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 pi 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.
1.1 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.
2.1 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 รท 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).
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.
2.1 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.
3.1 Use variables in expressions describing geometric
quantities (e.g., P = 2w + 2l, A = 1/2bh,
C = pd - the formulas for the
perimeter of a rectangle, the area of a triangle, and the circumference
of a circle, respectively).
3.2 Express in symbolic form simple relationships arising from
geometry.
1.1 Understand the concept of a constant such
as p; know the formulas for the circumference
and area of a circle.
1.2 Know common estimates of p (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 x height); compare these formulas
and explain the similarity between them and the formula for the
volume of a rectangular solid.
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).
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)
provides the most useful information in a given context.
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.
3.1 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.
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.
2.1 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.
3.1 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 understanding 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.