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School District of South Orange and MaplewoodMathematics Department

Appendix C: Level Distinctions in Algebra 1

Leveling is not a function of intelligence or mathematical talent or the ability to learn. Leveling in math begins with a consideration of the mathematical content that has to be developed, takes a measure of students’ prior learning, and enacts a plan to maximize learning across a spectrum of student achievement.

This appendix addresses the first of two categorical distinctions regarding the variation in students’ preparation to do mathematics in the various levels. This appendix describes the parameters of content by level for each learning objective based on the content outline. At each level, students will address every learning objective prescribed by State Standards. Variations and modifications in the content outline are based on evidence of foundational knowledge, the instructional time required for students to obtain mastery of essential aspects of content, and the opportunity to make mathematical decisions as the content is developed. The mathematics identified here establishes the expectations for mastery in each level.

In a broader sense, the second distinction addresses the Mathematical Practices, that is, the processes and strategic competencies students need to use in order to think mathematically and develop mathematical concepts and skills. The curriculum document which will be submitted for adoption in July will describe those processes and instructional approaches and generally describe differences within and across levels. The Mathematical Practices are defined and required by national and state standards.

Mathematics Curriculum: Algebra

Objective Level 2 Level 3 Level 4 Content Outline

1. Approach solving equations as a process of reasoning and explaining the reasoning. Solve single-variable linear equations with rational coefficients. Construct viable arguments to justify solution methods. Recognize, express and solve problems that can be modeled using single-variable linear equation

NJCCSS: N-Q-1,2,3; A-SSE-1,2;A-CED-1,2,3; A-REI-1,3

*1-2 weeks spent generating the knowledge of “doing-undoing” via tasks with increasingly higher levels of demand. *1-2 weeks spent working with equivalence and balance to conceptual fair-moves in algebraic manipulation.*1-2 weeks time practicing the skill of solving such equations, with connections to above concept attainment.*1 week on word problem, initially with repeated problems with similar to contexts as an access point to the skill.

Assess prior knowledge of each topic listed in the content outline through problems, examples and discussion of concepts and skills.For example, as a student solves linear equations, their justification for actions refers to maintaining the balance of equality on both sides of the equation. Instructional time spent only where gaps or weaknesses are identified.

Prior mastery of topic expected.

There is no instruction planned for this objective.

1. Represent real world situations with linear equations.

2. Explain solutions in the context of the problem.

3. Solve simple linear equations.4. Compare and contrast solution

methods; make decisions as to which is preferable for given types of problems.

6. Solve equations using combining like terms and the distributive property.

7. Solve equations with rational coefficients.

8. Solve equations with variables on both sides, including those with no and infinite solutions.

2. Recognize and solve real world problems involving proportional reasoning and rational numbers, including percent problems.

NJCCSS: N-Q 1

* 1 week on real world applications and other word problems with emphasis on setting up the proportions using language that evidences proportional reasoning. Repeated practice for problems using similar contexts. This provides access for some students who need to

Recognize / discuss proportionality in real situations and determine appropriate contexts for applying percent/proportionality in the real world. Translate from situations or other representations to symbolic form,

Prior mastery of topic expected.

There is no instruction planned for this objective.

1. Represent, apply and explain methods for solve problems involving rates and ratios, proportions and percents.

2. Apply proportions to similar figures

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see more examples of abstraction.

formulating an appropriate algebraic representation for the identified contexts.


3. Reason quantitatively and use units to solve problems.Solve equations involving several variables for one variable in terms of the others.

NJCCSS: N-Q-1,2,3 A-CED-4

*1-2 weeks As with all word problems, emphasis is mathematical structure. At first, no extra computation is required to provide the expected unit type. Several examples towards the end of the week include the problems for which extra computation/ conversion is required. Conceptual understanding of proportion, as developed Obj 2, and unit conversion go together.*1 week – Transforming formulas is always grounded in context. Before transforming, students become familiar with formulas, and they build relational thinking and flexibility through practice. E.g. the relationship between radius and circumference must be well understood before transforming. Then, while making connections to conceptual understandings of ‘fair-moves’, the transformed formula is grounded in context. E.g., “Now, this is the formula that describes circumference in terms of the radius.”

Use previously discussed, accepted means of algebraic manipulation regardless of the manipulated terms being number or variable.(Typically a big obstacle for students)

Use formulas students have previously used to give purpose to the variable manipulation(What if I want to know the base of my rectangle? How can I write a formula to give me that knowing area and height?)

Transform a variety of formulas and other equations to solve for one variable in terms of the others.

Example:

Given V=13π r2h,

solve for r.

1. Transform equations a. define appropriate quantities for the purpose of descriptive modeling2. Work with formulas a. Use units as a way to understand problems and to guide the solution

of multi-step problems b. interpret units consistently in a formula c. Choose a level of accuracy appropriate to the limitations on measurement when reporting quantities.

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As before, practice ensues.

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4. (A) Model and explain the concept of a function using situations, graphs, tables, and functional notation.(B) Interpret functions that arise in applications in terms of the context. (C) Analyze functions using different representations.

Graph linear functions given an equation or table and interpret the graph in the context of the problem. Recognize slope as rate of change and use the slope and intercepts to answer questions about a problem with or without context.

NJCCSS: F-IF 1 - 6, 9A-REI-10

*1-2 weeks Develop flexibility among representations with repeated use of contextualized prompts that students can do using more than one model. E.g., “Use the table to show how much it costs when you get in a taxi before it starts to move.” Deliver these prompts in context for sense making of the process. The repeated questions then elicit understanding that slope and intercepts are represented in table, equation and graph (and again made sense of with context/situation)-Review coordinate plane and plotting points.-Limited exposure to f(x), mostly use y=- Limited content and application to straightforward wording. Many times give students the equations “You are helping to plan an awards banquet for your school, and you need to rent tables to seat 180 people. Tables come in two sizes. Small tables seat 4 people, and large tables seat 6 people. This situation can be modeled by the equation 4x+6y=180. Where x is the

Determine domain and range from a table of values and a graph.

Distinguish between examples of and non-examples of functions based on relationships depicted in tables and graphs.

Determine rate of change and y-intercepts from tables and graphs. (Varied intervals for x and y).

Graphing lines given in any form using rational slopes and rational y-intercepts.

Translate between the forms of linear equations and recognize their equivalence.

Interpret x- and y- intercepts in the context of a

Find f ( c ) for a given f ( x )h ( t )=−16 t2+14 t+7 means height is a function of timeGiven a standard form word problem, write and equation, graph using correct domain and range, find and interpret intercepts in context. Interpret domain and range in context.Given a linear function in any form, rewrite in any form.

A. 1. Demonstrate understanding/apply and explain the concept of a function (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.) 2. Use function notation to…a. evaluate functions for inputs in their domainsb. interpret statements that use function notation in terms of a contextc. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. B 1. Interpret, connect and analyze different representations for functions.a. Graph linear functions using a table. b. Write and graph functions for horizontal and vertical lines.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 it describes. Key features include:a. intercepts; b. intervals where the function is increasing/decreasing, positive/ negative; c. slope2. Relate the domain of a function to its graph and, where possible, to the quantitative relationship it describes.Represent situations using standard form or slope-intercept form of an equation. Find pairs of values for the linear relationship 4. Interpret x and y intercept in the context of the problem. Use these values to graph equations in standard form.1. Find and interpret constant rate of change (slope) with or without context.2. Graph equations w/ the slope and y-intercept.3. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the

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Objective 4 cont’d

number of small tables and y is the number of large tables.”

-Graphing linear relationships using a table, standard form and, Slope intercept form-Describe, recognize, and be able to distinguish characterize of slope with or without context.

Develop a flexibility among representations through repeated prompting of context questions that students answer with the use of more than one representation. E.g., “Use the table to tell me how much it costs when you get in a taxi, before it moves.” Deliver these prompts in context for sense making of the process. The repeated contextual questions then elicit understanding that slope and intercepts are represented in table, equation and graph

problem. What happens in the context of the function (what happens when the input is zero, what happens when the output is zero?)

Describe how and why application or contexts can limit a domain of their functional representations.

coordinate plane, often forming a curve.

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5. Recognize, describe and represent linear relationships using words, tables, numerical patterns, graphs, and equations. Translate among these representations.

Describe, analyze and use key characteristics of linear functions and their graphs

Interpret and compare linear models for data that exhibit a linear trend including contextual problems. NJCCSS : S-ID-6, 8G-GEPE-5

2-3 weeks – Consistent practice moving among the four representations of function. Recognizing the variety of problems, for example linear equations in slope-intercept versus standard form, requires a concerted effort, either through days designated for each type of problem, or doing at least one of each type of problem a day. Encouragement of appropriate table use. -Point-slope form-Write for parallel lines

- Exposure to converting from slope-intercept to standard form (limit intercepts to integers) -Interpret and analyze representations of data on a scatter plot and line of best fit. (Students are given data and asked to predict a trend leading to line of best fit)

-Consistent practice moving among the four representations of functions. Recognizing the variety of problems, for example linear equations in y-intercept form versus standard form, requires a concerted effort, either through days designated for each type of problem, or doing at least one of each type of problem a day.

- Recognizing the variety of problems, for example linear equations in y-intercept form versus standard form, requires a concerted effort, either through days designated for each type of problem, or doing at least one of each type of problem a day. Encouragement of appropriate table use,

Write the equation of a line given the graph of the line: lines with integer and rational slopes y-intercepts, graphs where the y-intercept is not shown.

Write equation of a line given two points, a point and the slope, a point and a parallel line, a point and a horizontal or vertical line, both intercepts, or a point and an intercept, find an equation for the line in slope-intercept form, point-slope form or standard form

Given a linear equation or graph of a line, identify the slope of a perpendicular line.

Application Problems – Identify the most appropriate from of a line to represent a given situation. Identify the meaning of the slope and y-intercept in the context of a problem. For example: Given the cost of a gym membership after two months and after four months, identify the slope and what it represents, identify the intercept and what it represents. Identify the most appropriate from of a line to represent a given situation.

Create scatterplots, analyze correlation of data, find, write an equation for, and use trend lines

Given a graph, two points, a point and the slope, a point and a perpendicular or parallel line, a point and a horizontal or vertical line, both intercepts, or a point and an intercept, find an equation for the line in slope-intercept form, point-slope form or standard form.

Create scatterplots, analyze correlation of data, find, write an equation for, and use trend lines for

A. Write linear equations in slope-intercept form, point-slope form (level 4 only) and standard form, and use these equations to solve problems. B. Write equations for parallel and perpendicular lines, interpret in context.C. 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)D. Scatterplots 1. Plot bivariate data with a scatterplot and describe the nature of any possible linear trends: positive or negative, weak or strong.2. Approximate a trend line by drawing on the scatterplot through the cloud of points. Give the equation for this line.3. Use these trend lines to predict values.4. Compute (using technology) and interpret the correlation coefficient of a linear fit.

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such as a y=0, 1, 2 table vs a x= 0, y=0 table.

for prediction. prediction.


6. Solve inequalities and graph solutions on a number line.Model real world situations with inequalities and explain solutions in the context of the problem.

NJCCSS A-CED-1,3 A-REI-3

*2-3 Weeks. Use this unit as an opportunity to build on previous knowledge. That is, every day a student learns about solving and modeling linear inequalities, the student will also review the concepts and skills of linear equations.Significant work in building the concept of an algebraic inequality also requires bolstering up a conceptual appreciation for inequality. Possible activities include comparing quantities and describing the possibilities for x when x > 5.- - -

-Logically interpret and identify equivalent inequalities

(i.e. is the same as )-Limit solutions to common fractions and integers. -Explain solutions in context to the problem.-Exploring why the inequality changes when you divide or multiply by a negative. -Students should realize that there are an infinite number of solutions to statements

such as -Also focus on the difference between a greater than and a greater than or equal to. -If students are asked to write inequalities that use common phrases like “more than”, “product”, “is”, and “difference”

Reinforce logical interpretation of equivalent inequalities.Solve multistep inequalities, including those with infinite solutions or no solution. Graph solutions on a number line where possible. Interpret solutions in context.Use any rational number in the expression of the inequality or its solution.

Solve multistep inequalities, including those with infinite solutions or no solution. Graph solutions on a number line where possible. Interpret solutions in context.Level 4 OnlyRepresent real world problems with compound inequalities. Solve compound inequalities.Interpret solutions in context.

1. Represent real world situations with linear inequalities2. Explain solutions in the context of the problem.3. Solve simple linear inequalities6. Solve inequalities using combining like terms and distributive law.7. Solve inequalities with rational coefficients. 8. Solve inequalities with variables on both sides, including those with no and infinite solutions.Solve compound inequalities (Level 4 only)

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7. Solve equations involving the absolute value of a linear expression. NJCCSS A-REI-1; F-IF-1

*2-3 Weeks. Use this unit as an opportunity to build on previous knowledge. That is, every day a student learns about solving absolute value equations, the student will also review the concepts and skills of linear equations.Significant work in building the concept of absolute value may be required.----Re-explain meaning of absolute value (distance away form zero). -Emphasize that absolute value cannot be negative. (Bring up distance, “Even if you walk backwards, you are still moving a distance”)

-All word problems are limited

to either giving student the

equation or in the form

Reinforce general descriptions or student definitions for absolute valueSolve multi-step equations with absolute value expressions. Distinguish when absolute value equations have no solution

Solve multi-step equations with absolute value expressions, including expressions with no solution.Review the objective through error analysis, and pattern comparisons.

1. Solve equations with absolute value expressions.

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8. Graph and analyze the graph of the solution set of a two-variable linear inequality.

NJCCSS F-IF-1,5

*1-2 weeks – Meaning should drive the graphing of two-variable inequalities. For many students, this may require consistent use of the table for graphing the line as well as using the test-point for shading as follows: Plug in the point into the inequality; Determine if the inequality is satisfied (“Yes” or “No”); Write “Yes” or “No” on the graph where the test-point appears; Write the opposite word “Yes” or “No” on the other sides of the line; Shade on the side of the line where you wrote “Yes.” Such a scripted procedure should be developed via meaning making of the inequality

-Review how to graph a line -Relate how > looks on a number line verse how it looks in a linear equations (open circle is going to represent dashed line). -Limited exposure to word problems and real world situations.

Distinguish between notation and visual representation (<,dashed; &solid)

Use a test point to determine direction to shade for solution.

Contrast the differences between the solution to a linear inequality and a linear equation.

Write the linear inequality given its graph.

Emphasis is on writing and graphing linear inequalities.

Analyze and solve real world problems that can be represented with a two-variable linear inequality, and interpreting solutions in context.

Write the linear inequality given its graph. Interpret the graph in context.

1. Explain how any point in the shaded region satisfies the equation.2. Model a real world situation with a two-variable linear inequality and explain the solution set in the context of the problem.3. Graph a linear equality with correct shading and dotted/dashed lines.

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9. Solve systems of linear equations in two variables using algebraic and graphic procedures.

Recognize, express and solve problems that can be modeled using one or two-variable inequalities; or two variable systems of linear equations. Interpret their solutions in terms of the context of the problem.

NJCCSS A-REI-6

1 week – This topic is less essential than others in the curriculum. In order to spend more time on the developmental pieces in Algebra 1, here the topic is reduced to problems that only require the knowledges from above topics. I.e., students use their skills to graph two-variable inequalities (see previous) to draw two of these on one coordinate plane. There is little time for modeling with this technique.

- Review how to graph including a coordinate plane review. - Solve by graphing, substitution and elimination.

Graphing: -Equations are mostly are in slope intercept form. -All solutions are integers

Substitution: - One or both of the equations are x=, y= - If both are in standard form one of the variables have a coefficient of 1. -All coefficient are integers-To facilitate checking of solutions, All solutions are common fractions or integers

Elimination:- All coefficient are integers-To facilitate checking of solutions, All solutions are common fractions or integers.

Applications/Word Problems: -Make problems with situations students can relate to (money problems). -Stress to students they need two linear equations with two variables.

Solving Graphically: Solutions limited to integer coordinates. Lines given in any form.

Substitution and EliminationEquations are written in any form. Solutions are rational numbers to facilitate checking of solutions.Determine if the system of equations has no solution, one solution, or infinite solutions from solving by all three methods.Given real world situations, write and solve the system. Interpret the solutions in context.

Determine the solution of a system of linear inequalities

Given a system, choose and justify an approach for solving. Solve systems in any form including those with no solution or infinite solutions.

Given an equation, write another equation to make a system with a given characteristic (example: given 3 x+2 y=7 write another equation so that the system has no solution. Justify your answer.)

Write a system with a given solution (eg:(4,7)).

Given real world situations, write and solve the system. Interpret solutions in context.

…..

1. Explain that the solution to a system of linear equations is the ordered pair that satisfies both equations.2. Use tables to confirm the solution to a system.3. Solve systems graphically by finding the point of intersection, both with technology and by hand.4. Confirm that a solution satisfies both equations algebraically.5. Solve systems using algebraic techniques.6. Decide which method is best to solve a given system.7. Given an algebraic representation, represent the solution set to a system of linear inequalities graphically.8. Model real world situations using systems of linear equations and inequalities.

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10. Summarize, represent, and interpret data on a single count or measurement variable(1-4), or on 2categorical and quantitative variables(5-6).Interpret linear models(7-9)

2-3 days. This topic is less essential to development of algebraic thinking and therefore should be satisfied with experiential activities that require students to represent data using dotplots, histograms and scatterplots. They should be instructed in finding means, medians and modes. Students may be assigned a small, in-class project as an assessment, rather than a test.

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

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.

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

4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

5. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

9. Distinguish between correlation and causation.

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11. Determine and evaluate random processes underlying statistical experiments(1-2).

Make inferences and justify conclusions from sample surveys, experiments and observational studies(3-6).

NJCCSS S-IC-1-6

2-3 days: Again, keeping this limited due to time constraints, students learn about sample surveys through doing an in-class project assignment. They are coached regarding effective questioning and data tabulation and presentation.

Instruction limited to randomization, a few simulations, and a sampling survey to estimate population mean or proportion

A brief review and extensions, as required, for each aspect of content outline.

1. Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

2. Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.

3. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

4. Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

5. Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

6. Evaluate reports based on data.

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12. Apply probability concepts to determine the likelihood an event will occur in practical situations. NJCCSS S-CP-1-7

Find the probability of a single event or compound event occurring with or without replacement.

Content outline1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”). 2. Demonstrate or explain that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

(continued in last column)

Construct sample space.

Distinguish between independent and dependent events.

Distinguish between overlapping and mutually exclusive events -For example: addition rule

Differentiate between experimental and theoretical probability.

Define a sample space

Distinguish between independent and dependent events.

Distinguish between mutually exclusive and overlapping events in order to determine the probability of an occurrence

Differentiate between and calculate experimental and theoretical probabilities

Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).

Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

Determine the probability of mutually exclusive or overlapping events

Find the probability of a single event or compound event occurring with or without replacement.

3. Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. 4. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. 5. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. 6. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. 7. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.

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13. Use counting principles to determine the number of ways an event can occur. Interpret and justify solutions.

NJCCSS S-CP-9

1. Determine the number or outcomes by applying the counting principle.

1. Find the number of possible outcomes for single or compound events- selected situations

2. Apply counting principle

1. Find the number of possible outcomes for single or compound events- challenging situations

2. Distinguish between a combination and permutation and use these to solve real world problems.

3. Use permutations and combinations to compute probabilities of compound events and solve problems.

1. Find the number of possible outcomes for single or compound events.

2. Distinguish between a combination and permutation and use these to solve problems

3. Use permutations and combinations to compute probabilities of compound events and solve problems.

14. Apply the laws of exponents to numerical and algebraic expressions with integral exponents to rewrite them in different but equivalent forms or to solve problems.

NJCCSS N-RN-2

2 weeks: Laws of exponents are developed with consistent reference to the definition of exponent (as repeated multiplication). E.g., student outcome is to describe the product of powers rule with an example.Automaticity with exponent rules should also be the focus, and several days are devoted to practice with these rules.

Investigate exponential relationships to determine the properties of integer exponents

Use properties of exponents to simplify/evaluate expressions

Apply appropriate properties of exponents to solve equations

Scientific notation is reviewed as an application of exponents

Review properties of integer exponents.

Use these properties to simplify complex expressions.

Introduce rational exponents.

Review scientific notation.

1. Use product and quotient properties to rewrite exponential expressions2. Rewrite expressions so that all exponents are positive (Equivalent forms)3. Solve problems using exponents4. Convert between standard form and scientific notation

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15. Model and solve problems involving exponential growth and decay.

NJCCSS A-SSE-1,2,3

1 week:Repeated use of simple contexts that are modeled with exponential growth and decay.

Work simultaneously with contexts that are NOT modeled with exponential growth/decay (like linear and quadratic).

Consistent use of the table, context and graph as with other modeling units as above and below.

Determine the characteristic differences between linear and exponential functions (represented in tables/graphs/equations)

Analyze graphs or tables to determine the equation for an exponential function

Apply exponential functions to real world scenarios (Interest on a bank account/half-life of substance/etc.)

Determine the behavior/characteristic differences between exponential growth and decay

Review recursive routines for exponential functions. Connect the recursive routine to an explicit formula. Find an equation for a exponential function given points or a graph or a table of values..

Predict a function to exponential vs linear given a table of values.

Use exponential functions to model real world situations involving growth and decay given growth or decay factors or rates of increase or decrease.

Analyze the graph of an exponential function for domain and range, and key features such as increasing, decreasing, and y-intercept.

1. Model exponential function form y = abx

2. Analyze y-intercept and constant multiple to write a function rule

3. Graph exponential functions as continuous and positive

4. Model and/or solve real world problems

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16. Add, subtract and multiply polynomial expressions with or without a context.

NJCCSS A-APR-1,2

1 week – With multiplying, make connections to the distributive property.

For some students, the use of a multiplication matrix may be helpful, especially for multiplying a trinomial against a binomial.

Review the concepts of combining like terms and the distributive property

Identify like terms in polynomials based on their integer power

Apply distributive property to polynomial multiplication (requires extension to multiplying more than just monomial times binomial)

Apply newly learned concepts to simplify polynomial expressions

Investigate common patterns in polynomial multiplication

Identify like terms for powers of variables greater than 2.

Add, subtract and multiply polynomial expressions with or without a context.

Extended distribution i.e. A binomial and a trinomial.

Simplifying multi-term polynomial expressions

1Classify, add, subtract, & multiply polynomials2Use polynomials to represent real world situations; solve problems using polynomials3 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication.

17. Factor simple polynomial expressions with or without context.

Solve factored polynomial equations with or without context.

NJCCSS A-APR-3

2 weeks: The majority of problems require factoring out a common monomial and/or a quadratic when a=1. Consistent emphasis on the connection to multiplication (as the inverse). Also, direct guidance, such as requiring students to write out "Find the pair of factors of c that add to b," before finding the pair and writing the factored polynomial.

Review what it means to be a “factor”

Apply concept of a factor to identify a greatest common monomial factor of a polynomial expression

Solve equations using the “zero-product property”( in factored form or by algebraic manipulation to appear in factored form)

Determine the factors of a trinomial through various strategies (i.e. recognition of special patterns)

1. Increase knowledge of greatest monomial factor by using multi-variable/power polynomials.2. Factor polynomials by understanding and recognizing the patterns of special factoring as well i.e. difference of two squares. perfect square trinomials, factor by grouping.3. Solving real world problems in context which could be representative of the product of binomials for example area, vertical motion models, and other polynomial applications.4. Interpret and apply the zero product property to the solution of a factored quadratic trinomial with or without context. 5.Recognize and interpret extraneous/non-extraneous solutions after solving a quadratic equation.

1. Identify and factor out the GCF.2. Factor trinomials. Use products of binomials to represent area.3. Identify and factor the difference of squares.4. Use the zero-product property to solve quadratic equations.

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18. Recognize, describe, represent and analyze a quadratic function using words, tables, graphs and equations.

Analyze a table, numerical pattern, graph, equation or context to determine whether a linear or quadratic relationship could be represented.

Recognize and solve problems that can be modeled using a quadratic function. Interpret the solution in terms of the context of the original problem.

NJCCSS A-SSE-3,A-REI-4,7

1 week:Repeated use of simple contexts that are modeled with quadratic functions.

Work simultaneously with contexts that are NOT modeled with quadratic functions (like linear and exponential).

Consistent use of the table, context and graph as with other modeling units as above.

Recognize a parabola as the graphic representation of a quadratic function

Determine the symmetry of a quadratic function

Identify the characteristics of a parabola and their algebraic means for calculation based on the equation

Investigate the changes in the graph of a quadratic function based on alterations made to the coefficients of its equation

1. Identify the attributes of a parabola: the shape, the location of the axis of symmetry and its vertex.

2. Referencing the quadratic function of the parabola, recognize the significance of the coefficients of a, b, and for example if a is positive or negative based on the shape of the parabola, and the value of c from the graph and that it represents the y intercept.

3. Analyze a quadratic trinomial for graphing attributes: opening up/down, x and y intercepts, axis of symmetry, location of vertex (above or below x axis),

4. Given an equation, describe transformations to the parent function y = x2 . Compare and contrast different graphs of quadratic functions, noting differences in width and how this will relate to their relative equations.

5. Calculate the axis of symmetry and vertex of the parabola given the equation.

6. Identify equations whose graphs will be symmetric to the y-axis (b=0).

7. Model real world situations that provide a given context to address pertinent questions relative to quadratic equations including vertical motion model and area.

8 . Determine the number of solutions of a quadratic equation by graphing.

1. Find the value of the vertex from a table (noting the symmetry).

2. Determine positive or negative value of a based on the graph. Determine the value of c from the graph.

3. Compare and contrast different graphs of quadratic functions, noting differences in width and how this will relate to their relative equations.

4. Identify equations whose graphs will be symmetric to the y-axis (b=0).

5. Model real world situations with a quadratic equation, including vertical motion and area.

6. Find the vertex from the equation of a parabola.

7. Use the quadratic model of a given context to answer contextual questions.

8. Distinguish between functions with a maximum and minimum value, and this may relate to a given context.

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19. Solve quadratic equations.

NJCCSS A-SSE-3, A-REI-4,7

1 week:Introduce through quadratic contexts.

Tables and graphs are juxtaposed with linear and exponential so as to highlight quadratic features.

Determine the significance of a solution to a quadratic equation (numerically and graphical implications)

Use the quadratic formula

1. Solve quadratic equations using appropriate methods to determine the number and nature of the solutions for example: by factoring, inverse operations, quadratic formula and completing the square.

2. Use and understand multiple terms pertaining to solving quadratic equations, for example finding roots, finding zeros and x intercepts of a quadratic function.

3. Use the discriminant to determine number of solutions. Understand the discriminant’s predictive ability as the square root piece of the quadratic formula.

4. Determine the number of the solutions from the discriminant.

5. .Anticipate the location of the vertex of the parabola, above or below the x axis, based on analyzing the equation for a minimum or maximum vertex in conjunction with the discriminant and analyzing the number of solutions. 6 Solve graphically, by hand (approximation) and with technology.

7. Model real world problems.

8. Interpret solutions in the context of the problem.

1. Solve using inverse operations when b=0. 2. Solve using the quadratic formula for all quadratic equations.3. Solve a binomial square by taking the square root of both sides. (level 4 only)4. Solve graphically, by hand (approximation) and with technology.5. Interpret solution in the context of the problem.

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Objective Level 2 Level 3 Level 4 Content outline20. Analyze a table, numerical pattern, graph, equation or context to determine whether a linear, quadratic or exponential relationship could be represented. Or, given the type of relationship, determine the elements of the table, numerical pattern or graph. NJCCSS A-REI-11 F-IF-4-7,9

2 weeks:This topic is worked on throughout the year. When students learn linear functions, they are given quadratic and exponential problems as well so as to distinguish the liinear features. Again during the quadratic unit, quadratic contexts/tables/graphs are juxtaposed with linear and exponential so as to highlight quadratic features. Similar approach with exponential.

Analyze the characteristics of a given representation to indicate which category of functions the representation can be applied.

1. Use the shape of the graph and/or a table of values to distinguish between linear, quadratic, exponential and radical functions.2. Given a function select the appropriate domain to construct a graph as it relates to solving real world problems.

1. Use the shape of the graph to distinguish between functions.2. Analyze the change in the y-values to distinguish between functions.3. Complete a table given the type of function. 4. Find other points given the type of function.5. Solve real world problems.

21. Use the properties of radicals to rewrite numerical and algebraic expressions containing square roots in different but equivalent forms or to solve problems.

NJCCSS N-RN-1,2; A-APR-6; A-REI-2

1 week: This topic is less connected to others in the Algebra 1 curriculum (because of a lack of attention to square root functions, contexts, graphs and tables) and is also heavily emphasized in the Algebra 2 curriculum. Only an introduction to the topic is included, specifically introducing that the square root is the inverse of squaring, and simplifying only stand-alone square roots (no addition, subtraction, multiplication of radicals).

As it is in Level 2, this objective is introductory versus a broader ranged development. All descriptive described in Level 2 are appropriate for Level 3 content development for this objective.

Perform operations with radicals which includes simplifying, rationalizing the denominator, adding, subtracting, multiplying and dividing and solving real world problems.

1. Write square root radicals in simplest form

2. Find products and quotients of radicals

3. Rationalize the denominator

4. Add, subtract and multiply radical expressions

5. Solve real world problems

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