Deana Lorenson, Heather Arnesen, Charlie Merher
Problem of the Week
Our Number Sense Project will be incorporated into our classrooms as a problem of the week. We
will post the problem on the board on Monday and throughout the week, students can write up their
solutions and strategies for finding the solution. On Fridays, we will talk about the solutions,
proving correct and incorrect answers, talking about the strategies used to solve them. At this time
we would look at the math behind the problem.
Handshake Problem
Every student shakes another student’s hand. How many handshakes occurred?
Solution
If 28 students:
• 27 + 26 + 25 + …..
• 28 * 27/2
• Pairs
27 + 1 26 + 2 25 + 3….How many pairs of 28? 28* number of pairs
Assessment
You have just moved to a new school and the class you are in has 122 students. The teacher has
decided that a great way for you to get to know people and them to know you, you are going to
shake everyone’s hand in the class. How many handshakes are there going to be? How did you
solve your problem?
Locker Problem:
There are 750 lockers. It is a tradition that on the last day of school, the first student goes and opens
every locker. The second person changes every second locker; the third person changes every third
locker… How many lockers will be open after all 750 students follow this pattern?
Solution
1
2
3
4
5
open
closed
closed
open
closed
6
7
8
9
10
closed
closed
closed
open
closed
11
12
13
14
15
closed
closed
closed
closed
closed
16
17
18
19
20
open
closed
closed
closed
closed
21
22
23
24
25
closed
closed
closed
closed
open
Notice the pattern of closed lockers; 2, 4, 6 …
1-4= +3; 4-9= +5; 9-16= +7
Odd number of factors which means it must be a perfect square. The square root of 750= 27.3
27 lockers will be open.
Assessment
I want to find out how many numbers I need to do the multiples of to find out all the prime numbers
to 2153. I need to check all the multiples of ( 2,3,5…… etc up to what number?).
Flashlight Problem
The Beatles have only 17 minutes left to get to their van so they can make it to their next concert.
To get to their van, they have to cross a narrow bridge that only two of them can be on at a time. To
make matters even worse, it is pitch black out and they only have a single flashlight. Any party of
two who crosses the bridge must have the flashlight with them; The flashlight must be walked
across the bridge: it cannot be thrown.
It takes each of the Beatles a different amount of time to cross the bridge, as noted;
Paul’s Rate 1 minute
George’s Rate 10 minutes
John’s Rate 2 minutes
Ringo’s Rate 5 minutes
Any pair of Beatles who walk across must walk at the slower one’s pace. How can the “Fab-Four”
get across the bridge in 17 minutes in order to make it on time for their concert?
Solution
Paul & John
2 minutes
Paul walks back the flashlight
1 minute
George & Ringo
10 minutes
John walks the flashlight back
2 minutes
John & Paul
2 minutes = 17 minutes
Assessment
While hiking a family (grandma, grandpa, mom, dad, and two school-aged children) come to a
small lake. In the middle of the lake is an island that would be prefect for their picnic. On the shore
is a boat with one paddle. Next to the boat is a sign that says visitors are welcome to use the boat as
long as it gets returned to the same spot when done. The boat, however, is so small that only one
adult and one child or two children can go in it at one time. Is their a way they can use the boat to
get everyone to the island for their picnic? Show how this could be done using words/and or
pictures. ( AIMS Education Foundation. “Puzzle Play” pg 63
Solution hint: Sometimes the children will cross the lake together sometimes alone. And there are
multiple solutions.
1. Both children row across. One stays one rows back.
2. Father rows to island the child on the island rows back to shore.
3. Both children row to island, one stays one rows back to shore.
4. Mom rows to island child on the island rows back to shore.
5. Both children get into boat and row to island. One stays one rows back.
6. Grandma rows to island. Child on island rows back to shore.
7. Both children in boat row to island. One stays one rows back.
8. Grandpa rows to island. Child on island rows back to shore.
9. The child on shore gets in and they both go back to island. YUMM.
Extension pg 64-67 “The frustrated Farmer” answer pg 77
Frustrated Farmer
How can a farmer get a cat, a mouse, and a block of cheese safely across the river in a boat by
taking only one thing across at a time?
Taken from “Puzzle Play” a 2001 AIMS publication, pg. 64-67.
Magic Square 3x3
Hint: You may want to use some type of manipulative. Tiles with the digits written on them or
post it notes will help alleviate the students frustration.
Arrange the numbers 1-9 into the grid so that each row, column and diagonal will equal the same
sum.
Arrange the numbers 0-8 into the grid so that each row, column and diagonal will equal the same
sum.
Arrange the numbers 0-9 (and not using one of the tiles) into the grid so that each row, column and
diagonal will equal the same sum.
Arrange the numbers 2, 4, 6, 8, 10, 12, 14, 16, 18 into the grid so that each row, column and
diagonal will equal the same sum.
Arrange the numbers -9 through -1 into the grid so that each row, column and diagonal will equal
the same sum.
Arrange the numbers -4 through 4 into the grid so that each row, column and diagonal will equal the
same sum.
Arrange the numbers x, x+1, x+2, x+3, x+4, x+5, x+6, x+7, x+8 into the grid so that each row,
column and diagonal will equal the same sum.
Solution
X+1
X+8
X+3
X+ 6
X+4
X+2
X+5
X
X+7
Magic Square 4x4
Arrange the numbers 1-16 into the grid so that each row, column and diagonal will equal the same
sum.
You can do the extensions like those listed in the Magic Square 3x3.
Possible Solutions:
14
12
5
3
11
1
16
6
7
13
4
10
2
8
9
15
1
15
14
4
12
6
7
9
8
10
11
5
13
3
2
16
Assessment Magic Squares 12x12
Break the 12x12 into a total of 16 3x3 grids that would follow the pattern of the 3x3 and 4x4.
Possible Solution
119
126
121
101
108
103
38
45
40
20
27
22
124
122
120
106
104
102
43
41
39
25
23
21
123
118
125
105
100
107
42
37
44
24
19
26
92
99
94
2
9
4
137
144
139
47
54
49
97
95
93
7
5
3
142
140
138
52
50
48
96
91
98
6
1
8
141
136
143
51
46
53
56
63
58
110
117
112
29
36
31
83
90
85
61
59
57
115
113
111
34
32
30
88
86
84
60
55
62
114
109
116
33
28
35
87
82
89
11
18
13
65
72
67
74
81
76
128
135
130
16
14
12
70
68
66
79
77
75
133
131
129
15
10
17
69
64
71
78
73
80
132
127
134
Column Math
Arrange the tiles 1-9 into the format so that no numbers are used more than once.
ABC
+DEF
GHI
Solution: There are several possible solutions. One possible solution is:
127
368
495
ABC + DEF = GHI discussion points: Why can’t 1,2,3 be in the G’s place? Do we need to carry?
Do we carry only once? Can we carry more than once?
•
All solutions sums must equal 18.
•
1, 2, 3, can not be used in the hundreds place in the solution.
•
2 odds and 1 even in answer.
•
Only carry once.
See handout by Don S. Balka “Digit Delight: Problem-solving Activities Using 0 through 9.
How Many Factors?
How many factors are in 1,000,000?
1
10
100
1000
10000
100000
1000000
1
4
9
16
25
36
49
Sideways Math
SUN (136)
+ SUN (136)
SWIM (1072)
elf + elf = fool (721 +721 = 1442)
Yoyo – pop = pop (1010 – 505 = 505)
She * he = true (a=5; h=1; r=7; s=3; t=4; u=2)
Super bonus: straw * to = chairs (a=7; c=3; h=0; i=6; o=4; r=9; s=2; t=1; w=8)
Counterfeit Coin
You are given 9 coins but one of them is counterfeit. The counterfeit coin is heavier than the other
coins. What is the least amount of times must you weigh the coins to guarantee that the coin is
counterfeit?
Solution
Weigh 3 coins on each side of the balance.
If not equal weigh the heavier side coins against each other.
If equal take and weigh the remaining coins against each other.
Extension 12 coins but you don’t know if it is heavy or lighter? (It should be done in 3 steps)
The Four 4’s
Arrange four 4’s in a number sentence using addition, subtraction, multiplication, division, square
roots, factorials etc. (as long as a number, other than a 4 is not used (example squaring))
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1000 Dollar Bill Problem
You have 1000 one dollar bills and 10 envelopes. How can you arrange the money so that you
would be able to give any dollar amount to 10000?
envelope
1
2
3
4
5
6
7
8
9
10
amount
1
2
4
8
16
32
64
128
245
500
If you have 1 dollar you can make change for 1 dollar. Next you need to be able to make change for
$2. With the 1 & 2 you can make change up to $3. So you need to make change for a $4. With
the 1, 2, 4, you can make change up to $7. So now you need an $8, which allows you to make
change up to $15……The sum of the powers of 2 will equal 1 less than the next power of 2.
Break the Code
Deana wrote you a message in secret code. “ I will give you a hint. First I assigned a number to
each letter of the alphabet starting with A=1, B=2, …. Z=26. Then I squared each of the numbers
of the letters A- Z. Then I multiplied each one by a secret number. Finally I subtracted a second
secret number. The two secret numbers I used are factors of 221.”
Decipher the code: 1036 1855 2908 6275 308 2180 9 5183 815
13 and 17 A=1 2*13 – 17 = 9
B= 2 4*13 – 17 = 35
C= 3 9*13 – 17 = 100 etc… I LOVE MATH
Follow up extension of codes:
Charlie wrote you a message in secret code. “I will give you a hint. I first assigned an even number
to each letter of the alphabet starting with A= 2, B=4…. Then I multiplied each of the numbers of
the letters A-Z by a secret number. Then I divided each by 2. Finally I added another secret
number to the answer. The two secret numbers I used are factors of the number 64. The message
below is scrambled, unscramble the letters to make the six words.
Decipher the message: 68, 112, 64, 80, 52, 44, 44, 64, 56, 36, 92, 104, 48, 52, 120, 80, 132, 48, 76,
88, 36, 52, 44, 68, 36, 52,
Used * 4 + 32 should say “I have finally cracked the code”
Water Challenge
You have a 2 gallon and 5 gallon bucket. How can you measure to get exactly 1 gallon left?
Solution
Fill the 5 gallon bucket and pour 2 gallons into the 2 gallon bucket. Pour out the water from the 2
gallon bucket and fill it again from the 5 gallon bucket. You should have 1 gallon remaining in the
5 gallon bucket.
Take the 2 gallons and add to the 5 gallon. Fill the 2 gallon bucket and add to the 5 gallon bucket.
Fill the 2 gallon bucket and add until the 5 gallon bucket if full. One gallon is remaining in the 2
gallon bucket.
Fill the 2 gallon bucket and tilt the 2 gallon until half full????
1-2-3-4 Challenge
Use all four numbers: 1, 2, 3, 4 and any of the four operations ( + - * ÷) to form a number sentence
that equals the following:
1=
2=
3=
4=
5=
6=
7=
8=
9=
10=
11=
12=
An extension would be to add the use of ( ).
Assessment
Need to use the four operations +, – , *, ÷
Using the numbers 1, 2, 3, 4 and any of the four operations. Find an equation that equals 8. Then 9,
10, 11, and 12. You may not use parentheses.
Super Calculator
Amaze your students with your super calculator abilities!!! Have the students pick 3 numbers. The
numbers must be 6 digits, with no repeating digits and they can not have a 9 in the ten thousands
column. You will pick each of your 2, 5 digits numbers so that they match one of the students
numbers add together to make 9’s in each column. Then take the remaining 6 digit number and add
a 2 in the millions column and subtract 2 from the ones column to come up with the sum in a split
second.
Example:
Your number that matches to make 9's
Your number that matches to make 9's
Add a 2 in front and minus 2 in the ones
867530
132469
542907
457092
258637
2258635
You may want to have the students pick their 3 numbers first to help hide the pattern. You could do
this each day of the week and if they have not found the pattern, you can do it so that they pick a
number and then you give them your number that counters it.
Why does this work? Each pair makes 1,000,000 – 1 which makes both pairs make 2,000,000 – 2.
Happy Numbers
Happy numbers are 1 or 2 digit numbers that equal 1 when you take the sum of the tens digit
squared and the ones digit squared. If the sum is not 1, continue this until it gets to 1 or it repeats.
Example
2= 02 + 22=4
4= 02 + 42= 16
16= 12 + 62=37
Continue until you get a sum of 1 or until it repeats.
Solution:
1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100.
Found by Sophie Germain. She is a woman that you may want to do some research on.
Assessment
Remembering back to happy numbers, what were some of the numbers you hit that you knew
would not make a happy number. Did you have to check all the numbers until they stopped?
Did some numbers never end? Were there any patterns you found?
New Bill Design Challenge
You have been commissioned to create a system that would allow people to know what the value
of the bill without looking. You have a machine that will cut off the corners of the bill. You can
cut any amount of the corners of the bills. You will need to find a system for 1, 2, 5, 10, 20, 50,
and 100.
Solution
100 – no cuts
50- cut 1 corner
20- cut both corners on left side
10- cut opposite corners
5- cut both top corners
2- cut 3 corners
1- cut all 4 corners
Assessment: You have made seven new bills. Is it possible to cut one of your bills to make it worth
more in your new system? The bills need to me made so that if I was to cut a piece off it would not
increase the value of the bill.
Mystery Numbers
“Distinguishing Digit Cards” taken from CMP 6th grade Bits and Pieces I, pg. 166-171.
Assessment
____ ____ ____ . ____ ____
Clue 1. The digit in the tens and tenths places are the same, odd digit.
Clue 2. The digit in the ones place is the sum of the digits in the hundreds, tens, tenths, and
hundredths place.
Clue 3. The digit in the hundreds place is 2 times the digit in the tens place.
Clue 4. The digits to the right of the decimal point are consecutive (like 3&4, 7&8)
Clue 5. The digit in the hundreds and hundredths places are the same, even digits.
Clue 6. The digit in the ones place is 5 more than the digit in the tenths place.
The Fascinating Triangle 3 digits per side
Taken from” Just for the fun of it!” a 1999 AIMS publication, pg. 4-10 and 100-105.
There are many variations of numbers that can be used with this puzzle.
Magic Triangles 4 digits per side
Take the digits 0-8 or 1-9 and arrange them into a triangle with 4 digits per side.
What sums can you make on each side?
Possible solutions:
2
9
8
5
4
1 6
7
3 = 17
1
9
3
5
4
8
2
6
7 = 19
7
9
4
1
2
5
8
6
3 = 19
8
9
4
1
3
2
6
7
5 = 20
6
7
1
2
9
5
8
3
4 = 20
8
5
6
1
7
4
9
2
3 = 21
3
4
2
8
6
7
5
1
9 = 21
8
3
4
5
7
2
6
1
9 = 23
Corners of the triangle must add to a multiple of 3 so that the remainder of the 45 can be subtracted
by the sum of the corners is divisible by 3.
See handout Digit Delight: Problem-solving Activities Using 0 through 9 by Don S. Balka.
Magic C, I, T etc.
See handout Digit Delight: Problem-solving Activities Using 0 through 9 by Don S. Balka.
That’s Sum Face
This is a cube that follows the pattern of the magic squares, triangles, etc. Each side must add up to
the same sum.
Taken from” Just for the fun of it!” a 1999 AIMS publication, pg. 105-116.
Biscuit Question: I have four dogs. There names are; Aries, Lucky, Yankee, and Murray. One
day I went out and left the dogs alone at home. When I got home I realized I had forgotten to put
away their treats before I had left. The box was on the floor empty and this is what happened.
Aries ate half the box of treats and three more and was full. Lucky came along and ate half the box
of treats and three more and was full. Yankee ate half the box of treats and three more and was also
full. Finally Murray came along ate half the box of treats and three more. When I picked the box
up off the floor it was empty. How many biscuits did each dog eat? How many treats were in the
box to begin with?
Solution
90 in the box Aries ate 48, 42 in the box. Lucky ate 24, 18 in the box. Yankee ate 12 six in the box.
Murray ate 6 and the box is empty.
Total Count-Ability
How many different answers can you find and justify for the nursery rhyme Going to St. Ives?
As I was going to St. Ives,
I met a man with seven wives,
Every wife had seven sacks,
Every sack had seven cats,
Every cat had seven kits,
Kits, cats, sacks and wives,
How many were going to St. Ives?
Write how many are going to St. Ives and what assumptions you were using to get this number.
Taken from” Just for the fun of it!” a 1999 AIMS publication, pg. 1-3.
Deals on Wheels
What are all of the possible combinations of bicycles, tricycles, and wagons if there are 17 wheels
total?
Taken from” Just for the fun of it!” a 1999 AIMS publication, pg. 29-32
Penny Patterns
How can you arrange five pennies in a five by five grid so that no two pennies are in the same row,
column, or diagonal?
Taken from” Just for the fun of it!” a 1999 AIMS publication, pg. 45-49.
That’s Sum Challenge
What sums from one to 25 can be obtained by adding two, three, four, five or six consecutive
numbers?
Taken from” Just for the fun of it!” a 1999 AIMS publication, pg. 50-57.
Cookie Combos
How many ways can you count the number of cookies in the arrangement?
Taken from” Just for the fun of it!” a 1999 AIMS publication, pg. 12-16.
Calendar Capers
How many patterns can you find on a page from a calendar?
Taken from” Just for the fun of it!” a 1999 AIMS publication, pg. 16-24.
Balance Baffler
Using only a balance, how can you find the canister that is lighter than the others in the fewest tries?
Taken from” Just for the fun of it!” a 1999 AIMS publication, pg. 58-61.
Desert Crossings
What is the maximum number of watermelons you can get to market?
Taken from” Just for the fun of it!” a 1999 AIMS publication, pg. 62-64.
Cube Construction
How many unique arrangements can you make using two to four wooden blocks when at least one
face of each block must touch the face of another block?
Taken from” Just for the fun of it!” a 1999 AIMS publication, pg. 65-69.
Cutting Corners
What are all the different-sized square-bottomed boxes that can be made from a sheet of centimeter
grid paper?
Taken from” Just for the fun of it!” a 1999 AIMS publication, pg. 91-99
Slides and Jumps
What mathematical patterns can you discover in this puzzle?
Taken from” Just for the fun of it!” a 1999 AIMS publication, pg. 117-125.
Toothpick Teasers
How can you make exactly three congruent squares from the arrangement of toothpicks by moving
and/or removing different numbers of toothpicks?
Taken from “Puzzle Play” a 2001 AIMS publication, pg. 2-4.
Toothpick Puzzlers
How can you make different numbers of squares from an arrangement of 12 toothpicks by moving
or removing some of the toothpicks?
Taken from “Puzzle Play” a 2001 AIMS publication, pg. 5-6.
Square Pickings
How can you arrange eight sticks given to form exactly three squares?
Taken from “Puzzle Play” a 2001 AIMS publication, pg. 7-8.
Flipping Fish
How can you make the toothpick fish face a different direction by moving the fewest number of
toothpicks?
Taken from “Puzzle Play” a 2001 AIMS publication, pg. 9-10.
9-Square Toothpick Challenge
How can you change the number of squares by moving or removing a given number of toothpicks
from the original arrangement?
Taken from “Puzzle Play” a 2001 AIMS publication, pg. 11-12.
Penning a Half-Dozen
How can you create six equal-sized shapes using 12 toothpicks without cutting or overlapping any
of the toothpicks?
Taken from “Puzzle Play” a 2001 AIMS publication, pg. 13-14.
A Timely Question
When does 9 + 4 = 1?
Taken from “Puzzle Play” a 2001 AIMS publication, pg. 60-61
Royal Riddle
How can you get each royal couple across the gorge without leaving any prince alone with a
princess who is not his own?
Taken from “Puzzle Play” a 2001 AIMS publication, pg. 68-70.
Trouble in Paradox
How can you explain the paradox of the extra room?
Taken from “Puzzle Play” a 2001 AIMS publication, pg. 73-74
Cab Conundrum
How can you explain the paradox of the missing dollar?
Taken from “Puzzle Play” a 2001 AIMS publication, pg. 75-76
Number Sense and Number Theory
Games from Class
This is a list of games played in class and will be included in our curriculum for the upcoming year.
In-Between
The player on the left board starts on 80 and 20. The player on the right board starts on 4 and 1.
You can move only one bean and must move to an unoccupied square that keeps the sum of the
squares “in-between” your prior sum and the sum of your partners until someone is unable to move.
80
40
8
4
20
10
2
1
Keys:
You want to get to 50 and 30 to trap your partner.
Tic Tac Times
This game helps reinvigorate basic multiplication facts. The game requires the students to work
backward to determine which factors are available that will enable them to with the game. This
same idea can be used to block opponents on the game board. Observing the students playing the
game will informally assess this game.
Pig
This is a great adding and probability game. Student practice their basic addition skills (the goal is
to be the first to 100) and begin to understand the probability of rolling a one or a pair of ones in a
pair of dice. Observing the students playing the game will informally assess this game.
The Sum What Dice Game
This is a basic addition facts game where students must use mental arithmetic to solve the problems.
The strategy in the game is to find the best sum to cover on your digit card from each roll of the
dice. Observing the students playing the game will informally assess this game.
Contig
This game is to help students with utilizing number operations to get a number. The strategy comes
into play when the student must attempt to make a number for points. 2 to 5 people can play the
game. Observing the students playing the game will informally assess this game.
Hang Math
This is a variation on the original hangman and the students must make guesses as to the numbers in
the problem. The strategy of the game is to use what you know about number operations to help
solve the puzzle. Observing the students playing the game will informally assess this game.
What’s My Remainder?
This game is for two players and focuses on the relationship among the dividend, divisor, and the
remainder. The game helps the students make a connection to the type of answer they will get and
is the strategy for getting the most points. The assessment for this game is simulate a game
situation on the test and see if they can determine the type of answer they get when they divide the
numbers.
Nimble Calculator
This game will help the students develop strategies, use patterns as clues, practice addition and
subtraction, and learn calculator operation skills. There are 6 games on the activity sheet from the
class. A discussion of the strategies should occur after that games have been played. The
assessment would include simulations from the game on the test and the students would determine
which player would win.
Calculator Gulf
This is another estimation game that utilizes a calculator. The activity sheet is a gulf course where
students need to estimate a number to fit the equation and they have so many shots on each to make
par. The game has several types of equations with unknowns and can be easily changed to meet the
needs of the class. The assessment for this game would be to take holes (problems) from the game
and put them on the unit test.
Calculator Paths
The name is misleading because although a calculator is used for the several games the real goal is
to practice estimation and mental calculation. The calculator path games cover addition,
subtraction, division, and multiplication. These games will be informally assessed in the class
through observation of students playing the games and attempting to win.
Standards that are integrated into this unit:
7th Grade
1. Represent rational numbers as fractions, mixed numbers, decimals or percents and convert among
various forms as appropriate.
2. Use scientific notation with positive powers of 10, with appropriate treatment of significant
digits, to solve real-world and mathematical problems.
3. Locate and compare positive and negative rational numbers on a number line.
1. Add, subtract, multiply and divide fractions and mixed numbers.
2. Use the inverse relationship between extracting square roots and squaring positive integers to
solve real-world and mathematical problems.
3. Calculate the percentage of increase and decrease of a quantity in real-world and mathematical
problems.
4. Convert among fractions, decimals and percents and use these representations for estimations and
computations in real-world and mathematical problems.
5. Understand and compute positive integer powers of nonnegative integers and express examples as
repeated multiplication such as 3^4 = 3 x 3 x 3 x 3 = 81.
6. Apply the correct order of operations and grouping symbols when using calculators and other
technologies.
7. Know, use and translate calculator notational conventions to mathematical notation.
8. Understand that use of a calculator requires appropriate mathematical reasoning and does not
replace the need for mental computation.
1. Demonstrate, numerically and graphically, an understanding that rate is a measure of change of
one quantity per unit change of another quantity in real-world and mathematical problems.
1. Apply the correct order of operations including addition, subtraction, multiplication, division and
grouping symbols to generate equivalent algebraic expressions.
1. Calculate the radius, diameter, circumference and area of a circle given any one of these.
8th Grade
1. Assess the reasonableness of a solution by comparing the solution to appropriate graphical or
numerical estimates or by recognizing the feasibility of a solution in a given context.
2. Appropriately use examples and counterexamples to make and test conjectures, justify solutions
and explain results.
3. Translate a problem described verbally or by tables, diagrams or graphs, into suitable
mathematical language, solve the problem mathematically and interpret the result in the original
context.
4. Support mathematical results by explaining why the steps in a solution are valid and why a
particular solution method is appropriate.
5. Determine whether or not relevant information is missing from a problem.
6. Use accurately common logical words and phrases such as “and,” “or,” “if … then …,” “unique,”
“only if.”
1. Represent and compare rational and irrational numbers symbolically and on a number line.
2. Use rational and irrational numbers to solve real-world and mathematical problems.
3. Use scientific notation with positive and negative powers of 10, with appropriate treatment of
significant digits, to solve real-world and mathematical problems.
4. Classify numbers as rational or irrational.
1. Use calculator approximations of irrational and rational numbers in multi-step real-world and
mathematical problems.
2. Find integer approximations of square roots of positive integers without a calculator.
3. Multiply and divide expressions involving exponents with a common base.
4. Use the inverse relationship between nth roots and nth powers of rational numbers to solve realworld and mathematical problems.
5. Apply the correct order of operations and grouping symbols when using calculators and other
technologies.
6. Know, use and translate calculator notational conventions to mathematical notation.
7. Understand that use of a calculator requires appropriate mathematical reasoning and does not
replace the need for mental computation.
4. Apply the correct order of operations including addition, subtraction, multiplication, division,
grouping symbols and powers, to simplify and evaluate algebraic expressions.
Dialing Digits Game
Grade Level: Grade 7
Resource: Connected Mathematics Project: Data Around Us pp 23
The Dialing Digits Game is an activity that asks students to think about strategies for playing the
game based on the concept of place value.
To introduce the game have write the number $7,813,684,697.929.84 on the board, which is the
estimated 2005 national debt of the U. S. government in dollars? (An updated debt calculation can
be found daily at the following URL: http://www.brillig.com/debt_clock/)
Ask the students to read the number. Several will try but it will begin a necessary review of place
value for the students. The next important question is to ask about the magnitude of the number.
This is hard for many students to visualize so examining the current population of the U.S. and
dividing that number will yield the debt incurred by each and every citizen of the United States.
After this discussion introduce the game and as a class play a few rounds. Then distribute a spinner
and game cards to each player or team.
Rules for the game:
Materials: Dialing Digits Spinner and Dialing Digits Game Card
Directions:
Players take turns spinning the spinner. After each spin, each player must write the digit spun in
one of the empty blanks for that game. Players should keep their game cards hidden from each
other until the end of the game.
The game proceeds until all the blanks have been filled. The player that has placed the digits to
make the greatest number wins. However, before earning the win, the player must correctly read the
number that has been produced.
Students should play several rounds with different opponents. They should record any strategies
that they find to help them win.
Discussion:
Let students share the strategies they found. Challenge some students to play against the teacher.
Raise the issue of probability of winning the game based on the strategies presented.
Extensions:
Play two rounds of the game, and have each player add his or her numbers for the two rounds. The
player with the greatest sum wins.
Play to rounds of the game, and have each player subtract the number for round 2 from the number
for round 1. The team with the greatest difference wins.
Assessment
This question would appear on the unit test.
Suppose you play a two-digit version of Dialing Digits
a. What is the greatest number you could create?
b. What is the probability of getting this number on two spins?
c. If the result of the first spin is recorded in the first blank, what is the probability of getting a
two-digit number in the forties?
The Mystery Spinner Game
Grade Level 7
Resource: MathScape Chance Encounters pp 37
Students will try to solve mystery clues to create circular spinners. The sets of clues describes the
probabilities of spinning each part in words and numbers. As the students read and analyze clues to
create the spinner, they are exploring relationships among verbal , numerical, and visual
representations of probability.
Rules of the Game
1. Each player gets one clue. Players read the clues aloud to the group. They cannot show
their clues to one another.
2. The group draws one spinner that matches all the players’ clues.
3. The group labels the parts of the spinner with fractions, decimals, or percentages.
4. The group checks to make sure the spinner matches all of the clues.
Mystery Spinner Game Clue Sets
A full version can be found in the reproducible section of the MathScape Chance Encounters
Teachers Edition.
Example:
Clue 1: The five vehicles on this spinner are bike, bus, boat, train, and plane. You are four times as
likely to get a train as a plane.
Clue 2: You have a 1 in 12 probability of getting a plane. The chances of spinning something that
begins with B are 7/12.
Clue 3: The chances of getting a bike are half the chances of getting a bus.
Clue 4: In 60 spins, bus will probably be spun 20 times.
Discussion:
Ask the students what strategy they used to make a spinner that matched all the clues.
Is the first clue the best clue to start your spinner?
How can you prove that your spinner works with all of the clues?
When they have discussed their strategies the game can be concluded with the class working
together on clues, which have a mistake and seeing if the class can find the problem and fix the
clues so the spinner can be built.
Example
Clue 1: Car has the same chances as skateboard. Motorcycle has the same chances as rollerblades.
Clue 2: The chances of winning a skateboard are 1/16.
Clue 3: You have the highest chance of winning a bike.
Clue 4: You have a 50% chance of winning rollerblades.
Assessment
The following question would be included on a unit test after the unit is completed.
Design a spinner to match the following clues. Label the parts with fractions.
•
There are four pizza toppings.
•
You will get pepperoni 25% of the time.
•
You will get pineapple about 30 times in 180 spins.
•
You are twice as likely to get onion as pineapple.
•
You have the same chances of getting mushroom as pepperoni.
Pythagorean Theorem: Using Geoboards or dot paper if you don’t have access to a geoboard.
Create a line that is one unit long. Create a unit two long. Create a line that is four units. What
happens when you connect dots or pegs diagonally? Is the measure still just one unit? Can you
make a length of 2 ? Try to come up with lengths with the square roots of 1-50 ex. 1,
2, 3 , …….50. Are they all possible which ones are or are not possible.
World of pi
Grade Level: 6-8
Resource: Number Sense and Number Theory Class BSU 2005
This game introduces the concept of pi to students who may have never been exposed to irrational
numbers. The game tests their measuring skills against other teams of students in the class. The
strategy of the game is to find a best method for accuracy in measuring the various circles
throughout the class.
Rules of the game
•
•
•
•
•
•
Materials: Pencil, scissor, string, and ruler
The students working as teams will measure three circular objects in the class.
They will record the following information in a table below.
Once the data is collected the will complete the table.
Take the last column and find the average.
Each group will record their measurements for object 1 and their average from the last
column on the board.
Discussion:
Students should compare the answers found by the other groups.
Do we agree with the measurements by the other teams?
What could have happened to cause differences in the measurements?
What team is the closest to the average or pi?
The winners will be the team with the best answer and the students who has the best answer.
Assessment
Students will see a similar problem on the unit test where they will be asked if the answer could
possibly be correct.
Game Board for World of pi
Circumference
Object #1
Object #2
Object #3
Find the average of the last column.
Diameter
Circumference ÷
diameter
Dialing digits game card and spinner
Dialing Digits
Game 1 ____ ____ ____, _____ _____ _____, _____ _____ _____
Game 2 ____ ____ ____, _____ _____ _____, _____ _____ _____
Game 3 ____ ____ ____, _____ _____ _____, _____ _____ _____
Game 4 ____ ____ ____, _____ _____ _____, _____ _____ _____
0
1
2
9
3
8
4
5
7
6
Cover Up Game
Grade Level: 7
Resource: MathScape Chance Encounters
How many spins will it take to cover the game card? This is a question students ask as they begin
the Cover Up Game, played with a circular spinner that has unequal parts and a game card that has
equal parts. Students will learn to distinguish events that have equal probabilities from events that
have unequal probabilities. The game will also reinforce the idea that the parts of circular spinners
with larger areas tend to be spun more often than the parts with smaller ones.
Rules of the Game
1. Play with a partner using a spinner provided. You will also need to make a game Card and an
Extras Table (see example).
Game Card
B
B
R
R
Y
Y
Extras Table
B
B
R
Y
R
B
R
Y
Y
2. Take turns spinning a color. On your game card, cover a box that color with an X. If all boxes of
that color are covered, mark a tally in the extras table.
3. The game ends when every box on you Game Card is covered with an X. The goal is to make an
X in all the boxes with as few spins as possible.
Take the class results and help the students create a class frequency graph to compare the number of
spins it took to finish the game. When analyzing the graph ask the following questions:
What is the range of the number of spins in the class? How are they distributed? Are they clumped
together or spread out?
How does your game results compare with your classmates results? Of which color did you get the
most extras? How does this compare with your classmates?
New Game:
Have students create new game cards to improve their chances of finishing the game in fewer
spins. Have the students make predictions as to how many spins it will take to fill up their game
card.
Students need to use the same spinner and record their results as they did before keeping track of
extras and the number of spins.
Discussion:
Display several of the game cards that emphasize different comparisons.
Did anyone else finish in the same number of spins as you? How do the numbers of colors on their
game cards compare with yours? If their game card is different than yours, why do you think they
finished in the same number spins as you?
How do your results on the new card compare with the old card?
Did anyone do worse on the new card? Why do you think that happened? If you played again how
would you fill out your game card?
Would a game card that had 12 blues give you a good chance of finishing in the fewest spins? How
would you create a game card for 24 boxes, 48 boxes, or 100 boxes?
Assessment:
Suppose you are playing the Cover Up Game with the spinner shown below. Each time the spinner
lands on a color, players make an X in the box of that color on their game cards.
1. Find the probability of getting each color on the front of the spinner?
2. Make a 24-box game card that would give you a good chance of finishing the game in the
fewest possible spins.
3. Make a 24-box game card that would give you a poor chance of finishing the game in the
fewest possible spins?
Red
Green
Blue
Pink
Yellow
The Cover Up Game Spinner
Blue
Red
Blue
Yellow
The Power Up Game
Grade Level: 6 or 7
Resource: MathScape Language of Numbers pp 28
This game gives students practice in evaluating expressions with exponents and helps them to
develop some number sense about the size of different numbers. If you use calculators to play the
game, you will have to show the students how to use the yx button on their calculators.
Rules for the game
1. Each player rolls a number cube four times and records the digit rolled each time. If the
same digit is rolled more than two times, the player rolls again.
2. Each player chooses three of the four digits rolled to fill in the boxes in this arithmetic
expression: (A + B)C.
3. Players then evaluate their expressions. The player whose expression equals the greater
number gets one point.
As a class do three practice rounds and then have the students play the game. Students should play
10 rounds.
Students who understand the game and finish early can play again but under the following rules:
The winner of the round will win additional two points each turn when they make the highest
possible number from the three of the four digits.
Discussion
In order for the whole class to understand the game some important questions need to be asked.
1. What strategy did you use for finding the highest number? Did your strategy work?
2. What tip would you give your brother or sister if they were about to play the game for the
first time?
3. You roll two numbers, L and S: L is larger number, and S is the smaller number. Is it
always true that SL will be larger than LS? Can the two answers ever be the same?
Assessment
Questions that will be included on the unit test:
Arrange each set of numbers to make the largest and the smallest values for each expression.
1. 3, 6, 5, 2
largest (____ + ____)_______
smallest (____ + ____)_______
2. 7, 6, 6, 5
largest (____ + ____)_______
smallest (____ + ____)_______
3. 1, 2, 3, 4
largest (____ + ____)_______
smallest (____ + ____)_______