Warm Up
• Write down objective and homework in agenda
• Lay out homework (Distance & PT worksheet)• Homework (Volume worksheet)
• Get a calculator!!!
Unit 1 Common Core Standards• 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 real-
world 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• 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• 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
Vocabulary
Area of a circle Area of a circle is given by times the square of the radius
CircumferenceCircumference is the perimeter of or distance around a circle given by times the diameter of the circle.
cone a solid, 3-dimensional figure with one vertex and one circular base.
CylinderA solid, 3-dimensional figure with a curved side and two circular, congruent bases that are in parallel planes
Sphere A three dimensional solid that is perfectly round, ex. A ball.
volume The number of unit cubes or cubic units needed to fill the space inside a three-dimensional figure
What do cones, pyramids, rectangular prisms and
cylinders have in common?
• http://www.phschool.com/atschool/cmp2/active_math/site/Grade7/pouring/index.html
VOLUME OF PRISMS AND CYLINDERSWords Numbers Formula
Prism: The volume V of a prism is the area of the base B times the height h.
Cylinder: The volume of a cylinder is the area of the base B times the height h.
B = 2(5)= 10 units2
V = 10(3)
= 30 units3
B = p(22)= 4p units2
V = (4p)(6) = 24p 75.4 units3
V = Bh
V = Bh
= (pr2)h
Find the volume of the figure to the nearest tenth.
B. 4 in.
12 in.
= 192 p 602.9 in3
B = p(42) = 16p in2
V = Bh
= 16p • 12
Example
Area of base
Volume of a cylinder
Find the volume of the figure to the nearest tenth.
B. 8 cm
15 cm
B = p(82)
= 64p cm2
= (64p)(15) = 960p
3,014.4 cm3
Try This
Area of base
Volume of a cylinderV = Bh
A pyramid is named for the shape of its base. The base is a polygon, and all of the other faces are triangles. A cone has a circular base. The height of a pyramid or cone is measured from the highest point to the base along a perpendicular line.
VOLUME OF PYRAMIDS AND CONES
(22)
Example: Finding the Volume of Pyramids and Cones
Find the volume of the figure.
13V = • 14 • 6
V = 28 cm3
A.
V = Bh13
B = (4 • 7) = 14 cm2 12
Example 2: Finding the Volume of Pyramids and Cones
13V = • 9 • 10
V = 30 94.2 in3
B.
V = Bh13
B = (32) = 9 in2
Use 3.14 for .
Find the volume of the figure.
Example 3: Finding the Volume of Pyramids and Cones
13V = • 84 • 10
V = 280 m3
C.
V = Bh13
B = 14 • 6 = 84 m2
Find the volume of the figure.
Try This: Example 1A
13V = • 17.5 • 7
V 40.8 in3
A.
V = Bh13
B = (5 • 7) = 17.5 in2 12
Find the volume of the figure.
5 in.
7 in.
7 in.
13V = • 9 • 7
V = 21 65.9 m3
B.
V = Bh13
B = (32) = 9 m2
Use 3.14 for .
Find the volume of the figure.
Try This: Example 1B
7 m
3 m
13V = • 16 • 8
V 42.7 ft3
C.
V = Bh13
B = 4 • 4 = 16 ft2
Find the volume of the figure.
Try This: Example 1C
4 ft4 ft
8 ft
A sphere is the set of points in three dimensions that are a fixed distance from a given point, the center. A plane that intersects a sphere through its center divides the two halves or hemispheres. The edge of a hemisphere is a great circle.
Example 1: Finding the Volume of a Sphere
Find the volume of a sphere with radius 9 cm, both in terms of p and to the nearest tenth of a unit.
= 972p cm3 3,052.1 cm3
Volume of a sphere
Substitute 9 for r.
43
V = pr3
= p(9)343
Try This: Example 1
Find the volume of a sphere with radius 3 m, both in terms of p and to the nearest tenth of a unit.
= 36p m3 113.0 m3
Volume of a sphere
Substitute 3 for r.
43
V = pr3
= p(3)343
Extra Practice
• Volume of a cylinder: http://regentsprep.org/Regents/math/geometry/GG2/CylinderPage.htm
• Volume of a cone: http://regentsprep.org/Regents/math/geometry/GG2/ConePage.htm
• Volume of a sphere: http://regentsprep.org/Regents/math/geometry/GG2/SpherePage.htm
