Transcript My garden is filled with flowers

Math for Smart Kids
Book 1: Multiplication & Division
By Mor Harchol-Balter, Ph.D.
Copyright March 2003
Stars in my sky
My sky is filled with stars.
There are 4 rows, each with 10 stars.
The total number of stars is therefore 4 x 10.
There are 10 columns, each with 4 stars.
The total number of stars is therefore 10 x 4.
How many stars are there? ___________
Flowers in my garden
My garden is filled with flowers:
There are ______ rows, each with _______ flowers.
The total number of flowers is therefore _____ x _____.
There are ______ columns, each with _______ flowers.
The total number of flowers is therefore ______ x _____.
How many flowers are there? ___________
Sharing race cars
I have 18 toy race cars which I want to share:
If there are 3 kids in my class (including me) they can each have ______ cars,
which is the same thing as _______ rows. We write this as: 18 / 3 = 6.
If there are 6 kids in my class (including me) they can each have _____ cars,
which is the same thing as _______ columns. We write this as: 18 / 6 = 3.
If there are 2 kids in my class (including me) they can each have _____ cars,
which is the same thing as _______ columns. We write this as: ___ / ___ = ___.
If there are 9 kids in my class (including me) they can each have _____ cars.
We write this as: ___ / ___ = ___.
Tic-tac-toe for smartys
Play these games: Player 1 uses X’s. Player 2 uses O’s. Players alternate turns. On a turn,
a player places his marker in one of the empty boxes. The goal is to get 3 of your markers
in a row (row may be horizontal, vertical, or diagonal). In later games the goal is 4 in a row.
3-in-a-row wins
Number squares = ____ x ____ = _____
4-in-a-row wins
Number squares = ____ x ____ = _____
3-in-a-row wins
Number squares = ____ x ____ = _____
4-in-a-row wins
Number squares = ____ x ____ = _____
Flowers in my garden
My garden is filled with flowers and some bees:
How many flowers do you see
in the above garden? ______
How many flowers do you see
in the above garden? ______
How many flowers do
you see in the garden
to the left? ______
Flowers in my garden
Write the answers as a multiplication followed by a subtraction:
How many flowers do you see
in the above garden?
____ x ____ - _____ = ____
How many flowers do you see
in the above garden?
____ x ____ - _____ = ____
How many flowers do
you see in the garden
to the left?
_____ x _____ - _____ = ____
Division = Sharing
24 / 4 = _______
24 / 2 = _______
24 / 3 = _______
48 / 3 = _______
24 / 6 = _______
24 / 12 = _______
24 / 8 = _______
Factoring a chocolate bar
I have 24 squares of chocolate. There are so many
possible chocolate bars ...
Number squares = ___ x ____
Number squares = ___ x ____
Number squares = ___ x ____
Number squares = ___ x ____
How many rectangular chocolate bars can I make with 24 squares, where the number
of rows is smaller than the number of columns? _______________
Suppose the number of rows doesn’t have to be smaller than the number of columns,
now how many rectangular chocolate bars can I make? ___________
Factoring a chocolate bar
I have 20 squares of chocolate. How many rectangular chocolate bars can I make where
the number of rows is no larger than the number of columns? _______________
Number squares = ___ x ___
Number squares = ___ x ____
Number squares = ___ x ____
I have 16 squares of chocolate. How many rectangular chocolate bars can I make using
16 squares, where the number of rows is no larger than the number of columns?
_______________ Draw these chocolate bars below:
Prime chocolate bars
A number is called prime if there is only one way to make a
chocolate bar out of that number of squares (assuming number
of rows is no larger than the number of columns).
The number 13 is prime. Here’s the only rectangular chocolate bar with 13 squares:
Circle those numbers below which are prime numbers:
1
8
2
9
3
10
4
11
5
12
6
13
7
14
Theorems and Fallacies
A theorem is a statement that is always true.
A fallacy is a statement which may be true sometimes, but is also
sometimes false. Circle the THEOREMS on this page.
Any number which is
divisible by 2 (with no
remainder) is also
divisible by 4.
A chocolate bar with
2 rows, can always be
redrawn as a chocolate
bar with 4 rows.
Any number which is
divisible by 4 (with no
remainder) is also
divisible by 2.
A chocolate bar with
4 rows, can always be
redrawn as a chocolate
bar with 2 rows.
2-piece outfits
I have 4 shirts in my closet:
I have 3 pairs of shorts in my closet:
How many different outfits can I wear? _________
Suppose I had only 2 shirts in my closet, but still 3 pairs of shorts.
How many different outfits can I wear? _________
Suppose I have 8 shirts in my closet, and 3 pairs of shorts.
How many different outfits can I wear? _________
3-piece outfits
I have 4 shirts in my closet:
I have 3 pairs of shorts in my closet:
I have 2 hats in my closet:
How many different outfits can I wear? _______________
Secret Function 1
I have a secret coding box. When you give it a number on the left, it transforms
the number into a new number on the right. Can you figure out what my coding box is doing?
Look at the first 3 examples to figure out my secret transforming function.
Fill in
the blanks!
3
Secret
function
15
5
Secret
function
25
2
Secret
function
10
10
Secret
function
15
3
Secret
function
20
3
Secret
function
30
Fill in
the blanks!
Secret Function 2
OK, so that was too easy for a smartypants like you. Try this one!
Fill in
the blanks!
3
Secret
function
14
5
Secret
function
22
2
Secret
function
10
10
Secret
function
15
3
Secret
function
18
3
Secret
function
30
Fill in
the blanks!
Secret Function 3
This secret function takes 2 inputs and returns one output.
12
Secret
function
4
40
4
Secret
function
10
12
6
Secret
function
2
Secret
function
15
3
60
3
10
Fill in
the blanks!
15
15
2
Secret
function
2
Secret
function
4
Fill in
the blanks!
Secret Function 4
This secret function takes 2 inputs and returns one output.
12
Secret
function
5
40
4
Secret
function
11
12
6
Secret
function
3
Secret
function
15
3
60
3
10
Fill in
the blanks!
15
15
2
Secret
function
6
Secret
function
4
Fill in
the blanks!
The poison game -- level 1
Rules: You need 2 players. Youngest player chooses who goes first.
Players alternate turns. On each turn, a player is allowed
to eat 1 cookie. Player who eats the poison loses.
Game1:
Game2:
Game3:
Game4:
The poison game -- level 2
Rules: You need 2 players. Youngest player chooses who goes first.
Players alternate turns. On each turn, a player is allowed
to eat 1 or 2 cookies (player decides how many cookies to
eat). Player who eats the poison loses.
Game1:
Game2:
Game3:
Game4:
The poison game -- level 3
Rules: You need 2 players. Youngest player chooses who goes first.
Players alternate turns. On each turn, a player is allowed
to eat 1 or 2 or 3 cookies (player decides how many cookies to
eat). Player who eats the poison loses.
Game1:
Game2:
Game3:
Game4:
Building with blocks
I love to build with blocks. To make a stable tower, you want to make
sure that each block is supported by 2 blocks underneath, one on each side.
How many blocks are in a tower of height 2? ________
How many blocks are in a tower of height 3? ________
How many blocks are in a tower of height 4? ________
How many blocks are in a tower of height 5? ________
How many blocks are in a tower of height 6? ________
A tower of height 100 will have how many more blocks than
a tower of height 99? __________
My allowance
I made a deal with my parents. My allowance is only 1 cent this week,
but every week my allowance doubles.
In 1 week, my allowance will be 2 cents.
In 2 weeks, my allowance will be 4 cents.
In 3 weeks, my allowance will be _________
In 4 weeks, my allowance will be _________
In 5 weeks, my allowance will be _________
In 6 weeks, my allowance will be _________
In 7 weeks, my allowance will be _________
cents.
cents.
cents.
cents.
cents.
Cool, more
than a
buck a week
already!
In how many weeks will my allowance be over $10? ______________
In how many weeks will my allowance be over $100? _____________
Growing younger?
My mom is growing old. She’s already 35 years old,
and I’m only 5. That makes her 7 times my age!
5 years from now, my mom
will be 40, and I will be 10.
My mom will be how many
times my age? __________
10 years from now, my mom will be
how many times my age? ________
25 years from now, my mom will be
how many times my age? ________
Martian counting
Here on Earth we have 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
We make all our numbers out of these 10 digits.
Mars is so far away that they only got to hear about
the first 6 digits: 0, 1, 2, 3, 4, 5.
Martians need to make all their numbers out of these 6 digits.
1 flower
4 flowers
11 flowers
2 flowers
5 flowers
12 flowers
3 flowers
10 flowers
13 flowers
Try counting up to 100 in Martian... How long does it take you?
(Remember there’s no 6, 7, ...)
Earthlings versus Martians
I see
11 flowers
I see
15 flowers
I see
12 flowers
I see
20 flowers
I see
____ flowers
I see
____ flowers
I see
30 flowers
I see
_____flowers
Secret Function 5
What’s this secret function doing?
Fill in
the blanks!
6
Secret
function
10
24
Secret
function
40
25
Secret
function
41
30
Secret
function
15
3
Secret
function
30
36
Secret
function
15
Fill in
the blanks!
What is a square?
A square is a type of rectangle whose sides have the same length.
Sounds easy, huh? Cut out these 4 shapes
and fit them together to make a square.
What is a square number?
A number N is called a perfect square if one can arrange N chocolate squares
into a square shape.
For example, the number 16
is a perfect square, because
16 chocolate pieces can be
arranged in a square.
The number 20 is not a perfect square,
because 20 chocolate pieces can not be
arranged into a square. This is as
close as you get:
The numbers: 1, 4, 9, 16, 25 are all perfect squares.
What are the next couple perfect squares? ______, _____
Instant Lawn
My dad spends a lot of time talking about our neighbor’s lawn. Our neighbor’s lawn
grew out of a truck. The truck driver brought in 70 squares of grass. Each square
of grass measured one yard on each side (that’s called a square yard).
1 yard wide
1 yard long
1 square yard
The driver arranged all the squares into a huge 7 x 10 rectangle. The area
is the space the lawn takes up -- that’s 70 square yards. My dad says the
perimeter of the lawn is the number of yards of fence our neighbor needs
to buy to surround his beautiful lawn.
The perimeter of our neighbor’s
lawn is __________ yards.
The area our neighbor’s lawn
is __________ square yards.
Instant Lawn
The perimeter of this
lawn is __________ yards.
The area this lawn is
_______ square yards.
The perimeter of this
lawn is __________ yards.
The area this lawn is
_______ square yards.
Hint: Think about
multiple rectangles
Notice how the perimeter number seems to always be less than the area number?
Is this always true, or are there some lawns for which the perimeter number
is greater than the area number? ____________________________________
Multiplying in the big leagues
What’s 7 x 13?
7 x 13 =
7 x 10 = 70
7 x 3 = 21
TOTAL = 91
Multiplying in the big leagues
What’s 6 x 14?
6 x 14 =
___ x ___ = ___
___ x ___ = ___
TOTAL = _____
ANSWERS
Stars in my sky
There are 4 rows, each with 10 stars.
The total number of stars is thus 4 x 10.
There are 10 columns, each with 4 stars.
The total number of stars is thus 10 x 4.
How many stars are there?
40
Tic-tac-toe for smartys
3-in-a-row wins
O X X
O
X
O
# Squares =
3x3=9
4-in-a-row wins
# Squares =
5 x 5 =25
# Squares =
4 x 4 =16
4-in-a-row wins
# Squares =
24 / 2 =
24 / 3 =
48 / 3 =
6
12
8
16
24 / 6 =
7 flowers.
The total number of flowers is thus 4 x 7.
There are 7 columns, each with 4 flowers.
The total number of flowers is thus 7 x 4 .
How many flowers are there? 28
There are
4
rows, each with
How many flowers?
17
columns.
If there are 9 kids, each gets
14
2
2 x 12
# squares =
3x8
# squares =
4x6
1 x 24
# rectangular chocolate bars with 24 squares, where the
number of rows is smaller than the number of columns?
4
If number rows doesn’t have to be smaller than number of
columns, then can make
8
chocolate bars.
4 x 5 - 3 = 17
37
Factoring a chocolate bar
# squares =
18 / 2 = 9.
9 cars, which
equals
3
2 cars. 18 / 9 = 2.
Write the answers as a multiplication followed by a subtraction:
How many flowers?
# squares =
3
If there are 2 kids, each gets
Flowers in my garden
How many flowers?
4
24 / 8 =
column.
Flowers in my garden
6 x 6 =36
24 / 12 =
6 cars, which equals 1 row.
If there are 6 kids, each gets 3 cars, which equals 1
If there are 3 kids, each gets
My garden is filled with flowers and some bees:
Division = Sharing
24 / 4 =
I have 18 toy race cars which I want to share:
My garden is filled with flowers:
My sky is filled with stars.
3-in-a-row wins
Sharing race cars
Flowers in my garden
4 x 4 - 2 = 14
4 x 10 - 3 = 37
Factoring a chocolate bar
For 20 squares of chocolate. How many chocolate bars can I
make where the # rows is no more than # columns?
16 squares
# squares =
1 x 20
# squares =
2 x 10
# squares =
4x5
3 chocolate bars
3
Prime chocolate bars
Circle those numbers below which are prime numbers:
1
8
2
9
3
A theorem is a statement that is always true.
Circle the THEOREMS on this page.
Any number which is
divisible by 2 (with no
remainder) is also
divisible by 4.
10
4
Any number which is
divisible by 4 (with no
remainder) is also
divisible by 2.
11
5
How many different outfits?
12
6
A chocolate bar with
4 rows, can always be
redrawn as a chocolate
bar with 2 rows.
A chocolate bar with
2 rows, can always be
redrawn as a chocolate
bar with 4 rows.
13
7
14
Secret Function 1
3-piece outfits
3
Secret
function
15
5
Secret
function
25
2
Secret
function
10
Secret
function
50
10
How many outfits?
4 x 3 x 2
Secret Function 3
12
3
Secret
function
40
4
Secret
function
10
12
6
Secret
function
2
60
3
10
5
8
2
Secret
function
2-piece outfits
Theorems and Fallacies
4
20
Secret
function
2
Secret
function
4
Secret
function 3
divides
top input
by
bottom
input.
3 x 4 = 12
If only 2 shirts, how many outfits?
3 x 2= 6
If only 8 shirts, how many outfits?
3 x 8 = 24
Secret Function 2
Secret
function 1
multiplies
its inputs
by 5.
3
Secret
function
14
5
Secret
function
22
2
Secret
function
10
10
Secret
function
42
4
Secret
function
20
4
Secret
function
18
6
Secret
function
30
7
Secret
function
30
Secret Function 4
12
3
Secret
function
5
40
4
Secret
function
11
12
6
Secret
function
3
60
3
10
2
6
2
Secret
function
21
Secret
function
6
Secret
function
4
Secret
function 4
divides
top input
by
bottom
input, then
adds 1.
Secret
function 2
multiplies
its inputs
by 4
and then
adds 2.
Building with blocks
# blocks in tower of height 2?
3
# blocks in tower of height 3?
6
# blocks in tower of height 4?
10
# blocks in tower of height 5?
15
# blocks in tower of height 6?
21
# blocks in tower of height 100 minus # blocks in tower
of height 99?
100
My allowance
Growing younger?
In 1 week, my allowance will be 2 cents.
In 2 weeks, my allowance will be 4 cents.
5 years from now, my mom will be how
In 3 weeks, my allowance will be 8 cents.
many times my age?
In 4 weeks, my allowance will be 16 cents.
In 5 weeks, my allowance will be 32 cents.
In 6 weeks, my allowance will be 64 cents.
Over a
buck a week
already!
In 7 weeks, my allowance will be 128 cents.
In how many weeks will my allowance be over $10?
In how many weeks will my allowance be over $100?
10
14
Earthlings versus Martians
I see
11 flowers
I see
15 flowers
I see
12 flowers
I see
20 flowers
I see
18
flowers
I see
29
flowers
I see
30 flowers
I see
45 flowers
What is a square number?
A number N is called a perfect square if one can
arrange N chocolate squares into a square shape.
The numbers: 1, 4, 9, 16, 25 are all perfect squares.
What are the next couple perfect squares?
36, 49.
Martian counting
4
10 years from now, my mom will be how
many times my age?
Secret Function 5
6
10
24
Secret
function
40
25
Secret
function
41
30
Secret
function
50
Secret
function
30
Secret
function
100
18
36
11 flowers
2 flowers
5 flowers
12 flowers
3 flowers
10 flowers
13 flowers
1, 2, 3, 4, 5, 10, 11, 12, 13, 14, 15,
20, 21, 22, 23, 24, 25, 30, 31, 32, 33,
34, 35, 40, 41, 42, 43, 44, 45, 50, 51,
52, 53, 54, 55, 100.
2
Secret
function
4 flowers
Counting to 100 in Martian:
3
25 years from now, my mom will be how
many times my age?
1 flower
What is a square?
Secret
function 5
transforms
Earthling
numbers
to the
equivalent
Martian
numbers.
Instant Lawn
The driver arranged all the squares into a huge 7 x 10 rectangle.
The area is the space the lawn takes up -- that’s 70 square yards.
My dad says the perimeter of the lawn is the number of yards of
fence our neighbor needs to buy to surround his beautiful lawn.
Instant Lawn
Perimeter is 42 yards.
Area is
68 sq. yards.
Perimeter of our
neighbor’s lawn is
Perimeter is 58 yards.
yards.
Area of our
neighbor’s lawn is
Area= (8 x 17) - (4 x 9)
7 + 10 + 7 + 10 = 34
7 x 10 = 70
square yards.
Area is
Perimeter can be larger than area:
Example:
100 sq. yards.
Multiplying in the big leagues
What’s 6 x 14?
6 x 14 =
6 x 10 = 60
6 x 4 = 24
TOTAL = 84