Squares and quadratic formulas

Squares
Handleiding bij:
Patterns in Sports  Squares and quadratic formulas
Claire Linders 04-04-05
Dit pakket is een introductie op de kwadratische formule. Het pakket is in het Engels,
enkele moeilijke woorden zijn vertaald. Deze woorden zijn herkenbaar aan de
voetnoten, de vertalingen staan echter niet onderaan op de pagina, maar op de
laatste bladzijde, omdat bepaalde woorden meerdere keren voorkomen.
Aan de hand van verschillende contexten, maken de leerlingen opdrachten. Het is de
bedoeling dat de leerlingen het pakket zelfstandig (bv. in kleine groepjes)
doorwerken. De context uit de eerste opdracht, komt aan het einde weer terug, met
vragen die de leerlingen in het begin nog niet konden beantwoorden.
De tweede opdracht is een introductie op kwadraten, vervolgens wordt er met deze
kwadraten geoefend en ook de rekenmachine komt hier aan bod.
De optionele derde opdracht werkt met de applet “Geometric Algebra 2D”. De relatie
tussen kwadraat en vierkant is duidelijk zichtbaar te maken met deze applet. Ook
wordt hier de eerste aanzet tot formules gegeven. Waarschijnlijk kost deze opdracht
veel tijd, daarom is deze optioneel. Zonder deze opdracht, kunnen de andere
opdrachten ook gemaakt worden.
Opdracht 4 werkt ook met een applet; “Spotting Numbers Problems”. Bij deze applet
stellen de leerlingen, met hulp van een aantal vragen, een kwadratische formule op
bij een patroon van stippen. Deze applet kost minder tijd dan het zusje “Spotting
Numbers” waarbij de leerlingen zelf patronen moeten tekenen. De opgave over
“Spotting Numbers” is dan ook optioneel.
Opdracht 5 geeft de aanzet tot de formule van het figuur van opgave 1. Het
voorbeeld in opdracht 5 is kleiner dan in opgave 1 en daardoor makkelijker om een
formule bij te maken.
Aan het eind van het pakket komen de leerlingen weer terug bij opdracht 1, waar ze
als het goed is nu een formule voor kunnen maken. Er is nog een uitbreiding op
opdracht 1, waarbij de leerlingen het wedstrijdschema van het “FIFA World Youth
Championship 2005” bestuderen.
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Squares
Patterns in Sports – Squares and quadratic formulas
Problem 1.
a)
b)
c)
d)
e)
Chess and Checker - Squares
On the right you see a chessboard1, how many fields
are on the board?
_______________________________________
Did you count all the fields? Or did you use another
method?
_______________________________________
The second picture is a checkerboard2. How many
fields are on the checkerboard?
_______________________________________
Explain how you can calculate this quickly. Without
counting all the fields.
_________________________________________
_________________________________________
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,
because 25 = 5 x 5
(NL: 25 is het kwadraat van 5)
5 squared is 25,
because 5 x 5 = 25
(NL: 5 kwadraat is 25)
Notation:
5² = 25
(5² = 5 x 5 = 25)
f)
Calculate:
1²
=
3²
=
_______________
_______________
6²
7²
=
=
_______________
_______________
1
Squares
Calculator:
You can calculate a square on your calculator too.
You have to use the button that looks like:
12 square:
g)
h)
i)
j)
Calculate with your calculator:
12²
= ________________
13²
= ________________
32²
= ________________
63²
= ________________
Make Numbers and Space, ch. 8, exercises 4 and 5.
Use your calculator for:
-7²
= ________________
and
(-7)² = ________________
Explain this
_____________________________________________________________
_____________________________________________________________
Make Numbers and Space, ch. 8, exercise 13 (a – e).
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.
4. The area of
this square.
2. Drag
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?
________________________________________
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 = _______________________________________________
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Squares
Problem 3.
Spotting Numbers
In the figure, you see a pattern of squares.
a)
n=1
n=2
n=3
n=4
What is the formula for the number of
dots in this pattern?
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
b)
c)
n=2
n=3
Which number of a figure on the right belongs5 to the drawing on the left?
___________________________________________________________
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.
Formula: ____________________________________________________
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: _____________________________
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 in your class have the same formula?
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?
_____________________________________________________________
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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 4? 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 up11 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 brackets if
you need them!
7
Squares
______________________________________________________________
____________________________________________________________
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
9