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1st
2nd
3rd
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Geometry
The
Number System
Expressions and Equations% Eighth Grade Mathematics
Skills Sheet[ Know that there are numbers that are not rational, and
approximate them by rational numbers M . 8 .
K n o w
t h a t
n u m b e r s
t h a t
a r e
n o t
r a t i o n a l
a r e
c a l l e d
i r r a t i o n a l .
U n d e r s t a n d
i n f o r m a l l y
t h a t
e v e r y
n u m b e r
h a s
a
d e c i m a l
e x p a n s i o n ;
f o r
r a t i o n a l
n u m b e r s
s h o w
t h a t
t h e
d e c i m a l
e x p a n s i o n
r e p e a t s
e v e n t u a l l y ,
a n d
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a
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e x p a n s i o n
w h i c h
r e p e a t s
e v e n t u a l l y
i n t o
a
r a t i o n a l
n u m b e r . ( Work with radicals and integer exponentsƒ M . 8 .
P e r f o r m
o p e r a t i o n s
w i t h
n u m b e r s
e x p r e s s e d
i n
s c i e n t i f i c
n o t a t i o n ,
i n c l u d i n g
p r o b l e m s
w h e r e
b o t h
d e c i m a l
a n d
s c i e n t i f i c
n o t a t i o n
a r e
u s e d .
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s c i e n t i f i c
n o t a t i o n
a n d
c h o o s e
u n i t s
o f
a p p r o p r i a t e
s i z e
f o r
m e a s u r e m e n t s
o f
v e r y
l a r g e
o r
v e r y
s m a l l
q u a n t i t i e s
( e . g . ,
u s e
m i l l i m e t e r s
p e r
y e a r
f o r
s e a f l o o r
s p r e a d i n g ) .
I n t e r p r e t
s c i e n t i f i c
n o t a t i o n
t h a t
h a s
b e e n
g e n e r a t e d
b y
t e c h n o l o g y . Z Understand the connections between proportional
relationships, lines, and linear equations
M . 8 .
U s e
s i m i l a r
t r i a n g l e s
t o
e x p l a i n
w h y
t h e
s l o p e
m
i s
t h e
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p o i n t s
o n
a
n o n - v e r t i c a l
l i n e
i n
t h e
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t h e
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a x i s
a t
b . M Analyze and solve
linear equations and pairs of simultaneous linear equationsX M . 8 .
G i v e
e x a m p l e s
o f
l i n e a r
e q u a t i o n s
i n
o n e
v a r i a b l e
w i t h
o n e
s o l u t i o n ,
i n f i n i t e l y
m a n y
s o l u t i o n s ,
o r
n o
s o l u t i o n s .
S h o w
w h i c h
o f
t h e s e
p o s s i b i l i t i e s
i s
t h e
c a s e
b y
s u c c e s s i v e l y
t r a n s f o r m i n g
t h e
g i v e n
e q u a t i o n
i n t o
s i m p l e r
f o r m s ,
u n t i l
a n
e q u i v a l e n t
e q u a t i o n
o f
t h e
f o r m
x
=
a ,
a
=
a ,
o r
a
=
b
r e s u l t s
( w h e r e
a
a n d
b
a r e
d i f f e r e n t
n u m b e r s ) . ½ M . 8 .
S o l v e
l i n e a r
e q u a t i o n s
w i t h
r a t i o n a l
n u m b e r
c o e f f i c i e n t s ,
i n c l u d i n g
e q u a t i o n s
w h o s e
s o l u t i o n s
r e q u i r e
e x p a n d i n g
e x p r e s s i o n s
u s i n g
t h e
d i s t r i b u t i v e
p r o p e r t y
a n d
c o l l e c t i n g
l i k e
t e r m s . @ M . 8 .
A n a l y z e
a n d
s o l v e
p a i r s
o f
s i m u l t a n e o u s
l i n e a r
e q u a t i o n s . Ñ M . 8 .
U n d e r s t a n d
t h a t
s o l u t i o n s
t o
a
s y s t e m
o f
t w o
l i n e a r
e q u a t i o n s
i n
t w o
v a r i a b l e s
c o r r e s p o n d
t o
p o i n t s
o f
i n t e r s e c t i o n
o f
t h e i r
g r a p h s ,
b e c a u s e
p o i n t s
o f
i n t e r s e c t i o n
s a t i s f y
b o t h
e q u a t i o n s
s i m u l t a n e o u s l y .
Functions' Define, evaluate, and
compare functionsà M . 8 .
U n d e r s t a n d
t h a t
a
f u n c t i o n
i s
a
r u l e
t h a t
a s s i g n s
t o
e a c h
i n p u t
e x a c t l y
o n e
o u t p u t .
T h e
g r a p h
o f
a
f u n c t i o n
i s
t h e
s e t
o f
o r d e r e d
p a i r s
c o n s i s t i n g
o f
a n
i n p u t
a n d
t h e
c o r r e s p o n d i n g
o u t p u t . 7 Use
functions to model relationships between quantities
M . 8 .
D e s c r i b e
q u a l i t a t i v e l y
t h e
f u n c t i o n a l
r e l a t i o n s h i p
b e t w e e n
t w o
q u a n t i t i e s
b y
a n a l y z i n g
a
g r a p h
( e . g . ,
w h e r e
t h e
f u n c t i o n
i s
i n c r e a s i n g
o r
d e c r e a s i n g ,
l i n e a r
o r
n o n l i n e a r ) .
S k e t c h
a
g r a p h
t h a t
e x h i b i t s
t h e
q u a l i t a t i v e
f e a t u r e s
o f
a
f u n c t i o n
t h a t
h a s
b e e n
d e s c r i b e d
v e r b a l l y . ` Understand congruence and similarity using physical
models, transparencies, or geometry softwareX M . 8 .
V e r i f y
e x p e r i m e n t a l l y
t h e
p r o p e r t i e s
o f
r o t a t i o n s ,
r e f l e c t i o n s ,
a n d
t r a n s l a t i o n s . W M . 8 .
L i n e s
a r e
t a k e n
t o
l i n e s ,
a n d
l i n e
s e g m e n t s
t o
l i n e
s e g m e n t s
o f
t h e
s a m e
l e n g t h . 6 M . 8 .
A n g l e s
a r e
t a k e n
t o
a n g l e s
o f
t h e
s a m e
m e a s u r e . 2 M . 8 .
P a r a l l e l
l i n e s
a r e
t a k e n
t o
p a r a l l e l
l i n e s .
M . 8 .
U n d e r s t a n d
t h a t
a
t w o d i m e n s i o n a l
f i g u r e
i s
c o n g r u e n t
t o
a n o t h e r
i f
t h e
s e c o n d
c a n
b e
o b t a i n e d
f r o m
t h e
f i r s t
b y
a
s e q u e n c e
o f
r o t a t i o n s ,
r e f l e c t i o n s ,
a n d
t r a n s l a t i o n s ;
g i v e n
t w o
c o n g r u e n t
f i g u r e s ,
d e s c r i b e
a
s e q u e n c e
t h a t
e x h i b i t s
t h e
c o n g r u e n c e
b e t w e e n
t h e m . • M . 8 .
D e s c r i b e
t h e
e f f e c t
o f
d i l a t i o n s ,
t r a n s l a t i o n s ,
r o t a t i o n s ,
a n d
r e f l e c t i o n s
o n
t w o - d i m e n s i o n a l
f i g u r e s
u s i n g
c o o r d i n a t e s . ! M . 8 .
U n d e r s t a n d
t h a t
a
t w o - d i m e n s i o n a l
f i g u r e
i s
s i m i l a r
t o
a n o t h e r
i f
t h e
s e c o n d
c a n
b e
o b t a i n e d
f r o m
t h e
f i r s t
b y
a
s e q u e n c e
o f
r o t a t i o n s ,
r e f l e c t i o n s ,
t r a n s l a t i o n s ,
a n d
d i l a t i o n s ;
g i v e n
t w o
s i m i l a r
t w o d i m e n s i o n a l
f i g u r e s ,
d e s c r i b e
a
s e q u e n c e
t h a t
e x h i b i t s
t h e
s i m i l a r i t y
b e t w e e n
t h e m . , Understand and apply
the Pythagorean TheoremC M . 8 .
E x p l a i n
a
p r o o f
o f
t h e
P y t h a g o r e a n
T h e o r e m
a n d
i t s
c o n v e r s e . ž M . 8 .
A p p l y
t h e
P y t h a g o r e a n
T h e o r e m
t o
d e t e r m i n e
u n k n o w n
s i d e
l e n g t h s
i n
r i g h t
t r i a n g l e s
i n
r e a l - w o r l d
a n d
m a t h e m a t i c a l
p r o b l e m s
i n
t w o
a n d
t h r e e
d i m e n s i o n s . d M . 8 .
A p p l y
t h e
P y t h a g o r e a n
T h e o r e m
t o
f i n d
t h e
d i s t a n c e
b e t w e e n
t w o
p o i n t s
i n
a
c o o r d i n a t e
s y s t e m . \ Solve real-world and mathematical
problems involving volume of cylinders, cones, and spheres… M . 8 .
K n o w
t h e
f o r m u l a s
f o r
t h e
v o l u m e s
o f
c o n e s ,
c y l i n d e r s ,
a n d
s p h e r e s
a n d
u s e
t h e m
t o
s o l v e
r e a l - w o r l d
a n d
m a t h e m a t i c a l
p r o b l e m s . x M . 8 .
U s e
r a t i o < › n a l
a p p r o x i m a t i o n s
o f
i r r a t i o n a l
n u m b e r s
t o
c o m p a r e
t h e
s i z e
o f
i r r a t i o n a l
n u m b e r s ,
l o c a t e
t h e m
a p p r o x i m a t e l y
o n
a
n u m b e r
l i n e
d i a g r a m ,
a n d
e s t i m a t e
t h e
v a l u e
o f
e x p r e s s i o n s
( e . g . ,
À ² ) .
F o r
e x a m p l e ,
b y
t r u n c a t i n g
t h e
d e c i m a l
e x p a n s i o n
o f
"2 ,
s h o w
t h a t
"2
i s
b e t w e e n
1 a n d
2 ,
t h e n
b e t w e e n
1 . 4
a n d
1 . 5 ,
a n d
e x p l a i n
h o w
t o
c o n t i n u e
o n
t o
g e t
b e t t e r
a p p r o x i m a t i o n s . ·
Example with infusion:
Using significant numbers from our faith and/or the Bible, determine its
place in the number system, i.e., real, integer, rational, irrational,
whole, natural.
’ M . 8 .
K n o w
a n d
a p p l y
t h e
p r o p e r t i e s
o f
i n t e g e r
e x p o n e n t s
t o
g e n e r a t e
e q u i v a l e n t
n u m e r i c a l
e x p r e s s i o n s .
F o r
e x a m p l e ,
3 2
x
3
5
=
3
3
=
1 / 3 3
=
1 / 2 7 .
M . 8 .
U s e
s q u a r e
r o o t
a n d
c u b e
r o o t
s y m b o l s
t o
r e p r e s e n t
s o l u t i o n s
t o
e q u a t i o n s
o f
t h e
f o r m
x 2
=
p
a n d
x 3
=
p ,
w h e r e
p
i s
a
p o s i t i v e
r a t i o n a l
n u m b e r .
E v a l u a t e
s q u a r e
r o o t s
o f
s m a l l
p e r f e c t
s q u a r e s
a n d
c u b e
r o o t s
o f
s m a l l
p e r f e c t
c u b e s .
K n o w
t h a t
"2
i s
i r r a t i o n a l .
| M . 8 .
U s e
n u m b e r s
e x p r e s s e d
i n
t h e
f o r m
o f
a
s i n g l e
d i g i t
t i m e s
a n
i n t e g e r
p o w e r
o f
1 0
t o
e s t i m a t e
v e r y
l a r g e
o r
v e r y
s m a l l
q u a n t i t i e s ,
a n d
t o
e x p r e s s
h o w
m a n y
t i m e s
a s
m u c h
o n e
i s
t h a n
t h e
o t h e r .
F o r
e x a m p l e ,
e s t i m a t e
t h e
p o p u l a t i o n
o f
t h e
U n i t e d
S t a t e s
a s
3
x
1 0 8
a n d
t h e
p o p u l a t i o n
o f
t h e
w o r l d
a s
7
x
1 0 9 ,
a n d
d e t e r m i n e
t h a t
t h e
w o r l d
p o p u l a t i o n
i s
m o r e
t h a n
2 0
t i m e s
l a r g e r . •
Example with infusion: Find population of Catholics
in cities, countries, and the world and express them in scientific
notation.
1 M . 8 .
G r a p h
p r o p o r t i o n a l
r e l a t i o n s h i p s ,
i n t e r p r e t i n g
t h e
u n i t
r a t e
a s
t h e
s l o p e
o f
t h e
g r a p h .
C o m p a r e
t w o
d i f f e r e n t
p r o p o r t i o n a l
r e l a t i o n s h i p s
r e p r e s e n t e d
i n
d i f f e r e n t
w a y s .
F o r
e x a m p l e ,
c o m p a r e
a
d i s t a n c e - t i m e
g r a p h
t o
a
d i s t a n c e t i m e
e q u a t i o n
t o
d e t e r m i n e
w h i c h
o f
t w o
m o v i n g
o b j e c t s
h a s
g r e a t e r
s p e e d . •
Example with infusion: Graph the need for Catholic
parishes in relation to Catholic populations and compare them to actual
populations and parishes.
- M . 8 .
S o l v e
l i n e a r
e q u a t i o n s
i n
o n e
v a r i a b l e .
M . 8 .
S o l v e
s y s t e m s
o f
t w o
l i n e a r
e q u a t i o n s
i n
t w o
v a r i a b l e s
a l g e b r a i c a l l y ,
a n d
e s t i m a t e
s o l u t i o n s
b y
g r a p h i n g
t h e
e q u a t i o n s .
S o l v e
s i m p l e
c a s e s
b y
i n s p e c t i o n .
F o r
e x a m p l e ,
3 x
+
2 y
=
5
a n d
3 x
+
2 y
=
6
h a v e
n o
s o l u t i o n
b e c a u s e
3 x
+
2 y
c a n n o t
s i m u l t a n e o u s l y
b e
5
a n d
6 .
Example with infusion: Analyze the following Lenten fish fry
scenario where x = number of children attended at $2.00/child, y = number
of adults attended at $5.00/adult. 2x + 5y = $1,100
and
x + y = 250
attendees. How many adults and children attended the fish
fry?
[ M . 8 .
C o m p a r e
p r o p e r t i e s
o f
t w o
f u n c t i o n s
e a c h
r e p r e s e n t e d
i n
a
d i f f e r e n t
w a y
( a l g e b r a i c a l l y ,
g r a p h i c a l l y ,
n u m e r i c a l l y
i n
t a b l e s ,
o r
b y
v e r b a l
d e s c r i p t i o n s ) .
F o r
e x a m p l e ,
g i v e n
a
l i n e a r
f u n c t i o n
r e p r e s e n t e d
b y
a
t a b l e
o f
v a l u e s
a n d
a
l i n e a r
f u n c t i o n
r e p r e s e n t e d
b y
a n
a l g e b r a i c
e x p r e s s i o n ,
d e t e r m i n e
w h i c h
f u n c t i o n
h a s
t h e
g r e a t e r
r a t e
o f
c h a n g e . c M . 8 .
I n t e r p r e t
t h e
e q u a t i o n
y
=
m x
+
b
a s
d e f i n i n g
a
l i n e a r
f u n c t i o n ,
w h o s e
g r a p h
i s
a
s t r a i g h t
l i n e ;
g i v e
e x a m p l e s
o f
f u n c t i o n s
t h a t
a r e
n o t
l i n e a r .
F o r
e x a m p l e ,
t h e
f u n c t i o n
A
=
s 2
g i v i n g
t h e
a r e a
o f
a
s q u a r e
a s
a
f u n c t i o n
o f
i t s
s i d e
l e n g t h
i s
n o t
l i n e a r
b e c a u s e
i t s
g r a p h
c o n t a i n s
t h e
p o i n t s
( 1 , 1 ) ,
( 2 , 4 )
a n d
( 3 , 9 ) ,
w h i c h
a r e
n o t
o n
a
s t r a i g h t
l i n e . „
Example with
infusion: Compare and contrast the number of priests, deacons, and
Catholic laity over time and show this by graphing.
Ÿ M . 8 .
C o n s t r u c t
a
f u n c t i o n
t o
m o d e l
a
l i n e a r
r e l a t i o n s h i p
b e t w e e n
t w o
q u a n t i t i e s .
D e t e r m i n e
t h e
r a t e
o f
c h a n g e
a n d
i n i t i a l
v a l u e
o f
t h e
f u n c t i o n
f r o m
a
d e s c r i p t i o n
o f
a
r e l a t i o n s h i p
o r
f r o m
t w o
( x ,
y )
v a l u e s ,
i n c l u d i n g
r e a d i n g
t h e s e
f r o m
a
t a b l e
o r
f r o m
a
g r a p h .
I n t e r p r e t
t h e
r a t e
o f
c h a n g e
a n d
i n i t i a l
v a l u e
o f
a
l i n e a r
f u n c t i o n
i n
t e r m s
o f
t h e
s i t u a t i o n
i t
m o d e l s ,
a n d
i n
t e r m s
o f
i t s
g r a p h
o r
a
t a b l e
o f
v a l u e s . Š
Example with infusion: Analyze the rise or decline
of Christianity in our world over the past two millennia as a functional
relationship.
˜ M . 8 .
U s e
i n f o r m a l
a r g u m e n t s
t o
e s t a b l i s h
f a c t s
a b o u t
t h e
a n g l e
s u m
a n d
e x t e r i o r
a n g l e
o f
t r i a n g l e s ,
a b o u t
t h e
a n g l e s
c r e a t e d
w h e n
p a r a l l e l
l i n e s
a r e
c u t
b y
a
t r a n s v e r s a l ,
a n d
t h e
a n g l e - a n g l e
c r i t e r i o n
f o r
s i m i l a r i t y
o f
t r i a n g l e s .
F o r
e x a m p l e ,
a r r a n g e
t h r e e
c o p i e s
o f
t h e
s a m e
t r i a n g l e
s o
t h a t
t h e
s u m
o f
t h e
t h r e e
a n g l e s
a p p e a r s
t o
f o r m
a
l i n e ,
a n d
g i v e
a n
a r g u m e n t
i n
t e r m s
o f
t r a n s v e r s a l s
w h y
t h i s
i s
s o . w
Example with infusion: Investigate
different Catholic symbols and their changes when you reflect, dilate,
rotate, etc.
•
Example with infusion: Use the Pythagorean
Theorem in parish facilities to determine right triangles and/or 90
degree corners.
t
Example with infusion: Compare volume of
church candles and evaluate as to how long they burn, cost savings, etc.
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