Download Squares and quadratic formulas

Squares
Patterns in Sports – Squares and quadratic formulas
Problem 1.
Chess and Checker - Squares
a)
On the right you see a chessboard1, how many fields
are on the board?
b)
Did you count all the fields? Or did you use another
method?
c)
The second picture is a checkerboard2. How many
fields are on the checkerboard?
d)
Explain how you can calculate this quickly. Without
counting all the fields.
e)
These numbers are all called “square numbers” in
English. Why do you think that these numbers are
called square3?
Square numbers:
(NL: kwadraten)
25 is the square of 5,
5 squared is 25,
because 25 = 5 x 5
because 5 x 5 = 25
(NL: 25 is het kwadraat van 5)
(NL: 5 kwadraat is 25)
Notation:
5² = 25
(5² = 5 x 5 = 25)
f)
Calculate:
1² =
6² =
3² =
7² =
g)
1
Squares
Calculator
You can calculate a square on your calculator too, with this button
,
on the windows calculator (switch on “wetenschappelijk” in the
Or this one
“beeld” menu).
12 square:
h)
Calculate with your calculator:
12²
=
13²
=
32²
=
63²
=
-7²
(-7)²
=
=
i)
Use your calculator for:
and
Explain this
j)
Do questions a t/m h of the next exercise.
2
Squares
Extra (optional):
Problem 2.
Geometric Algebra 2D
Open the applet Geometric Algebra 2D
1. You can change the length
of the arrow by clicking on a
number on the number line.
2. Drag the arrows
onto the worksheet.
3. Connect two arrows
– with the red dots –
to get a square.
a)
b)
4. The area of
this square.
2. Drag
Find out how you can draw squares with the applet. Use the figure above if
you need help.
What is the area (NL: oppervlakte) of a square with arrows of
length 2?
c)
What is the area of a square with arrows of length 4?
d)
What is the area of a square with arrows of length –3?
e)
Fill in the table, you may use your calculator or the applet:
number
square
f)
-6
-5
-4
-3
-2
1
0
1
2
3
4
5
6
When you click on the button with “x” on the bottom of the screen, you can
draw arrows with length x.
What is the area of a square with sides x?
area =
3
Squares
Problem 3.
Spotting Numbers
In the figure, you see a pattern of squares.
a)
n=1
What is the formula for the number of
dots in this pattern?
n=2
n=3
n=4
number of dots =
Piles of cans
In the drawing you see a boy with a pile of cans. The height of the pile is 3, because
there are 3 levels of cans. In the figure on the right you see a simple version4 of piles
of cans, it is drawn with the applet Spotting Numbers.
n=1
n=2
n=3
b)
Which number of a figure on the right belongs5 to the drawing on the left?
c)
Open the applet Spotting Numbers Problems.
Choose the problem triangular numbers 3 of level 2.
Answer all the questions. Write down the formula of the last question.
number of dots =
The formula of question c looks a lot like the formula of question a. We call the
numbers of dots – for these formulas – square numbers.
Square numbers are not the only shape6 numbers we know. In question d we will see
triangular numbers and oblong numbers.
d)
Choose the problems oblong numbers and triangular numbers 1.
Answer all the questions.
Write down the formulas:
name: square numbers
name: oblong numbers
name: triangular numbers
– formula:
– formula:
– formula:
Aantal stippen =
Aantal stippen =
Aantal stippen =
Extra:
Open the applet Spotting Numbers and draw your own pattern of dots.
Ask a classmate7 to make the formula with your pattern.
Draw the figures 2 and 3 below and write down the formula:
formula: _____________________________________
4
Squares
Problem 4.
Sports Day – Quadratic formulas
At sports day at school, there is a hockey competition. There are 4 teams
participating. Each team plays against each other team twice8, a home match and an
away match.
a)
b)
1
Fill in the table on the right.
Make up9 your own results!
2
3
4
Team 1
Team 2
How many fields are in this table?
Team 3
Team 4
c)
Why can you not put any results along the diagonal?
d)
How many fields are along the diagonal in this table?
e)
How many football matches are played at sports day?
f)
How many matches are played, with 8 teams?
And with 100?
Show your calculations!
g)
Do you play sports yourself?
Which?
How many teams are in your competition?
How many matches are played in a season?
h)
Explain in words how you would calculate the number of matches with 128
teams.
i)
What is the formula for the number of matches, when there are n teams?
Matches =
j)
Does everyone the same formula? Ask three other students.
If someone has a different10 formula, write it down here.
Check if this formula gives you the same answers!
5
Squares
Problem 5.
Soccer matches
In the table you see the results in the Premier League. Each result is from one match;
the result in the yellow circle means: Aston Villa – Portsmouth: 3-0
You have already answered the first 3 questions on page 1:
a) What is the result of Arsenal – Middlesbrough?
b) What is the result of Tottenham Hotspur – Everton?
c) How many teams are participating in the Premier League?
d)
What is the formula for the number of matches when n is the number of
teams?
Matches =
e)
How many matches are played in a whole season in the Premier League?
f)
The Holland Casino Eredivisie has 18 participants. How many matches are
played in a season?
g)
Scotland has its own competition, there are 12 clubs participating. How many
matches are played in Scotland?
6
Squares
Fifa world youth championship
This summer the world championships football for players
under 21 years old, are being held in the Netherlands.
At the website: http://www.fifa.com you can find the match
schedule.
h)
How many group matches (first 9 days) are
played?
i)
How many matches are played in group 1?
j)
How many participants are in group 1?
k)
Can you use the formula of question d? Why or why not?
l)
The match schedule of the first round of this tournament is called a half
competition. Explain why.
m) Make a table for the results in group 1 of the Championship, like the table of
Sports Day. It is a half competition, so you will only use half of the table.
Make up your own results!
n)
Make a spotting numbers drawing too.
Hint: you will only use half of the figure of question 4g; use the diagonal to
divide the figure into two halves.
o)
p)
Which shape number is this?
What is the formula of a half competition with n participants? Use
parenthesis11 if you need them!
7
Squares
Summary
5² = 5 × 5 = 25
5 squared is 25
and
25 is the square of 5
For the pattern below, the formula is:
number of dots = n² + 2
n=1
n=2
n=3
n=4
n=5
8
Squares
Translations
1
chessboard
= schaakbord
2
checkerboard
= dambord
3
square
= vierkant (óf kwadraat)
4
version
= versie
5
to belong to
= behoren tot
6
shape
= vorm
7
classmate
= klasgenoot
8
twice
= twee keer
9
to make up
= verzinnen, bedenken
10
different
= verschillend, anders
11
parenthesis
= haakjes (,)
9