Algebra I

Units

1. 1

   Relationships between quantities

   5 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.
   5. Exponent rules, radicals and rational exponents; sums and products of rational and irrational numbersSimplify expressions with exponents and radicals, then justify whether exact results are rational or irrational.
2. 2

   Linear equations and inequalities

   10 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.
3. 3

   Systems of equations

   5 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. Systems of a linear and a quadratic equationFind and check every point shared by a line and a parabola; use the same substitution idea with a circle.
4. 4

   Graphs of equations and inequalities

   3 lessons

   1. 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.
   2. 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.
   3. Graphing linear inequalities and systems of inequalities as half-planesGraph each inequality and identify the points that satisfy every rule in a system.
5. 5

   Functions

   10 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.
6. 6

   Linear and exponential models

   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.
7. 7

   Quadratic expressions and equations

   13 lessons

   1. Adding, subtracting and multiplying polynomialsAdd, subtract and multiply polynomials; use special products and identify a product’s degree.
   2. Factoring out the GCF and factoring x^2 + bx + cFactor a quadratic completely by taking out the greatest common factor, then finding the pair with the right product and sum, and read the zeros from the factors.
   3. Factoring ax^2 + bx + c by groupingFactor a trinomial whose x² term has a coefficient by splitting the middle term with the product ac, grouping, and checking by multiplying back.
   4. Special products: difference of squares and perfect square trinomialsSpot a difference of squares or a perfect square trinomial, factor it without a search, and factor completely when a common factor or a repeated pattern is hiding.
   5. Solving quadratic equations by factoringSolve a quadratic equation by making one side zero, factoring, and checking every solution.
   6. Solving quadratics by inspection and square rootsSolve by isolating a squared expression, finding all real solutions, and simplifying exact roots.
   7. Completing the square to reveal the vertex and solveRewrite a quadratic to find its vertex, then complete the square to solve an equation.
   8. The quadratic formula and the discriminantSolve quadratics with the formula and predict the number of real solutions from the discriminant.
   9. Choosing a method to solve a quadraticChoose an efficient method, solve exactly when possible, and check which solutions fit the question.
   10. Graphing quadratic functions: vertex, axis of symmetry and interceptsSketch a parabola from its opening, axis, vertex, intercepts and mirror points, and say what its minimum or maximum value is.
   11. Vertex form and maximum/minimum problemsRead the vertex from y = a(x − h)² + k, rewrite a quadratic into that form, and use it to find the best input and the best value in a situation.
   12. Projectile motion and other quadratic modelsRead a launch model h(t) = −16t² + v₀t + h₀, then find its peak, its landing time and the times it reaches a target height, keeping only answers that fit the story.
   13. Using properties of exponents to rewrite exponential expressionsRewrite an exponential model for a new time unit and interpret its growth rate.
8. 8

   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.