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MAKES SENSE STRATEGIES
4 Teaching and Learning
Math
Smart-sheets target…
Math vocabulary development
Math concepts
Math application processes
Math rules & theorems
Includes developmentally appropriate graphic
organizers & think-sheets
* Primary
* Intermediate
* Middle school
This software provides tools
designed to enhance an
existing math program.
* High school
Includes tools for strategic planning math instruction
* Math unit & lesson planning
* Scaffolding instruction
* Selection of developmentally appropriate practice materials
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MAKES SENSE STRATEGIES
4 Teaching and Learning
Math
Smart-sheets
The following provide several
examples of how teachers have
used Makes Sense Strategies
when teaching math
© 2007 www.MakesSenseStrategies.com P.O. Box 267 Lillian, Al 36549 (251) 961-2407
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ADVANCED ALG/TRIG
Chapter 11 – Sequences and Series
Sequences
(an ordered list of numbers)
Arithmetic
D = common
difference
Arithmetic mean =
sum of 2 numbers
divided by 2 (the
average)
Recursive
formula
an = an-1 + d; a1
given
Explicit formula
an = a1 + (n-1)d
Geometric
R = common ratio
Geometric mean =
square root of
product of 2
numbers
Recursive
formula
an = an-1  r; a1
given
Explicit formula
an = a1  r(n-1)
Series
(sum of terms in a sequence)
Arithmetic
Finite – ends
Infinite – does
not end…
Summation Notation
uses Sigma ; has lower
and upper limits
Sum of a Finite
Arithmetic
Series
S = n/2(a1 +
an)
Geometric
FINITE – ends; has
a sum
INFINITE
Converges when |r|<1;
approaches a limit
Diverges when |r|> 1;
does not approach a limit
Sum of a Finite
Geometric Series
Sn = a1(1-rn)
1-r
Sum of an Infinite
Geometric Series
Sn = a1
1-r
Cube
Main ideas
Square
Features
Features
Count the sides
Conclusion about this main idea
Sides
A cube has
______ sides.
Count the sides
Touch each corner
with your pencil
A square has
______ corners.
Conclusion about this main idea
Corners
Hold the block and
count the corners
A cube has
_______ corners.
What can you
do with
it?
A square has
______ sides.
The sides are
shaped like
squares
A cube has more
corners
Conclusion about this main idea
Build
something
Make a design on
paper
Conclusion about these features
Conclusion about these features
Cubes are
Three
Dimensional
Squares are
One
Dimensional
A square is easier
to draw
and use
Reducing Fractions
Is about …
How to tell if a fraction is reduced to lowest terms
Yes, the fraction is reduced 1/6
Is the numerator 1 less than the
denominator?
Yes, the fraction is reduced 5/6
No, keep going
No, keep going
Is the numerator a “1”?
2/4 = 1/2
Divide
Divide both numbers and start again
3 3-3 =1
6 6-3 =2
5/3 = 1 2/3
Is there a number that will divide into
both numbers?
No, the fraction is reduced 4
7
Yes, keep going
Graphing Linear Equations
Slope-intercept form:
y = mx + b
Standard Form:
Ax + By = C
Example: y= 2x - 1
Example: 3x – 2y = 6
1. Identify the slope
(m) and the yintercept (b).
Example:
m = 2, b = -1
3. Use the slope the
locate a second
point.
2. Graph the yintercept on the yaxis. Example:
4. Draw a line
through the two
points.
1. Find the xintercept. Let y=0
and solve the
equation for x.
Example:
3x – 2(0) = 6
3x = 6
x=2
3. Graph the xintercept on the xaxis and the yintercept on the yaxis.
2. Find the yintercept. Let x=0
and solve the
equation for y.
Example:
3(0) – 2y = 6
-2y = 6
y=-3
4. Draw a line
through the two
points.
Copyright 2005 Edwin Ellis
SYSTEMS OF
LINEAR INEQUALITIES
First
Inequality
1 Graph:
Second
Inequality
1 Graph:
a. Solve for y and identify the
slope and y-intercept.
b. Graph the y-intercept on the yaxis and use the slope to locate
another point.
-or-
find the x and y intercepts and
graph them on the x and y axis.)
2 Determine if the line is solid
or dashed.
3 Pick a test point and test it
in the original inequality.
• If true, shade where the point
is.
• If false, shade on the opposite
side of the line.)
a. Solve for y and identify the
slope and y-intercept.
b. Graph the y-intercept on the yaxis and use the slope to locate
another point.
-orfind the x and y intercepts and
graph them on the x and y axis.)
2 Determine if the line is solid
or dashed.
3 Pick a test point and test it
in the original inequality.
• If true, shade where the point
is.
• If false, shade on the opposite
side of the line.)
Solution
Check
1 Darken the area where the
shaded regions overlap.
1 Choose a point in the
darkened area.
2 If the regions do not
overlap, there is no solution.
2. Test in both original
inequalities.
3. Correct if both test true.
These are the steps to …
Ratio Method
Factor Ax2 + Bx + C
Step 1
Five Steps
Copyright 2003 Edwin Ellis
www.GraphicOrganizers.com
menu
Factor out the GCF
Example:
24m2
– 32m + 8
8(3m2
– 4m + 1)
Write two binomials
8(
) (
)
Signs: + C signs will be the same sign as the sign of b . - C negative and positive
8(
) (
)
Step 2
Find AC
AC =
3
Step 3
Find the factors of AC that will add or subtract (depends on the
sign of c) to give u B. 3 and 1 are the factors of AC that will add
(note c is +) to give B.
Why are these
steps important?
Following these steps will
allow one to factor any
polynomial that is not
prime.
Factoring a quadratic
trinomial enable one to
determine the xintercepts of the
parabola.
Factoring also enables
one to use the xintercepts to graph.
Step 4
Write the ratio A/ Factor. Write the ratio Factor / C.
Reduce.
3/1
1/1
are reduced.
Step 5
Write the first ratio in the first binomial and the second ratio in
the second binomial.
8(3x – 1) ( x – 1)
Check using FOIL.
Key Topic
Lesson by Tuwanna McGee
is about...
Divisibility
dividing numbers that don’t have remainders
Essential Details
two (2) - if a number ends with 0,2,4,6,8
five (5) - if a number ends with 0 or 5
ten (10) - if a number ends with 0
three (3) - if the sum of the digits can be divided by 3
nine (9) - if the sum of the digits can be divided by 9
So what? What is important to understand about this?
It’s an easier way to divide BIG numbers!
Word Walls
Definition
New Word
triangle
3 sides
3 angles
Picture
Knowledge
Connection
Looks like a
pyramid
“tri” means 3
quadrilateral
4 sides
4 angles
Looks like a box
“quad” means 4
pentagon
5 sided box;
5 angle “pent”
Looks like a
house
means 5
hexagon
6 sided box;
6 angles “hex”
means 6
Looks like a stop
sign
Graphing a Quadratic Function by Hand
Is about …
These options allow one to graph any quadratic function. A quadratic function models many of the physical, business, and area problems one see in
real world situation. For example: maximize height of a projectile; maximize profit or revenue; minimize cost; Maximize/ minimize area or volume
Main Idea
Main Idea
Option 1
Details
1Complete the square in x to write the
quadratic function in the form f(x) = a(x –
h)2 + k.
2Graph the function in stages using
transformations.
menu
Option 2
Details
1Determine the vertex ( -b/2a, f(-b/2a)).
2Determine the axis of symmetry, x = _b/2a
3Determine the y-intercept, f(0).
4a) If the discriminant > 0, then the graph of the
function has two x-intercepts, which are found by
solving the equation.
b) If the discriminant = 0, the vertex is the xintercept.
c) If the discriminant < 0, there are no xintercepts.
5Determine an additional point by using the yintercept the axis of symmetry.
6Plot the points and draw the graph.
menu
Exponential y = abx
b>1 = growth, 0<b<1 = decay
Logarithms Logb N = P
B = base, N = #, P = power
Common Log = base 10; log
Natural Log = base e; ln
E
X
P
R
E
S
S
I
O
N
S
E
Q
U
A
T
I
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S
Copyright 2003
Edwin Ellis
Graphicorganizers.com
Exponential y = abx
Take log of both sides.
Logarithms Logb N = P
Write in exponential form.
Use properties while
solving and simplify.
Advanced Alg/Trig Chapter 8
EXPONENTIALS AND LOGARITHMS
Product property
logbxy = logbx + logb y
Quotient property
logbx / logb y = logbx - logb y
Power property
logbN x = x logb N
P
R
O
P
E
R
T
I
E
S
G
R
A
P
H
S
Exponential y = abx
Asymptote = x-axis, y = 0
y-intercept (0,1)
Logarithms Logb N = P
Asymptote = y-axis, x = 0
x-intercept = (1,0)
For both graphs, relate to
parent function and label
intercepts.
Graphing Absolute Value Equations
Find the vertex and make a table.
General form: y = │mx + b│ + c
Translate the parent function.
Parent function: y = │mx │
Translated form: y = │mx ± h│ ± k
1Find the vertex using (-b/m, c).
2Make a table of values.
3Choose values for x to the left and to
the right of the vertex. Find the
corresponding values of y.
4Graph the function.
1Graph the parent function. Its
vertex is usually the origin.
2Translate h units left (if h is positive)
or right (if h is negative).
3Translate k units up (if k is positive) or
down (if k is negative).
Example: y =│2x - 4│+ 1
Vertex (-(-4)/2 , 1 ) = (2, 1)
x y
3
│2(3) - 4│+ 1=│2│+1=3
1
│2(1) - 4│+1=│-2│+1=3
Example: y = -│2x - 4│+ 1
Parent function: -│2x│
So what?
What is important to
understand about this?
The graph is always a “V”. A minus sign outside the absolute value bars cause the “V” to be flipped upside down.
The m value in the equation affects the slope of the sides of the “V”.
Exponents
Hot Dog
Gist & Details
© 2003 Edwin Ellis
www.GraphicOrganizers.com
Gist
Zero Exponents
Details
n0 = 1 -n0 = -1
Gist
Details
Gist
Details
Gist
(-n) 0 = 1
Multiplying Like Bases
am  an = am + n
Quotient to a Power
(a/b)n = an/ bn
Dividing Like Bases
Details
Gist
Details
n-1 = 1/n
Gist
Details
Gist
Details
Gist
Details
am = am-n
an
Negative Exponents
1/n-1 = n
Power to a Power
(am)n = amn
Product to a Power
(ab)n = an bn
Polynomials
Algebraic expressions of problems when the task is to determine the value of one unknown number… X
Main idea
Monomials
Main idea
Binomials
Main idea
Trinomials
Example:
4x3 Cubic
5
Constant
Example:
7x + 4
Linear
4
9x +11 4th degree
Example:
3x2 + 2x + 1
Quadratic
Number of terms:
One = “mon”
4x3
Number of terms:
Two = “bi”
7x + 4
Number of terms:
Three =”tri”
3x2 + 2x + 1
Degree:
Sum the exponents
of its variables
Degree:
The degree of
monomial with
greatest degree
Degree:
The degree of the
monomial with
greatest degree
Main idea
Polynomials
Example:
3x5 + 2x3 + 5x2 + x - 4
5th degree
Number of terms:
Many (> three)
=”poly”
Degree:
The degree of the
monomial with
greatest degree
So what? What is important to understand about this?
The concept is important for the AHSGE Objective I-2. You may add or subtract polynomials when
determining or representing a customer’s order at a store.
Factor A2 - C2
“DOTS” = Difference of Two Squares
Step 1
Example: 3x4 - 48
Factor out the GCF.
3( x4 – 16)
Step 2
Check for : 1.) 2 terms
2.) Minus Sign
3.) A and C are perfect squares
Step 3
Write two binomials
Signs: One + ; One -
Why are these
steps important?
Following these steps will
allow one to factor any
polynomial that is not prime.
Factoring a quadratic
trinomial enables one to
determine the x-intercepts
of a parabola.
Factoring also enables one to
use the x-intercepts to
graph.
3(
+
)(
-
)
Step 4
Write the numbers and variables before they were squared in the
binomials. (Note: Any even power on a variable is a perfect square. . . just
half the exponent when factoring) x2 + 4 ) ( x2 - 4)3(
Step 5
Check for DOTS in DOTS
Check using FOIL.
3(x2 + 4 ) ( x2 - 4)
3(x2 + 4 ) ( x + 2)(x - 2)
y=y
x
x
y = kx
If test is true, shade to
include point
Direct
Variation
Graphing TwoVariable
Inequalities
Constant of variation, k
Graph line first – called
the boundary equation
Point-slope form
y – y1 = m(x – x1)
Slope-intercept form
y = mx + b
h is horizontal movement
and k is vertical movement
Linear
Equations
Absolute
Value
Functions
Standard form
Ax + By = C
y = |x – h| + k
Vertex (h,k)
Always graphs as a “V”
Equation of line of best
fit
Function notation
/evaluating functions
Function – vertical line
test
Test a point not on the
line – best choice is (0,0)
Relations and
Functions
Linear Models
Domain
Range
Line of best fit or trend
line
Scattergram
Functions, Equations, and Graphs
1 Graph the linear
equations on the same
coordinate plane.
2 If the lines intersect,
the solution is the
point of intersection.
3 If the lines are
parallel, there is no
solution.
4 If the lines coincide,
there is infinitely
many solutions.
Method 1
GRAPHING
1 Solve one of the linear
equations for one of
the variables (look for
a coefficient of one).
2 Substitute this
variable’s value into
the other equation.
3 Solve the new equation
for the one remaining
variable.
4 Substitute this value
into one of the original
equations and find the
remaining variable
value.
1 Look for variables with
opposite or same
coefficients.
2 If the coefficients are
opposites, add the
equations together. If
the coefficients are
the same, subtract the
equations, by changing
the sign of each term
in the 2nd equation and
adding.
3 Substitute the value
of the remaining
variable back into one
of the orig. equations
to find the other
variable.
Method 2
1 Choose a variable to
eliminate.
2 Look the coefficients
and find their LCD.
This is the value you
are trying to get.
3 Multiply each equation
by the needed factor
to get the LCD.
4 Continue as for regular
elimination
Method 3
SUBSTITUTION
ELIMINATION
Method 4
ELINIMATION
VIA
SUMMARY
MULTIPLICATION
The solution (if it has just one) is an ordered pair. This point is a solution to both equations
and will test true if substituted into each equation.
Topic
Solving Systems of Linear Equations
Finding solutions for more than one linear equation by using one of four methods.
Topic
Divisibility
Main Idea
Details
Two (2)
Dividing numbers that
don’t have remainders
Main Idea
Five (5)
Details
If a number ends with
O, 2, 4, 6, 8
Main Idea
Details
If a number ends with
O or 5
Ten (10)
Details
If a number ends with
O
Main Idea
Three (3)
Details
If the sum of the digits
can be divided by 3
Main Idea
Nine (9)
Details
If the sum of the digits
can be divided by 9
Exponent Rules
Zero
Exponents
n0 = 1
-n0 = -1
(-n) 0 = 1
Product to a
Power
(ab)n = an bn
Negative
Exponents
n-1 = 1/n
1/n-1 = n
Dividing Like
bases
am = am-n
an
Multiplying
Like Bases
am  an = am + n
Power to a
Power
(am)n = amn
Quotient to a
Power
(a/b)n = an/ bn
So what? What is important to understand about this?
Exponents are important for the graduation exam. Exponents are also used in exponential functions which
model population growth, compound interest, depreciation, radio active decay, and the list goes on an on. . . .
Topic
Comparing….
And
Main ideas
Synonyms
Notation
Graph
Intersection,
Overlap, Common,
Same
2x + 3 < 5 and 2x + 3 > -5
Topic
Or
Union, Every, All
3x – 2 > 13 or 3x – 2 < - 4
-5 < 2x + 3 < 5
----------│----------------│---------------
-4
0
1
----------│-----│----------│---------------
--2/3 0
5
So what? What is important to understand about this?
To solve real world problems involving chemistry of swimming pool
water, temperature, and science.
TOPIC
Order of Operations - What we should do in an equation situation…
AHSGE: I-1 Apply order of operations
Main Idea
PLEASE
Details
Parenthesis
Do all parentheses first.
DEAR
Main Idea
Details
Division
Do all division from left
to right next.
Main Idea
EXCUSE
Details
Exponents
Do all exponents second.
AUNT
Main Idea
Details
Addition
Do all addition from left
to right.
MY
Main Idea
Details
Multiplication
Do all multiplication
from left to right next.
Main Idea
SALLY
Details
Subtraction
Do all subtraction from
left to right.
Note: Complete multiplication and division from left to right even if division comes first.
Complete addition and subtraction from left to right even if subtraction comes first.
Polynomial Products
Multiplying Polynomials and Special Products
Main idea
Polynomial times
Polynomial
Monomial times a
Polynomial:
Use the distributive
property
Example:
-4y2 ( 5y4 – 3y2 + 2 )
= -20y + 12y –
Polynomial times a
Polynomial:
6
4
8y2
Use the distributive
Property
Example:
(2x – 3) (
+ x – 6) =
3
2
8x – 10x – 15x + 18
4x2
Main idea
Main idea
Main idea
Binomial times
Binomial
Square of a
Binomial
Difference of
Two Squares
FOIL:
F = First
O = Outer
I = Inner
L = Last
Example:
(3x – 5 )( 2x + 7)=
6x2 + 11 x - 35
(a + b) 2 =
a2 + 2ab + b2
(a – b) 2 =
a2 – 2ab + b2
1st term: Square the
first term
2nd term: Multiply two
terms and Double
3rd term: Square the last
term
DOTS:
(a + b) ( a – b) =
a 2 – b2
Example:
(t3 – 6) (t3 + 6) =
t6 – 36
Example:
(x + 6)2 =
x2 + 12 x + 36
So what? What is important to understand about this?
AHSGE Objective: I -3. Applications include finding area and volume. Special products are used in graphing
functions by hand. Punnett Squares in Biology.
These are the steps to …
Five Steps
Factor A2 - C2
“DOTS” = Difference of Two Squares
Copyright 2003 Edwin
Ellis
www.GraphicOrganizers.c
Name
om
Step 1
Example: 3x4 - 48
Factor out the GCF.
3(
x4
– 16)
AHSGE : I – 4
Factor Polynomials.
Five Steps
Step 2
Check for : 1.) Two terms
2.) Minus Sign
3.) A and C are perfect squares
Step 3
Write two binomials
Signs: One + ; One -
Why are these
steps important?
Following these steps
will allow one to factor
any binomial that is not
prime.
Factoring a quadratic
trinomial enables one
to determine the xintercepts of a
parabola.
3(
+
)(
-
)
Step 4
Write the numbers and variables before they were squared in the
binomials. (Note: Any even power of a variable is a perfect square.
Just half the exponent when factoring)
3(x2 + 4 ) ( x2 - 4)
Step 5
Check for DOTS in DOTS
Check using FOIL.
3(x2 + 4 ) ( x2 - 4)
3(x2 + 4 ) ( x + 2)(x - 2)
These are the steps to …
Five Steps
Factor Ax2 + Bx + C
Trial and Error.
Copyright 2003 Edwin
Ellis
www.GraphicOrganizers.c
Name
om
Step 1
Factor out the GCF
Example: 24m2 – 32m + 8
8(3m2 – 4 m + 1)
AHSGE: I – 4
Factor Polynomials.
Five Steps
Step 2
Write two binomials
Signs: +C -- signs will be the same sign as the sign of B
-C -- signs will be different: one negative and one positive
8( ) (
)
Step 3
List the factors of A and the factors of C.
A=3
C=1
1, 3
1, 1
Why are these
steps important?
Following these steps
will allow one to factor
any trinomial that is not
prime.
Factoring a quadratic
trinomial enables one
to determine the xintercepts of the
parabola.
Step 4
If C is positive, determine the factor combination of A and C that will
add to give B.
If C is negative, determine the factor combination of A and C that will
subtract to give B.
Since C is positive add to get B: 8 (3x – 1) (x – 1)
Step 5
Check using FOIL.
8(3x2 – 3x – x + 1)
=8(3x2 – 4x + 1)
=24x2 – 32 x + 8
These are the steps to …
Factoring Polynomials Completely
Five Steps
Copyright 2003 Edwin
Ellis
www.GraphicOrganizers.c
Name
om
Step 1
Factor out the GCF
Example: 2x3- 6x2= 2x2( x – 3)
Always make sure the remaining polynomial(s) are factored.
AHSGE: I – 4
Factoring Polynomials.
Five Steps
Step 2
Two Terms: Check for “DOTS”
A2 – C2 (Difference of Two Squares)
Example:
x2 – 4 = (x – 2 ) (x + 2)
See if the binomials will factor again. Check Using FOIL
Step 3
Three Terms: Ax2 + Bx + C
Check for “PST” m2 + 2mn + n2 or m2 – 2mn + n2.
Factor using short cut. No “PST”, factor using trial and error.
Example PST: c2 + 10 c + 25 = (c + 5)2
For an example of trial and error, see Trial/Error Method.
Why are these
steps important?
Following these steps
will allow one to factor
any polynomial that is
not prime.
Factoring allows one to
find the x-intercepts
and in turn graph the
polynomial.
Step 4
Four Terms: Grouping
Group two terms together that have a GCF.
Factor out the GCF from each pair. Look for common binomial.
Re-write with common binomial times other factors in a binomial.
Example:
5t4 + 20t3 + 6t + 24 = (t + 4) (5t3 + 6)
Step 5
Make sure that each polynomial is factored completely.
If you have tried steps 1 – 4 and the polynomial cannot be factored, the polynomial
is prime.
Big Air Rules
Snowboarding
Is about …
Vocabulary -- Define and give an example
When was the snowboard invented?
Projectile
What is “goofy foot”? What is regular foot?
Trajectory
What is “hang time”?
Gravity
Snowboarding and its relationship to math
More Vocabulary
Where does hang time occur on a parabola?
Acceleration due to gravity
Which motion is affected by gravity?
Parabola
Name sports that involve parabolas.
Fill in the blank. Horizontal and vertical motions are
___________( independent/ dependent) of each other.
Applications: Finding the center
of a circle, finding the perimeter
of a figure, finding the missing
side of a right triangle, solving
quadratic equations are all areas
where these problems are used
in real-world situations.
Standard form: ax2 + bx + c = 0
b  b2  4ac
2a
Quadratic Formula
Topic
Pythagorean Theorem
RADICAL FORMULAS
Most of these formulas involve
Distance Formula
simplifying a radical.
a 2  b2  c2
c = longest side
( x2  x1 )2  ( y2  y1 )2
Midpoint
Formula
Two points: (x1,y1) (x2,y2)
  x1  x2   y1  y2  
,


2
2


Two points: (x1,y1) (x2,y2)
Q: Why should I know how to use these formulas?
A: They are on the exit exam as well as in geometry and higher level math courses.
Definition
Term
Example
Symbolic form/read it
If an angle is a straight
angle, then its measure is
180.
p→q
If p, then q.
Definition
Example
Symbolic form/read it
Has the opposite meaning
as the original statement.
An angle is NOT a straight
angle.
Definition
Example
An if-then statement
Conditional
Term
Negation (of p)
Term
Inverse
Negates both the if and
then of a conditional
statement
Definition
Term
Contra-positive
Switches the if and then
and negates both.
If an angle is NOT a
straight angle, then its
measure is NOT 180.
Example
If an angle’s measure is
NOT 180, then it is NOT a
straight angle.
~p
NOT p.
Symbolic form/read it
~p → ~q
If NOT p, then NOT q.
Symbolic form/read it
~q → ~p
If NOT q, then NOT p.
Definition
Term
An if-then statement
Conditional
Name
Example
_____
If an angle is _____
a straight
_____
angle, then its
measure is
____
180.
Symbolic form/read it
The part following the if is the
hypothesis and the part
following the then is the
conclusion.
p→q
If p, then q.
Term
Converse
Term
Bi-conditional
Term
Definition
Example
Switches the if and then of
a conditional statement
If the measure of an angle
is 180, then it is a straight
angle.
Definition
Example
The combination of a
conditional statement and
its converse; usually
contains the words if and
only if.
An angle is a straight angle
if and only if its measure
is 180.
Definition
Picture
Symbolic form/read it
q→p
If q, then p.
Symbolic form/read it
p↔q
p if and only if q.
Knowledge Connection
Special Right Triangles
Is about …
30º- 60º- 90º
The ratio of the sides is 1 : 2: √3
Side opposite the 30º angle = ½ hypotenuse
30°
Side opposite the 60º angle = ½ hypotenuse times √3
The larger leg equals the shorter leg times √3
60°
45º- 45º- 90º
The ratio of the sides is 1 : 1 : √2
Side opposite the 45º angle = ½ hypotenuse times √2
45°
Hypotenuse = s √2 where s = a leg
45°
Title
Math Curse
by Jon Scieszka + Lane Smith
How do you get to school in the morning? What time do you leave
and what time do you arrive? If you were 15 minutes late leaving
your house for school, what time would you arrive? What would you
not have time to do if you were late and why?
1.
2.
3.
If there were approximately 1300 students in the school, how many
fingers would there be? How would you write that in scientific
notation?
If an M&M is about a centimeter long, then how many M&Ms long is
your foot?
If he bought an 80cent candy bar that was on sale for 25% off,
how much would he have to pay? Would it be more or less than the
candy bar that was on for sale for 50% off? Explain.
4.
Writing Linear Equations
…
Three Forms of a Linear Equation
Main Idea
Slope-Intercept
Form
Y = mx + b
Details
1. Given slope = m , and
y-int= b
2. Substitute m and b into the
equation.
3. Transform to Standard if
necessary.
Main Idea
Standard Form
Ax + By = C
Details
1. Given an equation in slopeintercept form.
a. Eliminate fractions
b. Add or Subtract
2. Given an equation in pointslope form.
a. Distribute
b. Eliminate Fractions
c. Add or Subtract
Main Idea
Point-Slope Form
y – y1 = m(x – x1)
Details
1. Given slope = m&point (x1, y1)
Substitute m and the
point into the equation.
2. Given 2 points
a. Find m
b. Substitute into point
slope form
c. Simplify/change to Slopeintercept or Standard
form if necessary.
So what? What is important to understand about this?
Many real life situations can be described by an equation, for instance, payroll deductions, temperature . . .