# Integrated Math 1

In Integrated Math 1, students build on the descriptive statistics, function, expressions and equations, and geometric work first encountered in the middle grades, taking the ideas further while using more formal reasoning and precise language. Specifically, students add to the statistical work from the middle grades by working with standard deviation, describing statistical distributions more precisely, and measuring goodness-of-fit with residuals and correlation coefficient. They formalize their concept of function and encounter exponential functions as well as other examples of non-linear functions. Explicit comparisons and contrasts are made between linear and exponential functions. They develop their abilities to see structure in expressions to show that expressions involving several operations are equivalent (for example, grasping that “substitution” works at various levels of complexity), and they solve equations and inequalities by writing a series of equivalent statements, justifying each step. In the Geometry units in this course, students develop precise definitions of shapes and relationships previously encountered. Instead of informal arguments, they create more rigorous proofs. Students use geometric transformations to define congruence and, once triangle congruence and criteria are established, they prove a variety of theorems and solve problems related to these ideas.

## Units

#### Summary

• Create dot plots, histograms, and box plots.

• Use available classroom technology to create histograms and box plots and calculate measures of center and spread.

• Use terms such as “flat,” “skewed,” “bell-shaped,” and “symmetric” to describe data distributions.

• Analyze and compare data sets.

• Understand relationships between mean and median for symmetrical and skewed data distributions.

• Recognize outliers when they exist, and know to investigate their source.

• Know that outliers affect the mean, but not the median of a data set.

• Describe variability by calculating deviations from the mean.

• Compare two data sets with the same means but different variabilities, and contrast them by calculating the deviation of each data point from the mean.

• Understand that IQR is a description of variability better suited to a skewed distribution.

• Work with two-way tables.

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#### Summary

• Explain each step in solving a simple equation in one variable. • Create and solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. • Model constraints and relationships between quantities by equations and inequalities, and by systems of equations and inequalities, and interpret solutions. • Solve systems of linear equations approximately by graphing and exactly by algebraic methods. • Understand the principles behind the method of elimination. • Graph the solution set to a linear inequality in two variables as a half-plane. • Graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

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#### Summary

• Represent data on two quantitative variables on a scatter plot. • Describe how two quantitative variables on a scatter plot are related. • Interpret the slope and the intercept of a linear model in the context of the data. • Use available technology to find lines of best fit. • Assess the goodness of fit of a line to a small data set by plotting and analyzing residuals. • Fit a linear function for a scatter plot that suggests a linear association. • Use available technology to compute correlation coefficients. • Understand that the correlation coefficient measures the “tightness” of a line fitted to data. • Understand that correlation does not necessarily imply causality.

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#### Summary

• Interpret key features of graphs in terms of the quantities represented. • Sketch graphs showing key features of the graph by hand and using technology. • Understand that a function from one set (the domain) to another set (the range) assigns to each element of the domain exactly one element of the range. • Use function notation. • Interpret statements that use function notation in various contexts. • Work with graphs of piecewise-defined functions, including step functions. • Relate the domain of a function to its graph. • Relate the domain of a function to the quantitative relationship it describes. • Calculate and interpret the average rate of change of a function over a specified interval. • Estimate the average rate of change of a function from its graph. • Solve for x such that f(x) = c, when f is a linear function. • Write an expression for the inverse of a linear function.

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#### Summary

• Distinguish between the growth laws of linear and exponential functions and recognize when a situation can be modeled by a linear function versus an exponential function. • Graph exponential functions and understand how changing by a constant factor over equal intervals affects the graph. • Model situations of growth and decay with exponential functions expressed in various different forms given a graph, a description of the situation, or two input-output pairs (including reading these from a table) • Understand that over time a quantity increasing exponentially will eventually exceed a quantity increasing linearly. • Understand the form of different expressions for exponential functions in terms of change by a constant factor over equal intervals.

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#### Summary

• Know and be able to use precise definitions of geometric terms. • Make formal geometric constructions by hand and using geometry software. • Given a geometric figure and a rotation, reflection, or translation draw the transformed figure. • Develop definitions of rotation, reflection, and translation. • Represent transformations in the plane; describe transformations as functions whose inputs and outputs are points in the plane. • Describe the rotations and reflections that carry a given quadrilateral or regular polygon onto itself. • Prove that the measures of the interior angles of a triangle have sum 180°.

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#### Summary

• Specify sequences of rigid motions that will carry a figure onto another. • Understand that there can be more than one sequence of rigid motions that carries a figure onto another figure. • Use the definition of congruence in terms of rigid motions to decide if two figures are congruent. • Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent (CPCTC). • Be able to explain how the criteria for triangle congruence follow from the definition of congruence in terms of rigid motions. • Prove theorems about lines and angles. • Prove theorems about parallelograms. • Prove base angles of isosceles triangles are congruent.

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