Big Idea: In this unit, we will explore sequences and learn how to describe and predict them. As we move through the unit we will determine whether a sequence is arithmetic (with a constant difference), geometric (with a constant ratio), or neither. We will also develop an understanding of recursive and explicit patterns, as we work to interpret what these sequences could be used for in real-world contexts.
Students will be able to:
Understand Sequences:
Target 1: Find the terms of a sequence given a variety of information.
This includes a visual pattern, a numeric pattern, an explicit formula, or a recursive formula.
Target 2: Determine when a sequence is arithmetic, geometric, or neither, and identify key information for each.
Use and Interpret Sequences:
Target 3: Write arithmetic and geometric sequences in explicit form given a variety of information.
This includes visual patterns, graphs, verbal descriptions or tables.
Target 4: Model situations using arithmetic and geometric sequences.
Big Idea: In the previous unit, we learned to describe and predict patterns in arithmetic and geometric sequences using equations. Building on that understanding, this unit focuses on linear equations and inequalities. We will solve linear equations with one or more variables, compare and justify the steps taken to solve, and interpret what the solutions mean in real-life situations. We will also learn to connect equations, tables, and graphs that represent linear relationships. In the second half of this unit we will apply this understanding to linear inequalities, as we focus on different constraints that might apply in real-world problems.
Students will be able to:
Solve Equations:
Target 5: Solve equations and identify their solution type -- one solution, no solution, or infinite solutions.
Target 6: Isolate a given variable in an equation in order to highlight a quantity of interest.
Target 7: Write and graph linear equations and use their graphs to identify solutions.
This includes writing equations from a graph, verbal description, or table. It also includes equations written in standard form.
Solve Inequalities:
Target 8: Write and solve rational inequalities and graph their solutions.
This includes absolute value and compound inequalities
Target 9: Write and graph a linear inequality and use the graph to identify solutions.
Big Idea: Now that we have used graphs, tables, and equations to represent and solve equations and inequalities, we can extend those skills to data. In this unit we will learn to display data using two-way tables (categorical data), as well as dot plots, histograms, and box plots (numerical data). We will describe and compare data sets by examining their shape, center, and spread, and interpret what those characteristics mean in context. We will also look at data that is best modeled within a scatterplot. When that is the case, we will learn to draw and analyze a line of best fit and use it to make predictions.
Students will be able to:
Use Two-Way Tables to Interpret Categorical Data
Target 10: Interpret a two-way frequency table summarizing data.
This includes interpreting joint, marginal and conditional relative frequencies.
Interpret Numeric Data given in Various Representations:
Target 11: Analyze and interpret data given on a dot plot, histogram or box plot.
Target 12: Use the center of a set of data (median, mean, or mode) to compare two or more different data sets.
Target 13: Use the spread of data (interquartile range, standard deviation) to compare two or more different data sets.
Model Data using Scatter Plots and Linear Regressions:
Target 14: Construct a scatterplot of bivariate data and describe how the variables are related.
This includes using a table of data or data in context.
Target 15: Create, interpret, and use a function that models the linear relationship in a scatter plot.
This includes both estimation and using technology. Interpreting includes slope, y-intercept, and the correlation coefficient.
Target 16: Determine and justify whether a relationship represents correlation or causation.
Big Idea: With linear equations and inequalities as our foundation, we can start to deepen our understanding of functions. In this unit we will determine whether a relationship is a function, use and evaluate function notation, and compare key features of functions -- maximum & minimum, symmetries, end behavior, and more. This will be our first introduction to domain, range, and average rate of change, each of which are concepts that will be addressed throughout high school and will help us to better understands real constraints around mathematical situations. We will also write, evaluate, and graph complex functions, including piecewise and absolute value functions, and identify various transformations of functions from the graph and within their equations.
Students will be able to:
Understanding Functions:
Target 17: Explain why a relation is or is not a function and represent it in function notation.
Target 18: Evaluate a function at a given input.
This include interpreting a statement given in function notation in the context of a given situation.
Interpret Functions:
Target 19: Interpret and/or compare key features of a function given a table, graph, or verbal description.
This includes intercepts; intervals of increasing/decreasing, positive/negative; max & min; symmetries; and end behavior.
Target 20: Identify the domain, range, and average rate of change of a function given its graph and interpret their meaning in the context of a given situation.
Target 21: Graph functions from their symbolic representation and show key features of the graph.
This includes piecewise functions and includes both graphing by hand and using technology.
Use Various forms of Function Equations:
Target 22: Convert between different equivalent forms of a function.
This includes converting between recursive and explicit forms of sequences.
Target 23: Analyze the effect of translations on functions.
This includes describing the effect of transformations on the graph of 𝑓(𝑥) by 𝑘𝑓(𝑥), 𝑓(𝑥) + 𝑘, 𝑓(𝑥 + 𝑘) for specific values of k(any real number).
Big Idea: In the previous unit, we explored functions and learned how to describe, compare, and model real-world situations using function notation. Building from that understanding, this unit focuses on systems of equations and inequalities, which are used when we have two or more equations or inequalities that are working together in a given scenario. We will learn to solve systems of linear equations by graphing, or solve algebraically using substitution or elimination. A big focus in this unit will be interpretation -- interpreting what the solutions mean in context, understanding what constitutes a solution vs. a non-solution, and expanding all of this interpretation to include systems of inequalities — graphing them, identifying solutions, and writing systems to model multiple constraints.
Students will be able to:
Solve Systems of Equations:
Target 24: Solve systems of linear equations algebraically using elimination.
Target 25: Solve systems of linear equations algebraically using substitution.
Target 26: Solve systems of linear equations graphically and identify both solutions and non-solutions.
Solve Systems of Inequalities:
Target 27: Solve systems of linear inequalities graphically and identify both solutions and non-solutions.
Target 28: Use the solutions from a system of inequalities to optimize a situation.
Big Idea: While all of our units leading up to this one have focused on linear equations and inequalities in various forms, this unit will expand our understanding of functions by introducing exponential functions. We will create equations, tables, and graphs to represent both linear and exponential functions, in order to better understand the differences. Within this unit we will learn how exponential functions model situations involving percent increase or decrease, as well as interest rates with different compounding intervals, and we will learn to fit exponential models to data.
Students will be able to:
Understand Exponential Equations:
Target 29: Interpret and/or compare features of an exponential function equation in context.
This includes interpreting the parameters of the function in terms of the context.
Target 30: Write and/or use an equation modeling an exponential function to solve problems.
This includes writing equations from graphs, verbal descriptions, or tables, as well as questions about percent increase & decrease.
Target 31: Graph an exponential function.
Use Exponential Equations:
Target 32: Compare an exponential and linear functions.
This includes comparison over given intervals or at specified points.
Target 33: Convert between expressions with radicals and those with rational exponents.
Target 34: Create, interpret, and use a function that models the exponential relationship in a scatter plot.
This includes both estimation and using technology.
Big Idea: Like we did in Unit 6, we will again expand our understanding of functions by working with quadratics. In this unit we will learn to justify whether a function is linear, quadratic, exponential, or none of these, and identify key features of quadratic functions using graphs, tables, and equations. We will write and interpret quadratic equations in different forms and compare properties of quadratic functions. We will also also describe how changing parts of a quadratic function’s equation, especially when written in vertex form, so we can understand and predict transformations of parabolas on a graph.
Students will be able to:
Understand Quadratic Relationships:
Target 35: Classify a relationship as linear, exponential or quadratic using patterns, tables, and equations.
Target 36: Identify and/or compare key features of a quadratic relationship from a table, graph or equation in factored form.
This includes the vertex, the intercepts, and whether the parabola is concave up or down.
Connect Quadratic Equations and Graphs:
Target 37: Graph a quadratic function in two variables.
Target 38: Use key features of a quadratic function to write an equation.
Manipulate of Quadratic Equations:
Target 39: Analyze the effect of translations and scale changes on quadratic functions.
This includes describing the effect of transformations on the graph of 𝑓(𝑥) by 𝑘𝑓(𝑥), 𝑓(𝑥) + 𝑘, 𝑓(𝑥 + 𝑘) for specific values of k (any real number).
Target 40: Compare and contrast quadratic functions written in factored, standard, and vertex forms.
This includes finding the vertex, intercepts, concavity in each form.
Target 41: Write and/or use an equation modeling a quadratic function to solve problems.
This includes writing equations from graphs, verbal descriptions, tables, or data sets and includes evaluating using order of operations and finding x- and y-intercepts.
Big Idea: In the previous unit, we explored quadratic functions by identifying their key features, comparing them to linear and exponential functions, graphing them, and modeling real-world situations. Building on that, this unit focuses on deepening our understanding of quadratic equations and how we can solve them. We will practice converting quadratic expressions between factored and standard forms, and use factored form to find the zeros of quadratic functions. We will compare different strategies for solving quadratic equations, like factoring. We will also learn to complete the square and using the quadratic formula. By the end of this unit we will identify quadratic equations that have no solution, one solution, or two solutions, interpret those solutions in context, and complete the square to reveal key features like the vertex of the parabola to better understand the solution type found.
Students will be able to:
Manipulate Polynomials:
Target 42: Combine polynomials through addition, subtraction and multiplication.
Target 43: Factor polynomials.
This includes dividing by a monomials (GCF) & factoring completely. "
Solve Quadratics:
Target 44: Identify or solve for the zeros of a quadratic function by factoring.
Target 45: Identify or solve for the zeros of a quadratic function using the square root property.
Target 46: Identify or solve for the zeros of a quadratic function by completing the square.
Target 47: Identify or solve for the zeros of a quadratic function using the quadratic formula.
Target 48: Identify quadratic equations with each solution type -- two solutions, one solution, or no real solution -- and interpret the solution type in the context of the problem.
Solve Systems with Quadratics:
Target 49: Solve systems of equations that include quadratics.