Ratios, Relationships, and the Number System
This is the year your child learns to think in ratios instead of just counting differences — "for every 3 of these, 2 of those" instead of "3 more of these." That one idea (a multiplicative relationship between two quantities) gets used all year: to divide fractions, to make sense of negative numbers, to write algebra expressions, to compute area and volume, and finally to describe a set of data. By June they should be comfortable with signed numbers, one-step equations, decimals in all four operations, basic area/volume/surface area, and describing a data set with center and spread — not as nine separate topics, but as one idea applied nine times.
Unit 1: Ratios and Rate ReasoningThis is where your child learns that comparing two quantities by 'how many times as many' gives different information than comparing by 'how many more.' They'll build ratio tables, double number lines, and move toward unit rates and percents — all as the same underlying idea shown different ways.9 skills ▸close ▾
- Writing ratios two waysGiven a real-world description (e.g. 'for every 3 cups of flour, use 2 cups of sugar'), write the relationship as a ratio using at least two notations (a:b and 'a to b').
- Comparing by ratio or by differenceExplain why a ratio and a raw count (difference) give different information about the same two quantities, using a specific example.
- Ratio tables and equivalent ratiosComplete a ratio table by finding missing values, given one complete row and the multiplicative relationship between rows.
- Double number lines for ratiosConstruct a double number line to represent a ratio relationship and use it to find an unknown quantity.
- Comparing rates in different formsCompare two rates presented in different units or formats (e.g. a table vs. a sentence) and determine which is the better buy or faster rate.
- Ratio, rate, or unit rateClassify a given statement about a ratio relationship as expressing a ratio, a rate, or a unit rate.
- Percent as a rate per 100Convert a ratio to a percent and interpret percent as a rate per 100 in a context never used during instruction (e.g. a sports statistic or a survey result from a different domain).
- Combining two ratios into oneGiven a non-routine problem where two ratios must be reasoned about simultaneously (e.g. mixing two batches with different ratios into one), plan a solution strategy and justify why it works.
- Spotting equivalent ratios in any tableGiven two ratio tables built from different contexts, identify the structural feature (constant multiplicative factor) that makes both equivalent ratios, and generalize a rule for recognizing equivalence in any table.
Unit 2: Dividing FractionsYour child already multiplies fractions and used splitting pictures for ratios. Now they learn what dividing by a fraction actually means — two different meanings, in fact — using bar models before any shortcut rule, and only later the 'multiply by the reciprocal' trick, which gets explained rather than handed down.9 skills ▸close ▾
- Dividing whole numbers by unit fractionsGiven a whole number divided by a unit fraction (e.g. 4 ÷ 1/2), draw a bar model showing how many groups of that fraction fit, and state the quotient.
- Dividing fractions by fractions with modelsDraw a bar or number-line model for a fraction ÷ fraction problem (e.g. 2/3 ÷ 1/6) and use it to find the quotient by counting groups.
- Dividing fractions with common denominatorsRewrite two fractions with a common denominator and divide by comparing numerators, explaining why this gives the same quotient as the bar model.
- Dividing fractions using the reciprocalExecute the standard algorithm (multiply by the reciprocal) to compute a quotient of two fractions or mixed numbers.
- Why dividing by fractions gives moreExplain, using a partitioning picture, why dividing by a fraction less than 1 produces a quotient larger than the original number.
- Two kinds of division problemsClassify a word problem as requiring measurement division or partitive division, and select the matching equation.
- Fraction division in ratio problemsSolve a multi-step word problem combining fraction division with a ratio comparison from Unit 1, and justify the choice of operation.
- Fraction division in new situationsGiven a fraction division scenario set in an unfamiliar context (e.g. recipe scaling, fabric cutting, fuel consumption) never used in class examples, construct an original bar model and equation to solve it.
- Why both fraction division methods agreeCompare the bar-model justification and the common-denominator justification for the same fraction division problem, and identify what structural feature both share.
Unit 3: The Decimal SystemFull standard algorithms for adding, subtracting, multiplying, and dividing decimals — including long division with decimal divisors — plus the discovery that every decimal is really a fraction, some ending, some repeating forever.10 skills ▸close ▾
- Adding and subtracting decimalsGiven two decimals to the thousandths, add or subtract them correctly by aligning place value, including cases with different numbers of decimal digits.
- Why decimal multiplication works differentlyExplain why decimal points must be aligned for addition/subtraction but the decimal point's final position is instead found by counting digits for multiplication.
- Multiplying decimals with the algorithmExecute the standard algorithm to multiply two multi-digit decimals and place the decimal point correctly by counting total decimal digits.
- Dividing decimals by groupingDivide a decimal by a decimal by reframing the problem as whole-number division, using the 'how many groups fit' reasoning from Unit 2 fraction division.
- Long division with decimalsFluently execute the standard long-division algorithm to divide a multi-digit decimal dividend by a multi-digit decimal divisor.
- When decimals end or repeatClassify a given rational number's decimal expansion as terminating or repeating by inspecting the prime factors of the fraction's denominator in lowest terms.
- Decimals to fractions using place valueConvert a terminating decimal to an equivalent fraction in lowest terms using place value of the last digit, and explain why the denominator before simplifying must be a power of ten.
- Turning repeating decimals into fractionsGenerate and justify a method for converting a repeating decimal into an exact fraction, using an algebraic manipulation not directly demonstrated for that specific case.
- Estimating with decimalsEstimate the reasonableness of a decimal computation's result using front-end or compatible-number estimation before or after computing exactly.
- Word problems with several decimal stepsGiven a real-world context requiring at least two different decimal operations chained together (e.g. unit pricing with tax and a discount), plan and execute the correct sequence of operations.
Unit 4: Negative Numbers and the Coordinate PlaneThe number line now extends below zero, and points can live in all four quadrants of a coordinate grid, not just the corner your child has used before. Absolute value gets introduced as distance from zero, and reflections and distances-between-points close the unit.10 skills ▸close ▾
- Negative numbers in real lifeGiven a real-world context (temperature, elevation, account balance), write a signed number to represent the quantity and explain what zero means in that context.
- Number lines below zeroPlot integers and rational numbers, including negatives, on a horizontal or vertical number line.
- Opposites and zeroExplain why two numbers that are opposites are the same distance from zero but on different sides of it.
- Ordering positive and negative numbersCompare and order rational numbers, including negatives and negative decimals, using a number line.
- Absolute value as distanceInterpret absolute value as distance from zero and distinguish it from the sign of a number.
- Comparing absolute value and orderUse absolute value and order of rational numbers together to solve a real-world comparison problem, such as ranking account balances or elevations.
- Plotting points in four quadrantsPlot ordered pairs with positive and negative coordinates in all four quadrants of the coordinate plane.
- Reflecting points across the axesPredict the coordinates of a point reflected across the x-axis or y-axis without plotting first, then verify by plotting.
- Horizontal and vertical distanceFind the distance between two points that share an x-coordinate or y-coordinate by reasoning about absolute value, not by counting grid squares alone.
- Design and describe a coordinate mapDesign a coordinate-plane map (e.g., a treasure map or city map) using all four quadrants, and write directions using reflections and distances that a partner can follow without seeing the original map.
Unit 5: Expressions with VariablesYour child moves from arithmetic with known numbers to arithmetic with a variable — a letter standing for a number that can change. They'll translate word phrases into symbols, evaluate expressions with signed and fractional values, meet exponents as repeated multiplication, and use the distributive property to build and prove equivalent expressions.9 skills ▸close ▾
- Coefficients, variables, and constantsIdentify the coefficient, variable, and constant term in a given one- or two-term expression.
- Turning words into expressionsTranslate a word phrase describing a real quantity (e.g., 'five more than three times a number') into an algebraic expression.
- Substituting negative numbers and fractionsEvaluate an algebraic expression by substituting a given negative integer or fraction for the variable and applying order of operations.
- What exponents really meanExplain why an exponent represents repeated multiplication rather than repeated addition or multiplication by the exponent.
- Exponents and order of operationsEvaluate numerical expressions involving whole-number exponents, including within a larger expression using order of operations.
- Using the distributive propertyGenerate an equivalent expression for a given expression by applying the distributive property to a sum or difference inside parentheses.
- Testing whether expressions are equivalentDetermine whether two given expressions are equivalent by evaluating both at several values, including a negative and a non-integer value, and by comparing their structure.
- Proving expressions are always equivalentConstruct a general argument, using an area model or algebraic reasoning, that a(b+c) and ab+ac are equivalent for any value of the variable, not merely for values tested.
- Why squaring does not distributeDetermine whether the claim '(x+y)^2 = x^2+y^2' is true for all values of x and y, using a counterexample and an area-model explanation of the error.
Unit 6: Equations and Proportional RelationshipsAn equation is a balance — whatever you do to one side, you must do to the other. Your child learns to solve one-step equations with all four operations, translate word problems into equations, and recognize when a table of values represents a proportional relationship, writing it as y = kx and graphing it.10 skills ▸close ▾
- Solving addition and subtraction equationsGiven a one-step addition or subtraction equation, the student solves for the unknown by applying the same inverse operation to both sides.
- Solving multiplication and division equationsGiven a one-step multiplication or division equation, the student solves for the unknown by applying the same inverse operation to both sides.
- Why equations must stay balancedThe student explains why performing an operation on only one side of an equation breaks equality, using the balance model.
- Writing equations from word problemsGiven a short word problem describing a real quantity and an unknown, the student writes a one-step equation that represents the relationship.
- Avoiding reversed comparison equationsThe student identifies which of two equations correctly models a comparison word problem prone to a reversal error, and justifies the choice by substituting a numeric value.
- Testing tables for constant ratiosGiven a table of paired values, the student determines whether the relationship is proportional by checking whether the ratio y/x is constant across multiple rows.
- Independent and dependent variables on graphsThe student distinguishes the independent variable from the dependent variable in a given real-world scenario and assigns each to the correct axis.
- The constant of proportionalityGiven a proportional relationship in context, the student writes the equation y = kx and states what k represents in that context.
- Graphing proportional relationships to predictGiven a proportional relationship, the student graphs it on a coordinate plane and uses the graph to predict an untabulated value.
- Same constant, same graphThe student generalizes across two different proportional contexts to explain why relationships with the same constant of proportionality produce identical graphs regardless of context.
Unit 7: Area, Surface Area, and VolumeFinding area of triangles, parallelograms, and trapezoids by breaking them into shapes already known, unfolding 3D solids into flat nets to find surface area, and finding volume of rectangular prisms with fractional edges. The same move — break it down into something familiar — repeats throughout.12 skills ▸close ▾
- Triangle area from rectangles and parallelogramsFind the area of a triangle by decomposing or rearranging it into a rectangle or parallelogram of known area.
- Why rearranging pieces keeps areaExplain why decomposing a shape into pieces and rearranging those pieces never changes the shape's total area.
- Finding the area of trapezoidsFind the area of a trapezoid by decomposing it into a rectangle and two triangles, or into two congruent trapezoids forming a parallelogram.
- Polygon area on the coordinate planeFind the area of an irregular polygon plotted on a coordinate plane by decomposing it into triangles and rectangles using the vertex coordinates.
- Planning surface area for combined solidsGiven the surface area of an unfamiliar composite solid (not a simple prism) built from known solids, plan a decomposition strategy and justify it before computing.
- Building nets of rectangular prismsConstruct a net that represents the faces of a given right rectangular prism, correctly matching each face's dimensions.
- Finding surface area from netsCalculate the surface area of a right rectangular prism or triangular prism by summing the areas of all faces shown in its net.
- Designing nets for a target surface areaDesign a net for a right rectangular prism that meets a specified total surface area target, and justify that the design is correct.
- Volume with fractional edge lengthsCalculate the volume of a right rectangular prism with fractional edge lengths using the formula V = l x w x h.
- Predicting how fractional edges affect volumeCompare a prediction about how fractional edge lengths affect volume to the actual computed volume, and reconcile any mismatch.
- Choosing area, surface area, or volumeClassify a real-world measurement scenario as requiring area, surface area, or volume based on what physical quantity is being measured.
- Surface area with an unknown edgeWrite and evaluate an algebraic expression for the surface area of a rectangular prism with one unknown edge length.
Unit 8: Statistical DistributionsThe year closes with describing a set of data by its shape, center, and spread together, not just one number. Your child will build dot plots, histograms, and box plots, compute mean, median, range, and MAD, and use all of it to compare two data sets with the same average but very different stories.9 skills ▸close ▾
- Spotting statistical questionsGiven a list of questions, classify each as a statistical question or not, justifying the classification by naming expected variability in possible answers.
- Making dot plots from dataConstruct a dot plot from a given small data set (9-15 values), correctly plotting each value above a labeled number line.
- Finding the medianCompute the median of an ordered data set, correctly applying the different rule for odd versus even counts of values.
- Mean absolute deviation and spreadCompute the mean absolute deviation (MAD) of a data set and explain what the resulting number indicates about spread around the mean.
- Comparing histograms and box plotsGiven a histogram and a box plot of the same data set, compare the shape of the distribution described by each, identifying clusters, gaps, and skew.
- Using spread to judge consistencyGiven two data sets with equal means but different spreads, construct an argument for which set is more consistent, using a computed measure of variability as evidence.
- When averages mislead readersGiven an unfamiliar real-world data set (e.g. city rainfall totals) never discussed in class, decide which measure of center would most mislead a reader and justify the choice.
- Comparing two data sets fullyGiven a real or provided data set, produce two different graphical representations (from dot plot, histogram, box plot) and a written comparison of two distributions using center and spread.
- Why one average is not enoughGiven a claim like 'these two classes performed the same because they have the same mean,' identify what information is missing and explain why the claim is incomplete.
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