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Course Description
Concepts covered in the course include: numbers and their relationship to each other, identity, number systems and theory, why and how computations work, estimation, proportion, probability, statistics, linear equations, mathematical connections, patterns, graphs, and measurement. Students will also utilize mathematical reasoning to solve various problems and to enhance their comprehension of the material.
Credits: 5
Released: 2010
Course Content
Unit 1 — Numbers and Number Relationships
California Mathematics Content Standards:
- Algebra I 1.0 Students identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable
- Mathematical Reasoning 2.5 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning.
- Measurement and Geometry 1.2 Construct and read drawings and models made to scale
- Measurement and Geometry 2.0 Students compute the perimeter, area, and volume of common geometric objects and use the results to find measures of less common objects. They know how perimeter, area, and volume are affected by changes of scale:
- Number Sense 1.0 Students know the properties of, and compute with, rational numbers expressed in a variety of forms.
- Number Sense 1.1 Compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line.
- Number Sense 1.1 Read, write, and compare rational numbers in scientific notation with approximate numbers using scientific notation.
- Number Sense 1.3 Convert fractions to decimals and percents and use these representations in estimations, computations, and applications.
- Number Sense 1.7 Solve problems that involve discounts, markups, commissions, and profit and compute simple and compound interest.
- Number Sense 2.0 Students use exponents, powers, and roots and use exponents in working with fractions.
- Number Sense1.4 Differentiate between rational and irrational numbers.
Unit Objectives
Upon completion of this Unit, students will be able to:
- Identify sets of numbers.
- Display sets of numbers with Venn Diagrams.
- Make a chart of the number system displaying the categories of numbers that are used in everyday life.
- Express numbers in decimal, exponential and scientific notation.
- Calculate percentages.
- Differentiate between rational and irrational numbers.
- Use ratios to solve real life problems.
Unit 2 — Number Systems and Number Theory
California Mathematics Content Standards:
- Algebra and Functions 1.2 Use a letter to represent an unknown number; write and evaluate simple algebraic expressions in one variable by substitution.
- Algebra and Functions 1.2 Use the correct order of operations to evaluate algebraic expressions such as 3(2x + 5)2
- Algebra and Functions 1.3 Apply algebraic order of operations and the commutative, associative, and distributive properties to evaluate expressions; and justify each step in the process
- Algebra and Functions 1.3 Simplify numerical expressions by applying properties of rational numbers and justify the process used.
- Algebra and Functions 1.4 Solve problems manually by using the correct order of operations or by using a scientific calculator.
- Algebra and Functions 1.5 Algebra and Functions 1.5 Understand that an equation such as y = 3 x + 5 is a prescription for determining a second number when a first number is given.
- Algebra I 1.0 Students identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable.
- Number Sense 1.0 Students know the properties of, and compute with, rational numbers expressed in a variety of forms.
- Number Sense 1.0: Students compare and order positive and negative fractions, decimals, and mixed numbers. Students solve problems involving fractions, ratios, proportions, and percentages.
- Number Sense 1.1 Read, write, and compare rational numbers in scientific notation (positive and negative powers of 10) with approximate numbers using scientific notation
- Number Sense 1.3 Convert fractions to decimals and percents and use these representations in estimations, computations, and applications.
- Number Sense 1.3: Use proportions to solve problems. 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.
- Number Sense 1.4 Determine the prime factors of all numbers and write the numbers as the product of their prime factors by using exponents to show multiples of a factor
- Number Sense 1.4 Differentiate between rational and irrational numbers
- Number Sense 1.5 Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers.
- Number Sense 1.7 Write the fraction represented by a drawing of parts of a figure; represent a given fraction by using drawings; and relate a fraction to a simple decimal on a number line.
- Number Sense 2.0 Students use exponents, powers, and roots and use exponents in working with fractions.
- Number Sense 2.4 Determine the least common multiple and the greatest common divisor of whole numbers; use them to solve problems with fractions.
Unit Objectives
Upon completion of this Unit, students will be able to:
- Describe the difference between prime numbers and composite numbers.
- Calculate the prime factors of any given number.
- Determine the Greatest Common Factor between two fractions.
- Determine the Least Common Multiple between two numbers.
- Know the rules for combining positive and negative numbers with multiplication, division, addition and subtraction.
- Solve simple linear equations with one variable.
- Recite and use correctly the Order of Operations.
- Solve equations using the Identity Principle, the Multiply by One Principle, and the Balance Concept.
- Use Algebra to correctly change from one unit of measurement to another.
Unit 3 — Computation and Estimation
California Mathematics Content Standards:
- Algebra and Functions 1.3 Apply algebraic order of operations and the commutative, associative, and distributive properties to evaluate expressions; and justify each step in the process
- Algebra and Functions 1.4 Solve problems manually by using the correct order of operations or by using a scientific calculator.
- Algebra I 1.0 Students identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable.
- Mathematical Reasoning. 2.1 Use estimation to verify the reasonableness of calculated results.
- Measurement and Geometry 1.0Students choose appropriate units of measure and use ratios to convert within and between measurement systems to solve problems
- Number Sense 1.0 Students know the properties of, and compute with, rational numbers expressed in a variety of forms.
- Number Sense 1.0 Students know the properties of, and compute with, rational numbers expressed in a variety of forms.
- Number Sense 1.1 Compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line.
- Number Sense 1.1 Estimate, round, and manipulate very large and very small numbers.
- Number Sense 1.4 Determine the prime factors of all numbers and write the numbers as the product of their prime factors by using exponents to show multiples of a factor
- Number Sense 2.4 Determine the least common multiple and the greatest common divisor of whole numbers; use them to solve problems with fractions
- Probability and Statistics 6.0Students know the definitions of the mean, median, and mode of a distribution of data and can compute each in particular situations.
- Statistics, Data Analysis, and Probability 1.1 Compute the range, mean, median, and mode of data sets.
- Statistics, Data Analysis, and Probability 1.2 Understand how additional data added to data sets may affect these computations of measures of central tendency.
- Statistics, Data Analysis, and Probability 1.3 Understand how the inclusion or exclusion of outliers affects measures of central tendency.
- Statistics, Data Analysis, and Probability 1.4 Know why a specific measure of central tendency (mean, median) provides the most useful information in a given context.
- Statistics, Data Analysis, and Probability 1.0 Students compute and analyze statistical measurements for data sets.
Unit Objectives
Upon completion of this Unit, students will be able to:
- Recite and correctly use the Order of Operations.
- Round fractional and whole numbers to any place.
- Reduce fractions to their simplest forms.
- Use mental math for quick estimation.
- Use estimation to verify calculated results.
- Determine the appropriate measurement to use for various measures.
- Use proportions to solve problems.
- Represent all possible outcomes for compound events in an organized way.
- Interpret and use ratios.
- Represent probabilities as ratios and as percents.
- Explain the meaning of multiplication of positive fractions.
- Compute the range, mean, median and mode of data sets.
- Understand how the inclusion or exclusion of data affects the average.
Unit 4 — Patterns, Functions, and Mathematical
Connections
California Mathematics Content Standards:
- Algebra and Functions 1.0: Students use variables in simple expressions, compute the value of the expression for specific values of the variable, and plot and interpret the results.
- Algebra and Functions 1.0: Students write verbal expressions and sentences as algebraic expressions, solve simple linear equations, and graph and interpret their results.
- Algebra and Functions 1.1: Students write and solve one-step linear equations in one variable.
- Algebra and Functions 1.5: Solve problems involving linear functions with integer values, write the equation; and graph the resulting ordered pairs of integers on a grid.
- Algebra and Functions 3.0: Students graph and interpret linear and some non-linear functions.
- Algebra and Functions 3.0: Students investigate geometric patterns and describe them algebraically.
- Algebra and Functions 4.0: Students solve simple linear equations over the rational numbers.
- Algebra I 1.0 Students identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable:
- Algebra I 17.0: Students determine the domain of independent variables and the range of dependent variables defined by a graph, a set of ordered pairs, or a symbolic expression.
- Math Standard Goals, 5th paragraph: Students make connections among mathematical ideas and between mathematics and other disciplines.
- Mathematical Reasoning 1.0 Students make decisions about how to approach problems.
- Mathematical Reasoning 1.1 Students analyze problems by identifying relationships, identifying missing information, observing patterns.
- Number Sense 1.0 Students know the properties of, and compute with, rational numbers expressed in a variety of forms.
- Number Sense 1.0 Students solve problems involving fractions, ratios, proportions, and percentages.
- Number Sense 1.3 Convert fractions to decimals and percents and use these representations in estimations, computations, and applications.
- Number Sense 1.3: Use proportions to solve problems. 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.
- Number Sense 2.0 Students use exponents, powers, and roots and use exponents in working with fractions.
Unit Objectives
Upon completion of this Unit, students will be able to:
- Identify Patterns.
- Anticipate the next value in a patterned sequence.
- Make T-Charts to display patterns.
- Create algebraic expressions that correctly represent pattern behavior.
- Read and interpret coordinate systems and plot data points on a coordinate system.
- Correctly use the x and y axis on a coordinate system for independent and dependent variables.
- Write and solve one step linear equations in one variable.
- Write and evaluate an algebraic expression for a two variable situation.
- Graph most linear equations and some non linear equations.
- Recognize multiple uses of mathematics in other fields.
- Develop generalizations of results obtained.
Unit 5 — Measurement
California Mathematics Content Standards:
- Algebra and Functions 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.
- Algebra I 1.0 Students identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers.
- Algebra I 17.0: Students determine the domain of independent variables and the range of dependent variables defined by a graph, a set of ordered pairs, or a symbolic expression.
- Geometry 8.0 Students know, derive, and solve problems involving the perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures.
- Geometry 9.0 Students compute the volumes and surface areas of prisms, pyramids, cylinders, cones, and spheres; and students commit to memory the formulas for prisms, pyramids, and cylinders.
- Geometry 11.0 Students determine how changes in dimensions affect the perimeter, area, and volume of common geometric figures
- Mathematical Reasoning 1.0 Students make decisions about how to approach problems.
- Mathematical Reasoning 1.1 Students analyze problems by identifying relationships, identifying missing information, observing patterns.
- Measurement and Geometry 1.0 Students deepen their understanding of the measurement of plane and solid shapes and use this understanding to solve problems.
- Measurement and Geometry1.0 Students choose appropriate units of measure and use ratios to convert within and between measurement systems to solve problems.
- Measurement and Geometry 2.0 Students compute the perimeter, area, and volume of common geometric objects and use the results to find measures of less common objects.
- Measurement and Geometry 2.0 Students identify and describe the properties of two-dimensional figures:
- Measurement and Geometry 2.1 Use formulas routinely for finding the perimeter and area of basic two-dimensional figures and the surface area and volume of basic three-dimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders.
- Measurement and Geometry 2.2 Know that the sum of the angles of any triangle is 180° and the sum of the angles of any quadrilateral is 360° and use this information to solve problems.
- Measurement and Geometry 2.3 Compute the length of the perimeter, the surface area of the faces, and the volume of a three-dimensional object built from rectangular solids. Understand that when the lengths of all dimensions are multiplied by a scale factor, the surface area is multiplied by the square of the scale factor and the volume is multiplied by the cube of the scale factor.
- Measurement and Geometry 2.4 Relate the changes in measurement with a change of scale to the units used and to conversions between.
- Measurement and Geometry 3.0 Students investigate geometric patterns and describe them algebraically:
- Number Sense 1.0 Students know the properties of, and compute with, rational numbers expressed in a variety of forms.
- Number Sense 1.0 Students solve problems involving fractions, ratios, proportions, and percentages.
- Number Sense 1.3 Convert fractions to decimals and percents and use these representations in estimations, computations, and applications.
- Number Sense 1.3: Use proportions to solve problems. Use cross-multiplication as a method for solving such problems.
- Number Sense 1.4 Determine the prime factors of all numbers and write the numbers as products of their prime factors.
- Number Sense 2.0 Students use exponents, powers, and roots and use exponents in working with fractions.
- Number Sense 2.4 Determine the least common multiple and the greatest common divisor of whole numbers.
Unit Objectives
Upon completion of this Unit, students will be able to:
- Understand the concept of a constant.
- Know formulas for the circumference and area of a circle.
- Know the common estimates of p and use these estimates to calculate the area of circles.
- Know and use the formulas for the volume of triangular prisms and cylinders and compare these formulas to the volume of a rectangular solid.
- Identify some of the relationships between the angles in a rectangle and a triangle.
- Compare geometric measures within and between measurement systems.
- Construct drawings and models made to scale.
- Use measures expressed as rates.
- Use formulas routinely for finding the perimeter and area of basic two dimensional figures.
- Calculate the surface area and volume of basic three dimensional figures including prisms and cylinders.
- Compute the length of the perimeter, the surface area of the faces, and the volume of a three dimensional object built from rectangular solids.
- Understand that when the lengths of all dimensions are multiplied by a scale factor the surface area is multiplied by the square of the scale factor and the volume is multiplied by the cube of the scale factor.
- Relate changes in measurement with a change of scale to the units used.
- Know what congruent figures are.
- Construct two dimensional patterns for three dimensional models such as cylinders and prisms.
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