JRMF x Variable Algebra Squaring and Subtracting Rectangle 2015

7/23/2019 JRMF x Variable Algebra Squaring and Subtracting Rectangle 2015

http://slidepdf.com/reader/full/jrmf-x-variable-algebra-squaring-and-subtracting-rectangle-2015 1/14

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

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 .

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 .

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 .

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

e c t a n

g l e s

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 .

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 .

1 0 6 7

C h a l l e n g e 6

b i g g e r n u m b e r s

, b u t t h e

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 .

C h a l l e n g e 7

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 ?

y x + y

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

R e c t a

n g l e s

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 ,

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 .

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 ,

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 .

A l g e b r a

c t a n g l e s

C h a l l e n g e

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

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 .

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/)

JRMF x Variable Algebra Squaring and Subtracting Rectangle 2015

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