Topic: Solving Simple Equations – One Variable



Plumsted Township Public Schools

Mathematics Program

New Egypt High School – Algebra I Curriculum

Department Philosophy

“To compete in today’s global, information-based economy, students must be able to solve real problems, reason effectively, and make logical connections.”[1] The mathematics department of the Plumsted Township School District believes that math is an essential part of society. Our math program offers a variety of experiences which correlate to authentic situations that exist in our community and beyond. The mathematics curriculum supports all students’ academic needs and provides a meaningful, thorough, and efficient education through the development of critical thinking and problem solving skills. Content provided in our program is congruent to the National (NCTM) and State Core Content Curriculum Standards (NJCCCS). In addition, our courses are designed to be flexible in addressing multiple assessments and cross content curriculum integration.

Goals:

The learners will:

• Devise strategies to solve meaningful problems that relate to authentic situations in society

• Understand and use mathematical processes to solve problems

• Use technology to address tasks within the mathematics domain

• Foster logical reasoning skills

• Develop conclusions using problem solving skills

• Use critical thinking skills to construct theories based on real life applications

• Identify the importance of mathematics and its benefits within human, physical, environmental, and social systems

• Model problems to find solutions within the mathematics domain

Assessment

The Plumsted Township School District mathematics program exposes students to a challenging sequence of courses which are aligned to the mission of the school district, as well as state and national standards. Students are given various learning assessments throughout their courses that measure student learning and identify deficiencies which target areas of individualized and program development. Curriculum documents were created by teachers and administrators to ensure integration of our commitment to implementing cutting-edge pedagogical and educational methodology within the academic program. Students in 3rd and 4th grade take the New Jersey Assessment of Skills and Knowledge (NJASK), students in the 8th grade take the Grade Eight Proficiency Assessment (GEPA), and students in the eleventh grade take the High School Proficiency Assessment (HSPA). Results are used to address needs within the curriculum.

New Jersey Core Content Curriculum Standards

4.1. Number and Numerical Operations

A. Number Sense

B. Numerical Operations

C. Estimation

4.2. Geometry and Measurement

A. Geometric Properties

B. Transforming Shapes

C. Coordinate Geometry

D. Units of Measurement

E. Measuring Geometric Objects

4.3. Patterns and Algebra

A. Patterns and Relationships

B. Functions

C. Modeling

D. Procedures

4.4. Data Analysis, Probability, and Discrete Mathematics

A. Data Analysis (Statistics)

B. Probability

C. Discrete Mathematics--Systematic Listing and Counting

D. Discrete Mathematics--Vertex-Edge Graphs and

Algorithms

4.5. Mathematical Processes

A. Problem Solving

B. Communication

C. Connections

D. Reasoning

E. Representations

F. Technology

Scope and Sequence

I. Solving Equations – One Variable

a. Properties of Operations

b. One step equations

c. Multi-step equations

d. Solving for variables on both sides of the equation

e. Formula’s – For science

II. Ratios

a. Rational Numbers

b. Proportional Reasoning

c. Percents

d. Percent of Change

e. Direct and inverse variations

III. Geometry

a. Angles and Triangles

b. Pythagorean Theorem

c. Similar Triangles

d. Trigonometric Ratios

IV. Graphing

a. Coordinate Plane

b. Domain & Range (Mapping, Functions

c. Patterns

V. Slope

a. Must solve m = (y2 – y1) / (x2 – x1) for any variable

b. Must find slope and graph line from y = mx + b.

c. Graphing Calculator Lab

d. Parallel and perpendicular

e. Scatter plot/ regression/ line of best fit

VI. Linear Inequalities

a. Linear Inequalities

b. Compound inequalities

VII. Systems of Equations

a. Graphing Systems of Equations

b. Substitution/Elimination Method

c. Graphing Systems of Inequalities

VIII. Statistics

a. Central Tendency

b. Weighted Averages

c. Probabilities and Odds

IX. Monomials

a. Multiplying/ Dividing

b. Exponential Properties

c. Introduction to Radicals

X. Polynomials

a. Add/Sub

b. Multiply/ Distribution

c. Factoring

XI. Quadratics

a. Graph

b. Finding Roots

▪ Graphing

▪ Factoring

▪ Quadratic Formula

|Topic: Solving Equations – One Variable |

|Enduring Understanding: The Learner Will Be Able To: |

|Demonstrate the crucial skills and concepts involved in solving equations. |

|Suggested |Concepts |Objectives |Conceptual Understanding(s) |NJCCCS (CPI) |

|Time | | | | |

| | Properties of Operations |TLW: |Evaluate the following expression. Name each property used in each step. | |

| |Essential Question: |Recognize each property of operations: |(8 x 3 – 19 + 5) + (3² + 8 x 4) | |

| |Why do we need mathematical properties? |- Additive Identity |Simplify 4(3x + 2) + 2(x + 3) | |

| |Why does 0 not have a multiplicative identity? |- Multiplicative Identity |Name the property used in each step: | |

| |Can 1 be the additive identity? |- Multiplication Property of Zero |ab(a + b) = (ab)a + (ab)b | |

| |Does the commutative property work with all |- Substitution |= a(ab) + (ab)b | |

| |operations? |- Multiplicative Inverse |= (a x a)b + a(b x b) | |

| | |- Reflexive |= a²b + ab² | |

| | |- Symmetric |4. Suppose the operation * is defined for all numbers a and b as a * b =| |

| | |- Transitive |a + 2b. Is the operation * commutative? Give examples to support your | |

| | |- Associative |answer. | |

| | |- Distributive | | |

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| | |Apply properties to simplified expressions and equations | | |

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| |One step equations |TLW: |Solve each equation: | |

| |Essential Question: |Analyze components of a one-step equation. |m – (-13) = 37 | |

| |What effect does information have on a situation? |Evaluate one-step equations using addition and |y = (-7) = -19 | |

| |How do we justify an answer? |subtraction. |-4r = -28 | |

| | |Evaluate one-step equations using multiplication and |2/5 t = -10 | |

| | |division. |Write an equation and solve. In 1995, the long distance company Sprint | |

| | |Solve any one-step equation. |introduced Sprint Sense, a plan in which long distance calls placed on | |

| | | |weekends cost only $0.10 per minute. | |

| | | |How long could you talk for $2.30? | |

| | | |What would be the cost of an 18 minute call? | |

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| |Multi step equations |TLW: |Solve each equation: | |

| |Essential Question: |Apply order of operations in multi-step equations. |5n + 6 = -4 | |

| |What effect does information have on a situation? |Evaluate equations involving more than one operation. |-9 – p/4 = 5 | |

| |How do we justify an answer? |Model a real-life situation with a multi-step equation. |3/2 a – 8 = 11 | |

| | |Evaluate model by working backwards. |(4t – 5)/-9 = 7 | |

| | | |Find three consecutive odd integers whose sum is 117. | |

| | | |The lengths of the sides of a quadrilateral are consecutive even integers.| |

| | | |Twice the length of the shortest side plus the length of the longest side | |

| | | |is 120 inches. Find the lengths of all the sides. | |

| |Solving for variables on both sides of the equation |TLW: |Solve each equation: | |

| |Essential Question: |Evaluate equations with the variable on both sides. |8y – 10 = -3y + 2 | |

| |What is the difference between an equation whose |Analyze equations containing grouping symbols. |5x – 7 = 5(x – 2) + 3 | |

| |solution set is the empty set and an equation that | |3/2 x – x = 4 + ½ x | |

| |is an identity? | |2. Write an equation and solve. One fifth of a number plus five times | |

| | | |that number is equal to seven times the number less 18. Find the number. | |

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| |Formula’s – for science |TLW: |Solve each formula for the variable specified: | |

| |Essential Question: |Evaluate equations and formulas for a specified variable. |d = rt, for r | |

| |Why would a scientist want/need to change a formula?| |d = vt + ½ at², for v | |

|Topic: Ratios |

|Enduring Understanding: The Learner Will Be Able To: |

|Suggested |Concepts |Objectives |Conceptual Understanding |NJCCCS |

|Time | | | |(CPI) |

| |Rational Numbers |TLW: |Write the numbers in each set in order from least to greatest. | |

| |Essential Question: |Label components of ratios: num./den. => $/hr. => |6/7, 2/3, 3/8 | |

| |What purpose do rational numbers serve? |mi./gal. |6.7, -5/7, 6/13 | |

| |How do you apply ratios and use them with |Compare and contrast values in rational form. |4/5, 9/10, 0.7 | |

| |mathematical operations? |Addition/Subtraction |Find a number between 2/5 and 7/2. | |

| | |Multiplication/Division |Evaluate the following: | |

| | | |¾ + (-4/5) + 2/5 = _____ | |

| | | |-1/2 – 2/3 = _____ | |

| | | |5/6(-2/5) = _____ | |

| | | |-9 ÷ (-10/27) = _____ | |

| | | |In one week, the Intel Corporation’s stock dropped 2 ¼ points. By | |

| | | |December 13, the stock had dropped another 2 1/8 points. How many | |

| | | |points did the stock drop in this time period? (4 3/8 pts) | |

| | | |The length of a flag is called the fly, and the width is called the | |

| | | |hoist. The blue rectangle in the United States flag is called the | |

| | | |union. The length of the union is 2/5 of the fly, and the width of | |

| | | |the union is 7/13 of the hoist. If the fly of the United States flag| |

| | | |is 3 feet, how long is the union? (1 1/5 feet) | |

| |Proportional Reasoning |TLW: |x/9 = -7/16 | |

| |Essential Question: |Solve a proportion. |(m + 9)/5 = (m – 10)/11 | |

| |What applications would you use proportions for and |Compare a proportion to a multi-step equation. |When a pair of blue jeans is made, the leftover denim scraps can be | |

| |justify? |Analyze use of proportions from real-life models. |recycled to make stationary, pencils, and more denim. One pound of | |

| |What is the justification for the connection of | |denim is left after making every 5 pairs of jeans. How many pounds | |

| |proportional reasoning to mathematical equations? | |of denim would be left from 250 pairs of jeans? (50 lbs.) | |

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| |Percents |TLW: |Write each ratio as a percent:\ | |

| |Essential Question: |Manipulate ratios into percentages. |2/3 | |

| |What methods are used in finding pieces of a whole? |Solve simple interest tasks. |5/6 | |

| |How? | |6/20 | |

| | | |35 is what percent of 70? | |

| | | |What number is 25% of 56? | |

| | | |Which earns more interest; $1500 at 10% interest for one year or $500| |

| | | |at 4% interest for 10 years? | |

| |Percent of Change |TLW: |The original selling price of a new sports video was $65.00. Due to | |

| |Essential Question: |Solve problems involving percent of increase/decrease. |demand, the price was increased to $87.75. What was the percent of | |

| |What are the applications and real-world usage of |Solve problems involving discounts or sales tax. |increase? (35%) | |

| |percentages? | |A sweater that costs $55.00 is discounted 15%. The sales tax is 6%. | |

| |How do you find these changes? | |What is the final price of the sweater? | |

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| | Direct and Inverse Variations |TLW: |Julio’s wages vary directly as the number of hours that he works. If| |

| |Essential Question: |Define constant of variation. |his wages for 5 hours are $29.75, how much will they be for 30 hours?| |

| |How do direct and inverse variations differ? |Develop methods to solve direct and inverse variations. |The length of a violin string varies inversely as the frequency of | |

| | | |its vibrations. A violin string 10 inches long vibrates at a | |

| | | |frequency of 512 cycles per second. Find the frequency of an 8-inch | |

| | | |string. | |

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|Topic: Geometry |

|Enduring Understanding: The Learner Will Be Able To: |

|Suggested |Concepts |Objectives |Core Activities |NJCCCS |

|Time | | | |(CPI) |

| |Angles and Triangles |TLW: |The measure of an angle is three times the measure of its supplement.| |

| |Essential Question: |Find the complement and supplement of an angle. |Find the measure of each angle. | |

| |What are the algebraic components of triangles? |Find the measure of the third angle of a triangle given |The measure of an angle is 34º greater than its complement. Find the| |

| |How can they be manipulated? |the measures of the other 2 angles. |measure of each angle. | |

| | |Classify various triangles. |What are the measures of the base angles of an isosceles triangle in | |

| | | |which the vertex angle measures 45º? | |

| | | |The measures of the angles of a triangle are given as xº, 2xº, and | |

| | | |3xº. What are the measures of each angle? Classify the triangle. | |

| |Pythagorean Theorem |TLW: |Find the length of the hypotenuse of a right triangle if the leg | |

| |Essential Question: |Label sides of a right triangle. |measurements are 5 and 12. | |

| |What information identifies right triangles and how |Apply the Pythagorean Theorem to solve problems. |Find the length of side a if b = 9 and c = 21. | |

| |can you use that information to solve for each part? | |Would a triangle with side lengths 6, 8, and 10 form a right | |

| | | |triangle? | |

| | | |On a baseball diamond, the distance from one base to the next is 90 | |

| | | |feet. What is the distance from home plate to second base? | |

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| |3. Similar Triangles | | | |

| |Essential Question: |TLW: | | |

| |Can you justify the relationship between ratios and | | | |

| |similar triangles? Explain. | | | |

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| |4. Trigonometric Ratios |TLW: |Example: | |

| |Essential Question: |Label sides of right triangle based on selected angle. |Given three pipe cleaners how many different triangles can you make, | |

| |Is there a relationship between sides and angles? |Define trigonometric functions as ratios. |using the entire length of each pipe cleaner as one side? (one) | |

| |Explain. |Evaluate measurement of angles and sides. |Compare the triangle you constructed with the one your neighbor | |

| | | |constructed. What do you notice about the angles in each triangle? | |

| | | |What was the common link before you made the triangles? (the side | |

| | | |lengths were all the same) Why do you think then that the angles | |

| | | |came out to be the same? | |

| | | |Be sure to develop the fact that sides and angles have a | |

| | | |relationship, before the relationship is defined. | |

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| | | |Given a right triangle ABC where angle A = 90 deg. and angle B is (. | |

| | | |Label the sides of the triangle as opposite, adjacent, hypotenuse. | |

| | | |Then write the trigonometric functions, which can be described by | |

| | | |using the above information. | |

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| | | |Note: Gather some standard right triangles and have students become | |

| | | |comfortable using the trig functions and their calculators to solve | |

| | | |for missing sides. Then develop a working understanding of the | |

| | | |inverse trig function such that a student would be able to use a | |

| | | |calculator and solve for a missing angle. | |

|Topic: Graphing |

|Enduring Understanding: The Learner Will Be Able To: |

|1. Apply algebraic concepts and connect them to their geometric representation. |

|Suggested |Concepts |Objectives |Core Activities |NJCCCS |

|Time | | | |(CPI) |

| |Coordinate Plane |TLW: | | |

| |Essential Question: |Identify and graph ordered pairs on a coordinate plane. |Plot the polling points on a coordinate plane: (-3, -5), (6, 0), (0,| |

| |How is the coordinate plane useful in representing |Translate graphs into ordered pairs. |-10), (4, -2). | |

| |data? |Reconstruct the coordinate plane to solve given task. | | |

| |Can you interpret the meaning of points in a plane? |Summarize and support independent and dependent axes. |Given a grid develop a student understanding of location. | |

| |How can you tell which quadrant a point is in by | | | |

| |just looking at the signs of the coordinate? | |Redefine the understanding with mathematical terms and notation…origin, | |

| | | |x-axis, y-axis, (x,y), etc. | |

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| | | |Students must then be able to read a graph. Given a graph students should| |

| | | |identify the location of the vertices. | |

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| | | |Example: | |

| | | |Small Wonder is a robot. She is standing in a room with a grid on the | |

| | | |floor and will only walk forward and step on the intersections of the | |

| | | |grid. Use a coordinate plane to map her path marking each stop with a | |

| | | |vertex. She walks up three steps and stops. Then she turns left and | |

| | | |walks 4 steps and stops. Small Wonder turns left again and walks 5 steps | |

| | | |before stopping. She then turns left and takes 7 steps and stops. Lastly| |

| | | |she turns left and takes 2 steps to her final destination. | |

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| | | |Graph her progress. | |

| | | |Label each vertex | |

| | | |Make a table of x values and corresponding y values. | |

| | | |What would have been a better way for Small Wonder to get where she was | |

| | | |going? | |

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| | | |Example: | |

| | | |Andy sells widgets. On Monday he sold 8 widgets. On Tuesday and Thursday | |

| | | |he sold 10 widgets each day. Wednesday Andy went to the dentist and | |

| | | |didn’t sell a single widget, but on Friday he sold 15. | |

| | | |Construct a graph using the coordinate plane and clear labels so that Beth| |

| | | |can analyze Andy’s sales per day. | |

| | | |In general if a mathematician is graphing information they will organize | |

| | | |the data into independent and dependant information. Independent data is | |

| | | |then generally graphed on the x-axis and will change in constant | |

| | | |intervals, which are not affected by any other information. What | |

| | | |information here is independent and why? (days of the week) | |

| | | |Therefore dependant data is generally found on the y-axis and is | |

| | | |information that is affected by the other information that is given. | |

| | | |Which information is the dependant data? Why? | |

| |Domain and Range -> Mapping , Functions |TLW: |Examples of independent domains: | |

| |Essential Question: |Define domain (indep) and range (dep) for a given |Time | |

| |Why would you need to manipulate data from one form |relation. |Sample Size (# of times a coin is flipped) | |

| |to another? |Generate tables, mappings, graphs, and ordered pairs from |Days of a week | |

| |Can one domain produce two different ranges? |any given information. |Examples of dependant ranges | |

| |Justify. |Define a relation as a function. |Money | |

| | |Introduce functional notation. |Successful outcomes (# of times a coin lands heads up) | |

| | |Evaluate the domain of a function to develop a range. |Sales | |

| | | |Therefore when combined you would have $/hr., Successful outcomes/Sample | |

| | | |size, and Sales/Day respectively. | |

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| | | |Information from any of these sources or others need to be charted in an | |

| | | |x,y table, mapped from a set x to a set y, and lastly graphed in the x,y | |

| | | |plane. | |

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| | | |Now that students have a working understanding of Independent (x, domain) | |

| | | |and Dependant(y, range) information the formal terms of relationship, | |

| | | |function, and function notation are needed to refine student | |

| | | |understandings. | |

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| | | |Maria earns ten dollars a day plus $6.75 per hour. | |

| | | |Create an x, y table to show a five hour day. | |

| | | |Convert the table to a mapping. | |

| | | |Graph the relationship. | |

| | | |Write a function to find out how much Maria will earn in x hours. | |

| |Patterns |TLW: | | |

| |Essential Question: |Write an equation in functional notation for a relation | | |

| |How does a pattern relate to a function? |given in a table. | | |

| |How are patterns related to the real world? | | | |

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|Topic: Slope |

|Enduring Understanding: The Learner Will Be Able To: |

|Analyze and conceptualize the behavior of linear equations. |

|Suggested |Concepts |Objectives |Core Activities |NJCCCS |

|Time | | | |(CPI) |

| |Solve m=(Y2-Y1)/ (X2-X1) for any variable |TLW: |1. What does it mean for a variable to have a subscript? i.e. y2 (The| |

| |Essential Question: |Know formula for slope. |subscript informs the reader that a specific value of that variable is | |

| |What is slope? |Define slope as the ratio of independent –vs- dependent |being used and that variable may be treated as a constant number.) | |

| |Why is slope important? |variables. | | |

| |What are the practical uses of slope? |Manipulate slope formula. |2. Given a graph that shows the money Gerald earns per hour worked, | |

| |Why do vertical lines have no slope? |Positive/negative/zero/no slope. |and the slope formula is | |

| | | |m = (Y2-Y1)/ (X2-X1). | |

| | | |a. What is the meaning of subtracting Y2-Y1 | |

| | | |in this situation? | |

| | | |b. What is the meaning of subtracting X2-X1 | |

| | | |in this situation? | |

| | | |c. In this situation what are the units of slope? | |

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| | | |3. The slope formula is m = (Y2-Y1)/ (X2-X1). | |

| | | |a. Why do the x’s and y’s have subscripts? | |

| | | |b. What would the purpose of switching the | |

| | | |notation to m = (Y - Y1)/ (X - X1), and list | |

| | | |all constant terms? (constant terms are | |

| | | |m, y1, and x1) | |

| | | |c. If the formula is then multiplied on both | |

| | | |sides by (X – X1) you receive | |

| | | |m(X - X1) = (Y - Y1) what is the name of | |

| | | |the slope formula in this form? | |

| | | |Note: Now that the student has begun to manipulate the slope formula | |

| | | |and has identified that once points are determined on a line that the | |

| | | |slope becomes constant the jump to | |

| | | |y = mx – mx1 + y1 is not a far one and as long as the understanding of | |

| | | |constant pieces has been formed the development of y = mx + b notation | |

| | | |where b = – mx1 + y1 is attainable. | |

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| | | |4. Graph the following equations. | |

| | | |a. y = x | |

| | | |b. y = 3x | |

| | | |c. y = -x | |

| | | |d. y = (1/3)x | |

| | | |e. y = 3 | |

| | | |f. 3 = x | |

| | | |Explain how a,b differ? | |

| | | |a,c? a,d? a.e? a,f? | |

| |Find slope and graph line from y=mx+b |TLW: |1. Given a graph of velocity (on the y-axis) versus time (on the | |

| |Essential Question: |Identify point-slope form, slope-intercept form. |x-axis). What is the meaning of the y-intercept? (it is the starting | |

| |Why are algebraic manipulations essential to formula|Define intercept. |speed because time or x = 0) What is the meaning for an x value that | |

| |development? |Deconstruct formulas to obtain points, slope. |crosses the x-intercept? (it tells you at what time the object has | |

| |How are each of the forms applied? |Graph a line given any linear equation. |stopped.) | |

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| | | |2. Given y = (2/3)x + 5. Find the x and y intercepts. Create an x | |

| | | |and y table with 5 points. Graph the line. | |

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| | | |3. Graph the following. | |

| | | |a. y = 2x – 4 | |

| | | |b. y – 4 = 2(x – 1) | |

| | | |c. 2/7 = (y -3)/(x + 1) | |

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| |Graphing Calculator Lab |TLW: | | |

| |Essential Question: |Compare and contrast various lines equations to understand| | |

| |How does slope affect the behavior of the line of a |how different functions behave. | | |

| |function? |Explore and discover linear behaviors. ( x = 3, y = 7) | | |

| |What are the characteristics of two lines? |Predict behaviors based on given conditions. | | |

| |What is the algebraic representation of | | | |

| |perpendicular lines? | | | |

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| |Parallel/Perpendicular Lines |TLW: | | |

| |Essential Question: |Formally define parallel and perpendicular lines. | | |

| |How are parallel and perpendicular lines |Generate parallel and perpendicular based on given | | |

| |constructed? |information. | | |

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| |Scatterplots/Regression/Line of Best Fit |TLW: | | |

| |Essential Question: |Graph and interpret data. | | |

| | |Estimate a linear relationship among the data. | | |

| | |Solve problems by using models | | |

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|Topic: Linear Inequalities |

|Enduring Understanding: The Learner Will Be Able To: |

|Translate knowledge of linear equations to justify solutions to linear inequalities. |

|Suggested |Concepts |Objectives |Conceptual Understanding |NJCCCS |

|Time | | | |(CPI) |

| | Linear Inequalities |TLW: |1. Graph the following inequality as if it were an equality. y < 3x | |

| |Essential Question: |Apply the knowledge of linear equations to synthesize the |-2. | |

| |What relationships exist between linear equations |graph of a linear inequality |Using your knowledge of algebra plug the following points into your | |

| |and linear inequalities? |Use graphical representations to construct algebraic |inequality and identify if they are on the line. | |

| | |representations. |(3,7) | |

| | |Develop the correlation to real-world tasks. |(2,5) | |

| | | |(2,2) | |

| | | |Which point makes the statement true? | |

| | | |If it were an inequality of one variable and it was of the form | |

| | | |then we left an open circle. What should you do to the line to show | |

| | | |that the line itself is not part of the solution? | |

| | | |What should you do with all of the points that make the statement true?| |

| | | |How do you suggest to do this? | |

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| | | |2. A forensic scientist is analyzing a car accident. She is able to | |

| | | |determine approximately how fast a car was traveling based upon the | |

| | | |length of the skid marks. If a stopped car creates no skid mark and a | |

| | | |car that is traveling at least 80 km/hour creates an 3 meter skid mark | |

| | | |then write an inequality to describe the relationship. Graph that | |

| | | |inequality and explain that Bill must have been speeding down Evergreen| |

| | | |road to leave a 2.3 meter skid mark before the stop sign. | |

| |Compound inequalities |TLW: |It is important to focus on the skills used in solving the inequalities| |

| |Essential Question: |Use skills of inequalities to develop methods for Boolean |and then examining the answer to develop it’s meaning. | |

| |How do you justify your solutions? |algebra. | | |

| | |Solve compound inequalities and graph their solution sets.|4 < x < 10 translates to an and task. Where the solution falls | |

| | |Solve real-world tasks that involve compound inequalities.|between the two values. | |

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| | | |x < - 4, x > 8 translates to an or situation where the student must | |

| | | |make a judgment as to which answer makes sense. i.e. If the above | |

| | | |statement was referring to an area of a backyard than x less then -4 | |

| | | |makes no sense therefore the solution must be x > 8. | |

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|Topic: Systems of Equations |

|Enduring Understanding: The Learner Will Be Able To: |

|Suggested |Concepts |Objectives |Conceptual Understanding |NJCCCS |

|Time | | | |(CPI) |

| |Graphing Systems of Equations |TLW: |1. On one coordinate plane please graph the following two lines. | |

| |Essential Question: |Solve systems of equations by graphing. |y = x + 5 | |

| |What does it mean to solve a system of equations? |Determine whether a system of equations has one solution, |y = 3x + 1 | |

| | |no solution, or infinitely many solutions by graphing. |Do the lines intersect? If so how many times? | |

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| | | |2. Sketch three pictures, each containing an image of one type of | |

| | | |solution. Then in a few words describe each image to justify your | |

| | | |solutions. | |

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| | | |3. Applying what you know about slope. Explain the relevance the | |

| | | |information about slope plays in predicting a solution. | |

| |Substitution/Elimination |TLW: |1. Use similar questions to drive home an algebraic approach to | |

| |Essential Question: |Solve systems of equations by using the |finding the solution. | |

| |Does it matter which variable is |substitution/elimination methods. | | |

| |substituted/eliminated? |Explore the inaccuracies of graphing through the |2. Given: y = (1/5)x + 2, y = (-1/7)x + 8 | |

| |Can you visualize a feasible answer? |sub./elim. methods. |Graph the system. | |

| | |Predict and carry out optimal solution methods. |How many values of x and y will make both statements true? | |

| | | |How would you go about finding the exact point(s) of intersection? | |

| | | |Explain why that is the best way to find the solution. | |

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| |Graphing Systems of Inequalities |TLW: |1. Find the feasible solution set. Given: | |

| |Essential Question: |Combine knowledge of graphing linear inequalities and |y >= 0, x >= 0, y < 4x, y < 5, and x < 7. | |

| |How does the inequalities compare to the system of |systems of linear equations to graph and solve systems of | | |

| |equations? |inequalities. |2. Find the feasible solution set and the values of the corner points.| |

| |Why is this important? | |Given: | |

| |What does it mean to have no solution? | |x >= 0, y ................
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