Grade 6 · Christian · NGSS/CCSS-aligned
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.
The whole year, in plain English. Tap any unit to see every skill inside, nothing is hidden.
This 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.
Given 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').
Explain why a ratio and a raw count (difference) give different information about the same two quantities, using a specific example.
Complete a ratio table by finding missing values, given one complete row and the multiplicative relationship between rows.
Construct a double number line to represent a ratio relationship and use it to find an unknown quantity.
Compare 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.
Classify a given statement about a ratio relationship as expressing a ratio, a rate, or a unit rate.
Convert 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).
Given 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.
Given 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.
Your 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.
Given 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.
Draw 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.
Rewrite two fractions with a common denominator and divide by comparing numerators, explaining why this gives the same quotient as the bar model.
Execute the standard algorithm (multiply by the reciprocal) to compute a quotient of two fractions or mixed numbers.
Explain, using a partitioning picture, why dividing by a fraction less than 1 produces a quotient larger than the original number.
Classify a word problem as requiring measurement division or partitive division, and select the matching equation.
Solve a multi-step word problem combining fraction division with a ratio comparison from Unit 1, and justify the choice of operation.
Given 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.
Compare the bar-model justification and the common-denominator justification for the same fraction division problem, and identify what structural feature both share.
Full 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.
Given two decimals to the thousandths, add or subtract them correctly by aligning place value, including cases with different numbers of decimal digits.
Explain why decimal points must be aligned for addition/subtraction but the decimal point's final position is instead found by counting digits for multiplication.
Execute the standard algorithm to multiply two multi-digit decimals and place the decimal point correctly by counting total decimal digits.
Divide 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.
Fluently execute the standard long-division algorithm to divide a multi-digit decimal dividend by a multi-digit decimal divisor.
Classify a given rational number's decimal expansion as terminating or repeating by inspecting the prime factors of the fraction's denominator in lowest terms.
Convert 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.
Generate and justify a method for converting a repeating decimal into an exact fraction, using an algebraic manipulation not directly demonstrated for that specific case.
Estimate the reasonableness of a decimal computation's result using front-end or compatible-number estimation before or after computing exactly.
Given 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.
The 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.
Given a real-world context (temperature, elevation, account balance), write a signed number to represent the quantity and explain what zero means in that context.
Plot integers and rational numbers, including negatives, on a horizontal or vertical number line.
Explain why two numbers that are opposites are the same distance from zero but on different sides of it.
Compare and order rational numbers, including negatives and negative decimals, using a number line.
Interpret absolute value as distance from zero and distinguish it from the sign of a number.
Use absolute value and order of rational numbers together to solve a real-world comparison problem, such as ranking account balances or elevations.
Plot ordered pairs with positive and negative coordinates in all four quadrants of the coordinate plane.
Predict the coordinates of a point reflected across the x-axis or y-axis without plotting first, then verify by plotting.
Find 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 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.
Your 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.
Identify the coefficient, variable, and constant term in a given one- or two-term expression.
Translate a word phrase describing a real quantity (e.g., 'five more than three times a number') into an algebraic expression.
Evaluate an algebraic expression by substituting a given negative integer or fraction for the variable and applying order of operations.
Explain why an exponent represents repeated multiplication rather than repeated addition or multiplication by the exponent.
Evaluate numerical expressions involving whole-number exponents, including within a larger expression using order of operations.
Generate an equivalent expression for a given expression by applying the distributive property to a sum or difference inside parentheses.
Determine 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.
Construct 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.
Determine 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.
An 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.
Given a one-step addition or subtraction equation, the student solves for the unknown by applying the same inverse operation to both sides.
Given a one-step multiplication or division equation, the student solves for the unknown by applying the same inverse operation to both sides.
The student explains why performing an operation on only one side of an equation breaks equality, using the balance model.
Given a short word problem describing a real quantity and an unknown, the student writes a one-step equation that represents the relationship.
The 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.
Given 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.
The student distinguishes the independent variable from the dependent variable in a given real-world scenario and assigns each to the correct axis.
Given a proportional relationship in context, the student writes the equation y = kx and states what k represents in that context.
Given a proportional relationship, the student graphs it on a coordinate plane and uses the graph to predict an untabulated value.
The student generalizes across two different proportional contexts to explain why relationships with the same constant of proportionality produce identical graphs regardless of context.
Finding 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.
Find the area of a triangle by decomposing or rearranging it into a rectangle or parallelogram of known area.
Explain why decomposing a shape into pieces and rearranging those pieces never changes the shape's total area.
Find the area of a trapezoid by decomposing it into a rectangle and two triangles, or into two congruent trapezoids forming a parallelogram.
Find the area of an irregular polygon plotted on a coordinate plane by decomposing it into triangles and rectangles using the vertex coordinates.
Given 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.
Construct a net that represents the faces of a given right rectangular prism, correctly matching each face's dimensions.
Calculate the surface area of a right rectangular prism or triangular prism by summing the areas of all faces shown in its net.
Design a net for a right rectangular prism that meets a specified total surface area target, and justify that the design is correct.
Calculate the volume of a right rectangular prism with fractional edge lengths using the formula V = l x w x h.
Compare a prediction about how fractional edge lengths affect volume to the actual computed volume, and reconcile any mismatch.
Classify a real-world measurement scenario as requiring area, surface area, or volume based on what physical quantity is being measured.
Write and evaluate an algebraic expression for the surface area of a rectangular prism with one unknown edge length.
The 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.
Given a list of questions, classify each as a statistical question or not, justifying the classification by naming expected variability in possible answers.
Construct a dot plot from a given small data set (9-15 values), correctly plotting each value above a labeled number line.
Compute the median of an ordered data set, correctly applying the different rule for odd versus even counts of values.
Compute the mean absolute deviation (MAD) of a data set and explain what the resulting number indicates about spread around the mean.
Given 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.
Given 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.
Given 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.
Given 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.
Given 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.
From the parent guide
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 · what to expect
This 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.
The full guide covers all 8 units: where kids get stuck, what to say, and how to tell it's working. Included with the course.
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