Grade 6
Math curriculum
By the end of grade six, students have mastered the four arithmetic operations
with whole numbers, positive fractions, positive decimals, and positive and
negative integers; they accurately compute and solve problems. They apply their
knowledge to statistics and probability. Students understand the concepts of
mean, median, and mode of data sets and how to calculate the range. They
analyze data and sampling processes for possible bias and misleading conclusions;
they use addition and multiplication of fractions routinely to calculate the
probabilities for compound events. Students conceptually understand and work
with ratios and proportions; they compute percentages (e.g., tax, tips, interest).
Students know about π and the formulas for the circumference and area of a
circle. They use letters for numbers in formulas involving geometric shapes and
in ratios to represent an unknown part of an expression. They solve one-step
linear equations.
1.0 Students compare and order positive and negative fractions, decimals, and
mixed numbers. Students solve problems involving fractions, ratios, proportions,
and percentages:
1.1 Compare and order positive and negative fractions, decimals, and mixed numbers
and place them on a number line.
1.2 Interpret and use ratios in different contexts (e.g., batting averages, miles per hour)
to show the relative sizes of two quantities, using appropriate notations (a/b, a to b, a:b).
1.3 Use proportions to solve problems (e.g., determine the value of N if 4/7 = N/21,
find the length of a side of a polygon similar to a known polygon). Use cross
multiplication as a method for solving such problems, understanding it as the
multiplication of both sides of an equation by a multiplicative inverse.
1.4 Calculate given percentages of quantities and solve problems involving discounts
at sales, interest earned, and tips.
2.0 Students calculate and solve problems involving addition, subtraction,
multiplication, and division:
2.1 Solve problems involving addition, subtraction, multiplication, and division of
positive fractions and explain why a particular operation was used for a given
situation.
2.2 Explain the meaning of multiplication and division of positive fractions and
perform the calculations (e.g., 5/8 ÷ 15/16 = 5/8 x 16/15 =2/3
2.3 Solve addition, subtraction, multiplication, and division problems, including those
arising in concrete situations, that use positive and negative integers and combinations
of these operations.
2.4 Determine the least common multiple and the greatest common divisor of whole
numbers; use them to solve problems with fractions (e.g., to find a common
denominator to add two fractions or to find the reduced form for a fraction).
Algebra and Functions
1.0 Students write verbal expressions and sentences as algebraic expressions and
equations; they evaluate algebraic expressions, solve simple linear equations,
and graph and interpret their results:
1.1 Write and solve one-step linear equations in one variable.
1.2 Write and evaluate an algebraic expression for a given situation, using up to three
variables.
1.3 Apply algebraic order of operations and the commutative, associative, and distributive
properties to evaluate expressions; and justify each step in the process.
1.4 Solve problems manually by using the correct order of operations or by using a
scientific calculator.
2.0 Students analyze and use tables, graphs, and rules to solve problems involving
rates and proportions:
2.1 Convert one unit of measurement to another (e.g., from feet to miles, from
centimeters to inches).
2.2 Demonstrate an understanding that rate is a measure of one quantity per unit value
of another quantity.
2.3 Solve problems involving rates, average speed, distance, and time.
3.0 Students investigate geometric patterns and describe them algebraically:
3.1 Use variables in expressions describing geometric quantities (e.g., P = 2w + 2l,
A =⁄2 bh, C = π d—the formulas for the perimeter of a rectangle, the area of a tri
angle, and the circumference of a circle, respectively).
3.2 Express in symbolic form simple relationships arising from geometry.
Measurement and Geometry
1.0 Students deepen their understanding of the measurement of plane and solid
shapes and use this understanding to solve problems:
1.1 Understand the concept of a constant such as π; know the formulas for the
circumference and area of a circle.
1.2 Know common estimates of π (3.14; 22⁄7) and use these values to estimate and
calculate the circumference and the area of circles; compare with actual measurements.
1.3 Know and use the formulas for the volume of triangular prisms and cylinders (area
of base × height); compare these formulas and explain the similarity between them
and the formula for the volume of a rectangular solid.
2.0 Students identify and describe the properties of two-dimensional figures:
2.1 Identify angles as vertical, adjacent, complementary, or supplementary and provide
descriptions of these terms.
2.2 Use the properties of complementary and supplementary angles and the sum of the
angles of a triangle to solve problems involving an unknown angle.
2.3 Draw quadrilaterals and triangles from given information about them (e.g., a
quadrilateral having equal sides but no right angles, a right isosceles triangle).
Statistics, Data Analysis, and Probability
1.0 Students compute and analyze statistical measurements for data sets:
1.1 Compute the range, mean, median, and mode of data sets.
1.2 Understand how additional data added to data sets may affect these computations
of measures of central tendency.
1.3 Understand how the inclusion or exclusion of outliers affects measures of central
tendency.
1.4 Know why a specific measure of central tendency (mean, median, mode) provides
the most useful information in a given context.
2.0 Students use data samples of a population and describe the characteristics
and limitations of the samples:
2.1 Compare different samples of a population with the data from the entire population
and identify a situation in which it makes sense to use a sample.
2.2 Identify different ways of selecting a sample (e.g., convenience sampling, responses
to a survey, random sampling) and which method makes a sample more representative
for a population.
2.3 Analyze data displays and explain why the way in which the question was asked
might have influenced the results obtained and why the way in which the results
were displayed might have influenced the conclusions reached.
2.4 Identify data that represent sampling errors and explain why the sample (and the
display) might be biased.
2.5 Identify claims based on statistical data and, in simple cases, evaluate the validity
of the claims.
3.0 Students determine theoretical and experimental probabilities and use these
to make predictions about events:
3.1 Represent all possible outcomes for compound events in an organized way
(e.g., tables, grids, tree diagrams) and express the theoretical probability of each
outcome.
3.2 Use data to estimate the probability of future events (e.g., batting averages or
number of accidents per mile driven).
3.3 Represent probabilities as ratios, proportions, decimals between 0 and 1, and
percentages between 0 and 100 and verify that the probabilities computed are
reasonable; know that if P is the probability of an event, 1-P is the probability of an
event not occurring.
3.4 Understand that the probability of either of two disjoint events occurring is the sum
of the two individual probabilities and that the probability of one event following
another, in independent trials, is the product of the two probabilities.
3.5 Understand the difference between independent and dependent events.
Mathematical Reasoning
1.0 Students make decisions about how to approach problems:
1.1 Analyze problems by identifying relationships, distinguishing relevant from
irrelevant information, identifying missing information, sequencing and
prioritizing information, and observing patterns.
1.2 Formulate and justify mathematical conjectures based on a general description
of the mathematical question or problem posed.
1.3 Determine when and how to break a problem into simpler parts.
2.0 Students use strategies, skills, and concepts in finding solutions:
2.1 Use estimation to verify the reasonableness of calculated results.
2.2 Apply strategies and results from simpler problems to more complex problems.
2.3 Estimate unknown quantities graphically and solve for them by using logical
reasoning and arithmetic and algebraic techniques.
2.4 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables,
diagrams, and models, to explain mathematical reasoning.
2.5 Express the solution clearly and logically by using the appropriate mathematical
notation and terms and clear language; support solutions with evidence in both
verbal and symbolic work.
2.6 Indicate the relative advantages of exact and approximate solutions to problems
and give answers to a specified degree of accuracy.
2.7 Make precise calculations and check the validity of the results from the context
of the problem.
3.0 Students move beyond a particular problem by generalizing to other
situations:
3.1 Evaluate the reasonableness of the solution in the context of the original situation.
3.2 Note the method of deriving the solution and demonstrate a conceptual under
standing of the derivation by solving similar problems.
3.3 Develop generalizations of the results obtained and the strategies used and apply
them in new problem situations.