Mathematics Expectations: Pre-Algebra

Mathematics Expectations: Pre-Algebra (75K PDF)

Content Overview: By the end of eighth grade, the student will select and justify the type of number used in a given situation, choosing from whole, rational, decimal, or integer. The student will order and compare these numbers symbolically or figuratively on a number line or using symbols for equality and inequality. The student will identify the properties for operations, including additive inverse and multiplicative inverse, for all decimal numbers. The student will add, subtract, multiply, and divide with integers and combine integers with other forms of number in other computations. The student will use whole number exponents to discuss squares and cubes and discuss orders of magnitude for large numbers in scientific notation. The student will find the least common multiple and greatest common divisor/factor for a pair of whole numbers. The student will understand the relationship between squaring a number and finding a number's square root. The student will estimate the values of computations involving integers and other forms of numbers. The student will understand the impact of choice of unit on measurement and find an approximation of the effect of error on a calculation involving measurements. The student will develop and apply the measurement formulas for both surface area and volume of rectangular solids, prisms, pyramids, cylinders, cones, and spheres. In all of these areas, the student should be able to represent the concepts and skills both concretely and with symbolic notation.

The student will describe, classify, and contrast geometric planar figures and solids. The student will describe and sketch the cross section of a 3-dimensional figure cut by a plane. The student will sketch a 2-dimensional representation of a 3-dimensional shape, from different points of view. The student will use coordinate systems to display and investigate the relationships between lines in the plane. The student will use proportions to examine the relative measures of corresponding parts of similar figures and find other measures through the use of scaling with proportions or the Pythagorean theorem. The student is coming to understand the nature of geometry as a deductive system. The student will examine a simple game to determine if the outcomes are equally likely or if the game is unfair. The student will be able to make and study lists of possible outcomes for an experiment to determine probability in settings involving "and," "or," and "not." With data, the student will examine the mean, median, mode, and range for a set of univariate data and study the spread of the data through box-and-whisker plots. The student will collect, record, organize, and analyze two variable data through scatter plots and informal trend lines. The student will translate among graphical, tabular, symbolic, and verbal expressions representing quantities using variables and expressions. The student will use equivalent expressions to simplify expressions and solve linear equations. The student will have an intuitive understanding of slope and its interpretation as a rate of change. As in number and operation areas, students should have the ability to represent and model these understandings in a concrete, as well as symbolic, way.

Process Overview: By the end of the eighth grade, The student will solve one- and two-step problems using equations, translating patterns based on data, verbal explanations, or graphs into expressions or equations for solutions. The student will examine patterns related to simple open-ended problems, selecting appropriate strategies, monitoring progress, and making adjustments based on the feedback by abandoning or revising approaches as necessary. The student will form a conjecture and investigate it, using inductive methods of thought and justification. The student will develop explanations for investigations and problem-solving activities, using manipulatives or appropriate technology for developing the explanation and illustrating the graphs, figures, or data involved. The student is growing in the ability to construct convincing arguments about statements formed from analysis of problems and other situations. The student should be able to summarize approaches and give verbal presentations, as well as listen and ask questions of those presented by peers. The student is learning to compare and contrast mathematical concepts, as well as apply mathematics from classes to the solution of problems in other disciplines. The student is learning to gather information, organize and interpret it, and then choose a representation to use in communicating the data’s meaning to others.

1. The student understands and applies the concepts and procedures of mathematics.

1.1 Number Sense

• 1.1.1 applies associative, commutative, identity, inverse, and distributive properties to simplify and complete rational number operations
• 1.1.2 translates between and orders simple fractions, mixed numbers, and improper fractions
• 1.1.3 orders sets of rational numbers, integers, or real numbers relative to their values
• 1.1.4 uses factors, multiples, and prime factorization to simplify and solve rational number computations (ex: writes 28/48 in reduced form)
• 1.1.5 uses fraction, decimal, ratio and percent equivalencies to solve problems
• 1.1.6 applies mental arithmetic to compute simple percentages (ex: 10%, 25%, 33.5%, 50%, 75%)
• 1.1.7 writes decimals in scientific notation, and recognizes and interprets numbers in scientific notation
• 1.1.8 computes in situations involving positive and negative rational numbers, decimals, integers, and real numbers
• 1.1.9 creates and solves whole number proportions
• 1.1.10 uses multiples of unit rate, or cost to estimate the result of a specified operation or purchase
• 1.1.11 gives estimates for values involving unit multiples using mathematics (ex: if 5 bottles cost $10, then 7 bottles cost $14)
• 1.1.12 determines the reasonableness of a calculation involving rational or decimal numbers
• 1.1.13 differentiates between situations where estimates are sufficient and those for which exact values are required

1.2 Measurement

• 1.2.1 measures angles to the nearest degree with a protractor and estimates angle measurements to the nearest ten degrees
• 1.2.2 knows the number of degrees in a circle, triangle, and quadrilateral
• 1.2.3 applies formulas for perimeter and area for triangles, standard quadrilaterals, and circles; surface area and volume for prisms, cylinders, and spheres
• 1.2.4 uses scales and ratios involving known measures to estimate or calculate measures of objects for which no direct information is given
• 1.2.5 converts between metric measures using powers of ten and movement of the decimal point (ex: 42.31 cm = 0.4231 m)

1.3 Geometric Sense

• 1.3.1 describes and classifies 3-dimensional figures using their defining attributes: faces, edges, angles, vertices, angle measures, and measures of faces
• 1.3.2 recognizes, sets up, and interprets proportions between similar figures
• 1.3.3 describes and constructs simple transformations for complex figures using combinations of translations, reflections, and rotations
• 1.3.4 models and sketches 2-D versions of 3-D figures and 3-D figures from 2-D views
• 1.3.5 displays points and lines in the coordinate plane

1.4 Probability and Statistics

• 1.4.1 discriminates between impossible and certain events, describing why they have the respective probabilities of 1 and 0
• 1.4.2 collects a random sample and describes the population it depicts
• 1.4.3 lists the possible outcomes for a simple event/experiment, uses them to calculate the probabilities of various outcomes, and understands the difference between independent and dependent events
• 1.4.4 describes changes in a graph from one reporting point to the adjacent reporting point
• 1.4.5 recognizes the type of data involved in a situation, counts or measures, and chooses the appropriate type of graph to represent it
• 1.4.6 calculates and applies the mean, median, mode, and range for a set of data

1.5 Algebraic Sense

• 1.5.1 identifies and extends numerical patterns based on constant additions (arithmetic) and constant multiplication (geometric), and develops an explicit expression involving a variable for the nth term of a pattern
• 1.5.2 finds the value associated with a variable in a formula given values for the other variables in the formula
• 1.5.3 finds the solution to a linear equation with integral coefficients (ex: solves 5x+2=37)
• 1.5.4 writes an equation representing a specified relationship between quantities (ex: what number when multiplied by 4 and then increased by 2 is 38)
• 1.5.5 graphs inequalities on the number line, and represents linear equations in tabular or graphical form

2. The student uses mathematics to define and solve problems.

2.1 develops, uses and applies strategies for solving multi-step problems

• 2.2 uses formulas, strategies, algebraic properties, and order of operations to analyze and solve problems
• 2.3 identifies the information needed to solve a problem
• 2.4 designs and conducts probability experiments

3. The student uses mathematical reasoning.

• 3.1 forms a conjecture based on data or a pattern, and uses examples to check reasonableness
• 3.2 relates meaning of a solution to conditions of original problem
• 3.3 develops a convincing argument for a conjecture based on evidence
• 3.4 creates a counterexample to a given invalid statement or justification for a true statement

4. The student communicates knowledge and understanding in both everyday and mathematical language.

• 4.1 gathers information from sources (formula, value, etc.) related to a problem at hand using reading, listening, and observation skills
• 4.2 organizes and represents data for use in determining patterns or solutions
• 4.3 communicates ideas orally and in writing using everyday and mathematical language

5. The student understands how mathematical ideas connect within mathematics, to other subject areas, and to real-life situations.

• 5.1 relates the characteristics of mathematical concepts or algorithms to one another, such as factors to multiples or parallelograms to rhombi
• 5.2 uses mathematical modeling in other disciplines
• 5.3 illustrates how mathematics can be used in other career areas