Subject: Algebra 1
|Subject: 8th/CC Math I |Timeframe Needed for Completion: 9 weeks |
|Grade Level: 8th | |
|Unit Title: Quadratic Functions, Pythagorean Theorem, Geometry |Grading Period: 4th nine weeks |
|Big Idea/Theme: Understandings: |
|Understand and apply the Pythagorean Theorem |
|Explain a proof of the Pythagorean Theorem and its converse |
|Find distance between two points using the Pythagorean Theorem |
|Interpret functions that arise in applications in terms of a context |
|Analyze functions using different representations |
|Build a function that models a relationship between two quantities |
|Build new functions from existing functions |
|Construct and compare linear, quadratic, and exponential models and solve problems |
|Write expressions in equivalent forms to solve problems |
|Solve equations and inequalities in one variable |
|Represent and solve equations and inequalities graphically |
|Experiment with transformations in the plane |
|Use coordinates to prove simple geometric theorems algebraically |
|Explain volume formulas and use them to solve problems |
|Essential Questions: |Curriculum Goals/Objectives |
|What are some real world applications which would use quadratic equations? | |
|Why is it important to understand different rates of increasing values ? |F.LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a |
|Does the intersection of two functions have any practical value in real world situations. Explain why |quantity increasing linearly, quadratically, or (more generally) as a polynomial function |
|or why not. |A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the |
|Why is it important to know the precise definitions for geometric terms? |quantity represented by the expression. Factor a quadratic expression to reveal the zeros of the |
|How caould you determine how tall a person is from a picture without a measuring device? |function it defines Complete the square in a quadratic expression to reveal the maximum or minimum |
|How do we use the Pythagorean Theorem in everyday life? |value of the function it defines. Use the properties of exponents to transform expressions for |
|Given a diagonal line segment, how can we use the Pythagorean Theorem to determine its’ length? |exponential functions. For example the expression 1.15t can be rewritten as (1.151/12 )12t = 1.01212t |
|Why is it important to know if lines are parallel or perpendicular? |to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. |
|Why is it important to know the perimeter or area of geometric figures in our everyday life? |G.CO.1: Know precise definitions of angle, circle, perpendicular line, parallel line and line segment, |
|Why is it important to know the volume of a container? |based on the undefined notions of a point, line, distance along a line, and distance around a circular |
| |arc. |
| |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. |
| |G.CPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or |
| |disprove that a figure defined by 4 given points in the coordinate plane is a rectangle; prove or |
| |disprove that the point |
| |( 1,square root 3) lies on the circle centered at the origin and containing the point (0, 2). |
| |G.CPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric |
| |problems (e.g. find the equation of a line parallel or perpendicular to a given line that passes through|
| |a given point). |
| |G.CPE.6 Find the point on a directed line segment between two given points that partitions the segment |
| |into a given ratio.. |
| |G.CPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles (e.g> |
| |using the distance formula |
| |G.GMD.1 Give an informal argument for the formulas for the circumference of a circle, volume of a |
| |cylinder, pyramid and cone. Use dissection arguments, Cavalieri’s principle, and informal limit |
| |arguments. |
| |G.GMD.3 Use volume formulas for cylinders, pyramids, cones and spheres to solve problems. |
|Essential Skills/Vocabulary: | |
|Vocabulary: | |
|completing the square | |
|minimum value | |
|maximum value | |
|vertex form | |
|square root | |
|successive approximations | |
|angle | |
|circle | |
|perpendicular lines | |
|parallel lines | |
|line segment | |
|point | |
|arc | |
|length of an arc | |
|right triangle | |
|hypotenuse | |
|legs | |
|Pythagorean Theorem | |
|Pythagorean Triple | |
|slope | |
|ratio | |
|rectangles | |
|triangles | |
|perimeter | |
|area | |
|volume | |
|cylinder | |
|pyramid | |
|cone | |
|sphere | |
| | |
|Essential skills: | |
|Understand through graphs and tables how quantities increase | |
|Factoring quadratic expressions | |
|Completing the square for quadratic functions | |
|Using properties of exponents | |
|Derive the quadratic formula by completing the square | |
|Understand why taking the square root of both sides of an equation yields two solutions | |
|Understand the intersection of two functions, of any type, is the solution of the equations | |
|Graph equations using the graphing calculator | |
|Explain a proof of the Pythagorean Theorem | |
|Determine the unknown lengths of a right triangle using the Pythagorean Theorem | |
|Apply the Pythagorean Theorem to determine distance in the coordinate plane | |
|Use coordinates to prove simple geometric theorems algebraically | |
|Use the slope formula to solve geometric problems | |
|Find the point on a line segment that divides the segment into a given ratio | |
|Use coordinates to compute perimeter and area | |
|Explain circumference and area formulas for a circle by determining the meaning of each term | |
|Explain volume formulas for a cylinder, pyramid and cone by determining the meaning of each term | |
|Using formulas solve problems for cylinders, pyramids, cones and spheres | |
| | |
|Guiding Questions: | |
|Which functions increase at a constant rate? | |
|Which functions increase at varied rates? | |
|Explain the meaning of the zeros in a quadratic function. | |
|Explain what the vertex represents, and how to determine minimum or maximum values. | |
|How do you determine growth or decay using properties of exponents? | |
|How do you use properties of exponents to rewrite exponential functions? | |
|What are the steps for completing the square? | |
|Can different type equations (linear, quadratic, exponential, etc.) have a common solution? Where is | |
|the solution? | |
|What is an angle? | |
|What is a circle? | |
|What are parallel lines? | |
|What are perpendicular lines? | |
|How can we tell if lines are parallel or perpendicular from their slopes? | |
|How do we determine the length of an arc on a circle? | |
|Given three lengths, how do you determine if they form a right triangle? | |
|Given points A and B on the coordinate plane, how do you determine the length of segment AB? | |
|Prove or disprove that a given point lies on a geometric figure | |
|Find equations parallel or perpendicular to a given line | |
|Using coordinate geometry and the distance formula how can you find the perimeters of polygons and the | |
|areas of rectangles and triangles | |
|How do you determine the circumference and area of a circle? | |
|Explain how the formulas for volume of a cylinder, pyramid and cone are developed. | |
|What is the volume relationship between a cylinder and a cone. | |
|What is the exponential term for perimeter, area and volume? Why is this exponent used in each case? | |
|Explain using units of measure. (ft, cm, etc.) | |
|Materials Suggestions: |
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|NCDPI Resources: |
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|National Library of Manipulatives |
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|NCTM Illuminations |
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|Lesson Plan sites and Activities: |
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|Math Graphic Organizers |
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|Problem Solving/Problem Websites |
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|Currituck County Schools – Common Core Resources |
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|AVID Library/Mathematics Write Path I and II |
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