Grade 6 Math: ratios, negative numbers, expressions, and data by talking it through

A taste of a lesson on Dividing fractions and fraction division word problems

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The 30 topics in Grade 6 Math

Each topic builds on the ones before it. The tutor checks what you already know and starts there.

1. Whole-number exponents and order of operations

   Exponent notation: a^n means n copies of a multiplied together, with base and exponent identified (e.g., 2^5 = 32, 10^3 = 1000). Students write and evaluate numerical expressions with exponents, applying the conventional order of operations: parentheses, exponents, multiplication and division left to right, then addition and subtraction left to right. Real understanding means explaining why 2+3^2 is 11, not 25.

2. Writing and reading expressions with variables

   A letter stands for an unknown or varying number. Students translate phrases into expressions (e.g., 'subtract y from 5' is 5 - y; 'twice the sum of n and 3' is 2(n + 3)) and read expressions back into words. They identify parts of an expression: terms (separated by + or -), factors (multiplied), coefficients (numerical factor of a term), sums, products, and quotients. They learn to view a part such as (8 + 7) in 2(8 + 7) as a single entity.

   Builds on Whole-number exponents and order of operations

3. Evaluating expressions and formulas

   Substitute specific values for variables and evaluate using the order of operations. Includes expressions arising from real-world formulas such as area of a rectangle A = lw, distance d = rt, or cost = 5h + 12. Students check that substitution gives a number and that the order of operations is followed, including exponents such as 3x^2 when x = 4.

   Builds on Writing and reading expressions with variables

4. Properties of operations and equivalent expressions

   Commutative, associative, and distributive properties let us rewrite expressions. Apply the distributive property to 3(2 + x) = 6 + 3x and to factor out common factors such as 6 + 9x = 3(2 + 3x). Combine like terms (e.g., 4y + 3y = 7y). Recognize that two expressions are equivalent when they name the same number for every substitution, e.g., y + y + y and 3y, and check equivalence by substituting values.

   Builds on Evaluating expressions and formulas

5. Equations and inequalities: testing solutions from a set

   Solving an equation means finding which values from a specified set make it true. Students use substitution to test whether a number such as 0, 1/2, 4, or 9 is a solution of an equation like 3x = 12 or an inequality like x + 5 > 8. Understand that an equation is a question and that the set of allowed values matters.

   Builds on Evaluating expressions and formulas

6. Solving one-step equations x + p = q and px = q

   Write and solve equations of the form x + p = q and px = q where p, q, and x are nonnegative rational numbers, including fractions and decimals. Use inverse operations (subtract p from both sides; divide both sides by p). Solve real-world problems by first writing the equation from the story, then solving and checking the answer in context.

   Builds on Properties of operations and equivalent expressions, Equations and inequalities: testing solutions from a set

7. Relationships between two varying quantities

   In a real-world situation where two quantities change together, identify the independent variable (the input chosen freely) and the dependent variable (the output that depends on it). Write an equation expressing the dependent variable in terms of the independent one, such as total cost c = 4n for n notebooks, or distance d = 60t, and make a table of values from it.

   Builds on Solving one-step equations x + p = q and px = q

8. Ratios and ratio language

   A ratio compares two quantities a:b (or a to b, a/b) with b not zero, e.g., 2:1 wings to beaks means for every 2 wings there is 1 beak. Students distinguish part-to-part and part-to-whole ratios, use ratio language correctly, and recognize that a ratio can be written in several equivalent forms such as 2:1, 2 to 1, and 2/1.

   Builds on Whole-number exponents and order of operations

9. Fluent multi-digit division (standard algorithm)

   Divide multi-digit whole numbers (e.g., 4,536 / 28) fluently using the standard long-division algorithm, estimating quotients first, handling remainders, and checking by multiplication. Connect the algorithm to place value and the distributive property of division over subtraction.

   Builds on Whole-number exponents and order of operations

10. Fluent operations with multi-digit decimals

    Add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. Place the decimal correctly in products and quotients by reasoning about place value (e.g., 0.6 x 0.4 = 0.24 because tenths times tenths are hundredths), and in division move the decimal in divisor and dividend to make the divisor whole.

    Builds on Fluent multi-digit division (standard algorithm)

11. Dividing fractions and fraction division word problems

    Interpret and compute quotients of fractions, such as (2/3) / (3/4) = 8/9, by using visual fraction models and multiplying by the reciprocal. Understand why dividing by a fraction asks how many groups of that size fit in the dividend, and solve word problems that involve division of fractions by fractions, creating story contexts such as how many 3/4-cup servings are in 2/3 cup.

    Builds on Fluent multi-digit division (standard algorithm)

12. Unit rates and rate language

    A unit rate a/b is the amount per one unit of b (e.g., 150 miles in 3 hours is 50 miles per hour). Students compute unit rates by division, including with fractions and decimals, and use rate language such as 'per', 'for each', and 'per hour'. Understand that a unit rate describes the ratio relationship at 1 unit of the second quantity.

    Builds on Ratios and ratio language, Fluent operations with multi-digit decimals, Dividing fractions and fraction division word problems

13. Percent as a rate per 100

    Percent means a rate per 100: 30% means 30 out of 100, or 30/100. Students convert between percents, fractions with denominator 100, and decimals (30% = 0.30), and understand percent as a ratio relationship to a whole of 100. They find a percent of a quantity as a rate per 100 (e.g., 30% of 80 = 30/100 x 80 = 24), and use 100 as the reference whole in tables and double number lines.

    Builds on Ratios and ratio language, Fluent operations with multi-digit decimals

14. Greatest common factor, least common multiple, and the distributive property in sums

    Find the greatest common factor (GCF) of two whole numbers up to 100 and the least common multiple (LCM) of two whole numbers up to 12. Use the distributive property to rewrite a sum of two whole numbers from 1 to 100 with a common factor, e.g., 36 + 8 = 4(9 + 2), by factoring out the GCF. Explain why factoring out the GCF produces an equivalent expression.

    Builds on Properties of operations and equivalent expressions, Fluent multi-digit division (standard algorithm)

15. Positive and negative numbers in context

    Positive and negative numbers describe quantities in opposite directions or values: temperature above and below zero, elevation above and below sea level, deposits and withdrawals, and positive and negative charge. Students read and write signed numbers in these contexts and explain what the sign and the size each mean.

    Builds on Whole-number exponents and order of operations

16. Rational numbers on the number line

    Every rational number (integers, fractions, decimals) is a point on the number line; negative numbers lie to the left of zero. Students find and position integers and other rational numbers on horizontal and vertical number lines. Recognize that opposite numbers lie on opposite sides of 0 at the same distance, and that -(-3) = 3 and -0 = 0.

    Builds on Positive and negative numbers in context

17. Writing and graphing inequalities x > c and x < c

    Write inequalities such as x > c or x < c to represent constraints in real-world situations (e.g., 'you must be at least 48 inches tall' as h >= 48, or 'at most 3 pounds' as w <= 3). Recognize that x > c has infinitely many solutions, graph them on a number line with an open or closed circle, and interpret the solution set in context.

    Builds on Equations and inequalities: testing solutions from a set, Rational numbers on the number line

18. Ordering rational numbers and absolute value

    Interpret inequality statements about rational numbers as relative positions on a number line (-3 > -7 means -3 is right of -7). Write, interpret, and explain order statements in context (-3 C is warmer than -7 C). Define absolute value as distance from 0, interpret it as magnitude in context (a balance of -30 dollars has absolute value 30), and distinguish statements about absolute value from statements about order (a balance less than -30 means a debt greater than 30).

    Builds on Rational numbers on the number line

19. The coordinate plane with four quadrants

    Extend the coordinate plane to four quadrants using ordered pairs (x, y) with signed coordinates. Know which quadrant each sign pattern lies in, and recognize that points differing only in signs are reflections across an axis or the origin. Plot points in all four quadrants.

    Builds on Rational numbers on the number line, Ordering rational numbers and absolute value

20. Graphing y = kx and other two-variable relationships

    Given a relationship between two varying quantities (e.g., y = 3x, or total cost c = 4n), make a table of (x, y) pairs from the equation, plot the pairs as points on the coordinate plane, and recognize that the points lie on a line through the origin when y = kx. Interpret points on the graph in context, including what the independent and dependent variables represent, and read values off the graph.

    Builds on Relationships between two varying quantities, The coordinate plane with four quadrants

21. Equivalent ratios: tables, tape diagrams, double number lines

    Build tables of equivalent ratios for quantities with whole-number measurements, find missing values, and use tape diagrams and double number line diagrams to reason about ratio relationships. Plot the pairs of values from a ratio table as points on the coordinate plane and use tables to compare ratios (which of two recipes is more concentrated?).

    Builds on Unit rates and rate language, The coordinate plane with four quadrants

22. Solving ratio and rate problems, including unit conversion

    Solve real-world problems using ratio and rate reasoning: unit price comparisons, constant-speed problems (7 hours to mow 4 lawns, how many lawns in 35 hours?), and finding the whole given a part and percent. Find a percent of a quantity as a rate per 100 (30% of 80 = 30/100 x 80 = 24) and find the whole given a part and percent. Convert measurement units using ratio reasoning (feet to inches, hours to minutes), multiplying or dividing by conversion factors correctly.

    Builds on Percent as a rate per 100, Equivalent ratios: tables, tape diagrams, double number lines

23. Coordinate plane: polygons, side lengths, and distances

    Draw polygons in the coordinate plane from given vertex coordinates. Find the length of a side joining two points with the same first coordinate (vertical) or the same second coordinate (horizontal) by subtracting coordinates and taking absolute values. Solve real-world problems such as the distance between two locations on a city grid.

    Builds on The coordinate plane with four quadrants

24. Area of triangles, quadrilaterals, and polygons

    Find the area of right triangles, other triangles, parallelograms, trapezoids, and polygons by composing shapes into rectangles or decomposing them into triangles and other shapes. Apply to real-world and mathematical problems, such as the area of a floor plan or a garden with an irregular outline.

    Builds on Evaluating expressions and formulas, Fluent multi-digit division (standard algorithm)

25. Volume of right rectangular prisms with fractional edges

    Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge length (e.g., 1/4-inch cubes), and show that the volume equals length times width times height, the same as multiplying edge lengths. Apply V = lwh and V = Bh in real-world problems.

    Builds on Evaluating expressions and formulas, Dividing fractions and fraction division word problems

26. Nets and surface area of three-dimensional figures

    Represent prisms and pyramids with nets made of rectangles and triangles, and use the net to find surface area by summing the areas of all faces. Apply to real-world problems such as wrapping paper for a box or paint for a pyramid-shaped tent.

    Builds on Area of triangles, quadrilaterals, and polygons

27. Statistical questions and variability

    A statistical question anticipates variability in the data; 'How old am I?' is not statistical, but 'How old are the students in my school?' is. Students write and recognize statistical questions and explain why the answers vary from one data set to another.

28. Describing distributions: center, spread, and shape

    A data set collected to answer a statistical question has a distribution described by its center, spread, and overall shape (symmetric, skewed, uniform, clusters, gaps, outliers). A measure of center (median, mean) summarizes all values with one number; a measure of variation (range, interquartile range, mean absolute deviation) describes how values vary with one number.

    Builds on Statistical questions and variability

29. Dot plots, histograms, and box plots on a number line

    Display numerical data on a number line using dot plots, histograms, and box plots. Build a box plot from the five-number summary (minimum, lower quartile, median, upper quartile, maximum), and read a dot plot or histogram for its shape, clusters, and gaps.

    Builds on Rational numbers on the number line, Describing distributions: center, spread, and shape

30. Summarizing data sets in context: center, variability, and choice of measures

    Summarize a numerical data set by reporting the number of observations, describing the attribute and how it was measured with its units, and giving median or mean and interquartile range or mean absolute deviation, along with any overall pattern and outliers. Relate the choice of center and variability measures to the data's shape and the context in which the data were gathered (median and IQR for skewed data with outliers; mean and MAD for symmetric data).

    Builds on Fluent operations with multi-digit decimals, Dividing fractions and fraction division word problems, Dot plots, histograms, and box plots on a number line

Practice questions in Grade 6 Math

A few of the kinds of questions you will work through, with answers.