Integrated Math I Page 1 - Wethersfield Public Schools Name: Integrated Math I Department: Mathematics Grade(s): 10-11 Level(s): 2 Course Number(s): 30602 Credits: 1 Course Description:

Wethersfield Public Schools Course Outline

Course Name: Integrated Math I Department: Mathematics Grade(s): 10-11 Level(s): 2 Course Number(s): 30602 Credits: 1 Course Description: This course description also appears in the course catalog. The course will revisit important algebraic topics such as solving equations, factoring, as well as graphing linear, quadratic and exponential equations. The course will also introduce important geometry concepts related to the Common Core to the students through a coordinate approach rather than a proof-based approach. Students with an A in Integrated Mathematics may take Algebra 2. All other students who pass Integrated Mathematics I will take Integrated Mathematics II. Required Instructional Materials: Name, author, date. (publisher and edition) Revised/Approval Date: April 18, 2013 Approved Administrative Team 10/9/13 Approved Student Programs & Services Committee 10/21/13 Authors/Contributors: Taryn Cutler, Michael Miller, Jeremy Sasur, Shannon Shouldice, Tara Tabellione, Jeffrey Weber Outline for Integrated Math 2: Intro to Polynomials Quad Functions & Equations Maximize Area Polynomial Functions (Exponent (with rational) rules to begin chapter) Exponents into Rational Expression & Equations Similarity

Integrated Math I Page 1

Simplify Radicals into Right Triangles & Trig Circles Probability Overarching Skills This section includes 21st Century skills and discipline focused skills such as inquiry skills, problem solving skills, research skills, etc. These objectives should be taught and assessed through the integration of the other units. This unit is not meant to be taught in isolation as a separate unit. Students will engage in tasks for which the solution method is not known in advance and build new mathematical knowledge through problem solving. Students will reason and think analytically to make and investigate mathematical conjectures. Students will learn from and work collaboratively with others in a spirit of mutual respect and open dialogue. Students will be able to express their understanding orally, in writing, and with models. Students will deepen their understanding by connecting mathematical ideas to each other and to real world applications of those ideas. Students will learn to use representation to understand and communicate mathematical ideas. Students will use technology as a tool to research, organize, evaluate and communicate information. Enduring Understandings Essential Questions • Patterns enable us to discover, analyze,

describe, extend, and formulate concrete understanding of mathematical and real-world phenomena.

• Algebra provides the underlying structure to make connection among all branches of mathematics, including measurement, geometry, calculus and statistics.

• Logic and reasoning are the foundations for developing arguments.

• Geometric terms can be proven or defined using formal geometric constructions with a variety of tools and methods.

• Innovations in technology allow users to explore and deepen their understanding of a new and long-standing mathematical concepts and applications of mathematics.

• What is a pattern? How do we find and show patterns? What can patterns reveal?

• How is thinking geometrically differently from thinking algebraically?

• How are the principles of algebra used to solve problems?

• What makes a solution reasonable? • How is the understanding of mathematics

enhanced by the use of technology? • How does the use of technology enrich

mathematics? • What criterion determines the most

efficient method of solving a problem?

Mathematical Practices (Show link to standards in parenthesis after objective) The student will: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure.


8. Look for and express regularity in repeated reasoning. Instructional Support Materials

• TI-84 Plus graphing calculator • TI-SmartView software • Geogebra and/or Geometer’s Sketchpad • iBooks / iPads

Suggested Instructional Strategies

• Modeling • Direct instruction • Guided practice • Interactive models • Differentiated tasks • Electronic demonstrations • Cooperative learning • Performance Tasks

Suggested Assessment Methods (Include use of school-wide analytic and course specific rubrics)

• Pre-assessments • Quizzes • Chapter Tests • Common summative assessments (midterm and final) • School-wide analytical rubrics


Unit 1: Modeling with Functions Time Frame: September Length of Unit: 4 weeks Enduring Understandings Essential Questions

• Functions can be represented in a variety of ways.

• Graphing functions gives us a visual model to study and interpret real-world situations.

• There are a variety of methods and techniques which can be used to graph functions.

• Real-world phenomena can be modeled by linear and exponential functions.

• Functions are used to describe different types of rates of change.

• In what ways does a relation differ from a function?

• How can functions be used to model real-world situations?

• What is a rate of change? • How is a linear rate of change different

from exponential rate of change? • How can it be determined whether a

function or situation is linear or exponential?

Priority (bold) and Supporting

CCSS Explanations and Examples (not all examples apply to

Algebra 1) 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.

In all problem situations the answer should be reported with appropriate units. In situations involving money, answers should be rounded to the nearest cent. When data sets involve large numbers (e.g. tables in which quantities are reported in the millions) the degree of precision in any calculation is limited by the degree of precision in the data. These ideas are introduced in the solution to contextual problems in Unit 2 and reinforced throughout the remainder of the course.

N.Q 2. Define appropriate quantities for the purpose of descriptive modeling.

N.Q 3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. F.IF 1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

The domain of a function given by an algebraic expression, unless otherwise specified, is the largest possible domain. Mapping diagrams may be used to introduce the concepts of domain and range. The vertical line test may be used to determine whether a graph represents a function.


F.IF 2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Examples: If f(x) = x2 + 4x – 12, find f(2) Let f(x) = 10x – 5; find f(1/2), f(-6), f(a) If P(t) is the population of Tucson t years after 2000, interpret the statements P(0) = 487,000 and P(10)-P(9) = 5,900.

F.IF 4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative....*

Students may be given graphs to interpret or produce graphs given an expression or table for the function, by hand or using technology. Examples: • A rocket is launched from 180 feet above the ground at time t = 0. The function that models this situation is given by h = – 16t2 + 96t + 180, where t is measured in seconds and h is height above the ground measured in feet. • What is a reasonable domain restriction for t in this context?

• Determine the height of the rocket 2 seconds after it was launched.

• Determine the maximum height obtained by the rocket.

• Determine the time when the rocket is 100 feet above the ground.

• Determine the time at which the rocket hits the ground.

• How would you refine your answer to the first question based on your

response to the fifth question?

• It started raining lightly at 5am, then the rainfall became heavier at 7am. By 10am the storm was over, with a total rainfall of 3 inches. It didn’t rain for the rest of the day. Sketch a possible graph for the number of inches of rain as a function of time, from midnight to midday.

F.IF 5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in factory, then the positive integers would be an appropriate domain for the function.*

Students may explain orally, or in written format, the existing relationships.

F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.*

The average rate of change of a function y = f(x) over an interval [a,b] is In addition to finding average rates of change from

functions given

symbolically, graphically, or in a table, Students may collect data from experiments or simulations (ex. falling ball, velocity of a car, etc.) and find average rates of change for the function modeling the situation. Examples:


• Use the following table to find the average rate of change of g over the intervals [-2, -1] and [0,2]:

x g(x) -2 2 -1 -1 0 -4 2 -10

• The table below shows the elapsed time when two different cars pass a 10, 20, 30, 40 and 50 meter mark on a test track. o For car 1, what is the average velocity (change in distance divided

by change in time) between the 0 and 10 meter mark? Between the 0 and 50 meter mark? Between the 20 and 30 meter mark? Analyze the data to describe the motion of car 1.

• How does the velocity of car 1 compare to that of car 2?

Car 1 Car 2 d t t

10 4.472 1.742 20 6.325 2.899 30 7.746 3.831 40 8.944 4.633 50 10 5.348

F-IF 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*

a. Graph linear ...functions and show intercepts..

Example: Find the x-intercept and the y-intercept for each of these lines. y = –3x + 18 y – 7 = 2/3(x + 4) 6x + 4y = 96

F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Example: Examine the functions below. Which function has the larger maximum? How do you know?

F.LE 1. Distinguish between situations that can be modeled with linear functions [and with exponential functions]. F.LE.1a. Prove that linear functions grow by equal differences over equal intervals... over equal intervals. F.LE.1b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another....

Given several tables of values, determine which represent linear functions, and explain why.


F.LE 3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly ...

Example: Contrast the growth of f(x)=3x + 20 and g(x)=3x, for x = 0, 1, 2, 3, 4, 5.

A.REI 10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

The graph below shows the height of a hot air balloon as a function of time. Explain what the point (50, 300) on this graph represents.

F.BF 3. Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology...

Students should recognize these transformations of the parent graph f(x)= x2: f(x)= x2 + k is a vertical translation f(x) = (x + k)2 is a horizontal translation f(x)= kx2 stretches the graph for k > 0 f(x)= –x2 reflects the graph over the x-axis.

Objectives (knowledge and skills) (Show link to standards in parenthesis after objective) The student will: 1.1 Model relationships between two quantities and interpret key features (F.IF.4). 1.2 Create a real-world scenario given a graphical representation or create a graph from a given

real-world scenario (F.IF.4, F.IF.5, N.Q.2, F.IF.9). 1.3 Compare and contrast continuous versus discrete functions (F.IF.4, F.IF.5). 1.4 Describe, from a graph, when it is increasing, decreasing and constant (F.IF.4, F.IF.5,

F.IF.9). 1.5 Determine which variable or circumstance refers to the independent and dependent variable

(F.IF.4, F.IF.5). 1.6 Choose and interpret the scale and origin of graphs and data displays (N.Q.1, N.Q.3). 1.7 Differentiate between a relation and a variety of functions, with an emphasis on linear

functions (N.Q.3, F.LE.1). 1.8 Identify, from choices, which graph represents a linear, quadratic, exponential, or absolute

value functions as well as non-functions (N.Q.3, F.IF.1, F.LE.1). 1.9 Draw a function, and determine which basic function models, based on a given situation

(F.IF.4).


1.10 Understand the relationship between the domain and range of a relation or function and write it as an inequality (F.IF.1).

1.11 Interpret the key features of graphs, including the rate of change, the zeros, and extrapolate their real-world meanings (F.IF.4, F.LE.3).

1.12 Determine the domain and range given a visual example or real-world situation (F.IF.5). 1.13 Compare properties of two functions each represented in a different way (F.IF.9). 1.14 Visualize vertical and horizontal changes in a graph given a parent function (emphasize

linear, but include those from 1.8) and a modified function (F.BF.3, F.IF.5). 1.15 Represent a function using a graph, expression, and/or table (F.IF.7, F.IF.9, A.REI.10). 1.16 Use function notation to evaluate linear equations/functions (F.IF.2). 1.17 Calculate and interpret the average rate of change of a function presented symbolically,

graphically, or as a table (F.IF.6, F.LE.1b). Instructional Support Materials

• iBook for iPad for entire unit • TI-84 Plus graphing calculator • TI-SmartView software • Geogebra and/or Geometer’s Sketchpad • Other supplemental resources as necessary


• Modeling • Direct instruction • Guided practice • Interactive models • Differentiated tasks • Electronic demonstrations • Cooperative learning • Performance tasks


• Pre-assessments • Quizzes • Unit tests • Projects / Mathematics labs & activities


Unit 2: Coordinate Geometry Time Frame: October & November Length of Unit: 5weeks Enduring Understandings Essential Questions

• Show that congruence is the geometric equivalence of equality.

• Coordinate Geometry depicts geometric relationships by using ordered pairs to represent points.

• Algebraic skills and concepts enable us to describe real world phenomena symbolically and graphically, and to model quantitative change.

• Rigid Motion, or isometries, (rotation, reflection, and translation) preserves distance and angle measures.

• The Pythagorean Theorem can be used to derive the Distance Formula.

• The various relationships between lines allows us to identify angles of a transversal creating opportunities to prove parallel and perpendicular lines.

• There are a variety of ways to find the measure of angles, segments, perimeter and areas.

• What is rigid motion? How is it used in Geometry?

• How can Coordinate Geometry be used to depict geometric relationships?

• How can Coordinate Geometry be used to describe various rigid motions?

• How can you prove lines are parallel on the coordinate plane?

• How can you prove lines are perpendicular on the coordinate plane?

• How do the properties of geometric figures influence their uses?

• What role does notation have in the construction and identification of geometric figures?

• How do we measure angles, segments, perimeter and area?


CCSS Explanations and Examples

G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Students may use geometry software and/or manipulatives to model transformations. Students may observe patterns and develop definitions of rotations, reflections, and translations.

G.CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines,

Students may use geometric simulations (computer software or graphing calculator) to explore theorems about lines and angles.


alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.

G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line

Students may use geometric software to make geometric constructions. Examples: ● Construct a triangle given the lengths of two sides and the measure of

the angle between the two sides.

● Construct the circumcenter of a given triangle.

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 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

For two-step equations, flow charts may be used to help students “undo” the order of operations to find the value of a variable. For example, this flow chart may be used to solve the equation 4x – 2 = 30.

Then students learn to solve equations by performing the same operation (except for division by zero) on both sides of the equal sign.

A.CED 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear ... functions

Here are some examples where students can create equations and inequalities.

a. (Two step equation) The bank charges a monthly fee of $2.25 for your Dad’s checking account and an additional $1.25 for each transaction with his debit card, whether used at an ATM machine or by using the card to make a purchase. He noticed a transaction charge of $13.50 on this month’s statement. He is trying to remember how many times he used the debit card. Can you use the information on the statement help him figure out how many transactions he made?

b. (Equations which require using the distributive property) Jessica

wanted to buy 7 small pizzas but she only had four $2 off coupons. So, she bought four with the discount and paid full price for the other three, and the bill came to $44.50. How much was each small


pizza?

c. (Inequality) The student council has set aside $6,000 to purchase the shirts. (They plan to sell them later at double the price.) How many shirts can they buy at the price they found online if the shipping costs are $14?

G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Students may use geometric simulation software to model figures or line segments. Examples: ● Given A(3, 2) and B(6, 11),

o Find the point that divides the line segment AB two-thirds of the way from A to B.

The point two-thirds of the way from A to B has x-coordinate two-thirds of the way from 3 to 6 and y coordinate two-thirds of the way from 2 to 11. So, (5, 8) is the point that is two-thirds from point A to point B.

o Find the midpoint of line segment AB.

Objectives (knowledge and skills) (Show link to standards in parenthesis after objective) The student will: 2.1 Derive the distance formula on the coordinate plane through the use of the Pythagorean

Theorem (review the coordinate plane and plotting points) (G.GPE.6, G.CO.1). 2.2 Show that congruence is the geometric equivalence of equality (G.CO.1). 2.3 Derive the segment bisector theorem on the coordinate plane through the use of the midpoint

formula (G.GPE.6, G.CO.1). 2.4 Use the distance formula to introduce and develop the Segment Addition Postulate and apply

to solve geometric problems (G.GPE.6, G.CO.1). 2.5 Investigate angle bisectors using prior knowledge (G.CO.12). 2.6 Develop and use the Angle Addition Postulate as well as adjacent angles (G.CO.1). 2.7 Construct intersecting lines (G.CO.4, G.CO.9). 2.8 Investigate linear pairs and vertical angles using the construction of intersecting lines using

protractors and technology (G.CO.4, G.CO.9). 2.9 Solve linear pair and vertical angle problems incorporating solving equations with variables

on both sides (A.REI.3, A.CED.1). 2.10 Discuss reflections and rotations of shapes in a coordinate plane (G.CO.4, G.CO.9). 2.11 Construct and write equations of parallel and perpendicular lines (lines rotated 90°), using

visual support (G.GPE.5, A.CED.1). 2.12 Discover the perpendicular bisector theorem using the coordinate plane (G.CO.9). 2.13 Discover the distance formula using the Pythagorean Theorem (G.CO.9, G.CO.12). 2.14 Investigate properties of parallel lines cut by a transversal using a protractor, coordinate

plane, and technology (G.CO.9).


Instructional Support Materials • TI-84 Plus graphing calculator • TI-SmartView software • Geogebra and/or Geometer’s Sketchpad • Algebra / Geometry text • Electronic textbook resources • Other supplemental resources as necessary






Unit 3: Systems of Linear Equations & Inequalities Time Frame: November & December Length of Unit: 4 weeks Enduring Understandings Essential Questions

• Systems can be solved using a variety of methods.

• Systems can be used to model relationships in the real world.

• Setting two functions equal to each other finds a common solution.

• Solutions of linear systems have a variety of real-world interpretations.

• What is a system? • How can the best method to solving a

system be determined? • Why are systems useful when analyzing

real world situations? • What is the difference between the

solutions of a system of equations versus inequalities?

• How can the use of systems be represented geometrically?


CCSS Explanations and Examples A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Examples: • −7

3𝑦 − 8 = 111

• 3𝑥 > 9 • 𝑎𝑥 + 7 = 12 • 3+𝑥

7= 𝑥−9

4

• Solve for x: 23𝑥 + 9 < 18

A-REI 5. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Example: Given that the sum of two numbers is 10 and their difference is 4, what are the numbers? Explain how your answer can be deduced from the fact that they two numbers, x and y, satisfy the equations x + y = 10 and x – y = 4.

A.REI 6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

The system solution methods can include but are not limited to graphical, elimination/linear combination, substitution, and modeling. Systems can be written algebraically or can be represented in context. Students may use graphing calculators, programs, or applets to model and find approximate solutions for systems of equations. Examples:

Your class is planning to raise money for a class trip to Washington, DC, by selling your own version of Connecticut Trail Mix. You find you can purchase a mixture of dried fruit for $3.25 per pound and a nut mixture for $5.50 per pound. The class plans to combine the dried fruit and nuts to make a mixture that costs $4.00 per pound, which will be sold at a higher price to make a profit. You anticipate you will need 180 pounds of trail mix. How many pounds of dried fruit and how many pounds of mixed nuts do you need?


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.*

Students need to understand that numerical solution methods (data in a table used to approximate an algebraic function) and graphical solution methods may produce approximate solutions, and algebraic solution methods produce precise solutions that can be represented graphically or numerically. Students may use graphing calculators or programs to generate tables of values, graph, or solve a variety of functions. Example: • Given the following equations determine the x value that results in an

equal output for both functions.

REI.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes

A.CED 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales

Students may collect data from water that is cooling using two thermometers, one measuring Celsius, the other Fahrenheit. From this they can create of the relationship and show that it can be modeled with a linear function.

A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

Example: • A club is selling hats and jackets as a fundraiser. Their budget is $1500

and they want to order at least 250 items. They must buy at least as many hats as they buy jackets. Each hat costs $5 and each jacket costs $8. o Write a system of inequalities to represent the situation. o Graph the inequalities. o If the club buys 150 hats and 100 jackets, will the conditions be

satisfied? o What is the maximum number of jackets they can buy and still

meet the conditions? G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. ONLY using this standard as the Pythagorean Theorem Inequality to classify triangles

Students may use graphing calculators or programs, tables, spreadsheets, or computer algebra systems to solve right triangle problems. Example: ● Find the height of a tree to the nearest tenth if the angle of elevation of

the sun is 28° and the shadow of the tree is 50 ft.


G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

Students may use geometric simulations (computer software or graphing calculator) to explore theorems about triangles.

Objectives (knowledge and skills) (Show link to standards in parenthesis after objective) The student will: 3.1 Write equations from application problems for systems of linear equations and inequalities

(A. CED.2). 3.2 Graph systems of linear equations and approximate its solution(s) (A.REI.6). 3.3 Solve systems of linear equations and calculate its solution(s) (A.REI.6). 3.3 Explain the real-world meaning of the point of intersection of two lines in a system of

equations (A.REI.11). 3.4 Interpret the solution to the systems of linear equations (A.CED.3). 3.5 Prove that, given a system of two equations in two variables, replacing one equation by the

sum of that equation and a multiple of the other produces a system with the same solutions (A.REI.5).

3.6 Graph linear inequalities in two variables (A.REI.12). 3.7 Create constraints from real-world problems of linear inequalities in two variables

(A.CED.3). 3.8 Interpret the real-world meaning of the solutions of systems of linear inequalities (A.CED.3). 3.9 Construct triangles in the coordinate plane and calculate the length of each side. From this,

students will classify triangles using the Pythagorean Theorem Inequality and the Triangle Inequality Theorem. See activity 2. (G.SRT.8, G.CO.11).

Instructional Support Materials

• TI-84 Plus graphing calculator • TI-SmartView software • Geogebra and/or Geometer’s Sketchpad • iBooks / iPads • Algebra / Geometry text • Electronic textbook resources • Other supplemental resources as necessary


Suggested Activities • Activity 1: Students will create a company and compare the selling price vs. the cost to

produce the item. They will write equations to represent their constraints. Then students will graph their constraints. After, students will discuss the interpretation of the intersection of their equations and its real-world meaning.

• Activity 2: Create three intersecting lines to form a triangle. Find the intersections

(vertices). Find the length of each side. Use the Pythagorean Theorem Inequality to classify the triangle. Use the Triangle Inequality Theorems to order the length of sides and the magnitude of its angles.






Unit 4: Triangles and Proofs Time Frame: December & January Length of Unit: 4 weeks Enduring Understandings Essential Questions

• Congruent figures are figures with congruent corresponding parts.

• Inductive and deductive reasoning are used to prove valid geometric statements true.

• A proof is a detailed description of the logical reasoning used to deduce a theorem from either definitions, postulates, or other previously proven theorems.

• Congruent figures are figures that have congruent corresponding sides are congruent and congruent corresponding angles are congruent.

• Congruent triangles are a specific form of similar triangles whose scale factor is one.

The properties, postulates, and theorems of triangles are essential in the development of coordinate geometry, additional polygons, and trigonometry. Two geometric objects are similar if one can be made congruent to the other using uniform scaling.

• Congruent triangles are a specific form of similar triangles whose scale factor is one. The Pythagorean Theorem and trigonometric ratios can be used to find a side length or angle measure of a right triangle.

• What does it mean for two figures to be congruent?

• How is rigid motion used to prove congruence? • How is coordinate geometry used to prove

congruence? • Why is it important to prove a statement true? • What information is essential to prove triangles

are congruent? • What is proof? Why do we need proofs? • How are segments and angles formed within

triangles related? • What similarities and differences exist between

triangles?


CCSS Explanations and Examples G.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

A rigid motion is a transformation of points in space consisting of a sequence of one or more translations, reflections, and/or rotations. Rigid motions are assumed to preserve distances and angle measures. Congruence of triangles Two triangles are said to be congruent if one can be exactly superimposed on the other by a rigid motion, and the congruence theorems specify the conditions under which this can occur.


G.CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

A rigid motion is a transformation of points in space consisting of a sequence of one or more translations, reflections, and/or rotations. Rigid motions are assumed to preserve distances and angle measures. Students may use geometric software to explore the effects of rigid motion on a figure(s).

G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions

G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

Students may use geometric simulations (computer software or graphing calculator) to explore theorems about triangles.

G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line

Students may use geometric software to make geometric constructions. Examples: ● Construct a triangle given the lengths of two sides and the measure of

the angle between the two sides.

● Construct the circumcenter of a given triangle.

G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Similarity postulates include SSS, SAS, and AA. Congruence postulates include SSS, SAS, ASA, AAS, and H-L. Students may use geometric simulation software to model transformations and demonstrate a sequence of transformations to show congruence or similarity of figures.

G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

Students may use geometric simulation software to make geometric constructions.


Objectives (knowledge and skills) (Show link to standards in parenthesis after objective) The student will: 4.1 Identify congruent figures using rigid motions (G.CO.6). 4.2 Define congruent figures and rigid motions (G.CO.7). 4.3 Given the coordinates of figures, students will determine whether figures are congruent using

the distance formula, protractors, and/or technology. If so, define which rigid motions were used (G.CO.7).

4.4 Formulize theorems about triangles including the Third Angles Theorem, Angle Sum Theorem, and Exterior Angle Theorem (G.CO.10).

4.5 Create 2-column triangle proofs using Side-Side-Side Theorem, Angle-Side-Angle Theorem, Side-Angle-Side Theorem, Angle-Side-Side Theorem, and Hypotenuse-Leg Theorem (G.SRT.5, G.CO.8, and G.CO.7).

4.6 Use technology, coordinate paper, and/or compasses to discover the inscribed/circumscribed circles of triangles (G.C.3).

4.7 Construct medians, angle bisectors, and perpendicular bisectors of triangles using technology and/or by hand (G.CO.12).

4.8 Create the midsegments of a triangle using technology and/or by hand (G.CO.10). 4.9 Interpret the medians, angle bisectors, perpendicular bisectors, midsegments and the point of

concurrency of a triangle and determine which point is the most important (G.CO.10, G.CO.12).

Instructional Support Materials

• TI-84 Plus graphing calculator • TI-SmartView software • Geogebra and/or Geometer’s Sketchpad • iBooks / iPads • Algebra / Geometry text • Graphing Calculator • Electronic textbook resources • Other supplemental resources as necessary


• Create a proof, cut out the steps, and have students reconstruct the proofs. • Modeling • Direct instruction • Guided practice • Interactive models • Differentiated tasks • Electronic demonstrations • Cooperative learning • Performance tasks


• Pre-assessments • Quizzes


• Unit tests • Projects / Mathematics labs & activities


Unit 5: Modeling Exponential Functions Time Frame: January & February Length of Unit: 4 weeks Enduring Understandings Essential Questions

• Exponents are used to represent complex expressions.

• Linear functions have a constant difference, whereas exponential functions have a constant ratio.

• How do exponential functions model real-world problems and their solutions?

• What characterizes exponential growth and decay?

• How can one differentiate an exponential model from a linear model given a real-world set of data?


CCSS Explanations and Examples F.IF 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. e. Graph exponential ... functions, showing intercepts and end behavior...

Exponential functions are similar to linear functions in that the y-intercept often represents a starting point. In exponential growth models, as the independent variable increases, the dependent variable increases at continually increasing rates. In exponential decay models, as the independent variable increases, the dependent variable approaches zero asymptotically.

F.IF 8b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02) t, y = (0.97)t, y = (1.01) 12t, y = (1.2) (t/10), and classify them as representing exponential functions.

All exponential functions may be written in the form f(x) = abx. In a later course students may learn that they may also be written in the form f(x) = aebx.

F.IF 6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.*

For linear functions, the rate of change is also the slope of the line. Example: A cell phone company uses the function y = 0.03x + 14.99 to determine the monthly charge, y, in dollars, for a customer using the phone of x minutes. Interpret the slope of this function in the context of this problem and indicate the appropriate units.

F.IF 3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers...

Recursive and explicit rules for sequences are first introduced in the context of chemistry. For example, the number of hydrogen atoms in a hydrocarbon is a function of the number of carbon atoms. This relationship may be defined by the recursive rule “add two to the previous number of hydrogen atoms” or explicitly as h = 2 + 2c. The function may also be represented in a table or a graph or with concrete models.


F.BF 2. Write ... geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.*

A recursive formula is An= r An-1. An explicit formula is An= A1 rn-1

F.LE 1. Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove ... that exponential functions grow by equal factors over equal intervals....

Example: Common differences for world population growth show an increasing trend, suggesting that an exponential model may be more appropriate than a linear one.

Years since 1980 Population (billions) Common differences

0 4.453 --- 1 4.529 .076 2 4.609 .080 3 4.690 .081 4 4.771 .081 5 4.852 .081 6 4.936 .084 7 5.022 .086 8 5.109 .087 9 5.196 .087

c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

Example: Technetium-99m is a drug taken by a patient and then used to study tumors in the brain, lungs and other parts of the body. A patient takes a 1000-mg pill. The data below shows how much active ingredient remains in the body over 6-hour time intervals.

Technetium-99m Decay # of 6-hour Time

Intervals Amount of Drug Remaining (mg)

0 1000 1 500 2 250 3 125

F.LE 2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

Example: Determine an exponential function of the form f(x) = abx using data points from the table. Graph the function and identify the key characteristics of the graph.

x f(x) 0 2 1 6 3 54

A.SSE 3c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as [1.15 (1/12)](12t) ≈1.012(12t) to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

In this example the annual growth factor is 1.15 and the monthly growth factor is 1.012.


F.IF 3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers...

Recursive and explicit rules for sequences are first introduced in the context of chemistry. For example, the number of hydrogen atoms in a hydrocarbon is a function of the number of carbon atoms. This relationship may be defined by the recursive rule “add two to the previous number of hydrogen atoms” or explicitly as h = 2 + 2c. The function may also be represent in a table or a graph or with concrete models.

F.BF 1. Write a function that describes a relationship between two quantities.* a. Determine an explicit expression, a recursive process, or steps for calculation from a context.

Example; You frequently go to the gym to work out lifting weights. You plan to gradually increase the size of the weights over the next month. You always put two plates that appear to be the same weight on each side of the bar. The plates are not labeled but you do know the bar weighs 20 kg. How can we express the total weight you lifted on any day?” Students will assign a variable for the weight of a plate (say w) and derive an expression for the total weight lifted, 4w +20 or its equivalent

F.BF 2. Write arithmetic and geometric sequences ... recursively and [arithmetic sequences] with an explicit formula, use them to model situations, and translate between the two forms.*

Arithmetic sequences may be introduced through geometric models. For example, the number of beams required to make a steel truss in the following pattern is an arithmetic sequence.

Arithmetic sequences are also found in patterns for integers which can be used to justify and reinforce rules for operations. For example in completing this pattern

5 * 4 = _______

5 * 3 = _______

5 * 2 = _______

5 * 1 = _______

5 * 0 = _______

5 * -1 = _______

5 * -2 = _______

Students will find an arithmetic sequence with a common difference of -5. This pattern illustrates that the product of a positive integer and a negative integer is negative. Compound interest is a good example of a geometric sequence. For example,

“You just won first prize in a poetry writing contest. If you take the $500 you won and invest it in a mutual fund earning 8% interest per year, about how long will it take for your money to double? “


Objectives (knowledge and skills) (Show link to standards in parenthesis after objective) The student will: 5.1 Graph exponential functions 𝑦 = 𝑎𝑏𝑥 and evaluate it form an equation (F.IF.7). 5.2 Identify the key features of exponential growth functions (F.IF.7). 5.3 Identify exponential functions in different representations (F.IF.7). 5.4 Compare linear and exponential functions (F.LE.1, F.LE.2). 5.5 Compare and contrast the features of exponential growth and decay functions (F.LE.1,

F.LE.2). 5.6 Compute simple and compound interest from real-world problems (A.SSE.3c). 5.7 Recognize geometric sequences from tables/ordered pairs (F.BF.3). 5.8 Identify and extend patterns using recursion (F.BF.2, F.LE.2, F.IF.3). 5.9 Write geometric sequences recursively with explicit formula (F.IF.3, F.BF.3, F.LE.2). 5.10 Compare geometric and arithmetic sequences (F.BF.2). 5.11 Interpret and compare the average rate of change for linear and non-linear functions

(F.IF.6). Instructional Support Materials

• TI-84 Plus graphing calculator • TI-SmartView software • Geogebra and/or Geometer’s

Sketchpad • iBooks / iPads • Algebra text • Other supplemental resources as

necessary

Suggested Instructional Strategies • Modeling • Direct instruction • Guided practice • Interactive models • Differentiated tasks • Electronic demonstrations • Cooperative learning • Performance tasks

Suggested Activities

• CHET activity to discover exponential growth Suggested Assessment Methods (Include use of school-wide analytic and course specific rubrics)



Unit 6: Properties and Areas of Polygons Time Frame: March Length of Unit: 4 weeks Enduring Understandings Essential Questions

• Regular polygons are seen in the real-world and their symmetric properties play an important role.

• There is a relationship between the interior angles and sides of a convex polygon.

• The largest set of a specific polygon is the quadrilaterals, their properties linking and differentiating the various quadrilateral types.

• How do the properties of polygons influence their uses?

• What similarities exist between a circle and a regular polygon?

• How are the area formulas of regular polygons determined?

• How do we find the area of an irregularly shaped shaded region?

• How can the properties of similarity and congruency be applied to polygons?

• How can the properties of parallelograms be used in real world situations?

• How can the knowledge of a basic parallelogram be used to find the perimeter and area of real-life objects?

• What properties can link various types of quadrilaterals together?

• What properties of triangles can be applied to other polygons? What properties can differentiate the various types of quadrilaterals?


CCSS Explanations and Examples

G.CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Students may use geometric simulations (computer software or graphing calculator) to explore theorems about parallelograms.

G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Students may use geometry software and/or manipulatives to model transformations.

G.GPE.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).

Lines can be horizontal, vertical, or neither. Students may use a variety of different methods to construct a parallel or perpendicular line to a given line and calculate the slopes to compare the relationships.


G.GPE.4 Use coordinate to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

Students may use geometric simulation software to model figures and prove simple geometric theorems. Example:

● Use slope and distance formula to verify the polygon formed by connecting the points (-3, -2), (5, 3), (9, 9), (1, 4) is a parallelogram.

G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Students may use geometric simulation software to model figures or line segments. Examples: ● Given A(3, 2) and B(6, 11),

o Find the point that divides the line segment AB two-thirds of the way from A to B.

The point two-thirds of the way from A to B has x-coordinate two-thirds of the way from 3 to 6 and y coordinate two-thirds of the way from 2 to 11.

So, (5, 8) is the point that is two-thirds from point A to point B.

o Find the midpoint of line segment AB.

A.REI 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

A.CED 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear ... functions

Objectives (knowledge and skills) (Show link to standards in parenthesis after objective) The student will: 6.1 Identify quadrilaterals based on their properties (G.CO.11). 6.2 Use coordinates to investigate properties of quadrilaterals (G.GPE.4, G.GPE.5). 6.3 Prove theorems about parallelograms (G.O.11). 6.4 Prove theorems about parallelograms using the coordinate plane (G.GPE.4, G.GPE.5). 6.5 Investigate the interior and exterior angles of polygons, discovering the rotational

measurement involved in producing congruent figures (G.CO.3). 6.6 Solve problems for various parts of polygons, including sides, angles, and diagonals using

linear equations (A.REI.3, A.CED.1).


6.7 Develop and apply area formulas for triangles and quadrilaterals (A.SSE.1, A.CED.4). 6.8 Solve problems involving perimeters and areas of triangles and quadrilaterals (A.SSE.1,

A.CED.4). 6.9 Determine and apply area formulas for a circle and regular polygons (G.MD.1). 6.10 Use the area addition postulate to find the areas of composite figures and use composite

figures to find the areas of irregular shapes (G.MG.3). 6.11 Find perimeters and areas of figures in the coordinate plane (G.GPE.7). 6.12 Describe the effects of perimeter and area when one or more dimensions of a figure are

changed (G.GPE.7). Instructional Support Materials

• TI-84 Plus graphing calculator • TI-SmartView software • Geogebra and/or Geometer’s Sketchpad • iBooks / iPads • Algebra / Geometry text • Other supplemental resources as necessary






Unit 7: Surface Area and Volume Time Frame: April & May Length of Unit: 4 weeks Enduring Understandings Essential Questions

• There are three different types of measurement based upon dimensionality: length, area, and volume.

• The intersection of a solid and a plane can be determined by visualizing how the plane slices the solid to form a two-dimensional cross section.

• Three dimensional geometric objects are closely related to real world objects and using the properties of similarity we can represent these real world objects by constructing scaled models.

• What types of three dimensional solids exist?

• How are do the areas of polygons related to theand the volumes of their respective solids compare?

• How do the properties of geometric solids influence their uses?

• How do the surface areas and volumes of similar solids compare?

Priority and Supporting CCSS Explanations and Examples*

G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.*

Missing measures can include but are not limited to slant height, altitude, height, diagonal of a prism, edge length, and radius.

G.GMD.A.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.

Cavalieri’s principle is if two solids have the same height and the same cross-sectional area at every level, then they have the same volume.

G.GMD.A.2 (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

G.GMD.A.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

G.GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Students may use geometric simulation software to model figures and create cross sectional views. Example: ● Identify the shape of the vertical, horizontal, and other cross

sections of a cylinder. G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).*

Students may use simulation software and modeling software to explore which model best describes a set of data or situation.


Priority and Supporting CCSS Explanations and Examples*

G.MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).*

Students may use simulation software and modeling software to explore which model best describes a set of data or situation.

G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.*

G.GMD.B.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Objectives (knowledge and skills) (Show link to standards in parenthesis after objective) The student will: 7.1 Classify and identify polyhedra and Platonic solids using their properties (G.MG.1). 7.62 Visualize and identify the cross-section of various three-dimensional figures. (G.GMD.4). 7.2 3 Define surface area, lateral area, volume, lateral faces, bases, altitude or height, vertex,

edges, slant (G.MG.1). 7.74 Use Cavalieri’s principle to calculate the volume of similar solids (G.GMD.1). 7.3 5 Calculate volume, surface area, and lateral area for cylinders, pyramids, cones, spheres, and

complex figures (that is, figures that contain more than one prism, pyramid, cylinder, or cone) (G.GMD.3).

7.4 6 Find missing parts of each formula including slant height, altitude, diagonals of prism, edge length and radius (G.GMD.3).

7.5 7 Make connections between one-dimensional, two-dimensional, and three-dimensional figures, such as the ratio of sides/perimeters/areas and volumes (G.GMD.3).

6.6 Visualize and identify the cross-section of various three-dimensional figures. (6.7 Use Cavalieri’s principal to calculate the volume of similar solids. (G.GMD.1)7.8 Use geometric software to apply concepts of density based on area and volume (G.MG.2). Instructional Support Materials

• TI-84 Plus graphing calculator • TI-SmartView software • Geogebra and/or Geometer’s Sketchpad • iBooks / iPads • Algebra / Geometry text • Other supplemental resources as necessary

Suggested Instructional Strategies • Modeling • Direct instruction • Guided practice • Interactive models • Differentiated tasks • Electronic demonstrations • Cooperative learning • Performance tasks


• Pre-assessments

Formatted: Indent: Left: 0", First line: 0"


• Quizzes • Unit tests • Projects / Mathematics labs & activities


Unit 8: Descriptive Statistics Time Frame: May & June Length of Unit: 4 weeks Enduring Understandings Essential Questions

• Exploratory analysis of data makes use of graphical and numerical techniques to study patterns and departures from patterns.

• Regression is an effective model for prediction.

• There is a difference between causation and correlation.

• What interpretations can be made from an organized set of data?

• What organizational strategy provides the best representation of data?

• To what extent can we predict the future? • Is correlation ever causation? • How can modeling data help us to

understand patterns?


CCSS Explanations and Examples (not all examples apply to Algebra 1)

S-ID 1 Represent data with plots on the real number line (dot plots, histograms, and box plots).

Students may compare and contrast the advantage of each of these representations.

S-ID 2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

Students may use spreadsheets, graphing calculators and statistical software for calculations, summaries, and comparisons of data sets.

Example: Given a set of test scores: 99, 96, 94, 93, 90, 88, 86, 77, 70, 68, find the mean, median and standard deviation. Explain how the values vary about the mean and median. What information does this give the teacher?

S-ID 3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

Example: After the 2009-2010 NBA season LeBron James switched teams from the Cleveland Cavaliers to the Miami Heat, and he remained the top scorer (in points per game) in his first year in Miami. Compare team statistics for Cleveland (2009-2010) and Miami (2010-2011) for all players who averaged at least 10 minutes per game. Using the 1.5 X IQR rule, determine for which team and year James’s performance may be considered an outlier.

S-ID 6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. S-ID 6a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data... Use given functions or choose a function suggested by the context. Emphasize linear and exponential models.

Students may use spreadsheets, graphing calculators, and statistical software to represent data, describe how the variables are related, fit functions to data, perform regressions, and calculate residuals. Example: Make a scatter plot of data showing the rise in sea level over the past century. Fit a trend line and use it to predict the sea level in the year 2020.


Priority (bold) and Supporting CCSS Explanations and Examples (not all examples apply to Algebra 1)

S.ID.6b Informally assess the fit of a function by plotting and analyzing residuals. S.ID.6c Fit a linear function for a scatter plot that suggest a linear association

S.ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

Students may use spreadsheets or graphing calculators to create representations of data sets and create linear models. Example: • Lisa lights a candle and records its height in inches every hour. The results recorded as (time, height) are (0, 20), (1, 18.3), (2, 16.6), (3, 14.9), (4, 13.2), (5, 11.5), (7, 8.1), (9, 4.7), and (10, 3). Express the candle’s height (h) as a function of time (t) and state the meaning of the slope and the intercept in terms of the burning candle.

S-ID 8. Compute (using technology) and interpret the correlation coefficient of a linear fit.

Example: Collect height, shoe-size, and wrist circumference data for each student. Determine the best way to display the data. Answer the following questions: Is there a correlation between any two of the three indicators? Is there a correlation between all three indicators? What patterns and trends are apparent in the data? What inferences can be made from the data?

S.ID.9 Distinguish between correlation and causation

Some data leads observers to believe that there is a cause and effect relationship when a strong relationship is observed. Students should be careful not to assume that correlation implies causation. The determination that one thing causes another requires a controlled randomized experiment. Example: Diane did a study for a health class about the effects of a student’s end-of-year math test scores on height. Based on a graph of her data, she found that there was a direct relationship between students’ math scores and height. She concluded that “doing well on your end-of-course math tests makes you tall.” Is this conclusion justified? Explain any flaws in Diane’s reasoning.

Objectives (knowledge and skills) (Show link to standards in parenthesis after objective) The student will: 8.1 Organize data in tables and graphs and choose a table and graph to display the data (S.ID.1). 8.2 Create stem-and-leaf plots, frequency tables and histograms (S.ID.1). 8.3 Describe the central tendency of a data set (S.ID.1, S.ID.2, S.ID.3). 8.4 Create and interpret box-and-whisker plots (S.ID.1, S.ID.2, S.ID.3). 8.5 Compare data sets and be able to summarize the similarities and difference between the shape and

measures of centers and spreads of the data sets (S.ID.2, S.ID.3). 8.6 Use linear and exponential regression models to determine the equation of best-fit for given data

(S.ID.6a, S.ID.6b, S.ID.6c).


8.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data (S.ID.7).

8.8 (Extension) Use technology to compute and interpret the correlation coefficient and the coefficient of determination (S.ID.8).

8.9 (Extension) Distinguish between correlation and causation (S.ID.9). Instructional Support Materials

• TI-84 Plus graphing calculator • TI-SmartView software • Geogebra and/or Geometer’s Sketchpad • iBooks / iPads • Algebra text • Other supplemental resources as necessary






Documents

Integrated Math I Page 1 - Wethersfield Public Schools Name: Integrated Math I Department: Mathematics Grade(s): 10-11 Level(s): 2 Course Number(s): 30602 Credits: 1 Course Description: