M.1HS: High School Mathematics I

High School Mathematics I

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High School Mathematics I covers linear and exponential relationships: interpreting and graphing functions, writing equations and inequalities, solving systems, and distinguishing linear from exponential growth patterns. Descriptive statistics and scatter plots provide tools for analyzing real data sets. The course also addresses congruence, transformations, and geometric constructions.

M.1HS.1Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.M.1HS.10Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, (e.g., using technology to graph the functions, make tables of values, or find successive approximations). Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value exponential, and logarithmic functions.M.1HS.11Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality) and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.M.1HS.12Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. 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. The graph of f is the graph of the equation y = f(x).M.1HS.13Use function notation, evaluate functions for inputs in their domains and interpret statements that use function notation in terms of a context.M.1HS.14Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n)+ f(n-1) for n ≥ 1.M.1HS.15For a function that models a relationship between two quantities, 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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.M.1HS.16Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. (e.g., If the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.)M.1HS.17Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.M.1HS.18Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.M.1HS.19Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (e.g., Given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.)M.1HS.2Define appropriate quantities for the purpose of descriptive modeling.M.1HS.20Write a function that describes a relationship between two quantities.M.1HS.21Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.M.1HS.22Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.M.1HS.23Distinguish between situations that can be modeled with linear functions and with exponential functions.M.1HS.24Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).M.1HS.25Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.M.1HS.26Interpret the parameters in a linear or exponential function in terms of a context.M.1HS.27Explain 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.M.1HS.28Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.M.1HS.29Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.M.1HS.3Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.M.1HS.30Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.M.1HS.31Represent data with plots on the real number line (dot plots, histograms, and box plots).M.1HS.32Use 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.M.1HS.33Interpret differences in shape, center and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).M.1HS.34Summarize 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.M.1HS.35Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.M.1HS.36Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.M.1HS.37Compute (using technology) and interpret the correlation coefficient of a linear fit.M.1HS.38Distinguish between correlation and causation.M.1HS.39Know precise definitions of angle, circle, perpendicular line, parallel line and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.M.1HS.4Interpret expressions that represent a quantity in terms of its context.*M.1HS.40Represent transformations in the plane using, for example, transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).M.1HS.41Given a rectangle, parallelogram, trapezoid or regular polygon, describe the rotations and reflections that carry it onto itself.M.1HS.42Develop definitions of rotations, reflections and translations in terms of angles, circles, perpendicular lines, parallel lines and line segments.M.1HS.43Given a geometric figure and a rotation, reflection or translation draw the transformed figure using, e.g., graph paper, tracing paper or geometry software. Specify a sequence of transformations that will carry a given figure onto another.M.1HS.44Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.M.1HS.45Use 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.M.1HS.46Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.M.1HS.47Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.M.1HS.48Construct an equilateral triangle, a square and a regular hexagon inscribed in a circle.M.1HS.49Use coordinates to prove simple geometric theorems algebraically. (e.g., 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).)M.1HS.5Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions and simple rational and exponential functions.M.1HS.50Prove 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.)M.1HS.51Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, (e.g., using the distance formula).M.1HS.6Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.M.1HS.7Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. (e.g., Represent inequalities describing nutritional and cost constraints on combinations of different foods.)M.1HS.8Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (e.g., Rearrange Ohm’s law V = IR to highlight resistance R.M.1HS.9Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).M.1HS.CAMCConnecting Algebra and Geometry through CoordinatesM.1HS.CPCCongruence, Proof, and ConstructionsM.1HS.DSDescriptive StatisticsM.1HS.LERLinear and Exponential RelationshipsM.1HS.REReasoning with EquationsM.1HS.RQRelationships between Quantities
Example Problems
The formula for average speed when traveling a distance of d over a time t is:


Bella says, 'I can run 8 miles per hour!' Winston says, 'I can run 700 feet per minute!'

Which one is faster?
Is the ordered pair a solution of the following equation?
Is a solution of ?


Determine if the given value of x is in domain of h or not:


Determine if the given value of x is in domain of f or not:
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