. What is the least
number of moves for a 3-tower game? A 4-tower game? Can you generalize?
Cookie Monster
The cookie monster sneaks into the kitchen and eats half a cookie; on the second day he
comes in and eats half of what remains of the cookie from the first day; on the third day
he comes in and eats half of what remains from the second day. If the cookie monster
continues this process for 7 days, how much of the cookie has he eaten? How much is
left? If the process continues, will he ever eat the entire cookie? Make a graph with day
on the horizontal axis and amount of cookie eaten on the vertical axis.
Subsets
1. List all the subsets of a set that contains 3 elements.
2. List all the subsets of a set that contains 4 elements.
3. List all the subsets of a set that contains n elements.
19
Fibonacci
1. In how many ways can you walk up a stairway if you can take either 1 or 2 steps at a
time?
2. Suppose you have a large supply of 1-cent and 2-cent Australian coins. How many
ways can you lay out each amount from 0 to 15 cents? (Count 1-1-2 as different from 12-1 and 2-1-1)
3. You have 2 kinds of stepping-stones, 1x1 and 1x2. Determine in how many ways can
you make a walkway (of different lengths) that is exactly 2 units wide.
4. Fact: Men do not like to use a urinal if the urinal next to them is also in use. We
need our space. So, how many different arrangements are possible in a row of n urinals?
What if you have a group of really shy men? They need no one within 2 urinals of them.
Now, how many different arrangements are possible? (from Bret Taylor )
5. Find out all you can about the Fibonacci Numbers. In particular, where do they arise
in nature? (pinecone, pineapples, petals and leaves on plants.)
Getting There
B
A
If you can move only right or up, find the number of paths from A to B.
Pascal’s Triangle
1. How many odd numbers are in the 100th row of Pascal’s triangle?
2. How many entries in the 100th row of Pascal’s triangle are divisible by 3? By 5?
3. When you divide a number by 2, the remainder is 0 or 1. Color the entries in Pascal’s
triangle according to this remainder. You get a beautiful visual pattern. Can you explain
it? Can you generate the pattern on a computer?
4. What about the patterns you get when you divide by other numbers? When you divide
a number by 3, the remainder is 0, 1, or 2. Divide the entries in Pascal’s triangle by 3 and
color them according to their remainder. Can you explain your picture?
Number Patterns
1. Phone trees.
a. Suppose every person makes 2 phone calls. Diagram and identify any patterns.
Suppose each call takes 1 minute. Make a diagram to show the number of people as a
function of time. Find an expression for the number of people (n) called at any minute.
Make a graph with time on the horizontal axis and the number of people who are called at
a given minute on the vertical axis. Find an expression for the total number of people
called after t minutes.
b. Suppose every person makes 3 phone calls. Diagram and identify any patterns.
Suppose each call takes 1 minute. Make a diagram to show the number of people as a
function of time. Make a graph of this relation.
c. Suppose that every person makes a phone call every minute. Make a diagram
to show the number of people as a function of time. Diagram and identify any patterns.
20
Let me count the ways
Holton has 19 toothpicks. How many ways can he arrange them into three piles so that
each pile contains an odd number of toothpicks? For what other numbers can you make 3
piles so that each pile contains an odd number of toothpicks? Is there a pattern to the
number of ways they can be arranged?
Digits
1.
a. Use the digits 1-9 to make two 3-digit numbers with a sum of 1332.
b. Use the digits 1-9 to make three 2-digit numbers so that the sum of 2
of them is the third number
c. Use the digits 1-9 to make three 2-digit numbers so that the difference
of 2 of them is the third number
d. Use the digits 1-9 to make three 2-digit numbers so that the sum of 2
of them is the third number AND as large as possible.
e. Use the digits 1-9 to make three 2-digit numbers so that the product
of 2 of them is the third number
2. Use the digits 0-5 to make two 3-digit numbers with the smallest difference.
3. Write down a number, find the sum of the squares of the digit to make a new number,
find the sum of the squares of the digits of the new number. Continue this process until
you reach a stopping point. Do this for the first 20 numbers. What do you observe? Try
using cubes instead of squares and see what happens.
4.
a. Write down a 2-digit number with the first digit larger than the
second. Reverse the digits for a second 2-digit number. Subtract
the smaller from the larger. Repeat several times with different
starting numbers. What do you notice about the results of the
subtraction? How are they related to the digits of the starting
number? Prove this with algebra.
b. Find all the 2-digit numbers that give a sum of 154 when the original
number and the number obtained by reversing its digits are added.
5. Write any three digits, then repeat those three digits to give a 6-digit number in the
form abc,abc. Divide your number by 7 and get a result. Is there a remainder? Divide
the result by 11: again was there a remainder? Take this second result and divide it by 13
and note the final result. Repeat for several other beginning three digits. Explain with
algebra.
6. Write the digits 1-8 in the squares so that no two consecutive numbers are next to each
other. Is there more than one solution?
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Other Topics
PROBLEMS 2012
Nice Math Problems and some short questions to motivate
1. If the three vertices A, B, and C in the three adjacent squares shown in the figure are
collinear, find the value of x. (DeLoyd Stertz, University of Wisconsin, Eau Claire, WI
54701.)
C
B
A
4
7
x
2. Write the numbers from 1 to 15 on a number line so that the sum of adjacent pairs is
always a prime number.
3. Which of the following cannot be the difference of two prime numbers? (Explain):
1, 2, 7, 8, 10
4. The ratio of a to b is 3/5 and the ratio of b to c is 10/13. Find the ratio of a to c.
5. The ratio of birds to dogs is 3:2. The ratio of cats to birds is 4:7. There are 87
animals. How many of each?
6. At a math contest 10 problems were given. Each participant received 5 points for
each correctly solved problem and lost 3 points for each incorrectly solved problem.
How many problems were correctly solved by a participant who received a final score of
34 points? of 10 points? What scores are possible?
7. How many cubes, each with a surface area of 54 square centimeters, are needed to
form 2 cubes each with a surface area of 216 square centimeters?
8. Construct (without measuring) a square with exactly twice the area of the one given.
9. The sum of 10 numbers is what percent of the average of the 10 numbers?
10. A farmer wants to enclose 100 square meters of land using a rectangle. What
dimensions should he make the enclosure to minimize the amount of fencing he has to
use. Make a graph of the possibilities (length, area).
11. Given a square with side s what should be cut off each corner to make it a regular
octagon?
12. Insert parentheses to make the following a true statement: 5 - 2 x 1 + 4 ÷ 6 = 5.
13. The perimeter of an isosceles right triangle is 2m. Find the area in terms of m.
14. A square inside a circle has an area that is half the area of the circle. Find the length
of the side of the square.
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15. Take two 8 by 10 sheets of paper. Using the first sheet join the two shorter sides
with tape and form the shape into a short wide cylinder. Take the second sheet and join
with tape the two longer sides to form a tall narrow cylinder. Which cylinder will have
the greater volume or will the volumes be the same? [From the Arithmetic Teacher,
February, 1992.]
16. Draw any square. Label it ABCD. Join A to the midpoint of BC, B to the midpoint
of CD, C to the midpoint of DA, and D to the midpoint of AB. What shape is formed in
the center? How is its area related to the area of the original? (First make a conjecture,
then test, and then prove.)
A
B
C
D
17. Two candles of equal length are lighted at the same time. One candle takes 9 hours
to burn out and the other takes 6 hours to burn out. After what length of time will the
slower-burning candle be exactly twice as long as the faster burning one? [From the
Arithmetic Teacher, February, 1992.]
18. Two shepherds count their respective flocks. Nathan says to Emi, "If I had one of
your sheep we would both have the same size flock." Emi says, "Yes, and if I had one of
your sheep, my flock would be twice the size of your flock." How many sheep does each
shepherd have in his or her flock?
19. Cut a circle into two parts that have the same area but are not congruent.
20. A trip will be 20,000 km. The driver has 5 tires she wishes to rotate so that each gets
the wear. How many km will each tire accumulate?
21. A tennis tournament has 60 players. How many games are needed to determine a
winner?
22. A university has 6 times as many students as professors. If S is the number of
students and P is the number of professors, write an equation expressing the relationship
between S and P.
23. A bottle of perfume costs 70 rand. The perfume in the bottle is 10 times the value of
the bottle. How much is the bottle worth?
24. A pen has goats and ducks. There are 44 feet and 30 eyes. How many goats are
there?
25. A group of 200 learners took a test and 98% were successful. How many more
successful learners would need to be added to this group in order to have a 99% success
rate?
26. Find two common fractions whose difference is 2/13. Now find two common
fractions that are in lowest terms and that have different denominators, whose difference
is 2/13.
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