M.TMS: Transition Mathematics for Seniors

Transition Mathematics for Seniors

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Transition Mathematics for Seniors bridges high school mathematics to college-readiness for students who need to consolidate and extend key skills before graduation. Algebra content spans linear equations, quadratic functions, polynomial and rational expressions, and exponential models, while geometry addresses coordinate methods, volume, and geometric modeling. Statistics and probability, including regression, data displays, and inference, prepare students for quantitative reasoning in post-secondary contexts.

M.TMS.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.TMS.10Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract and multiply polynomials.M.TMS.11Create 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.TMS.12Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.M.TMS.13Represent 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.M.TMS.14Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.M.TMS.15Solve simple rational and radical equations in one variable and give examples showing how extraneous solutions may arise.M.TMS.16Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.M.TMS.17Explain 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.TMS.18Solve quadratic equations in one variable. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)^2 = q that has the same solutions. Derive the quadratic formula from this form. Solve quadratic equations by inspection (e.g., for x^2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.M.TMS.19Prove 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.TMS.2Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.M.TMS.20Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.M.TMS.21Explain why the x-coordinates of the points where the graphs of the equation y = f(x) and y = g(x) intersect are the solution of the equation f(x) = g (x); find the solution approximately (e.g., using technology to graph the functions, make tables of values or find successive approximations).M.TMS.22Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.M.TMS.23Graph 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.TMS.24Understand 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.TMS.25Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.M.TMS.26Interpret the parameters in a linear or exponential function in terms of a context.M.TMS.27For 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.TMS.28Distinguish between situations that can be modeled with linear functions and with exponential functions.M.TMS.29Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line, give examples of functions that are not linear.M.TMS.3Solve quadratic equations with real coefficients that have complex solutions.M.TMS.30Describe qualitatively the functional relationship between two quantities by analyzing a graph.M.TMS.31Identify 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.M.TMS.32Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.M.TMS.33Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasingly linearly, quadratically, or (more generally) as a polynomial function.M.TMS.34Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.M.TMS.35Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).M.TMS.36Construct 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.TMS.37Write a function that describes a relationship between two quantities.M.TMS.38Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.M.TMS.39Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.M.TMS.4Use the structure of an expression to identify ways to rewrite it. For example, see x^4 – y^4 as (x^2)^2 – (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2 – y^2) (x^2 + y^2).M.TMS.40Use 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).M.TMS.41Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, (e.g., using the distance formula).M.TMS.42Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with topographic grid systems based on ratios).M.TMS.43Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. Interpret linear models.M.TMS.44Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.M.TMS.45Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.M.TMS.46Summarize 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.TMS.47Represent data with plots on the real number line (dot plots, histograms, and box plots).M.TMS.48Use 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.TMS.49Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).M.TMS.5Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.M.TMS.50Computer (using technology) and interpret the correlation coefficient of a linear fit.M.TMS.51Distinguish between correlation and causation.M.TMS.52Understand statistics as a process for making inferences about population parameters based on a random sample from that population.M.TMS.6Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.M.TMS.7Graph proportional relationships, interpreting the unit rates as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.M.TMS.8Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plan; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.M.TMS.9Solve linear equations in one variable.M.TMS.A-APRAlgebra – Arithmetic with Polynomials and Rational ExpressionsM.TMS.A-CEAlgebra – Creating EquationsM.TMS.A-REIAlgebra – Reasoning with Equations and InequalitiesM.TMS.A-SEAlgebra – Seeing Structure in ExpressionsM.TMS.F-BFFunctions – Building FunctionsM.TMS.F-IFFunctions – Interpreting FunctionsM.TMS.G-EGPGeometry – Expressing Geometric Properties with EquationsM.TMS.G-GMDGeometry – Geometric Measuring and DimensionM.TMS.G-MGGeometry – Modeling with GeometryM.TMS.NQ-CNSNumber and Quantity – The Complex Number SystemM.TMS.NQ-RNNumber and Quantity – The Real Number SystemM.TMS.SP-ICQStatistics and Probability – Interpreting Categorical and Quantitative DataM.TMS.SP-IJCStatistics and Probability – Making Inferences and Justifying Conclusions
Example Problems
Distribute to create an equivalent expression with the fewest symbols possible:
Solve for x.
Use the equation:


Complete the missing value in the solution to the equation:
Aaron has $30 to spend on chocolate bars and lollipops. He buys a box of lollipops for $4.

After buying the lollipops, he wants to spend the remaining money on chocolate bars that cost $2.75 each.

Write and solve an inequality to find the greatest number of chocolate bars that Aaron can buy.
What is the slope of the following line?
Goblins

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