Plumsted Township Public Schools

Mathematics Program

Algebra I Curriculum

Plumsted Township Public Schools

Mathematics Program

New EgyptHigh 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 PlumstedTownshipSchool 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 PlumstedTownshipSchool 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

1

Revised:: 2008

Plumsted Township Public Schools

Mathematics Program

Algebra I Curriculum

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

1

Revised:: 2008

Plumsted Township Public Schools

Mathematics Program

Algebra I Curriculum

Scope and Sequence

1

Revised:: 2008

Plumsted Township Public Schools

Mathematics Program

Algebra I Curriculum

I.Solving Equations – One Variable

  1. Properties of Operations
  2. One step equations
  3. Multi-step equations
  4. Solving for variables on both sides of the equation
  5. Formula’s – For science

II.Ratios

  1. Rational Numbers
  2. Proportional Reasoning
  3. Percents
  4. Percent of Change
  5. Direct and inverse variations

III.Geometry

  1. Angles and Triangles
  2. Pythagorean Theorem
  3. Similar Triangles
  4. Trigonometric Ratios

IV.Graphing

  1. Coordinate Plane
  2. Domain & Range Mapping, Functions
  3. Patterns

V.Slope

  1. Must solve m = (y2 – y1) / (x2 – x1) for any variable
  2. Must find slope and graph line from y = mx + b.
  3. Graphing Calculator Lab
  4. Parallel and perpendicular
  5. Scatter plot/ regression/ line of best fit

VI.Linear Inequalities

  1. Linear Inequalities
  2. Compound inequalities

VII.Systems of Equations

  1. Graphing Systems of Equations
  2. Substitution/Elimination Method
  3. Graphing Systems of Inequalities

VIII.Statistics

  1. Central Tendency
  2. Weighted Averages
  3. Probabilities and Odds

IX.Monomials

a.Multiplying/ Dividing

b. Exponential Properties

c. Introduction to Radicals

X.Polynomials

  1. Add/Sub
  2. Multiply/ Distribution
  3. Factoring

XI.Quadratics

  1. Graph
  2. Finding Roots
  • Graphing
  • Factoring
  • Quadratic Formula

1

Revised:: 2008

Plumsted Township Public Schools

Mathematics Program

Algebra I Curriculum

Topic: Solving Equations – One Variable
  1. Enduring Understanding: The Learner Will Be Able To:
Demonstrate the crucial skills and concepts involved in solving equations.
Suggested
Time / Concepts / Objectives / Conceptual Understanding(s) / NJCCCS (CPI)
  1. Properties of Operations
  • Essential Question:
  • Why do we need mathematical properties?
  • Why does 0 not have a multiplicative identity?
  • Can 1 be the additive identity?
  • Does the commutative property work with all operations?
/ TLW:
  • Recognize each property of operations:
- Additive Identity
- Multiplicative Identity
- Multiplication Property of Zero
- Substitution
- Multiplicative Inverse
- Reflexive
- Symmetric
- Transitive
- Associative
- Distributive
  • Apply properties to simplified expressions and equations
/
  1. Evaluate the following expression. Name each property used in each step.
(8 x 3 – 19 + 5) + (3² + 8 x 4)
  1. Simplify 4(3x + 2) + 2(x + 3)
  2. Name the property used in each step:
ab(a + b) = (ab)a + (ab)b
= a(ab) + (ab)b
= (a x a)b + a(b x b)
= a²b + ab²
4. Suppose the operation * is defined for all numbers a and b as a * b = a + 2b. Is the operation * commutative? Give examples to support your answer.
  1. One step equations
  • Essential Question:
  • What effect does information have on a situation?
  • How do we justify an answer?
/ TLW:
  • Analyze components of a one-step equation.
  • Evaluate one-step equations using addition and subtraction.
  • Evaluate one-step equations using multiplication and division.
  • Solve any one-step equation.
/
  1. Solve each equation:
  2. m – (-13) = 37
  3. y = (-7) = -19
  4. -4r = -28
  5. 2/5 t = -10
  6. Write an equation and solve. In 1995, the long distance company Sprint introduced Sprint Sense, a plan in which long distance calls placed on weekends cost only $0.10 per minute.
  7. How long could you talk for $2.30?
  8. What would be the cost of an 18 minute call?

  1. Multi step equations
  • Essential Question:
  • What effect does information have on a situation?
  • How do we justify an answer?
/ TLW:
  • Apply order of operations in multi-step equations.
  • Evaluate equations involving more than one operation.
  • Model a real-life situation with a multi-step equation.
  • Evaluate model by working backwards.
/
  1. Solve each equation:
  2. 5n + 6 = -4
  3. -9 – p/4 = 5
  4. 3/2 a – 8 = 11
  5. (4t – 5)/-9 = 7
  6. Find three consecutive odd integers whose sum is 117.
  7. 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.

  1. Solving for variables on both sides of the equation
  • Essential Question:
  • What is the difference between an equation whose solution set is the empty set and an equation that is an identity?
/ TLW:
  • Evaluate equations with the variable on both sides.
  • Analyze equations containing grouping symbols.
/
  1. Solve each equation:
  2. 8y – 10 = -3y + 2
  3. 5x – 7 = 5(x – 2) + 3
  4. 3/2 x – x = 4 + ½ x
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.
  1. Formula’s – for science
  • Essential Question:
  • Why would a scientist want/need to change a formula?
/ TLW:
  • Evaluate equations and formulas for a specified variable.
/
  1. Solve each formula for the variable specified:
  2. d = rt, for r
d = vt + ½ at², for v

Topic: Ratios

Enduring Understanding: The Learner Will Be Able To:
Suggested
Time / Concepts / Objectives / Conceptual Understanding / NJCCCS
(CPI)
  1. Rational Numbers
  • Essential Question:
  • What purpose do rational numbers serve?
  • How do you apply ratios and use them with mathematical operations?
/ TLW:
  • Label components of ratios: num./den. => $/hr. => mi./gal.
  • Compare and contrast values in rational form.
  • Addition/Subtraction
  • Multiplication/Division
/
  1. Write the numbers in each set in order from least to greatest.
  2. 6/7, 2/3, 3/8
  3. 6.7, -5/7, 6/13
  4. 4/5, 9/10, 0.7
  5. Find a number between 2/5 and 7/2.
  6. Evaluate the following:
  7. ¾ + (-4/5) + 2/5 = _____
  8. -1/2 – 2/3 = _____
  9. 5/6(-2/5) = _____
  10. -9 ÷ (-10/27) = _____
  11. 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)
  12. 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)

  1. Proportional Reasoning
  • Essential Question:
  • What applications would you use proportions for and justify?
  • What is the justification for the connection of proportional reasoning to mathematical equations?
/ TLW:
  • Solve a proportion.
  • Compare a proportion to a multi-step equation.
  • Analyze use of proportions from real-life models.
/
  1. x/9 = -7/16
  2. (m + 9)/5 = (m – 10)/11
  3. When a pair of blue jeans is made, the leftover denim scraps can be recycled to make stationary, pencils, and more denim. One pound of denim is left after making every 5 pairs of jeans. How many pounds of denim would be left from 250 pairs of jeans? (50 lbs.)

  1. Percents
  • Essential Question:
  • What methods are used in finding pieces of a whole? How?
/ TLW:
  • Manipulate ratios into percentages.
  • Solve simple interest tasks.
/
  1. Write each ratio as a percent:\
  2. 2/3
  3. 5/6
  4. 6/20
  5. 35 is what percent of 70?
  6. What number is 25% of 56?
  7. Which earns more interest; $1500 at 10% interest for one year or $500 at 4% interest for 10 years?

  1. Percent of Change
  • Essential Question:
  • What are the applications and real-world usage of percentages?
  • How do you find these changes?
/ TLW:
  • Solve problems involving percent of increase/decrease.
  • Solve problems involving discounts or sales tax.
/
  1. The original selling price of a new sports video was $65.00. Due to demand, the price was increased to $87.75. What was the percent of increase? (35%)
  2. A sweater that costs $55.00 is discounted 15%. The sales tax is 6%. What is the final price of the sweater?

  1. Direct and Inverse Variations
  • Essential Question:
  • How do direct and inverse variations differ?
/ TLW:
  • Define constant of variation.
  • Develop methods to solve direct and inverse variations.
/
  1. Julio’s wages vary directly as the number of hours that he works. If his wages for 5 hours are $29.75, how much will they be for 30 hours?
  2. 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.

Topic: Geometry

Enduring Understanding: The Learner Will Be Able To:
Suggested
Time / Concepts / Objectives / Core Activities / NJCCCS
(CPI)
  1. Angles and Triangles
  • Essential Question:
  • What are the algebraic components of triangles?
  • How can they be manipulated?
/ TLW:
  • Find the complement and supplement of an angle.
  • Find the measure of the third angle of a triangle given the measures of the other 2 angles.
  • Classify various triangles.
/
  1. The measure of an angle is three times the measure of its supplement. Find the measure of each angle.
  2. The measure of an angle is 34º greater than its complement. Find the measure of each angle.
  3. What are the measures of the base angles of an isosceles triangle in which the vertex angle measures 45º?
  4. 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.

  1. Pythagorean Theorem
  • Essential Question:
  • What information identifies right triangles and how can you use that information to solve for each part?
3. Similar Triangles
  • Essential Question:
  • Can you justify the relationship between ratios and similar triangles? Explain.
/ TLW:
  • Label sides of a right triangle.
  • Apply the Pythagorean Theorem to solve problems.
TLW: /
  1. Find the length of the hypotenuse of a right triangle if the leg measurements are 5 and 12.
  2. Find the length of side a if b = 9 and c = 21.
  3. Would a triangle with side lengths 6, 8, and 10 form a right triangle?
  4. 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?

4. Trigonometric Ratios
Essential Question:
  • Is there a relationship between sides and angles? Explain.
/ TLW:
  • Label sides of right triangle based on selected angle.
  • Define trigonometric functions as ratios.
  • Evaluate measurement of angles and sides.
/
  1. Example:
  2. Given three pipe cleaners how many different triangles can you make, using the entire length of each pipe cleaner as one side? (one) 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.
  1. 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.
  1. 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
Time / Concepts / Objectives / Core Activities / NJCCCS
(CPI)
  1. Coordinate Plane
  • Essential Question:
  • How is the coordinate plane useful in representing data?
  • Can you interpret the meaning of points in a plane?
  • How can you tell which quadrant a point is in by just looking at the signs of the coordinate?
/ TLW:
  • Identify and graph ordered pairs on a coordinate plane.
  • Translate graphs into ordered pairs.
  • Reconstruct the coordinate plane to solve given task.
  • Summarize and support independent and dependent axes.
/
  • Plot the polling points on a coordinate plane: (-3, -5), (6, 0), (0, -10), (4, -2).
  1. Given a grid develop a student understanding of location.
  1. Redefine the understanding with mathematical terms and notation…origin, x-axis, y-axis, (x,y), etc.
  1. Students must then be able to read a graph. Given a graph students should identify the location of the vertices.
  1. Example:
  2. 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.
  1. Graph her progress.
  2. Label each vertex
  3. Make a table of x values and corresponding y values.
  4. What would have been a better way for Small Wonder to get where she was going?
  1. Example:
  2. 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.
  3. Construct a graph using the coordinate plane and clear labels so that Beth can analyze Andy’s sales per day.
  4. 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)
  5. 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?

  1. Domain and Range -> Mapping , Functions
  • Essential Question:
  • Why would you need to manipulate data from one form to another?
  • Can one domain produce two different ranges? Justify.
/ TLW:
  • Define domain (indep) and range (dep) for a given relation.
  • Generate tables, mappings, graphs, and ordered pairs from any given information.
  • Define a relation as a function.
  • Introduce functional notation.
  • Evaluate the domain of a function to develop a range.
/ Examples of independent domains:
  • Time
  • Sample Size (# of times a coin is flipped)
  • Days of a week
Examples of dependant ranges
  • Money
  • Successful outcomes (# of times a coin lands heads up)
  • Sales
Therefore when combined you would have $/hr., Successful outcomes/Sample size, and Sales/Day respectively.
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.
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.
  1. Maria earns ten dollars a day plus $6.75 per hour.
  2. Create an x, y table to show a five hour day.
  3. Convert the table to a mapping.
  4. Graph the relationship.
  5. Write a function to find out how much Maria will earn in x hours.

  1. Patterns
  • Essential Question:
  • How does a pattern relate to a function?
  • How are patterns related to the real world?
/ TLW:
  • Write an equation in functional notation for a relation given in a table.

Topic: Slope

Enduring Understanding: The Learner Will Be Able To:
  1. Analyze and conceptualize the behavior of linear equations.

Suggested
Time / Concepts / Objectives / Core Activities / NJCCCS
(CPI)
  1. Solve m=(Y2-Y1)/ (X2-X1) for any variable
  • Essential Question:
  • What is slope?
  • Why is slope important?
  • What are the practical uses of slope?
  • Why do vertical lines have no slope?
/ TLW: