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Warm Up
• Write down objective and homework in
agenda
• Lay out homework (none)
• Homework (Distance & PT worksheet)
Unit 1 Common Core Standards
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8.EE.7 Solve linear equations in one variable.
a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or
no solutions. Show which of these possibilities is the case by successively transforming the given
equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results
(where a and b are different numbers).
b. Solve linear equations with rational number coefficients, including equations whose solutions
require expanding expressions using the distributive property and collecting like terms.
8.G.6 Explain a proof of the Pythagorean Theorem and its converse.
8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in realworld and mathematical problems in two and three dimensions.
8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate
system.
A-CED.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.
Note: At this level, focus on linear and exponential functions.
A-CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in
solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R. Note: At this
level, limit to formulas that are linear in the variable of interest, or to formulas involving squared or
cubed variables.
Unit 1 Common Core Standards
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A-REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at
the previous step, starting from the assumption that the original equation has a solution. Construct a viable
argument to justify a solution method.
A-REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x)
intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology
to graph the functions, make tables of values, or find successive approximations. Include cases where f(x)
and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Note: At
this level, focus on linear and exponential functions.
A-REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients
represented by letters.
A-SSE.1 Interpret expressions that represent a quantity in terms of its context.
a. Interpret parts of an expression, such as terms, factors, and coefficients.
b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For
example, interpret P(1+r)n as the product of P and a factor not depending on P.
Note: At this level, limit to linear expressions, exponential expressions with integer exponents and
quadratic expressions.
G-GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a
circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and
informal limit arguments.
Note: Informal limit arguments are not the intent at this level.
G-GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.*
Note: At this level, formulas for pyramids, cones and spheres will be given.
G-GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles,
e.g., using the distance formula.
Unit 1 Common Core Standards
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N-Q.1 Use units as a way to understand problems and to guide the solution of
multi-step problems; choose and interpret units consistently in formulas; choose
and interpret the scale and the origin in graphs and data displays.
N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when
reporting quantities.
N-RN.1 Explain how the definition of the meaning of rational exponents follows
from extending the properties of integer exponents to those values, allowing for a
notation for radicals in terms of rational exponents. For example, we define 51/3
to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (5 1/3)3
must equal 5.
N-RN.2 Rewrite expressions involving radicals and rational exponents using the
properties of exponents.
Note: At this level, focus on fractional exponents with a numerator of 1.
MP.1 Make sense of problems and persevere in solving them.
MP.2 Reason abstractly and quantitatively.
MP.4 Model with mathematics.
MP.7 Look for and make use of structure.
Warm Up
Warm Up
Vocabulary
Distance Formula
The distance d between any two points is given by the
formula d =
Hypotenuse
Longest side of a right triangle, opposite the right angle
Legs
The two sides of a right triangle that make up the right angle
Pythagorean Theorem
The Pythagorean Theorem describes the relationship of the
lengths of the sides of a right triangle where in any right
triangle, the sum of the squares of the lengths of the legs is
equal to the square of the length of the hypotenuse.
Pythagorean Triple
Three positive integers that make up the lengths of the sides
of a right triangle
right angle
Angle that measures 90 degrees
What is a right triangle?
hypotenuse
leg
right angle
leg
It is a triangle which has an angle that is
90 degrees.
The two sides that make up the right angle
are called legs.
The side opposite the right angle is the
hypotenuse.
The Pythagorean Theorem
In a right triangle, if a and b are the
measures of the legs and c is the
hypotenuse, then
a2 + b2 = c2.
Note: The hypotenuse, c, is always the
longest side.
Proof
Find the length of the
hypotenuse if
1. a = 12 and b2 = 16.
2
2
12 + 16 = c
144 + 256 = c2
400 = c2
Take the square root of both sides.
2
400  c
20 = c
Find the length of the hypotenuse if
2. a = 5 and b = 7.
5 2 + 7 2 = c2
25 + 49 = c2
74 = c2
Take the square root of both sides.
74  c
8.60 = c
2
Find the length of the hypotenuse given a =
6 and b = 12
1.
2.
3.
4.
180
324
13.42
18
Find the length of the missing side given a
= 4 and c = 5
1.
2.
3.
4.
1
3
6.4
9
Pythagorean Triple
• A pythagorean triple is made up of three whole
numbers that form the three sides of a right
triangle
• No decimal answers!!
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Is 3, 4, and 5 a pythagorean triple?
Is 5, 8, 12 a pythagorean triple?
Is 2, 5, 9 a right triangle?
Is 5, 12, 13 a right triangle?
Word Problem Practice
http://regentsprep.org/Regents/math/ALGEBRA
/AT1/PracPyth.htm
Extra Resources PT
• http://www.mathwarehouse.com/geometry/t
riangles/how-to-use-the-pythagoreantheorem.php
• http://videos.howstuffworks.com/hsw/25946discovering-math-pythagorean-theoremvideo.htm
Find the distance
Distance Formula
• Used to find the distance between two points
distance  ( x2  x1 )  ( y2  y1 )
2
2
Example
• Find the distance between A(4,8) and B(1,12)
A (4, 8)
B (1, 12)
distance  ( x2  x1 )  ( y2  y1 )
2
distance  (1  4)  (12  8)
2
distance  (3)  (4)
2
2
2
distance  9  16  25 
5
2
• Find the distance between:
– A. (2, 7) and (11, 9)
(9)  (2)  85
2
2
– B. (-5, 8) and (2, - 4)
(7)  (12)  193
2
2
• http://phschool.com/webcodes10/index.cfm?
fuseaction=home.gotoWebCode&wcprefix=ae
a&wcsuffix=1103
Find the Distance between each
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1) (7, 3), (−1, −4)
2) (3, −5), (−3, 0)
3) (6, −7), (3, −5)
4) (5, 1), (5, −6)
5) (5, −8), (−8, 6)
6) (4, 6), (−4, −3)
7) (−7, 0), (−2, −4)
8) (−4, −3), (1, 4)
9) (−2, 2), (−6, −8)
10) (6, 2), (0, −6)