Algebra 1 Part 1

High Priority Standards

CCSS.Math.Practice.MP1

Learning Goal

Students will be able to make sense of problems and persevere in solving them.

Proficiency Scale

4: In addition to score 3.0 performance, the student demonstrates an in-depth inference or advanced application, or innovates with the learning goal.

3: Student demonstrates mastery with the learning goal as evidenced by:

• Discussing, explaining, and solving a problem with multiple representations and in multiple ways.
• Struggling with various attempts over time. 
• Learning from previous solution attempts.
• Checking answers using a different method or strategy.

2: Student demonstrates they are nearing proficiency by:

• Explaining his/her thought processes when solving a problem and representing it in several ways.
• Trying several approaches in find a solution and seeking hints only if stuck.

1: Student demonstrates a limited understanding or skill with the learning goal by: 

• Explaining his/her thought processes when solving a problem one way.
• Staying with a challenging problem for more than one attempt.

Learning Targets

• Explain the meaning of a problem and look for efficient ways to solve it

• Use concrete objects or pictures to help conceptualize and solve problems

• Checks their thinking by asking themselves, “Does this make sense?” 

• Listens to the strategies of others and tries different approaches

• Uses a different strategies to check answers

• Takes time to thoughtfully consider problems

Learning Design

• Provides time and facilitates discussion in problem solutions
• Facilitates discourse in the classroom so that students UNDERSTAND the approaches of others
• Provides opportunities for students to explain themselves, the meaning of a problem, etc.
• Provides opportunities for students to connect concepts to “their” world
• Provides students TIME to think and become “patient” problem solvers
• Facilitates and encourages students to check their answers using different methods (not calculators)
• Provides problems that focus on relationships and are “generalizable”

High Priority Standards

CCSS.Math.Practice.MP2

Learning Goal

Students will be able to reason abstractly and quantitatively.

Proficiency Scale

4: In addition to score 3.0 performance, the student demonstrates an in-depth inference or advanced application, or innovates with the learning goal

3: Student demonstrates mastery with the learning goal as evidenced by 

• Converting situations into symbols to solve problems. 
• Converting mathematical equations into meaningful situations.

2: Student demonstrates they are nearing proficiency by translating situations into symbols to solve problems. 

1: Student demonstrates a limited understanding or skill with the learning goal by reasoning with models or pictorial representations to solve problems.
 

Learning Targets

• Recognize that a number represents a specific quantity 
• Connect the quantity to written symbols and create a logical representation of the problem at hand
• Consider both the appropriate units involved and the meaning of quantities
• Write simple expressions that record calculations with numbers and symbols
• Represent or round numbers using place value concepts

Learning Design

• Provides a range of representations of math problem situations and encourages various solutions
• Provides opportunities for students to make sense of quantities and their relationships in problem situations
• Provides problems that require flexible use of properties of operations and objects
• Emphasizes quantitative reasoning which entails habits of creating a coherent representation of the problem at
• hand; considering the units involved; attending to the meaning of quantities, not just how to compute them and/or rules; and knowing and flexibly using different properties of operations and objects

High Priority Standards

CCSS.Math.Practice.MP3

Learning Goal

Students will be able to construct viable arguments and critique the reasoning of others.

Proficiency Scale

4: In addition to score 3.0 performance, the student demonstrates an in-depth inference or advanced application, or innovates with the learning goal.  
3: Student demonstrates mastery with the learning goal as evidenced by:

• Justifying and explaining, with accurate language and vocabulary, why his/her solution is correct.
• Comparing his/her strategy to other students’ strategies, asking questions, and making connections with his/her own thinking.
• Explaining the reasoning of others.

2: Student demonstrates they are nearing proficiency by:

• Explaining his/her thinking and the thinking of others with accurate vocabulary.
• Explaining other students’ solutions and identifying strengths and weaknesses of the strategy. 

1: Student demonstrates a limited understanding or skill with the learning goal by: 

• Explaining his/her solution. 
• Discussing other ideas, approaches, and strategies. 

Learning Targets

• Construct arguments using concrete referents, such as objects, pictures, and drawings

• Refine their mathematical communication skills by answering questions like “How do you know?” and “Can you show me another way?”

• Refine their mathematical communication skills by asking others questions like “How do you know?” and “How did you get that?”

• Explain their thinking to others and respond to others’ thinking

Learning Design

• Provides ALL students opportunities to understand and use stated assumptions, definitions, and previously established results in constructing arguments
• Provides ample time for students to make conjectures and build a logical progression of statements to explore the truth of their conjectures
• Provides opportunities for students to construct arguments and critique arguments of peers
• Facilitates and guides students in recognizing and using counterexamples
• Encourages and facilitates students justifying their conclusions, communicating, and responding to the arguments of others
• Asks useful questions to clarify and/or improve students’ arguments

High Priority Standards

CCSS.Math.Practice.MP4

Learning Goal

Students will be able to model with mathematics.

Proficiency Scale

4: In addition to score 3.0 performance, the student demonstrates an in-depth inference or advanced application, or innovates with the learning goal.  

3: Student demonstrates mastery with the learning goal as evidenced by:

• Recognizing math in everyday situations.
• Using a variety of models, symbolic representations, and technology tools to represent the solution to a problem and accurately explain the solution representation.

2: Student demonstrates they are nearing proficiency by: 

• Recognize math in everyday situations, when prompted.
• Using models and symbols to represent and solve a problem. 

1: Student demonstrates a limited understanding or skill with the learning goal by using models to represent and solve a problem with teacher support.
 

Learning Targets

• Represents problem situations in multiple ways including numbers, words (mathematical language), drawing pictures, using objects, acting out, making a chart, list, or graph, creating equations, etc. and use all of these representations as needed
• Connect different representations and explain the connections
• Evaluate results in the context of the situation and reflect on whether the results make sense
• Evaluate the utility of models to determine which models are most useful and efficient to solve problems

Learning Design

• Provides problem situations that apply to everyday life
• Provides rich tasks that focus on conceptual understanding, relationships, etc.
   

High Priority Standards

CCSS.Math.Practice.MP5

Learning Goal

Students will be able to use appropriate tools strategically.

Proficiency Scale

4: In addition to score 3.0 performance, the student demonstrates an in-depth inference or advanced application, or innovates with the learning goal.  

3: Student demonstrates mastery with the learning goal as evidenced by combining various tools to explore and solve a problem as well as justifying his/her tool selection and problem solution.

2: Student demonstrates they are nearing proficiency by selecting from a variety of provided tools the ones that can be used to solve a problem and explaining his/her reasoning for the selection. 

1: Student demonstrates a limited understanding or skill with the learning goal by using the appropriate tool, when provided, to find a solution.
 

Learning Targets

• Consider the available tools (including, but not limited to, estimation, graph paper, manipulatives, table, list, etc.) when solving a mathematical problem and decide when certain tools might be helpful
• For example, they may use unit cubes to fill a rectangular prism and a ruler to measure the dimensions
• Use graph paper to accurately create graphs and solve problems or make predictions from real world data
   

Learning Design

• Provides a variety of tools and technology for students to explore to deepen their understanding of math concepts
• Provides problem solving tasks that require students to consider a variety of tools for solving (Tools might include pencil/paper, concrete models, manipulatives, ruler, protractor, calculator, spreadsheet, computer algebra system, statistical package, or dynamic geometry software, etc.)
   

High Priority Standards

CCSS.Math.Practice.MP6

Learning Goal

Students will be able to attend to precision.

Proficiency Scale

4: In addition to score 3.0 performance, the student demonstrates an in-depth inference or advanced application, or innovates with the learning goal.  

3: Student demonstrates mastery with the learning goal as evidenced by using appropriate symbols, vocabulary, and labeling to communicate effectively and exchange ideas.

2: Student demonstrates they are nearing proficiency by incorporating appropriate vocabulary and symbols in most mathematical communications.

1: Student demonstrates a limited understanding or skill with the learning goal by communicating his/her reasoning and solution to others, with support.
 

Learning Targets

• Use clear and precise language in their discussions with others and in their own reasoning
• Specify units of measure and state the meaning of the symbols used
• Report answers that appropriately address the context of a problem
   

Learning Design

• Facilitates, encourages and expects precision in communication
• Provides opportunities for students to explain and/or write their reasoning to others
   

High Priority Standards

CCSS.Math.Practice.MP7

Learning Goal

Students will be able to look for and make use of structure. 

Proficiency Scale

4: In addition to score 3.0 performance, the student demonstrates an in-depth inference or advanced application, or innovates with the learning goal.  

3: Student demonstrates mastery with the learning goal as evidenced by: 

• Noticing mathematical expressions as component parts.
• Using mathematical generalizations to identify the most efficient solution to mathematical tasks.

2: Student demonstrates they are nearing proficiency by composing and decomposing number situations and relationships in order to simplify solutions.

1: Student demonstrates a limited understanding or skill with the learning goal by looking for structure or patterns within mathematics to help him/her solve problems efficiently.
 

Learning Targets

• Look closely to discover a pattern or structure
  • For instance, students use properties of operations as strategies to add, subtract, multiply and divide with whole numbers, fractions, and decimals.
• Examine numerical patterns and relate them to a rule or a graphical representation
   

Learning Design

• Provides opportunities and time for students to explore patterns and relationships to solve problems
• Provides rich tasks and facilitates pattern seeking and understanding of relationships in numbers rather than following a set of steps and/or procedures

High Priority Standards

CCSS.Math.Practice.MP8

Learning Goal

Students will be able to look for and express regularity in repeated reasoning.

Proficiency Scale

4: In addition to score 3.0 performance, the student demonstrates an in-depth inference or advanced application, or innovates with the learning goal.  

3: Student demonstrates mastery with the learning goal as evidenced by:

• Connecting prior knowledge to an unfamiliar mathematical situation.
• Creating a model or equation that unifies the various aspects of a problem.
• Noticing patterns, making generalizations, and predicting patterns.

2: Student demonstrates they are nearing proficiency by finding and explaining patterns. 

1: Student demonstrates a limited understanding or skill with the learning goal by connecting prior knowledge to new situations and noticing patterns with prompting from a teacher or peer.
 

Learning Targets

• Notice repetitive actions in computation and look for more shortcut methods
• Use repeated reasoning to understand algorithms and make generalizations about patterns
   

Learning Design

• Provides problem situations that allow students to explore regularity and repeated reasoning
• Provides rich tasks that encourage students to use repeated reasoning to form generalizations and provides opportunities for students to communicate these generalizations
   

High Priority Standards

A1.IF.A.1 Understand that a function from one set (domain) to another set (range) assigns to each element of the domain exactly one element of the range. a.Represent a function using function notation. b.Understand that the graph of a function labeled 𝑓 is the set of all ordered pairs (𝑥, y) that satisfy the equation 𝑦=f(𝑥).

A1.IF.A.2 Use function notation to evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

A1.IF.B.3 Using tables, graphs and verbal descriptions,interpret key characteristics of a function that models the relationship between two quantities.

A1.IF.B.4 Relate the domain and range of a function to its graph and, where applicable, to the quantitative relationship it describes.

A1.IF.B.5 Determine the average rate of change of a function over a specified interval and interpret the meaning.

A1.IF.B.6 Interpret the parameters of a linear or exponential function in terms of the context.

A1.IF.C.7 Graph functions expressed symbolically and identify and interpret key features of the graph.

A1.IF.C.8 Translate between different but equivalent forms of a function to reveal and explain properties of the function and interpret these in terms of a context.

A1.IF.C.9 Compare the properties of two functions given different representations.

A1.BF.A.1 Analyze the effect of translations and scale changes on functions.

Learning Goal

Students will be able to graph and analyze linear, quadratic, and exponential functions.

Proficiency Scale

4: In addition to score 3.0 performance, the student demonstrates an in-depth inference or advanced application, or innovates with the learning goal.  

3: Student demonstrates mastery with the learning goal as evidenced by:

• Understanding that a function from one set (domain) to another set (range) assigns to each element of the domain exactly one element of the range.
• Representing a function using function notation.
• Using function notation to evaluate functions for inputs in their domains.
• Interpreting statements that use function notation in terms of a context.
• Determining the average rate of change of a function over a specified interval. 
• Interpreting the meaning of the average rate of change over a specified interval in a given context.
• Using tables, graphs and descriptions, interpret key characteristics of a function that models the relationship between two quantities.
• Interpreting the parameters of a linear in terms of the context.
• Graphing linear functions from their symbolic representation to show key features of the graph both by hand and by using technology.
• Translating between different forms of a function to reveal and explain different properties of the function and interpreting them in terms of a context (e.g. slope and intercepts).
• Comparing the properties of two linear functions given different representations (e.g. tables, graphs, equations, and in words).

2: Student demonstrates they are nearing proficiency by:

• Recognizing and recalling specific vocabulary such as:  input, output, domain, range, function, linear, quadratic, exponential, function notation f(x), intercepts, increasing, decreasing
• Performing processes such as: 
• Understanding that the graph of a function labeled f is the set of all ordered pairs (x, y) that satisfy the equation y=f(x).
• Relating the domain and range of a function to its graph and, where applicable, to the quantitative relationship it describes.
• Determining if a function is linear, quadratic, exponential.
• Determining if a relation is a function.
• Recognizing slope and y-intercepts from a graph.

 
1: Student demonstrates a limited understanding or skill with the learning goal by determining if a graph is a function using the vertical line test.
 

Learning Targets

• Understand the concept of a function and use function notation
• Understand that f(x) denotes the elements of the range of a function f that correspond to the to the elements of the domain and the graph of f is the graph of the equation y = f(x)
• Understand that the input and output values of a function correspond to (x, y) values on the Cartesian coordinate plane.
• Interpret functions that arise in applications in terms of a context
• Analyze functions using different representations
• Build a function that models a relationship between two quantities
• Interpret the parameters of functions in terms of the situation they model
• Translate between different but equivalent forms of a function to reveal and explain properties of the function and interpret these in terms of a context 
   

Learning Design

High Priority Standards

A1.CED.A.1 Create equations and inequalities in one variable and use them to model and/or solve problems.

A1.CED.A.2 Create and graph linear, quadratic and exponential equations in two variables.

A1.CED.A.3 Represent constraints by equations or inequalities and by systems of equations or inequalities, and interpret the data points as a solution or non-solution in a modeling context

A1.REI.A.1 Explain how each step taken when solving an equation or inequality in one variable creates an equivalent equation or inequality that has the same solution(s) as the original.

A1.REI.C.6 Explain that the graph of an equation in two variables is the set of all its solutions plotted in the Cartesian coordinate plane.

A1.REI.C.7 Graph the solution to a linear inequality in two variables.

Learning Goal

Students will be able to solve, write, and graph equations.

Proficiency Scale

.

4: In addition to score 3.0 performance, the student demonstrates an in-depth inference or advanced application, or innovates with the learning goal.  

3: Student demonstrates mastery with the learning goal as evidenced by:

• Creating and manipulating equations in multiple forms.
• Solving multi-step equations. 
• Creating linear equations in one variable to model and solve problems.
• Creating linear equations in two variables to model and solve problems.
• Graphing linear equations in two variables on the Cartesian coordinate plane with labels and scales.    
• Interpreting data points to determine if they are a solution or non-solution within a modeling context.
• Representing constraints with a linear equation within a modeling context. 

2: Student demonstrates they are nearing proficiency by:

• Recognizing and recalling specific vocabulary such as:  linear equation, equations, slope-intercept form, y-intercepts, x-intercepts, coefficient, constant, variable, coordinate plane, quadrants, distributive property, combining like terms, standard form
• Performing processes such as: 
  • Explaining how each step taken when solving a linear equation creates an equivalent equation that has the same solution(s) as the original.
  • Explaining that the graph of a linear or exponential equation in two variables is the set of all its solutions plotted on the Cartesian coordinate plane.  
  • Explaining that a point not on the graph of a linear or exponential equation in the Cartesian coordinate plane is not a solution.

1: Student demonstrates a limited understanding or skill with the learning goal by finding the slope and y-intercept from an equation in slope-intercept form.
 

Learning Targets

• Represent and solve equations graphically and explain that the graph is the set of all its solutions plotted on the coordinate plane.
• Interpret parts of an expression, such as terms, factors, and coefficients
• Create equations in one variable and use them to solve problems.
• Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales
• Represent constraints by equations and interpret the data points as a solution or non-solution in a modeling context.In exponential relationships when solving for the exponent, limit bases to 2, 3, 5, and 10 and limit exponents to 1, 2, 3, and 4.
• Example: Explain why 2x + 6 = 8 and 2x = 2 are equivalent and have the same solution.
• When creating and graphing equations, limit exponentials to the form y = ab^x, where b is rational and greater than zero.

Learning Design
 

High Priority Standards

A1.APR.A.1 Add, subtract and multiply polynomials, and understand that polynomials follow the same general rules of arithmetic and are closed under these operations.

A1.APR.A.2 Divide polynomials by monomials.

A1.SSE.A.2 Analyze the structure of polynomials to create equivalent expressions or equations.

Learning Goal

Students will be able to simplify exponential and polynomial expressions.  

Proficiency Scale

4: In addition to score 3.0 performance, the student demonstrates an in-depth inference or advanced application, or innovates with the learning goal.  

3: Student demonstrates mastery with the learning goal as evidenced by:

• Multiplying polynomials by polynomials.
• Dividing polynomials by monomials.
• Simplifying expressions using multiple rules of exponents.

2: Student demonstrates they are nearing proficiency by:

• Recognizing and recalling specific vocabulary such as: polynomials, monomials, binomials, trinomials, base, exponent, coefficient, constant, terms, like terms, expression, degree of term, degree of polynomial
• Performing processes such as: 
• Adding and subtracting polynomials.
• Multiplying a polynomial by a monomial.
• Writing a polynomial in standard form and determining the degree of polynomial.
• Simplifying expressions using a single rule of exponents.

1: Student demonstrates a limited understanding or skill with the learning goal.
 

Learning Targets

• Perform arithmetic operations on polynomials
• Multiplying polynomials limited to the product of a binomial and a trinomial (or fewer terms).
• Integer coefficients for all polynomial operations.
• When dividing, the monomial should be a factor of the polynomial.
• When factoring, include polynomials with a GCF that, when factored, results in a factorable quadratic expression.
• Analyze the structure of polynomials to determine an appropriate method for decomposing and composing to create equivalent equations.
   

High Priority Standards

A1.CED.A.4 Solve literal equations and formulas for a specified variable that highlights a quantity of interest.

A1.REI.A.1 Explain how each step taken when solving an equation or inequality in one variable creates an equivalent equation or inequality that has the same solution(s) as the original.

A1.SSE.A.1 Interpret the contextual meaning of individual terms or factors from a given problem that utilizes formulas or expressions.

A1.NQ.A.1 Explain how the meaning of rational exponents extends from the properties of integer exponents.

A1.NQ.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. Limit to rational exponents with a numerator of 1.

Learning Goal

Students will be able to manipulate equations.

Proficiency Scale

4: In addition to score 3.0 performance, the student demonstrates an in-depth inference or advanced application, or innovates with the learning goal.  

3: Student demonstrates mastery with the learning goal as evidenced by:

• Solving literal equations and formulas for a specified variable that highlights a quantity of interest.
• Interpreting the contextual meaning of individual terms or factors in a problem.
• Identifying which variable to solve for in a given situation.
• Rewriting equations in other forms (linear: standard and slope-intercept).

2: Student demonstrates they are nearing proficiency by:

• Recognizing and recalling specific vocabulary such as:  formula, variable, solve, function, simplify, equivalent expression, standard, slope-intercept, and vertex
• Performing a process such as 
• Isolating a variable in an equation, without completely simplifying.
• Explaining how each step when solving an equation creates an equivalent equation that has the same solution(s) as the original.

1: Student demonstrates a limited understanding or skill with the learning goal.
 

Learning Targets

• Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R
• Example: Compare how doubling the beginning principal affects the final amount as opposed to doubling the time or the rate. A = P(1 + r)^t
• Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales
• Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression
• Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function
   

High Priority Standards

A1.DS.A.1 Analyze and interpret graphical displays of data.

A1.DS.A.4 Summarize data in two-way frequency tables. Interpret relative frequencies in the context of the data, and recognize possible associations and trends in the data.

A1.DS.A.5 Construct a scatter plot of bivariate quantitative data describing how the variables are related; determine and use a function that models the relationship. a) Construct a linear function to model bivariate data represented on a scatter plot that minimizes residuals. b) Construct an exponential function to model bivariate data represented on a scatter plot that minimizes residuals.

A1.DS.A.6 Interpret the slope (rate of change) and the y-intercept (constant term) of a linear model in the context of the data.

A1.DS.A.7 Determine and interpret the correlation coefficient for a linear association.

A1.DS.A.8 Distinguish between correlation and causation.

Learning Goal

Students will be able to summarize, represent and interpret data.

Proficiency Scale

4: In addition to score 3.0 performance, the student demonstrates an in-depth inference or advanced application, or innovates with the learning goal.  

3: Student demonstrates mastery with the learning goal as evidenced by:

• Representing data on a scatterplot and determining the type of function that models the relationship.
• Construct a linear function to model the data on a scatterplot (line of best fit) that minimizes residuals.
• Construct an exponential function to model the data represented on a scatterplot that minimizes residuals.
• Interpreting the meaning of the slope and intercept of a linear model in the context of the data.
• Summarize data in two-way frequency tables.  Interpret relative frequencies in the context of the data.

2: Student demonstrates they are nearing proficiency by:

• Recognizing and recalling specific vocabulary such as:  positive correlation, negative correlation, no correlation, outlier, scatterplot, slope, intercept, line of best fit, and range 
• Performing processes such as: 
  • Recognizing possible associations and trends in the data.
  • Distinguish between correlation and causation.
  • Interpreting the correlation coefficient for a linear association in relation to data.
  • Calculating and comparing the measures of center (median, mean, mode) of two or more different data sets.
  • Interpreting differences in shape, center, and spread in the context of 3 different data sets, accounting for possible effects of outliers.

1: Student demonstrates a limited understanding or skill with the learning goal by analyzing and interpreting data points on a histogram, box plot, dot plot, or scatterplot.
 

Learning Targets

• Represent data with plots on the real number line (dot plots, histograms, and box plots)
• Use statistics appropriate to the shape of the data distribution to compare center (median, mean, and mode) and spread (interquartile range) of two or more different data sets
• Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers)
• Represent data on two quantitative variables on a scatter plot, and describe how the variables are related
• Fit a linear function for a scatter plot that suggests a linear association
• Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data
• Standard deviation should be limited to a small data set (less than or equal to ten data points) with an integral mean.
   

Learning Design

• Two-way frequency tables: http://mathbitsnotebook.com/Algebra1/StatisticsReg/ST2TwoWayTable.html