Pre-Algebra: Foundations for Algebra I
This is the year math stops being "follow the steps for this kind of problem" and starts being "figure out what kind of problem this even is." Your child will learn to handle really big and really small numbers with exponents, meet numbers that never end or repeat (like square roots that don't come out clean), solve equations that can honestly have no answer or infinite answers, understand what a function actually is, prove shapes are the same or similar instead of just eyeballing it, use the Pythagorean theorem, work with slope and straight-line graphs, solve two equations at once, and read real scattered data for a trend. By June this feeds directly into Algebra I, which assumes all of it is already solid — not something to re-teach.
Unit 1: Exponents, Powers, and Scientific NotationThis unit builds exponent notation from scratch: what a power actually means, the three rules for combining powers with the same base, then the strange-but-necessary extension to zero and negative exponents, and finally scientific notation for very big and very small numbers. It ends with comparing unfamiliar real-world quantities by their order of magnitude.9 skills ▸close ▾
- Evaluating expressions with exponentsEvaluate numerical expressions with positive integer exponents using order of operations.
- Simplifying with the exponent rulesApply the product-of-powers, quotient-of-powers, and power-of-a-power rules to simplify expressions with a single common base.
- When exponent rules do not applyDistinguish expressions where an exponent rule applies (same base) from superficially similar expressions where it does not (different bases, or addition instead of multiplication).
- Zero and negative exponents from patternsExplain why extending the pattern of decreasing exponents forces the definitions a^0 = 1 and a^-n = 1/a^n.
- Writing numbers in scientific notationConvert numbers between standard form and scientific notation for very large and very small numbers.
- Calculating with scientific notationPerform addition, subtraction, multiplication, and division of numbers expressed in scientific notation, including cases requiring re-normalization of the leading digit.
- Comparing quantities by order of magnitudeCompare the order of magnitude of two unfamiliar real-world quantities given in different units, without being told which quantities to compare or which operation applies.
- Why zero and negative exponents always workGeneralize the exponent rules to justify why a rule that holds for positive integer exponents must also hold for exponent zero and negative integers, using an argument that would apply to any base.
- Finding mistakes in exponent solutionsCritique a flawed worked example that misapplies an exponent rule (e.g. adding exponents with different bases, or mishandling a negative exponent's sign) and identify precisely where the reasoning fails.
Unit 2: The Real Number SystemStudents learn that not every number can be written as a clean fraction or a decimal that ends or repeats — these are the irrational numbers, and square roots of most whole numbers are their main example. Kids learn to estimate these roots without a calculator, place them accurately on a number line, and justify why a number is or isn't rational.9 skills ▸close ▾
- Perfect squares and their rootsGiven a whole number, identify whether it is a perfect square, and if so, state its square root.
- Square roots and cube rootsEvaluate square roots of perfect squares and cube roots of perfect cubes without a calculator.
- Why some square roots are irrationalExplain why the square root of a non-perfect-square must be irrational, using the definition of rational number as a ratio of integers.
- Sorting rational and irrational numbersClassify a given number as rational or irrational based on its decimal expansion or its form (fraction, perfect-square root, non-perfect-square root, known constant like pi).
- Estimating square roots to tenthsEstimate the value of a non-perfect-square root to the nearest tenth by reasoning between consecutive perfect squares.
- Irrational numbers on a number linePlace an unfamiliar irrational number on a number line and justify the placement in writing using bounding rational numbers.
- Ordering rational and irrational numbersCompare and order a mixed set of rational and irrational numbers (fractions, decimals, roots, pi) from least to greatest.
- Is this new number rationalGiven a real-world claim about an unfamiliar irrational number (e.g. a newly defined constant), determine whether it could be rational, and justify the determination without prior exposure to that specific number.
- What a calculator display cannot proveGiven only a decimal expansion with no visible pattern (e.g. from a calculator display truncated at 10 digits), decide whether the number could still be rational and identify what additional information would be needed to know for certain.
Unit 3: Linear Equations in One VariableKids extend equation-solving to messier equations — variables on both sides, distributing and combining like terms — and confront the fact that solving an equation can honestly end in 'no solution' or 'every number works,' not just a single number. The core image is a balance scale: whatever you do to one side, you do to the other.10 skills ▸close ▾
- One step and two step equationsGiven a one- or two-step equation like 3x + 5 = 20, execute the correct sequence of inverse operations to find x.
- Distributing and combining like termsGiven an equation requiring distribution and combining like terms, apply both procedures in the correct order to isolate the variable.
- Variables on both sidesGiven an equation with variables on both sides, transform it into an equivalent equation with the variable on one side.
- Why some equations have no solutionExplain why solving an equation to a false numerical statement (like 3 = 5) means the equation has no solution.
- One, none, or infinitely many solutionsClassify an equation as having one solution, no solution, or infinitely many solutions by examining its structure before fully solving.
- What no solution equations shareCompare two structurally different equations that both simplify to 'no solution' and identify what they share.
- Writing equations from real situationsTranslate a real-world scenario describing two changing costs or quantities into a linear equation and solve for the unknown.
- Tape overlap and how many solutionsGiven an unfamiliar word problem about strips of tape with unknown overlap, determine whether the situation forces a unique length, no possible length, or any length, and justify the classification.
- Spotting solution counts without solvingGiven a set of four equations with matched surface complexity, sort them by solution count using structural reasoning rather than full solving.
- Like terms in expressionsRecall the definition of like terms and identify them within a multi-term expression.
Unit 4: FunctionsThis is where 'a rule that takes an input and gives exactly one output' gets formal. Students learn to spot a function in a table, a graph, an equation, or a description, use the vertical line test, and — the hard part — recognize when a table and a graph are actually describing the exact same function.8 skills ▸close ▾
- One output for each inputGiven a set of ordered pairs or a table, determine whether it represents a function by checking whether any input repeats with a different output.
- The vertical line testApply the vertical line test to determine whether a graph represents a function, including jagged, piecewise, and curved graphs.
- Matching a table to a graphGiven the same function presented as a table and as a graph, determine whether they represent the same rule by comparing corresponding input-output pairs.
- Checking for a constant rate of changeGiven a function's equation, generate a table of input-output pairs and identify whether the rate of change between consecutive rows is constant.
- Comparing functions shown different waysGiven two functions in two DIFFERENT representations (e.g. one as a table, one as a verbal description), determine which has the greater rate of change by extracting rate of change and initial value from each.
- Building a piecewise rule from wordsGiven a real-world situation described in words with a changing rate at a threshold (e.g. a billing plan), construct a piecewise function rule and justify whether it is linear over its full domain.
- When change is not a functionExplain, using a specific counterexample, why a graph that co-varies smoothly can still fail the function definition.
- Linear or nonlinear in real situationsClassify a set of real-world scenario descriptions as representing linear or nonlinear functions based on whether their rate of change is constant.
Unit 5: Transformations, Congruence, Similarity, and the Pythagorean TheoremThis long unit replaces 'these shapes look the same' with an actual argument. Kids define congruent and similar precisely — congruent means one figure can be slid, flipped, or turned onto the other; similar adds resizing. They build up the Pythagorean theorem from an area diagram they construct themselves, then use it for distances, including between two points on a coordinate grid.12 skills ▸close ▾
- Translations with coordinate rulesGiven a figure on a coordinate grid and a translation rule, plot the image and state the coordinates of each vertex.
- Reflections with coordinate rulesState the coordinate rule for a reflection over the x-axis, the y-axis, or a given horizontal/vertical line, and apply it to a figure.
- Distance and angles under transformationsExplain why distance between vertices is preserved under translations, reflections, and rotations but not under dilations, while angle measures are preserved under all four transformation types.
- Congruence through rigid motionsGiven two figures on a grid, generate and describe a sequence of rigid motions that maps one onto the other, and use that sequence to justify that the figures are congruent.
- Finding the scale factor for similarityGiven two similar figures where the scale factor is not stated, determine the dilation scale factor and center that maps one to the other, then complete the congruence sequence to confirm similarity.
- Angle pairs with parallel linesUse informal angle arguments (vertical, corresponding, alternate interior) to find unknown angle measures when parallel lines are cut by a transversal.
- Why triangle angles sum to 180Construct an informal argument, using a transversal through a triangle's vertex, that the interior angles of any triangle sum to 180 degrees.
- Proving the Pythagorean theorem with areaUsing an area-based diagram (e.g. rearrangement of four congruent right triangles inside a square), explain why a squared plus b squared equals c squared for any right triangle.
- Real problems with the Pythagorean theoremApply the Pythagorean theorem to find an unknown leg or hypotenuse length in a right triangle presented in a real-world context.
- Testing for a right triangleDetermine whether a triangle with three given side lengths is a right triangle, using the converse of the Pythagorean theorem.
- Distance between points using trianglesCompute the distance between two points on the coordinate plane by constructing a right triangle from the segment and applying the Pythagorean theorem.
- Solving problems with no diagramGiven a real-world scenario with no diagram (e.g. two ships' positions given as coordinates, or a 3-D box diagonal), decide independently which combination of transformation and/or Pythagorean reasoning applies and solve it.
Unit 6: Linear Relationships: Slope, Proportionality, and SystemsSlope stops being 'rise over run' memorized as a ratio and becomes a proven fact: similar triangles cut from the same line always have proportional sides, which is why a straight line has one constant rate of change everywhere on it. From there, students graph lines using slope-intercept form, separate proportional relationships (which pass through the origin) from linear ones that don't, and take their first pass at systems of two equations — solving by graphing, then by substitution.13 skills ▸close ▾
- Similar triangles and constant slopeGiven two triangles formed under a line by drawing legs parallel to the axes at two different points, explain why the triangles are similar and why this forces the rise-over-run ratio to be constant.
- Using the slope formulaCalculate the slope of a line from two given points using the slope formula.
- Graphing proportional relationships and slopeGraph a proportional relationship from a table or equation and identify the slope as the unit rate.
- Comparing proportional relationships in different formsCompare two proportional relationships given in different representations (table, graph, equation, verbal description) to determine which has the greater rate.
- Graphing and interpreting slope and interceptDerive and use slope-intercept form y = mx + b to graph a line given its equation, and explain what m and b represent in a real context.
- Using the origin to test proportionalityClassify a given linear relationship as proportional or non-proportional by testing whether it passes through the origin.
- Intersection points as system solutionsEstimate the solution to a system of two linear equations by graphing both lines and reading the intersection point.
- What intersection points mean in contextExplain what the intersection point of two lines means in terms of the two real-world quantities each line represents.
- Solving linear systems using substitutionSolve a system of two linear equations algebraically using substitution, showing all steps.
- Choosing which variable to isolateGiven a system where neither equation has an isolated variable, decide which variable to isolate and justify the choice before solving by substitution.
- How many solutions a system hasDetermine whether a system has one, no, or infinitely many solutions by comparing slopes and intercepts, and connect each case to Unit 3's one-variable equation outcomes.
- Writing systems for unfamiliar situationsGiven a real-world scenario with two unstated relationships, drawn from a domain not used in this unit's pricing- and motion-based worked examples (e.g. population growth, mixture/concentration problems, or tank fill/drain rates), generate and solve a system of equations that models it, choosing an appropriate method and justifying the choice.
- Three lines and one shared pointGiven three graphed lines with no context, determine which two intersect at a point that also lies on the third, and explain how you know without computing coordinates.
Unit 7: Systems of Linear Equations: Modeling and ApplicationThis unit takes the systems skills from Unit 6 and points them at real situations — two constraints on the same two quantities at once. Students write systems from word problems, choose between graphing and substitution and defend the choice, and interpret what a system's solution (or lack of one) actually means in context.10 skills ▸close ▾
- Testing ordered pairs in systemsGiven a system of two linear equations, identify whether an ordered pair satisfies both equations by substitution and checking.
- Intersection points as system solutionsGraph two linear equations on the same coordinate plane and identify the intersection point as the system's solution.
- Substitution with one equation already solvedSolve a system of two linear equations using substitution when one equation is already solved for a variable.
- Choosing between graphing and substitutionCompare the graphing and substitution methods for a given system and justify which is more efficient for that specific system.
- Writing systems from real situationsWrite a system of two linear equations from a real-world context describing two constraints on the same two quantities.
- Interpreting solution points in contextInterpret the intersection point of a graphed system in terms of the original context, stating what quantity and value it represents.
- Predicting solutions from slopes and interceptsClassify a system as having one solution, no solution, or infinitely many solutions by comparing slopes and y-intercepts without fully solving.
- Meaning of no or infinite solutionsExplain, using context, why a system with no solution represents two constraints that can never both be true, and why infinitely many solutions represents the same constraint stated two ways.
- Modeling and solving with fractional answersGiven an unfamiliar two-constraint scenario with non-integer or fractional intersection values, construct, solve, and justify a system, choosing a method and defending that choice.
- No solution in standard formGeneralize the condition for a system having no solution to systems written in standard form, without being shown that form during instruction.
Unit 8: Bivariate Data and Lines of FitReal data is messy — points scattered around, not sitting neatly on a line. This unit teaches kids to spot a trend anyway, draw and compare lines that follow that trend, write the line's equation, use it to predict, and know that a prediction from a trend is not a guarantee for any one point. It also extends the same thinking to categorical data in two-way tables.12 skills ▸close ▾
- Reading association in scatter plotsGiven a scatter plot, classify the pattern as positive, negative, or no association.
- Describing scatter plots in wordsDescribe, in words naming both variables, whether a scatter plot shows a linear or nonlinear pattern and note clusters or outliers.
- Drawing a line of fitDraw a straight line that follows the trend of a scattered but linear-associated dataset.
- Judging which line fits betterCompare two candidate lines of fit on the same scatter plot and justify which follows the trend more closely, citing specific points.
- Writing the equation of a lineWrite the equation of a fitted line in slope-intercept form and identify the slope and y-intercept from a graph.
- What slope and intercept meanInterpret the slope and y-intercept of a fitted line as statements about the real-world variables, including units.
- Predicting with a line of fitGiven a new x-value within the data range, use a fitted line's equation to predict a y-value and judge whether the prediction is reasonable.
- Two-way tables and relative frequencyConstruct a two-way table from categorical data and calculate relative frequencies by row and column.
- Spotting links between categoriesUse relative frequencies in a two-way table to argue whether an association exists between two categorical variables.
- Fitting a line to new dataGiven an unfamiliar real dataset with two numeric variables never used in instruction, construct a scatter plot, fit a line, and justify the fit using evidence from the data.
- Checking another student's lineCritique a peer's completed line-of-fit analysis (plot, line, and interpretation) and identify a specific improvement, using no formula for a 'correct' answer.
- Trends versus exact rulesExplain why a line of fit predicts a trend rather than an exact value for any individual data point.
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