JRMF x Variable Algebra Squaring and Subtracting Rectangle 2015
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Transcript of JRMF x Variable Algebra Squaring and Subtracting Rectangle 2015
7/23/2019 JRMF x Variable Algebra Squaring and Subtracting Rectangle 2015
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Roland Brooks Arthur Stone
William TutteCedric Smith
Algebra on RectanglesThe quest for a square that could be tiled withsmaller squares of all different sizes started with
the discovery that it was possible to tile rectangleswith squares.
The square with a 21 in it means that this is a 21by 21 square. This rectangle tiling is great, butimagine if a square could be found that could be
tiled with little squares - all of different sizes.
That’s what the Cambridge University gentlemenon the left were driven to discover in the 1930s.I’m keeping it a secret if they were successful ornot, but during their hunt, they did discovermany rectangular tilings. These will be the basisfor the puzzles and problems at this station.squaring.net is a great place for the keen studentto explore further.
2014 JRMF
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C h a l l e n g e 1
T h i s b i g r e c t a n g l e h a
s b e e n t i l e d b y s q u a r e s . I n t h e m i d d l e t h e r e i s a 3 b y 3 s
q u a r e , a n d
t h e r e i s a l s o a 4 b y 4
a n d l i t t l e 1 b y 1 . W h a t s i z e s a r e t h e o t h e r s q u a r e s ? I f t w o s q u a r e s
h a v e t h e s a m e s i z e , c o l o r t h e m t h e s a m e c o l o r . H o p e f u l l y , s i m i l a r c o l o r s w i l l n o t t o u c h .
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2 0
1 4
1 7
4
1 6
C h a l l e n g e 2
J u s t l i k e t h e l a s t r e c t a n g l e , t h i s o n e h a s b e e n t i l e d
b y s q u a r e s . H o p e f u l l y t h i s t i m e , n o t w o
w i l l b e t h e s a m e s i z e
. I f t w o a r e t h e s a m e s i z e ,
w e ’ l l j u s t g i v e u p a n d g o o n
t o t h e n e x t
c h a l l e n g e .
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C h a l l e n g e 3
T h a t w a s n i c e , b u t w
e r e a l l y w a n t t o t i l e u s i n g d
i f f e r e n t s i z e s o f s q u a r e s . T
r y t o t i l e a
r e c t a n g l e u s i n g a l l d
i f f e r e n t s i z e d s q u a r e s .
T h i s 3 2 b y 3 3 r e c t a n g l e c a n b
e , b u t i t i s
d i f f i c u l t . I f y o u f a i l a t t h i s t a s k , d o n ’ t w o r r y , w e ’ l l f i n d a w a y t o s o l v e i t i n a l a t e r c h a l l e n g e .
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C o v e r t h i s 3 2 x 3 3 r e c t a n g l e u s i n g d i f f e r e
n t s i z e s o f n o n - o v e r l a
p p i n g s q u a r e s .
A l g e b
r a
o n R
e c t a n
g l e s
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1 4
1 9
C h a l l e n g e 4
T h i s b i g r e c t a n g l e i s
n e a r l y a s q u a r e , b u t n o t q u i t e . I t l o o k s l i k e i t h a s b e e n s
u c c e s s f u l l y
t i l e d w i t h e l e v e n s q u a
r e s o f d i f f e r e n t s i z e s i n c l u d i n g a 1 4 b y 1 4 a n d a 1 9 b y 1 9 s
q u a r e .
F i n d t h e s i z e s o f a l l t h
e o t h e r s q u a r e s .
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x
2 5
C h a l l e n g e 5
T h i s i s a n o t h e r r e c t a
n g u l a r t i l i n g , b u t t h i s t i m e w e a r e g i v e n f e w e r c l u e s . U s e t h e 2 5 b y
2 5 s q u a r e t o f i g u r e
o u t t h e d i m e n s i o n s o f a l l t h e o t h e r s q u a r e s - a n d t h e v a
l u e o f x i n
p a r t i c u l a r .
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8
1 0
1 3 8
8
1 0 6 7
7 1 3
x
C h a l l e n g e 6
T h
e s e
a r e
b i g g e r n u m b e r s
, b u t t h e
c h
a l l e n g e r e m a i n s t h e s a m e
- f i n d t h e
d i m e n s i o n s o f a l l s q u a r e s .
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3 3
x y
3 2
C h a l l e n g e 7
H e
r e w e h a v e t w o v a r i a b l e s . A g a i n - f i n d
t h e d i m e n s i o n s o f a l l s q u a r e s .
D o t h e s e
t h r e e p u z z l e s h a v e u n i q u e a n s w
e r s ?
1 4
4
x y
y x + y
3 x
4 x
5 x
6 x
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H e r e w e h a v e t w o v a r i a b l e s . A g a i n - f i n d t h e
d i m e n s i o n s o f a l l s q u a r e s . D o t h
e s e t h r e e
p u z z l e s h a v e u n i q u e a n s w e r s ?
A l g e b r a
o n
R e c t a
n g l e s
3 3
x y
3 2
1 4
4
x y
y x + y
3 x
4 x
5 x
6 x
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x y
x + y
C h a l l e n g e 8
R e m e m b e r b a c k i n c h a l l e n g e 3 , y o u t r i e d t h e v e r y d i f f i c u l t
t a s k o f t i l i n g a r e c t a n g l e w i t h s q u a r e s o f d i f f e r e n t
s i z e s . Y o u w e r e e v e n h e l p e d b y b e i n g g i v e n t h e d i m e n s i o n s o f t h e r e c t a n g l e . I t w a s s t i l l t o u g h ! I n c o n t r a s t ,
t h e f i g u r e s o n t h i s p a g e a r e e a
s y t o c r e a t e . T h e s e a p p e a r t o b e r e c t a n g u l a r t i l i n g s o f s q
u a r e s a n d r e c t a n g l e s ,
b u
t l o o k s c a n b e d e c e i v i n g ! W
e a r e g o i n g t o i m a g i n e t h a t a
l l o f t h e l i t t l e t i l i n g r e c t a n g l e s
a r e a c t u a l l y s q u a r e s .
A s b e f o r e , w e w i l l p u t a n u m b e
r o r v a r i a b l e i n e a c h t o r e p r e s e n t e d g e l e n g t h . I t l o o k s f u n
n y , b u t a f t e r w e s o l v e
t h e s e p u z z l e s w e w i l l b e a b l e t o s t r e t c h e a c h s q u a r e s o t h a t
i t ’ s t h e r i g h t s i z e . H e r e i s a n
e x a m p l e :
T h e s e c o n d l a s t d i a g r a m g i v e s
u s y - 2 x = y + 1 3 x = f r o m
w h i c h w e c o n c l u d e t h a t x = 0 a n d c a n s i m p l i f y
t h e d i a g r a m
b y s t r e t c h i n g a n d
c o n t o r t i n g t h e o t h e r r e c t a n
g l e s s o t h e y b e c o m e t h e s i x s q u a r e s i n t h e l a s t
d i a g r a m . H e r e , a l l t h e s q u a r e s
a r e n o t d i f f e r e n t s i z e s , b u t t w o o f t h e t i l i n g s b e l o w d o g i v e a n s w e r s i n w h i c h a l l
t h e s q u a r e s w i l l b e o f d i f f e r e n t s i z e s . O n e o f t h e s e w i l l b e t h
e s o l u t i o n t o c h a l l e n g e 3 . T h
e o t h e r s e v e n w i l l r u n
i n t o d i f f i c u l t i e s .
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R e m e m b e r i n t h e 1 s t c h a l l e n g
e , y o u t r i e d t h e v e r y d i f f i c u l t
t a s k o f t i l i n g a r e c t a n g l e w i t h s q u a r e s o f d i f f e r e n t
s i z e s . Y o u w e r e e v e n h e l p e d b y b e i n g g i v e n t h e d i m e n s i o n s o f t h e r e c t a n g l e . I t w a s s t i l l t o u g h ! I n c o n t r a s t ,
t h e f i g u r e s o n t h i s p a g e a r e e a
s y t o c r e a t e . T h e s e a p p e a r t o b e r e c t a n g u l a r t i l i n g s o f s q
u a r e s a n d r e c t a n g l e s ,
b u
t l o o k s c a n b e d e c e i v i n g ! W
e a r e g o i n g t o i m a g i n e t h a t a
l l o f t h e l i t t l e t i l i n g r e c t a n g l e s
a r e a c t u a l l y s q u a r e s .
A s b e f o r e , w e w i l l p u t a n u m b e
r o r v a r i a b l e i n e a c h t o r e p r e s e n t e d g e l e n g t h . I t l o o k s f u n
n y , b u t a f t e r w e s o l v e
t h e s e p u z z l e s w e w i l l b e a b l e t o s t r e t c h e a c h s q u a r e s o t h a t
i t ’ s t h e r i g h t s i z e . H e r e i s a n
e x a m p l e :
T h e s e c o n d l a s t d i a g r a m g i v e s
u s y - 2 x = y + 1 3 x = f r o m
w h i c h w e c o n c l u d e t h a t x = 0 a n d c a n s i m p l i f y
t h e d i a g r a m
b y s t r e t c h i n g a n d
c o n t o r t i n g t h e o t h e r r e c t a n
g l e s s o t h e y b e c o m e t h e s i x s q u a r e s i n t h e l a s t
d i a g r a m . H e r e , a l l t h e s q u a r e s
a r e n o t d i f f e r e n t s i z e s , b u t t w o o f t h e t i l i n g s b e l o w d o g i v e a n s w e r s i n w h i c h a l l
t h e s q u a r e s w i l l b e o f d i f f e r e n t s i z e s . O n e o f t h e s e w i l l b e t h
e s o l u t i o n t o c h a l l e n g e 3 . T h
e o t h e r s e v e n w i l l r u n
i n t o d i f f i c u l t i e s .
x y
x + y
A l g e b r a
o n
R e
c t a n g l e s
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C h a l l e n g e
9
A l g e b r a i c a l l y a s s u m e t h e s
e a r e s q u a r e s
a n d s o l v e .
C h a l l e n g e 1 1
T h e t i l i n g o n t h
e l e f t i s v e r y s p e c i a l . I t i s t h
e c u r r e n t r e c o r d h o l d e r
f o r t h e l o w e s t
r a t i o o f L a r g e s t S q u a r e t o S m a l l e s t S q u a r e . C o u l d
s o m e t i l i n g e x i s t w i t h a 2 : 1 r a t i o ? W h a t a b o u t 3 : 1 o r 4 : 1 ? W h a t i s
t h e l o w e s t r a t i o p o s s i b l e ?
M a t h P i c k l e . c o m
w i l l p a y $ 1 0 0 f o r a n
e x a m p l e o f a
4 : 1 t i l i n g o r a p u b l i s h e d p r o o f t h a t i t i s i m p o s s i b l e .
C o n t a c t g o r d @
m a t h p i c k l e . c o m f o r y o u r r e w
a r d o r t o c o m m i s e r a t e
a n d d i s c u s s t h i s d i f f i c u l t p r o b l e m .
C h a l l e n g e 1 0
S o
l v e a l g e b r a i c a l l y . T h i n k a n d
t h e n m a k e a s k e t c h o n
g r a p h p a p e r o f t h e g e o m e t r i c s o l u t i o n t h a t t h i s i m p l i e s .
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All MathPickle puzzle designs, including Algebra on
Rectangles, are guaranteed to engage a wide spectrum of student
abilities while targeting the following Standards for MathematicalPractice:
MP1 Toughen up!
This is problem solving where our students develop grit and
resiliency in the face of nasty, thorny problems. It is the most sought
after skill for our students.
MP3 Work together!
This is collaborative problem solving in which students discuss their
strategies to solve a problem and identify missteps in a failed
solution. MathPickle recommends pairing up students for all itspuzzles.
MP6 Be precise!
This is where our students learn to communicate using precise
terminology. MathPickle encourages students not only to use the
precise terms of others, but to invent and rigorously define their
own terms.
MP7 Be observant!
One of the things that the human brain does very well is identify
pattern. We sometimes do this too well and identify patterns thatdon't really exist.
Algebra on Rectangles targets the following Common Core
State Standards from K-12:
CCSS.MATH.CONTENT.1.OA.B.4
Understand subtraction as an unknown-addend problem. For
example, subtract 10 - 8 by finding the number that makes 10 when
added to 8.
CCSS.MATH.CONTENT.2.NBT.B.5
Fluently add and subtract within 100 using strategies based on place
value, properties of operations, and/or the relationship between
addition and subtraction.
CCSS.MATH.CONTENT.3.NBT.A.2
Fluently add and subtract within 1000 using strategies andalgorithms based on place value, properties of operations, and/or
the relationship between addition and subtraction.
CCSS.MATH.CONTENT.6.EE.A.2
Write, read, and evaluate expressions in which letters stand for
numbers.
CCSS.MATH.CONTENT.7.EE.B.4
Use variables to represent quantities in a real-world or mathematical
problem, and construct simple equations and inequalities to solve
problems by reasoning about the quantities.
CCSS.MATH.CONTENT.8.EE.C.7
Solve linear equations in one variable.
CCSS.MATH.CONTENT.HSA.CED.A.1
Create equations and inequalities in one variable and use them to
solve problems.Include equations arising from linear and quadratic
functions, and simple rational and exponential functions.
CCSS.MATH.CONTENT.HSA.CED.A.2
Create equations in two or more variables to represent relationships
between quantities
Standards for
Mathematical Practice
Common Core
State Standards
(http://www.corestandards.org/Math/Practice/)