Squares
Patterns in Sports – Squares and quadratic formulas
Problem 1. Chess and Checker - Squares
a)On the right you see a chessboard[1], 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 checkerboard[2]. 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 square[3]?
f)Calculate:
1² =6² = 3² =7² =
g)
Calculator
You can calculate a square on your calculator too, with this button ,
Or this one on the windows calculator (switch on “wetenschappelijk” in the “beeld” menu).
12 square:
h)Calculate with your calculator:
12²=13²=
32²=63²=
i)Use your calculator for: -7² =
and(-7)²=
Explain this
j)Do questions a t/m h of the next exercise.
Extra (optional):
Problem 2.Geometric Algebra 2D
Open the applet Geometric Algebra 2D
a)Find out how you can draw squares with the applet. Use the figure above if you need help.
b)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?
square
e)Fill in the table, you may use your calculator or the applet:
f)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?
Problem 3.Spotting Numbers
In the figure, you see a pattern of squares.
a)What is the formula for the number of dots in this pattern?
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 version[4] of piles of cans, it is drawn with the applet Spotting Numbers.
b)Which number of a figure on the right belongs[5] to the drawing on the left?
c)Open the applet Spotting Numbers Problems.
Choose the problem triangular numbers 3of level 2.
Answer all the questions. Write down the formula of the last question.
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 shape[6] 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–formula:
name: oblong numbers–formula:
name: triangular numbers–formula:
Extra:
Open the applet Spotting Numbers and draw your own pattern of dots.
Ask a classmate[7] to make the formula with your pattern.
Draw the figures 2 and 3 below and write down the formula:
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 twice[8], a home match and an away match.
1 / 2 / 3 / 4Team 1
Team 2
Team 3
Team 4
a)Fill in the table on the right.
Make up[9] your own results!
b)How many fields are in this table?
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?
j)Does everyone the same formula? Ask three other students.
If someone has a different[10] formula, write it down here.
Check if this formula gives you the same answers!
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?
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?
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: 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)Which shape number is this?
p)What is the formula of a half competition with n participants? Use parenthesis[11] if you need them!
Summary
5² = 5 × 5 = 255 squared is 25
and
25 is the square of 5
For the pattern below, the formula is:
number of dots = n² + 2
Translations
1
[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 (,)