1A: Standards and Big Ideas

This is the first trimester of Integrated Math 1. This course is comprised of three units.

Unit 1: Relationships and Reasoning with Equations

Timeline: Approximately 5 weeks

Big Ideas:

  • There are universal practices in graphing points and lines.  Those graphs help represent real data and functions.  
  • Solving equations and inequalities helps determine unknown variables and the functions themselves.
  • The different parts of expressions, equations and inequalities can represent certain values in the context of a situation and help determine a solution process.  
  • Relationships between quantities can be represented symbolically, numerically, graphically, and verbally in the exploration of real world situations.
  • Rules of arithmetic and algebra can be used together with notions of equivalence to transform equations and inequalities.
  • Equivalent forms of an expression can be found, dependent on how the expression is used

Essential Questions:

  • How do you graph points and values from a table?
  • How do you determine scale for the axes of a graph?
  • How does input/output relate to x/y and graphing?
  • How do you solve one and two step equations and inequalities?
  • How are equations and inequalities used to solve real world problems?  
  • When is it advantageous to represent relationships between quantities symbolically? numerically? graphically?  
  • Why are procedures and properties necessary when manipulating numeric or algebraic expressions?  
  • How can the structure of expressions/equations/inequalities help determine a solution strategy?
  • For a linear expression that represents a real-world context, what do the coefficient and the constant represent?  
  • How does changing (either) the coefficient/constant in a given equation affect the other terms?
  • Why do some word problems require linear equations, while other word problems require linear inequalities?  
  • How do you know if your solution to an equation or inequality makes sense for a given context?  
  • Why is it important to consider constraints when writing an equation for a context (word problem)?  
  • Why might you want to rearrange a given formula to highlight a specific variable or quantity? (For example, consider the distance equation D = rt. Why might you want to solve for r?)

Power Standards:

  • A.SSE.1 Interpret expressions that represent a quantity in terms of its context. ★
         a) Interpret parts of an expression, such as terms, factors, and coefficients.
         b) Interpret complicated expressions by viewing one or more of their parts as a
    single entity. For example, interpret 𝑃(1 + 𝑟) 𝑛 as the product of P and a factor
    not depending on P.
  • A.REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
  • A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.  
  • A.CED.1 Create equations and inequalities in one variable including ones with absolute value and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
  • A.CED.3  Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

Unit 2: Connecting Algebra and Geometry Through Coordinates

Timeline: Approximately 3 weeks

Big Ideas:

  • Geometric figures can be represented in the coordinate plane.  
  • Algebraic properties (including those related to the distance between points in the coordinate plane) may be used to prove geometric relationships.
  • The algebraic relationship between the slopes of parallel lines and the slopes of perpendicular lines.
  • The distance formula may be used to determine measurements related to geometric objects represented in the coordinate plane (e.g., the perimeter or area of a polygon).
  • The distance formula may be used to determine measurements related to geometric objects represented in the coordinate plane (e.g., the perimeter or area of a polygon).
  • Algebraic formulas can be used to find measures of distance on the coordinate plane.  
  • The coordinate plane allows precise communication about graphical representations.  
  • The coordinate plane permits use of algebraic methods to obtain geometric results.

Essential Questions:

  • How are basic geometric figures constructed in order to maintain their properties using a variety of tools?  
  • What is the relationship between the slopes of parallel lines and of perpendicular lines?  
  • Given a polygon represented in the coordinate plane, what is its perimeter and area?  
  • How can geometric relationships be proven through the application of algebraic properties to geometric figures represented in the coordinate plane?
  • Why are the slopes of parallel lines equal?  
  • Why do the slopes of perpendicular lines have a product of –1?  
  • Why is it useful to know the Pythagorean Theorem, if you can’t remember the distance formula?
  • If we are given the equation of a line parallel or perpendicular to a line we are trying to find, what do we know and what do we not know?  
  • What information would you need in order to prove/disprove that a triangle is a right isosceles triangle (given its coordinates on a coordinate plane)?
  • How can the distance between two points be determined?
  • How are the slopes of lines used to determine if the lines are parallel, perpendicular, or neither?
  • How do we write the equation of a line that goes through a given point and is parallel or perpendicular to another line?  
  • How can slope and the distance formula be used to determine properties of polygons and circles?
  • How can slope and the distance formula be used to classify polygons?
  • How do I apply what I have learned about coordinate geometry to a real–world situation?

Power Standards:

  • G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
  • G.GPE.5 Prove the slope criteria for parallel and perpendicular lines; 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).
  • G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Unit 3: Descriptive Statistics

Timeline: Approximately 3.5 weeks

Big Ideas:

  • Statistical patterns of change can be represented using data tables, graphs, and problem conditions.
  • Data can be represented and interpreted in a variety of formats.  
  • Extreme data points (outliers) can skew interpretations of a set of data.  
  • Synthesizing information from multiple sets of data results in evidence-based interpretation.  
  • Center and spread of a data set may be compared in multiple ways.
  • Data in a two –way frequency table can be summarized using relative frequencies in the context of the data.  
  • Making predictions for values within or near the data set is more reliable than for values far beyond the data set
  • Linear models can be created, used, and interpreted for real-life situations.

Essential Questions:

  • How is useful data collected, reported, and analyzed?
  • How can data be varied?
  • How do various representations of data lead to different interpretations of the data?  
  • When and how can extreme data points impact interpretation of data?  
  • Why are multiple sets of data used?  
  • How are center and spread of data sets described and compared?  
  • How is a data set represented in a two-way frequency table summarized?  
  • How do you determine which model is best to represent a given data set?  
  • How do outliers affect the center, shape, and spread of a data set?  
  • Which graphical representations display each of the following: mean, median, interquartile range, and standard deviation?  
  • How can you determine if the data is skewed?  
  • How can data be represented in order to promote a certain agenda?
  • What real world situations can be modeled by a linear relationship?
  • How can technology help to determine whether a linear model is appropriate in a given situation?

Power Standards:

  • S.ID.1 Represent data with plots on the real number line (dot plots, histograms & box plots).
  • S.ID.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.
  • S.ID.3 Interpret differences in shape, center and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
  • S.ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
  • S.ID.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.

 

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