Integrated Math I

Units

1. 1

   Relationships between quantities

   4 lessons

   1. Units as a guide: unit analysis and scalesConvert quantities and rates, check formula units, and compare graph rates using labelled scales.
   2. Defining quantities and choosing a sensible level of accuracyDefine quantities with units, choose relevant information, and report answers at a sensible precision.
   3. Interpreting the parts of an expressionInterpret an expression’s signed terms, factors, coefficients, and quantities, including a repeated growth factor.
   4. Using structure to write equivalent expressionsRewrite expressions using their structure and justify why the new forms have the same value.
2. 2

   Linear equations and inequalities

   11 lessons

   1. Solving as reasoning: justifying each stepSolve linear equations, justify each line, and check solutions in the original equation.
   2. Solving multistep linear equations, including special casesSolve multistep equations and distinguish one solution, no solution, and every real number as a solution.
   3. Writing equations and inequalities in one variable from situationsDefine an unknown, write a relationship from a situation, solve it, and interpret the answer.
   4. Rearranging formulas to highlight a quantityRearrange a formula to leave a requested variable alone, including when it appears in two terms.
   5. Solving and graphing linear inequalities, including compound inequalitiesSolve and graph single and compound inequalities, including reversing signs when dividing by a negative.
   6. Absolute value equations and inequalitiesSolve absolute value equations and inequalities, graph solutions, and check candidates in the original statement.
   7. Slope from two points, tables and graphsFind the slope of a line from two points, a table or a graph, and say what it means as a rate with units.
   8. Slope-intercept form: graphing and writing linesGraph a line from y = mx + b, write y = mx + b from a graph, a point and slope, or two points, and say what m and b mean in a story.
   9. Point-slope and standard form; converting between forms; horizontal and vertical linesWrite a line in point-slope form from one point and a slope, graph from standard form using intercepts, convert between all three forms, and choose the form the information suggests.
   10. Equations in two variables and modelling with constraintsWrite a two-variable equation, graph its intercepts, and identify pairs that meet every constraint.
   11. Linear, compound and absolute value inequalitiesSolve linear, compound and absolute value inequalities, graph each solution set on a number line, and write it in interval notation.
3. 3

   Linear and exponential relationships

   6 lessons

   1. Equal differences or equal factors: telling linear from exponentialClassify tables and situations by comparing changes over equal input intervals.
   2. Writing linear and exponential functions from graphs, tables, descriptions and two pointsWrite and check a linear or exponential rule from a graph, table, description or two points.
   3. Exponential growth eventually overtakes linear and quadratic growthCompare models with tables and growing steps to find when an exponential takes a lasting lead.
   4. Interpreting the parameters of linear and exponential models in contextInterpret each model number with its units, its meaning, and the context’s starting time.
   5. Graphing exponential functions: intercept, asymptote, growth and decaySketch y = a·bˣ from a table, name its y-intercept and asymptote, tell growth from decay by b, and say its domain and range.
   6. Percent growth and decay; simple and compound interestWrite a growth or decay model from a percent rate, tell simple interest from compound interest, use A = P(1 + r/n)^(nt), and estimate a doubling time.
4. 4

   Systems of equations

   7 lessons

   1. Systems of linear equations: solutions as intersectionsFind and check a shared point, then predict how many solutions two lines have.
   2. Solving systems by substitutionSolve a system by substitution, check its ordered pair, and recognize no solution or infinitely many solutions.
   3. Solving systems by elimination, and why it worksSolve systems by elimination and explain why scaling and reversible combinations preserve their solutions.
   4. Modelling with systems and choosing a methodModel two facts with a system, choose a solution method, and check the answer in context.
   5. Linear systems in two variables: graphing, substitution and eliminationSolve a pair of linear equations by graphing, substitution and elimination, and say when there is one solution, none or infinitely many.
   6. Linear systems in three variablesSolve three linear equations in three unknowns by eliminating one unknown twice, check the triple, and use the method to fit a parabola through three points.
   7. Systems of linear inequalities and linear programmingGraph a system of linear inequalities as a region, find its corner points, and maximise or minimise a linear quantity by comparing its value at the corners.
5. 5

   Functions and their graphs

   13 lessons

   1. What a function is: one output per input, function notation, domain and rangeTest whether a relation is a function, read function notation, and find domain and range in context.
   2. Sequences as functions: arithmetic and geometric, recursive and explicitIdentify a constant difference or ratio, write rules with the correct start, and find a specified term.
   3. Interpreting key features of graphs and tables in contextInterpret graph and table features with input and output units, then sketch a graph from a story.
   4. Average rate of change over an intervalFind and interpret average rates from graphs, tables, and formulas, with appropriate units.
   5. Graphing linear, quadratic, square root, cube root, absolute value, piecewise and step functionsSketch each family from a rule, labeling anchor points, key features and included endpoints.
   6. Equivalent forms that reveal properties: zeros, vertex, growth and decay ratesChoose a form to find a quadratic’s zeros, vertex, or y-intercept, and read an exponential rate per stated time unit.
   7. Comparing functions given in different representationsCompare the same property of two functions, even when one is an equation and the other is a table or graph.
   8. Building functions from contexts and combining standard functionsBuild functions from a situation, combine them, and use the result to answer a question.
   9. Transforming graphs: f(x) + k, k·f(x), f(kx), f(x + k), and even and odd functionsDescribe graph shifts and scales, find a multiplier from a point, and classify functions as even, odd, or neither.
   10. Inverse functions for linear functionsFind a linear function’s inverse, check it by a round trip, and identify its graph points.
   11. The graph of an equation is the set of all its solutionsDecide whether a point belongs on an equation’s graph and explain why the graph contains more than table points.
   12. Solving f(x) = g(x) by graphing: intersections, tables and successive approximationFind every input where two graphs meet, and approximate inputs that fall between gridlines.
   13. Graphing linear inequalities and systems of inequalities as half-planesGraph each inequality and identify the points that satisfy every rule in a system.
6. 6

   Describing data

   8 lessons

   1. Dot plots, histograms and box plots on the real number lineDraw three displays on an even number line and choose one to answer a data question.
   2. Comparing center and spread: median, mean, IQR and standard deviationCompare two groups using a center and a matching spread, with values and units.
   3. Shape, skew and outliers: how they move the center and spreadName a distribution’s shape, test for outliers with IQR fences, and choose a useful center and spread.
   4. Two-way tables, relative frequencies and associationCalculate relative frequencies, judge association, and distinguish association from cause.
   5. Scatter plots and fitting linear, quadratic and exponential modelsPlot paired data, choose a model from its pattern, and judge whether a prediction is within the data range.
   6. Residuals and residual plots: judging how well a model fitsCalculate residuals, plot them against x, and use their pattern to judge a model’s fit.
   7. Interpreting slope and intercept of a linear model in contextInterpret a fitted line’s slope and intercept with units and predicted language; decide whether its intercept is meaningful.
   8. Correlation coefficient and correlation versus causationInterpret r, recognize its limits, and explain why correlation alone does not establish causation.
7. 7

   Congruence and constructions

   15 lessons

   1. Precise definitions: segment, angle, circle, perpendicular and parallel linesDefine these figures and line relationships precisely, then test definitions against counterexamples.
   2. Rigid motions as functions: translations, reflections, rotations and sequencesMap points through rigid motions, describe an ordered sequence, and check every vertex of its image.
   3. Symmetry: the rigid motions that carry a figure onto itselfIdentify and justify the reflections and rotations that carry a figure onto itself.
   4. Congruence defined by rigid motions and corresponding partsUse rigid motions to establish congruence, read corresponding parts from letter order, and find missing triangle measures.
   5. Triangle congruence criteria: SSS, SAS, ASA (and AAS), and why SSA and AAA failDecide whether marked parts prove triangle congruence, and justify the decision.
   6. Conditional statements and the logic of proofWrite a claim as if-then, form its converse, inverse and contrapositive, disprove it with a counterexample, and tell a deduction from a pattern.
   7. Writing a two-column proofTurn a claim into Given and Prove, plan a proof backward, and write each statement with a reason that justifies it.
   8. Proving theorems about lines and anglesProve angle relationships and both directions of the perpendicular-bisector theorem with reasons for each step.
   9. Proving theorems about triangles: angle sum, isosceles, midsegment, mediansProve and apply four triangle theorems, including the exterior-angle result.
   10. Proving theorems about parallelogramsProve parallelogram properties, recognize special parallelograms, and justify tests for an unmarked quadrilateral.
   11. Formal constructions with compass and straightedgeConstruct bisectors, a parallel line and inscribed polygons; justify each using equal distances or congruent triangles.
   12. Indirect proof: assume the opposite and reach a contradictionWrite the opposite of a claim, assume it, find the clash with a given or a proven fact, and conclude the claim.
   13. The triangle inequality and the order of sides and anglesDecide whether three lengths make a triangle, find the possible third sides, and order the sides of a triangle from its angles.
   14. Interior and exterior angles of polygonsFind the interior angle sum of any convex polygon, the interior and exterior angles of a regular polygon, and a missing angle or number of sides.
   15. Quadrilaterals: trapezoids, kites and how the families nestUse the properties of trapezoids and kites, place every quadrilateral in the family tree, and classify a quadrilateral from the marks it has.
8. 8

   Connecting algebra and geometry

   4 lessons

   1. Slopes of parallel and perpendicular linesWrite a line through a point parallel or perpendicular to a given line, and justify its slope.
   2. Partitioning a directed segment in a given ratioFind a point that divides a directed segment in a given ratio.
   3. Proving theorems with coordinatesUse distances, slopes, and midpoints to prove claims about a figure; use letters to prove a claim for every triangle.
   4. Perimeter and area of polygons in the coordinate planeFind a polygon’s perimeter from coordinate distances and its area by enclosing and subtracting.