Secondary Mathematics Benchmarks Progressions, Grades 7–12: Number (N)
Number sense is the cornerstone for mathematics in everyday life. Comparing prices, deciding whether to buy or lease a car, estimating tax on a purchase or tip for a service, and evaluating salary increases in the context of annual inflation rates all require understanding of and facility with quantified information. Interpreting much of what appears in daily news releases relies on an ability to glean valid information from numerical data and evaluate claims based on data. Through the study and application of ratios, rates, and derived measures, students extend their sense of number to contextual situations, paying heed to units. They develop the capacity to work with precision and accuracy and to spot and minimize errors, an important skill in a world that increasingly relies on quality control. In all of these endeavors, electronic technology provides an accurate and efficient way to manage quantitative information. In addition, fluency and flexibility in the algorithms and properties that govern numerical operations are important for procedural computation; they lay conceptual foundations for the study of algebra and for reasoning in all areas of mathematics.
N.A.1 Rational numbers
a. Identify rational numbers, represent them in various ways, and translate among these representations.
Rational numbers are those that can be expressed in the form 
, where p and q are integers and q ≠ 0.
- Identify whole numbers, fractions (positive and negative), mixed numbers, finite (terminating) decimals, and repeating decimals as rational numbers.
The decimal form of a rational number eventually repeats. A decimal is called terminating if its repeating digit is 0. A fraction has a terminating decimal expansion if and only if its denominator in reduced form has only 2 and 5 as prime factors.
- Express a percent having a finite number of digits as a rational number by expressing it as a ratio whose numerator is an integer and whose denominator is 100 (or, more generally, whose denominator is a power of 10).
- Transform rational numbers from one form (fractions, decimals, percents and mixed numbers) to another.
b. Understand and use inequalities to compare rational numbers; apply basic rules of inequalities to transform expressions involving rational numbers.
For any rational numbers a, b, and c, a < b implies that a + c < b + c; further, a < b implies that –a > –b. For c > 0, a < b implies that ac < bc; for c < 0, a < b implies that ac > bc.
c. Locate rational numbers on the number line and explain the significance of these locations.
- Show that a number and its opposite are mirror images with respect to the point 0.
- Identify one or more rational numbers that lie between two given rational numbers and explain how this can be done no matter how close together the given numbers are.
d. Know and apply effective methods of calculation with rational numbers.
- Demonstrate understanding of algorithms for addition, subtraction, multiplication, and division of numbers expressed as fractions, terminating decimals, or repeating decimals by applying the algorithms and explaining why they work.
- Add, subtract, multiply, and divide (with a non-zero divisor) rational numbers and explain why these operations always produce another rational number.
- For any rational number r, determine its opposite, –r and its reciprocal, 1/r, if r ≠ 0. Explain why –r and 1/r are rational whenever r is rational.
- Extend the properties of computation with whole numbers (e.g., commutative property, associative property, distributive property) to rational number computation.
- Interpret parentheses and employ conventional order of operations in a numerical expression.
- Check answers by estimation or by independent calculations, with or without calculators and computers.
Task related to this benchmark: Autobahn, King's Deli, Talk Is Cheap
e. Recognize, describe, extend, and create well-defined numerical patterns.
Of special interest are arithmetic sequences, those generated by repeated addition of a fixed number, and geometric sequences, those generated by repeated multiplication of a fixed number.
f. Solve practical problems involving rational numbers.
Examples: Calculate markups, discounts, taxes, tips, average speed.
Tasks related to this benchmark: Autobahn, King's Deli, Talk Is Cheap
N.A.2 Absolute values
a. Know and apply the definition of absolute value.
The absolute value is defined by |a| = a if a ≥ 0 and |a| = –a if a < 0.
b. Interpret absolute value as distance from zero.
c. Interpret absolute value of a difference as "distance between."
Example: |5 – 1| = 4 is the distance between 5 and 1 on the number line.
N.A.3 Prime decomposition, factors, and multiples
a. Know and apply the Fundamental Theorem of Arithmetic, that every positive integer is either prime itself or can be written as a unique product of primes (ignoring order).
- Identify prime numbers; describe the difference between prime and composite numbers.
- Determine divisibility rules, use them to help factor composite numbers, and explain why they work.
- Write a prime decomposition for numbers up to 100.
b. Explain the meaning of the greatest common divisor (greatest common factor) and the least common multiple and use them in operations with fractions.
- Determine the greatest common divisor and least common multiple of two whole numbers from their prime factorizations.
- Use greatest common divisors to reduce fractions
and ratios n:m to an equivalent form in which gcd (n, m) = 1.Fractions
in which gcd (n, m) = 1 are said to be in lowest terms. - Add and subtract fractions by using least common multiple of denominators.
c. Write equivalent fractions by multiplying both numerator and denominator by the same non-zero whole number or dividing by common factors in the numerator and denominator.
N.A.4 Ratio, rates, and derived quantities
a. Interpret and apply measures of change such as percent change and rates of growth.
b. Calculate with quantities that are derived as ratios and products.
- Interpret and apply ratio quantities including velocity, density, pressure, population density.
Examples of units: Feet per second, grams per cc3, people per square mile.
- Interpret and apply product quantities including area, volume, energy, work.
Examples of units: Square meters, kilowatt hours, person days.
c. Solve data problems using ratios, rates, and product quantities.
- Convert measurements both within and across measurement systems.
Task related to this benchmark: Satellite
d. Create and interpret scale drawings as a tool for solving problems.
e. Use unit analysis to clarify appropriate units in calculations.
Example: The calculation for converting 50 feet per second to miles per hour can be checked using the unit calculation. 
yields the correct units since the units feet, seconds, and minutes all appear in both numerator and denominator.
f. Identify and apply derived measures.
Derived measures are quantities determined by calculation.
Examples: Percent change, density, the composite scale used for college rankings.
Task related to this benchmark: King's Deli
g. Use and identify potential misuses of weighted averages.
- Identify and interpret common instances of weighted averages.
Examples: Grade averages, stock market indexes, Consumer Price Index, unemployment rate.
- Analyze variation in weighted averages and distinguish change due to weighting from changes in the quantities measured.
Example: Suppose a company employed 100 women with average annual salaries of $20,000 and 500 men with average salaries of $40,000. After a change in management, the company employed 200 women and 400 men. To correct past inequities, the new management increased women's salaries by 25% and men's salaries by 5%. Despite these increases, the company's average salary declined by almost 1%.
Task related to this benchmark: Autobahn
N.B.1 Estimation and approximation
a. Use simple estimates to predict results and verify the reasonableness of calculated answers.
- Use rounding, regrouping, percentages, proportionality, and ratios as tools for mental estimation.
b. Develop, apply, and explain different estimation strategies for a variety of common arithmetic problems.
Examples: Estimating tips, adding columns of figures, estimating interest payments, estimating magnitude.
Task related to this benchmark: Autobahn, King's Deli
c. Explain the phenomenon of rounding error, identify examples, and, where possible, compensate for inaccuracies it introduces.
- Interpret apportionment as a problem of fairly distributing rounding error.
Examples: Analyzing apportionment in the U.S. House of Representatives; creating data tables that sum properly.
Task related to this benchmark: A Safe Load
d. Determine a reasonable degree of precision in a given situation.
- Assess the amount of error resulting from estimation and determine whether the error is within acceptable tolerance limits.
- Choose appropriate techniques and tools to measure quantities in order to achieve specified degrees of precision, accuracy, and error (or tolerance) of measurements.
Example: Humans have a reaction time to visual stimuli of approximately 0.1 sec. Thus, it is reasonable to use hand-activated stopwatches that measure tenths of a second but not hundredths.
Task related to this benchmark: Cones Launch
e. Interpret and compare extreme numbers (e.g., lottery odds, national debt, astronomical distances).
f. Apply significant figures, orders of magnitude, and scientific notation when making calculations or estimations.
g. In a problem situation, use judgment to determine when an estimate is appropriate and when an exact answer is needed.
N.B.2 Exponents and roots
a. Use the definition of a root of a number to explain the relationship of powers and roots.
If an = b, for an integer n ≥ 0, then a is said to be an nth root of b. When n is even and b > 0, we identify the unique a > 0 as the principal nth root of b, written 
.
- Use and interpret the symbols
and 
; know that 
, 
.By convention, for
is used to represent the non-negative square root of a.
b. Estimate square and cube roots and use calculators to find good approximations.
- Know the squares of numbers from 1 to 12 and the cubes of numbers from 1 to 5.
- Make or refine an estimate for a square root using the fact that if
0 ≤ a < n < b, then
; make or refine an estimate for a cube root using the fact that if a < n < b, then 
.
c. Evaluate expressions involving positive integer exponents and interpret such exponents in terms of repeated multiplication.
d. Convert between forms of numerical expressions involving roots and perform operations on numbers expressed in radical form.
Examples: Convert 
and use the understanding of this conversion to perform similar calculations and to compute with numbers in radical form.
Task related to this benchmark: Neighborhood Park
e. Interpret rational and negative exponents and use them to rewrite expressions in alternative forms.
Examples: 
- Convert between expressions involving rational exponents and those involving roots and integral powers.
Examples:
- Convert between expressions involving negative exponents and those involving only positive ones.
Examples:
- Apply the laws of exponents to expressions containing rational exponents.
Examples:
N.B.3 Real numbers
a. Categorize real numbers as either rational or irrational and know that, by definition, these are the only two possibilities.
- Locate any real number on the number line.
- Apply the definition of irrational numbers to identify examples and recognize approximations.
Square roots, cube roots, and nth roots of whole numbers that are not respectively squares, cubes, and nth powers of whole numbers provide the most common examples of irrational numbers. Pi (π) is another commonly cited irrational number.
- Know that the decimal expansion of an irrational number never ends and never repeats.
- Recognize and use
and 3.14 as approximations for the irrational number represented by pi (π). - Determine whether the square, cube, and nth roots of integers are integral or irrational when such roots are real numbers.
b. Establish simple facts about rational and irrational numbers using logical arguments and examples.
Examples: Explain why, if r and s are rational, then both r + s and rs are rational (for example, both ¾ and 2.3 are rational in 
, which is the ratio of two integers, hence rational); give examples to show that, if r and s are irrational, then r + s and rs could be either rational or irrational (for example, 
is irrational, whereas 
is rational).
c. Show that a given interval on the real number line, no matter how small, contains both rational and irrational numbers.
Example: To determine an irrational number between 
and 
, consider that ≈ 3.1622776 . . . , while = 
; so the number 3.17177177717 . . . , where the number of 7s in each successive set of 7s increases by one, is irrational and lies in this interval.
- Given a degree of precision, determine a rational approximation to that degree of precision for an irrational number.
d. Extend the properties of computation with rational numbers to real number computation.
Example: Example: If the area of one circle is 4π and the area of another, disjoint circle is 25π, then the sum of the areas of the two circles is 4π + 25π = (4 + 25)π = 29π, since the distributive property is true for all real numbers.
N.C.1 Number bases
a. Identify key characteristics of the base-10 number system and adapt them to other common number bases (binary, octal, and hexadecimal).
- Represent and interpret numbers in the binary, octal, and hexadecimal number systems.
- Apply the concept of base-10 place value to understand representation of numbers in other bases.
Example: In the base-8 number system, the 5 in the number 57,273 represents 5 x 84.
b. Convert binary to decimal and vice versa.
c. Encode data and record measurements of information capacity using various number base systems.
N.D.1 Complex numbers
a. Know that if a and b are real numbers, expressions of the form a + bi are called complex numbers, and explain why every real number is a complex number.
Every real number, a, is a complex number because it can be expressed as a + 0i. The imaginary unit, sometimes represented as 
, is a solution to the equation x2 = −1.
- Express the square root of a negative number in the form bi, where b is real.
Just as with square roots of positive numbers, there are two square roots for negative numbers; in
, 2i is taken to be the principal square root based on both the Cartesian and trigonometric representations of complex numbers.Examples: Determine the principle square root for each of the following:
.
b. Identify complex conjugates.
The conjugate of a complex number a + bi is the number a – bi.
c. Determine complex number solutions of the form a + bi for certain quadratic equations.
- Know that complex solutions of quadratic equations with real coefficients occur in conjugate pairs and show that multiplying factors related to conjugate pairs results in a quadratic equation having real coefficients.
The complex numbers
are the roots of the equation 
= x2 - 6x + 14 = 0, whose coefficients are real.
N.E.1 Computation with complex numbers
a. Compute with complex numbers.
- Add, subtract, and multiply complex numbers using the rules of arithmetic.
- Use conjugates to divide complex numbers.
Example:
This process can also be applied to the division of irrational numbers involving square roots, such as
.
N.E.2 Argand diagrams
a. Interpret complex numbers graphically using an Argand diagram.
In an Argand diagram, the real part of a complex number z = x + iy is plotted along the horizontal axis, and the imaginary part is plotted on the vertical axis. An Argand diagram enables complex numbers to be plotted as points in the plane just as the real line enables real numbers to be plotted as points on a line.