Mathematics 7: Proportional Reasoning and Rational Number Systems
This is the year math stops being "follow the recipe" and starts being "here's the rule, and here's why it's true." Your kid will spend the fall pinning down what "proportional" really means and learning to compute with negative numbers, fractions, and decimals the same way. In winter they turn that into algebra — expressions and equations they can solve and explain. In spring they point all of it at geometry (scale drawings, angles, circles, 3D shapes) and finish with probability and sampling, which is really just ratios and rates wearing a different hat. By June they should be able to look at a table, a graph, or a word problem and immediately ask "is this proportional?" and "what's actually equal to what here?" — and answer both.
Unit 1: Proportional RelationshipsThis unit is about one question asked five different ways: is the ratio between two quantities always the same? Your kid learns to check that using fraction unit rates, tables, graphs, and equations, then applies it to real percent problems like tax, tip, and discount.9 skills ▸close ▾
- Unit rate calculation with fractional quantitiesGiven a rate involving fractional quantities (e.g. 3/4 cup per 2/3 hour), compute the unit rate.
- Constant of proportionality k derived from a tableGiven a table of paired values, determine the constant of proportionality k by computing y/x for each row.
- Proportional versus non-proportional relationships identified across representationsGiven two representations of the same context (a table and a graph, or a table and a verbal description) that are NOT both provided directly, determine whether the underlying relationship is proportional by checking whether the ratio y/x is constant across all given pairs, not just the first pair or the visual shape of the graph.
- The role of the origin in distinguishing proportional from linear-but-non-proportional graphsExplain why a straight-line graph that does not pass through the origin represents a non-proportional relationship, using the constant-ratio definition rather than the visual shape alone.
- The equation y = kx as a representation of a proportional relationshipWrite the equation y = kx for a proportional relationship given a real-world context, a table, or a graph, identifying k correctly in each case.
- The point (1, r) as the unit rate on a proportional graphInterpret the point (1, r) on a graph of a proportional relationship as showing the unit rate r.
- Multi-step percent problems involving tax, tip, markup, and discountGiven a multi-step context involving tax, tip, markup, or discount, solve for the final amount, correctly identifying the original/base amount before calculating the percent.
- The non-commutativity of sequential percent changes applied to an unfamiliar order-of-operations scenarioGiven a two-step scenario where an item is marked up then discounted (or discounted then marked up) by the same percent, determine whether the final price is the same in both orders and justify the answer using the structure of percent change, without having been shown this specific comparison during instruction.
- Proportionality analysis applied to a pricing structure with a fixed cost, in a context not used during instructionGiven a completely unfamiliar real-world dataset (e.g. a utility bill with a flat fee plus a per-unit rate, presented as a table), determine whether it is proportional and, if not, identify what would need to change about the pricing structure to make it proportional.
Unit 2: Operations with Rational NumbersThis unit extends everything your kid knows about adding, subtracting, multiplying, and dividing to negative numbers, fractions, and decimals together. Every new move gets shown first as a movement on a number line before it becomes symbols. It ends with problems that deliberately mix all four operations, which will feel harder than anything before it — that's expected.9 skills ▸close ▾
- Additive inverse as a distance-preserving reflection across zero on the number lineGiven a point on a number line, state its additive inverse and explain why the two points are the same distance from zero.
- Addition of rational numbers as directed movement on the number lineCompute the sum of two rational numbers with the same or different signs using a number-line movement model.
- The equivalence of subtraction and addition of the inverse (p - q = p + (-q))Explain why subtracting a rational number is equivalent to adding its additive inverse, using two contrasted number-line diagrams.
- The sign rule for multiplication of rational numbers, derived from a repeated-addition patternDerive the rule for the sign of a product of two rational numbers by extending the pattern of repeated addition to negative factors.
- Multiplication and division of signed rational numbers, including fractions and decimalsApply the derived sign rules to compute products and quotients of rational numbers including fractions and decimals.
- Fraction-to-decimal conversion via long division, including repeating decimalsConvert a fraction to its decimal form, determining whether the result terminates or repeats, using long division.
- Multi-step rational-number word problems requiring selection of operations without contextual cuesSolve a multi-step word problem involving all four operations on rational numbers in a context not used during instruction (e.g., elevation change across multiple unrelated stops, or combined temperature and time-of-day shifts).
- Justification of a signed-number operation rule using a self-constructed number-line diagramJustify a chosen sign rule (for addition, subtraction, multiplication, or division) by constructing an original number-line diagram for a rational-number pair the student has not seen used with that rule before.
- The distinction between absolute distance between two rational numbers and signed change from one to anotherDistinguish problems that require finding a difference in distance (absolute value of a difference) from problems that require finding a signed change, given a set of contextual word problems.
Unit 3: Expressions and Multi-Step EquationsHere arithmetic becomes algebra. Your kid learns that an equation is a statement that two things are equal, and every legal move has to keep both sides equal — like a balance scale. They simplify expressions first, then solve increasingly complex equations, then translate word problems into equations, then do the same with inequalities.11 skills ▸close ▾
- Like terms in a linear expression with rational, including negative, coefficientsGiven a linear expression with rational coefficients, combine like terms correctly, including terms with negative coefficients.
- The distributive property applied with a negative multiplierApply the distributive property to expand an expression like -3(2x - 5), including when the multiplier is negative.
- Equivalence between an expanded and a factored form of the same linear expressionExplain why two differently-written expressions, one expanded and one factored, represent the same quantity for every value of the variable.
- Two-step linear equations with rational coefficients, form px + q = rSolve a two-step equation with rational coefficients by undoing operations in reverse order, recording each step as a balance move on both sides.
- Multi-step linear equations of form p(x + q) = r and equations with variables on both sidesSolve multi-step equations that require distributing and combining like terms before undoing operations, including equations with the variable on both sides.
- Translation of a word problem into a px + q = r equationGiven a verbal description of a real situation, decide which quantity is unknown and write an equation of the form px + q = r that models it.
- Arithmetic versus algebraic solution methods for the same word problemCompare an arithmetic (guess-check-reason) solution and an algebraic (equation) solution to the same word problem and judge which reveals the general relationship more clearly.
- One-variable linear inequalities requiring a sign reversalSolve a one-variable inequality of form px + q > r or px + q < r, correctly reversing the inequality symbol when multiplying or dividing by a negative number.
- The solution set of a linear inequality represented on a number lineGraph the solution set of a one-variable inequality on a number line and explain what the shading and endpoint mean about which numbers satisfy it.
- Selecting between an equation and an inequality model for a novel real-world constraint problemGiven a completely unfamiliar multi-step scenario with extraneous numeric information, decide whether it calls for an equation or an inequality, then solve it and justify the choice.
- Application of the equation-as-balance principle to an unfamiliar quantitative domainGiven a real-world balance scenario never used in instruction (e.g., a chemistry mixture or a multi-currency exchange), construct and solve an equation or inequality that models it and explain why the balance principle still applies.
Unit 4: Scale Drawings and Geometric ConstructionsThis is Unit 1's proportional relationships applied to shapes: a scale factor is just the constant of proportionality, applied to every length in a figure at once. The second half shifts to building triangles by hand with a compass, straightedge, and protractor, and figuring out when the given information locks in exactly one triangle.10 skills ▸close ▾
- Scale factor as a constant of proportionality applied to a single lengthGiven a scale drawing and its stated scale, compute the actual length of a labeled segment.
- The relationship between a linear scale factor and the resulting area, computed via actual lengths rather than by squaring the scale factor directlyGiven a scale drawing, compute the actual area of a region by first finding actual side lengths, then applying an area formula.
- The multiplicative (not additive) relationship between linear scale factor and area scale factorExplain why area scales by the square of the linear scale factor, using a grid-square count as evidence.
- The defining property of a scaled copy: every corresponding length pair shares the same constant ratioDistinguish a true scaled copy of a figure from a figure that has been distorted (e.g., stretched in one direction, or altered by addition rather than multiplication).
- Combining two scale ratios (original-to-actual and actual-to-new) to reproduce a figure at a new scaleGiven a scale drawing at one scale, produce an accurate redrawing of the same figure at a specified different scale.
- Deriving a scale factor between two representations with no scale given, using measured corresponding distancesGiven two maps or drawings of the same unfamiliar region at different, unstated scales with no shared labeled reference length, determine the scale relationship between them using only measured distances.
- The triangle inequality as the condition for SSS construction to succeedConstruct a triangle from three given side lengths using a compass and straightedge, and determine whether the construction is possible.
- The conditions (SSS, SAS, ASA) under which a triangle is uniquely determined, versus conditions that permit more than one triangle or noneGiven two side lengths and the included angle (SAS), or two angles and the included side (ASA), construct the triangle with a protractor and ruler and state whether the given conditions determine a unique triangle.
- Generalizing the existence/uniqueness rule for triangles beyond the SSS/SAS/ASA cases explicitly taughtGiven an unfamiliar set of three conditions on a triangle (a mix of side and angle constraints, including an SSA case not previously modeled), determine whether zero, one, or infinitely many triangles satisfy them, and justify the conclusion.
- Discriminating between scale-factor problems and triangle-construction problems based on structural features of the prompt, not its position in the unitGiven a scale-drawing redrawing task and a triangle-construction task presented in mixed, unlabeled order, select and apply the correct procedure for each.
Unit 5: Angle Relationships and AreaFour angle relationships — complementary, supplementary, vertical, adjacent — become sources of equations to solve, using the same balance idea from Unit 3. Then area work extends from grade-6 polygons to circles and composite shapes, ending with surface area of solids made from triangles and quadrilaterals.9 skills ▸close ▾
- Complementary and supplementary angle equationsGiven a diagram of two angles labeled as complementary or supplementary, write an equation using a variable for the unknown angle and solve it.
- Vertical and adjacent angle relationships in a crossing-lines figureGiven an unlabeled diagram of two lines crossing, identify which angle pairs are vertical and which are adjacent, and state the relationship (equal vs. sums to 180) that applies to each.
- Multi-step reasoning across combined angle relationships (vertical, adjacent, supplementary) in a single figureGiven a multi-angle diagram with several lines crossing at one point and only one angle measure provided, write and solve a chain of equations to find every unknown angle, citing the specific relationship used for each step.
- The invariance of the triangle angle sumExplain why the sum of the interior angles of any triangle is 180 degrees, using a torn-corner or parallel-line demonstration, rather than stating the rule as a memorized fact.
- Circle area and circumference formulasCalculate the area and circumference of a circle given its radius or diameter, choosing the correct formula for what is asked.
- Decomposition strategies for composite figures involving circlesGiven a composite figure combining a circle with triangles and/or quadrilaterals, decompose it into named component shapes two different valid ways and compute the total area from either decomposition.
- Surface area of polyhedra built from triangles and quadrilateralsCalculate the surface area of a three-dimensional figure composed of triangular and quadrilateral faces by identifying and summing the area of each distinct face.
- Transfer of angle-relationship reasoning to unfamiliar contextsGiven a real-world scenario not resembling any diagram used in instruction (e.g. the angle between clock hands, or a compass bearing problem), determine which angle relationship applies and solve for the unknown angle.
- Generalized decomposition strategy for surface area of scaled composite solidsGiven a novel composite solid combining a scaled copy of a shape from Unit 4 with new triangular faces, and no formula provided, determine a strategy to find its total surface area and justify why that strategy is valid for any solid built the same way.
Unit 6: Circles, Surface Area, and VolumeThis unit treats every geometry formula as a description of a physical action — going around a shape, wrapping it, or filling it up. It moves from circles (reconnecting pi to Unit 1's constant-ratio idea) through surface area built from actual paper nets, to volume with fractional dimensions, and closes with why volume scales by the cube of the scale factor for similar solids.10 skills ▸close ▾
- Circumference and area formulas for a circle (C = πd or 2πr; A = πr²)Given the radius or diameter of a circle, calculate its circumference and area using the correct formula for each.
- Pi as a fixed ratio of circumference to diameterExplain why pi is the same fixed ratio for every circle, regardless of size, by connecting it to the constant-of-proportionality idea from Unit 1.
- Surface area of prisms via netsGiven a net of a rectangular prism or triangular prism, calculate its total surface area by finding and summing the area of each face.
- Surface area of pyramids via netsGiven a net of a pyramid, calculate its total surface area, distinguishing the base from the lateral (triangular) faces.
- Volume of a rectangular prism with fractional edge lengthsGiven a rectangular prism with fractional edge lengths, calculate its volume by multiplying length, width, and height.
- The independence of surface area and volume as measures of a solidGiven two solids with equal volume but different dimensions, determine whether their surface areas are also equal and justify the conclusion by relating it to how each formula depends on the arrangement of faces.
- Solving V = lwh for an unknown dimensionGiven the volume of a rectangular prism and two of its three dimensions, solve an equation to find the missing dimension.
- Solving a surface-area equation for an unknown dimensionGiven the surface area of a rectangular prism and all but one dimension, solve an equation to find the missing dimension.
- The relationship between scale factor and volume ratio for similar solidsGiven two similar solids and their scale factor, predict the ratio of their volumes and explain why it is the cube of the scale factor rather than the scale factor itself.
- Multi-step application combining surface area, volume, and dimension-solving on composite or unfamiliar solidsGiven a real-world composite figure (e.g., a grain silo as cylinder-plus-cone-like shape simplified to prism-plus-pyramid), plan and carry out a multi-step solution combining surface area, volume, and equation-solving from this unit.
Unit 7: Introduction to ProbabilityProbability here is just a proportion — a long-run rate of favorable outcomes over all outcomes — which is why it leans hard on Unit 1's rate language and Unit 2's fractions. It moves from single-event probability to experimental probability (and why it doesn't match theory with small samples) to compound events using sample spaces and tree diagrams.9 skills ▸close ▾
- Theoretical probability of a single event expressed as fraction, decimal, and percentGiven a context with a stated ratio of outcomes, state the theoretical probability of a specified outcome as a fraction, decimal, and percent.
- The divergence between theoretical and experimental probability under small sample sizesExplain, using a specific numeric example from a class simulation, why an experimental probability computed from 20 trials can differ from the theoretical probability of the same event.
- A probability model with outcomes assigned probabilities summing to oneDevelop a uniform or non-uniform probability model for a given chance process by assigning probabilities to each outcome so they sum to 1.
- Agreement between a theoretical probability model and observed relative frequencyCompare a theoretical probability model against long-run relative frequencies collected from repeated trials of a chance process, and judge whether the model is a good fit.
- The sample space of a two-stage compound eventConstruct a complete sample space for a two-stage compound event using a tree diagram or organized list.
- Probability of a compound event as a fraction of the sample spaceCalculate the probability of a compound event from its sample space by counting favorable outcomes over total outcomes.
- The structural difference between combining independent stage probabilities correctly versus combining them by additionDifferentiate between a compound-event context that requires multiplying stage probabilities and one that has been miscombined by adding them.
- A simulation designed to estimate the probability of a novel compound eventDesign and carry out a simulation (using coins, number cubes, spinners, or a random-number generator) to estimate the probability of a compound event described in an unfamiliar real-world scenario.
- The relationship between sample size and confidence in a probability modelGiven two proposed probability models for the same context, judge which better matches collected experimental data and justify the choice using the size of the sample.
Unit 8: Sampling and Statistical InferenceThe year's closing unit. Instead of computing a known probability, your kid now uses sample data to make a claim about a whole population — and learns that a sample is only as trustworthy as the process that produced it. They generate multiple samples, watch estimates bounce around, and use that bounce to judge whether two groups are really different.9 skills ▸close ▾
- Random vs. biased sampling methods and who gets over/under-representedGiven a description of a sampling method (e.g. survey students exiting the gym after a game, vs. survey every 20th name on the school roster), classify it as likely to produce a representative sample or a biased sample, and name which group is over- or under-represented.
- The terms population, sample, random sampleState, from memory, the definitions of population, sample, and random sample using a given scenario.
- Point estimates of a population proportion or mean from a single random sampleGiven one random sample's data (e.g. proportion of tagged fish, mean bag weight), compute a point estimate of the population characteristic and explain what the estimate does and does not tell you about the exact population value.
- Sampling variability across repeated same-size samplesGiven a set of 8-10 simulated sample estimates from the same population, describe the spread of the estimates and explain why no single sample estimate should be treated as the exact population value.
- Generating repeated random samples to bound a population estimateGenerate multiple random samples of a stated size from a given population (physical or simulated), record each sample's estimate, and use the collection of estimates to state a plausible range for the population characteristic.
- Difference in means expressed as a multiple of MAD, for distributions with similar variabilityGiven two numerical data sets (e.g. heights of two grade levels) with similar variability, compare their distributions by expressing the difference in means as a multiple of the mean absolute deviation (MAD).
- Informal visual and numerical judgment of whether two groups differ meaningfullyGiven two dot plots with visibly overlapping distributions, judge informally whether the groups differ meaningfully, and justify the judgment using both center and spread.
- A complete original sampling plan: population, method, size, measurementDesign a complete sampling plan for a stated real-world question (define population, sampling method, sample size, and what will be measured) for a context never used during instruction.
- Critique of a stated survey claim's sampling methodGiven a claim in a news-style headline about a survey result (e.g. "9 out of 10 dentists agree"), critique the sampling method implied by the claim and propose a specific improvement.