 # Learning Standards for Mathematics

 Mathematical Reasoning Number and Numeration 1. Students use mathematical reasoning to analyze mathematical situations, make conjectures, gather evidence, and construct an argument. Students: • apply a variety of reasoning strategies. • make and evaluate conjectures and arguments using appropriate language. • make conclusions based on inductive reasoning. • justify conclusions involving simple and compound (i.e., and/or) statements. This is evident, for example, when students: use trial and error and work backwards to solve a problem. identify patterns in a number sequence. are asked to find numbers that satisfy two conditions, such as n > -4 and n ≤ 6. 2. Students use number sense and numeration to develop an understanding of the multiple uses of numbers in the real world , the use of numbers to communicate mathematically, and the use of numbers in the development of mathematical ideas. Students: • understand, represent, and use numbers in a variety of equivalent forms (integer, fraction, decimal, percent, exponential, expanded and scientific notation). • understand and apply ratios, proportions, and percents through a wide variety of hands-on explorations. • develop an understanding of number theory (primes, factors, and multiples). • recognize order relations for decimals, integers, and rational numbers. This is evident, for example, when students: use prime factors of a group of denominators to determine the least common denominator. select two pairs from a number of ratios and prove that they are in proportion. demonstrate the concept that a number can be symbolized by many different numerals as in: Sample Problems  Key ideas are identified by numbers (1). Performance indicators are identified by bullets (•). Sample tasks are identified by triangles ( )

Students will understand mathematics and become mathematically confident by
communicating and reasoning mathematically, by applying mathematics in
real-world settings, and by solving problems through the integrated study of
number systems , geometry, algebra, data analysis, probability, and trigonometry

 Operations Modeling/Multiple Representation 3. Students use mathematical operations and relationships among them to understand mathematics. Students: • add, subtract, multiply, and divide fractions, decimals, and integers. • explore and use the operations dealing with roots and powers . • use grouping symbols (parentheses) to clarify the intended order of operations. • apply the associative, commutative, distributive, inverse, and identity properties. • demonstrate an understanding of operational algorithms (procedures for adding, subtracting, etc.). • develop appropriate proficiency with facts and algorithms. • apply concepts of ratio and proportion to solve problems. This is evident, for example, when students: create area models to help in understanding fractions, decimals, and percents. find the missing number in a proportion in which three of the numbers are known, and letters are used as place holders. s arrange a set of fractions in order, from the smallest to the largest: 3 1 2 1 1 —, —, —, —, — 4 5 3 2 4 illustrate the distributive property for multiplication over addition, such as 2(a + 3) = 2a + 6. 4. Students use mathematical modeling/multiple representation to provide a means of presenting, interpreting, communicating, and connecting mathematical information and relationships. Students: • visualize, represent, and transform two- and three-dimensional shapes. • use maps and scale drawings to represent real objects or places. • use the coordinate plane to explore geometric ideas. • represent numerical relationships in one- and two-dimensional graphs. • use variables to represent relationships. • use concrete materials and diagrams to describe the operation of real world processes and systems. • develop and explore models that do and do not rely on chance. • investigate both two- and three-dimensional transformations. • use appropriate tools to construct and verify geometric relationships. • develop procedures for basic geometric constructions. This is evident, for example, when students: build a city skyline to demonstrate skill in linear measurements, scale drawing, ratio, fractions, angles, and geometric shapes. bisect an angle using a straight edge and compass. draw a complex of geometric figures to illustrate that the intersection of a plane and a sphere is a circle or point .

Sample Problems Measurement Uncertainty 5. Students use measurement in both metric and English measure to provide a major link between the abstractions of mathematics and the real world in order to describe and compare objects and data. Students: • estimate, make, and use measurements in real-world situations. • select appropriate standard and nonstandard measurement units and tools to measure to a desired degree of accuracy. • develop measurement skills and informally derive and apply formulas in direct measurement activities. • use statistical methods and measures of central tendencies to display, describe, and compare data. • explore and produce graphic representations of data using calculators/computers. • develop critical judgment for the reasonableness of measurement. This is evident, for example, when students: use box plots or stem and leaf graphs to display a set of test scores. estimate and measure the surface areas of a set of gift boxes in order to determine how much wrapping paper will be required. explain when to use mean, median, or mode for a group of data. 6. Students use ideas of uncertainty to illustrate that mathematics involves more than exactness when dealing with everyday situations. Students: • use estimation to check the reasonableness of results obtained by computation, algorithms, or the use of technology. • use estimation to solve problems for which exact answers are inappropriate. • estimate the probability of events. • use simulation techniques to estimate probabilities. • determine probabilities of independent and mutually exclusive events. This is evident, for example, when students: construct spinners to represent random choice of four possible selections. perform probability experiments with independent events (e.g., the probability that the head of a coin will turn up, or that a 6 will appear on a die toss). estimate the number of students who might chose to eat hot dogs at a picnic.

Sample Problems Students will understand mathematics and become mathematically confident by
communicating and reasoning mathematically, by applying mathematics in
real-world settings, and by solving problems through the integrated study of
number systems, geometry, algebra , data analysis, probability, and trigonometry.

Patterns/Functions

7. Students use patterns and functions to develop
mathematical power, appreciate the true beauty of
mathematics, and construct generalizations that
describe patterns simply and efficiently.

Students:
• recognize, describe, and generalize a wide variety of
patterns and functions.
• describe and represent patterns and functional
relationships using tables, charts and graphs , algebraic
expressions , rules, and verbal descriptions.
• develop methods to solve basic linear and quadratic
equations.
• develop an understanding of functions and functional
relationships: that a change in one quantity (variable)
results in change in another.
• verify results of substituting variables.
• apply the concept of similarity in relevant situations.
• use properties of polygons to classify them.
• explore relationships involving points, lines, angles, and
planes.
• develop and apply the Pythagorean principle in the
solution of problems.
• explore and develop basic concepts of right triangle
trigonometry.
• use patterns and functions to represent and solve
problems.

This is evident, for example, when students: find the height of a building when a 20-foot ladder reaches the
top of the building when its base is 12 feet away from the
structure. investigate number patterns through palindromes (pick a 2- digit
number, reverse it and add the two—repeat the process until a
palindrome appears)

palindrome palindrome s solve linear equations, such as 2(x + 3) = x + 5 by several
methods.

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