MATHEMATICS STANDARDS AND BENCHMARKS Grades K-12 Fairfield-Suisun Unified School District May 2000 District Mathematics Curriculum Committee Rob Buoncristiani, Director of Curriculum Elementary School Jerry Bernhardt Linda Flood Sharon Kane-Parisian Nancy Lamp Kathy LaRocco Kathlan Latimer Janice Napier Janis Okamoto Mary Paiko Cynthia Sheppard Kathrene Skaife Ruth Yerkes Middle School Julie Crozier Bill Doherty Del Fawcett Linda Krummes Sherry McCormick Walter Price Dennis Sherman High School Rick Bryan Pat Cleaver sh:wgmath FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS INTRODUCTION Mathematics is taking on an increasingly important role in today’s world. Mathematics is connected to other subjects and to life experiences. It is present all around us. It is hoped that the students also experience the enjoyment and fascination of mathematics as they gain mathematical power. To be successful, students need a strong, balanced curriculum that challenges them to make connections, solve problems, and see pattern and logic in the world around them. A knowledge of mathematics is critical for all students, not only for those whose careers will demand advanced mathematical preparation, but for all citizens. All students are capable of learning rigorous mathematical. All students includes those performing at, below, and above grade level; English language learners, special education students, and others with special learning needs. This curriculum is based upon The California Mathematics Academic Content Standards for grades K-12 (1998) which focuses on essential content for all students and intends to prepare them for the study of advanced mathematics. Students will: develop fluency with basic computational skills and procedural competencies develop understanding of mathematical concepts become mathematical problem solvers who can recognize and solve routine problems readily and can find ways to reach a solution or goal where no routine path is apparent communicate precisely about quantities, logical relationships and unknowns via the use of signs, symbols, models, graphs, and mathematical terms gather data, analyze evidence, and build arguments using mathematical reasoning to support or refute hypothesis; and make connections among mathematical ideas between mathematics and other disciplines. PAGE 3 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS GOALS TO BE ACHIEVED PAGE 4 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS THE STRANDS The strands in mathematics were developed to break the content into a small set of manageable and understandable categories. It is often difficult to restrict a particular mathematical concept or skills to a single strand. Because the content of mathematics builds and changes from grade to grade, the contents of any one strand change considerably over the course of K-7. Thus the strands serve only as an aid to organizing and thinking about the curriculum and no more. They describe the curriculum rather than define it. For the same reason, the identification of strands does not mean that each is to be given equal weight in each year of mathematics education. The general nature of each strand is described below. NUMBER SENSE The mathematics for this standard centers primarily on the development of number concepts, on computation with numbers (addition, subtraction, multiplication, division, finding powers and roots, etc.), on numeration (systems for writing numbers, including base ten, fractions, negative numbers, rational numbers, percents, scientific notation, etc.), and on estimation. At higher levels this strand includes the study of prime and composite numbers, of irrational numbers and their approximation by rationals, of real numbers, and of complex numbers. ALGEBRA AND FUNCTIONS This strand involves two closely related subjects. Functions are rules that assign to each element in an initial set an element in a second set. For example, as early as kindergarten, children take collections of colored balls and sort them according to color, thereby assigning to each ball its color in the process. Later, students work with simple numeric functions, such as unit conversions which assign, for example, 12 inches to each foot. Functions are, therefore, one of the key areas of mathematical study. As indicated, they are encountered informally in the elementary grades and grow in prominence and importance with the student’s increasing grasp of algebra in the higher grades. Algebra proper again starts informally. It appears initially in its proper form in the third grade as ―generalized arithmetic.‖ In later grades, algebra is the vital tool deeded for solving equations and inequalities, and using these as mathematical models of real situations. Students solve the problems that arise by translating from natural language—by which they communicate daily—to the abstract language of algebra, and conversely, from the formal language of algebra to natural language to demonstrate clear understanding of the concepts involved. MEASUREMENT AND GEOMETRY PAGE 5 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS Geometry is the study of space and the figures in space. In the early grades this strand includes the use of measuring tools such as rulers, and recognition of basic shapes such as triangles, circles, squares, spheres, and cubes. In later grades this extends to the study of area and volume, and the measurement of angles. In high school plane geometry is studied, both an introduction to the concept of mathematical proof and a fascinating structure which has profoundly influenced civilization for over 2000 years. MATHEMATICAL REASONING Whenever a mathematical statement is justified, mathematical reasoning is involved. Mathematical reasoning is involved in explaining arithmetic facts, in solving problems and puzzles at all levels, in understanding algorithms and formulas, and in justifying basic results in all areas of mathematics. Mathematical reasoning does not necessarily stand alone, but is inherently embedded in each of the other strands. STATISTICS, DATA, AND PROBABILITY This strand includes the definitions and calculations of various averages, the analysis of data by classification and by graphical displays, taking into account randomness and bias in sampling. This strand has important connections with the Algebra and Functions strand and the Number Sense strand in the study of permutations and combinations, and of Pascal’s triangle. In the elementary grades, effort is largely limited to collecting data and displaying it in graphs, in addition to calculating simple averages, and performing probability experiments. This strand becomes more important in the later grades when the students have gained the necessary skill with fractions, and algebraic concepts in general, so that statistics and their impact on daily life can be discussed with more sophistication than would have been possible earlier. PAGE 6 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS KINDERGARTEN By the end of kindergarten, students understand the consistency of small numbers, quantities and simple shapes in their everyday environment. They count, compare, describe and sort objects, and develop a sense about properties and patterns. Throughout this document denotes specific standards from the Mathematics Framework for California Public Schools Throughout this document denotes the most important standards at each grade level All other items are District Standards NUMBER SENSE 1.0 The student understands the relationship between numbers and quantities, (i.e., that a set of objects has the same number of objects in different situations, regardless of its position or arrangement). The student will: 1.1 1.2 1.3 1.4 1.5 Compare two or more sets (up to 10 objects in each group) and identify whether a set is equal to, more than, or less than the other. Count, represent, name and order numbers (to 30) using objects. Know that the larger numbers describe sets with more objects in them than smaller numbers. Understand ordinal numbers up to fifth place. Understand the concept of a unit and its subdivision into equal parts (e.g., one object, such as a candy bar and its division into equal parts to be shared among four people. Example: 2.0 Identify which halves make a whole ______ + ______ = The student understands simple addition and subtraction situations. Solve real-world addition and subtraction problems using up to ten concrete objects. The student will: 2.1 Example: Three animals, a goat, a pig, and a cow, were in a barnyard. The cow went into the barn to take a nap. How many animals were left in the barnyard? PAGE 7 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 3.0 The student uses estimation strategies in computation and problem solving with numbers that use the ones and tens places. Understand when an estimate is reasonable. Understand common terms used with estimation (e.g., ―about,‖ ―near,‖ ―closer to,‖ ―between,‖ ―a little less than‖). Demonstrate ―fair share‖ concept by distributing a number of objects equally among a group and then determine if each member of a group has the same number of objects The student will: 3.1a 3.1b 3.2 Example: The bear made four pancakes for his two best friends. He wants to give each of his friends the same number of pancakes. How many pancakes should he put on each friend’s plate? Use counters to find out. 3.3 Understand the relative value of pennies and nickels. ALGEBRA AND FUNCTIONS 1.0 The student understands a variety of patterns. The student will: 1.1 Use attributes to describe and sort objects (e.g., all these balls are green, those are red). Example: Describe how the following two objects are the same or different. Example: Show students stars sorted into three sets as shown and ask them to identify how stars were sorted. 1.2 1.3 Compile collections. Understand the vocabulary used to describe a collection (all, some, none). PAGE 8 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 1.4 Identify and extend simple patterns according to identified attributes (e.g., shape, size, color). MEASUREMENT AND GEOMETRY 1.0 The student understands that there are properties such as length, weight, capacity, and time and that comparisons can be made by using these properties. The student will: 1.1 Compare the length, weight, and capacity of objects by making direct comparisons or using reference objects (e.g., shorter/longer/taller, lighter/heavier, which holds more?). Example: Who is the tallest girl in the class? The shortest boy? Which weighs more – an elephant or a mouse? What animal has the longest neck – a giraffe or a cat? 1.2 Understand the concepts of time (e.g., morning, afternoon, evening, day, yesterday, tomorrow, week, year) including tools that measure time (e.g., clock, calendar). Example: 2.0 Tell what you did before you came to school today. What do you do at night? 1.3 1.4 Know the days of the week. Know the time (to the nearest hour) of everyday events. The student knows common geometric objects in their environment and describes their features. The student will: 2.1 Know the names of common geometric objects (e.g., circle, triangle, square, rectangle, cube, sphere, cone, cylinder). Example: Which of these is a square? 2.2 Compare familiar plane and solid objects by common attributes (e.g., position, shape, size, roundness, number of corners). PAGE 9 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS STATISTICS, DATA ANALYSIS AND PROBABILITY 1.0 The student collects information about objects and events in his or her environment. The student will: 1.1a 1.1b 1.2 Collect data and organize the results using objects, pictures, and picture graphs. Compare sets of data represented in graphs. Identify, describe and extend simple patterns involving shape, size, or color such as circle, triangle, or red, blue. 2.0 The student understands the basic concepts of probability. 2.1 Understand the concept of chance (e.g., spin spinner to see who gets the highest). MATHEMATICAL REASONING 1.0 The student makes decisions about how to set up a problem. The student will: 1.1 2.0 1.2 Select the appropriate approach, materials, and strategies to use to solve a problem. Use tools and strategies such as manipulatives or sketches to model problems. The student solves problems in reasonable ways and justifies reasoning. The student will: 2.1 Explain the reasoning used with concrete objects and pictorial representations. Example: Here are three straws. Put them in order from shortest to longest. Use real straws and glue them on a paper. 2.2 2.3 2.4 Make precise calculations and check the validity of the results from the context of the problem. Predict outcomes and make reasonable estimates. Understand connections between one problem and another. PAGE 10 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS GRADE 1 By the end of first grade, students understand and use the concept of "ones" and "tens" in the place value number system. They add and subtract small numbers with ease. They measure with simple units and locate objects in space. They describe data and analyze and solve simple problem situations. Throughout this document denotes specific standards from the Mathematics Framework for California Public Schools Throughout this document denotes the most important standards at each grade level All other items are District Standards NUMBER SENSE 1.0 The student understands and uses numbers up to 100. The student will: 1.1 1.2 Count, read, and write whole numbers to 100. Compare and order whole numbers to 100 using the symbols for less than, equal to, or greater than (<, =, >). Example: Which of the following are correct and which are incorrect? (a) 75 > 76 (b) 48 < 42 (c) 89 > 91 (d) 59 < 67 (e) 34 = 33 1.3 Understand equivalent forms of the same number (e.g., physical models, diagrams, number expressions (to 20) (i.e., 8 can be represented as 4 + 4, 5 + 3, 2 + 2 + 2 + 2, 10 - 2, 11 - 3). Count and group objects into ones and tens (e.g., 3 groups of ten and 4 more is 34 or 30 + 4) 1.4 Example: A certain brand of chewing gum has 10 pieces in each pack. If there are 14 students, what is the smallest number of packs we must buy to make sure each student gets at least one piece of gum? If there are 19 students? What about 21 students? There are 5 quarters, 9 dimes, 3 nickels, and 8 pennies. They are supposed to be put in piles of ten (coins). How many such piles can be formed by all these coins, and how many are left over? Example: PAGE 11 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 1.5 1.6 1.7 1.8 1.9 2.0 Know the value of coins and how different combinations of coins equal the same value (e.g., I have some pennies, nickels, and dimes in my pocket. I reach in and pull out three coins. How much money might I have? List all the possibilities). Order coins by their value (e.g., penny, nickel, dime, quarter). Understand that 100 can be broken down into tens and ones in several ways. Use mental addition up to 100 (e.g., "what is 25 and 1 more?"). Know numbers up to 100. The student understands the meaning of addition and subtraction and uses these operations to solve problems. The student will: 2.1 Example: Example: Know the addition and subtraction facts up to 20. I had 10 cupcakes, but I ate 3 of them. How many cupcakes do I have left? How many if I had 18 and ate 5? 2.2 2.3 2.4 Use the inverse relationship between addition and subtraction to solve problems. Understand whole number relations (e.g., identify one more than, one less than, ten more than, ten less than a given number). Count by 2s, 5s and 10s with numbers to 100. Which numbers are missing? 24, 26, 28, 30, ___, ___, 36, ___, 40 15, 20, 25, 30, ___, ___, 45, ___, 55 2.5 2.6 2.7 Understand the meaning of addition (i.e., putting together, increasing) and subtraction (i.e., taking away, comparing, finding the difference). Solve addition and subtraction problems with one- and two-digit numbers (e.g., 22 + 3 = __ ). Find the sum of three one-digit numbers. 3.0 The student uses estimation strategies in computation and problem solving that involve numbers that use the ones, tens, and hundreds places. The student will: 3.1 3.2 Make reasonable estimates when comparing larger or smaller numbers. Estimate the quantity of sets containing up to 20 objects. PAGE 12 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS ALGEBRA AND FUNCTIONS 1.0 The student uses number sentences to solve problems. The student will: 1.1 Use number sentences to model and solve problem situations that involve addition and subtraction. Example: Do the following problems in succession: 1. Marie had some pencils in her desk. She put 5 more in her desk. Then she had 14. How many pencils did she have in her desk to start with? 2. Eddie had 14 helium balloons. A number of them floated away. He had 5 left. How many did he lose? Nina had 14 seashells. That was 5 more than Pedro had. How many seashells did Pedro have? 5 + ( ) = 6? ( ) + 12 = 14? 3. 4. 1.2 1.3 1.4 Understand the meaning of the symbols +, -, =. Create problem situations that could lead to given number sentences involving addition and subtraction. relate problem situations and number sentences involving additional subtraction. MEASUREMENT AND GEOMETRY 1.0 The student uses direct comparisons and non-standard units to describe the measurements of objects (e.g., measures with unifix cubes, links, uses a balance scale). The student will: 1.1 Compare the length, weight, and volume of two or more objects using direct comparison or a non-standard unit. Example: Measure your desk by using the length of a ballpoint pen. How many ballpoint pens would be roughly equal to the length of your desk? The width of your desk? Which is longer? 1.2 Tell time to the nearest half hour and compare time related to events (e.g., before/after, shorter/longer). PAGE 13 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 1.3 1.4 2.0 Understand calendar units (day, week, month, year). Understand and identify halves of a whole. The student knows common geometric figures, classifies them by common attributes, and understands their relative position or location in space. The student will: 2.1 Example: 2.2 2.3 Know the properties and relationships of triangles, rectangles, squares and circles, including the faces of three-dimensional objects (e.g., make a picture of a house by using triangles, squares, and rectangles). Use common attributes to classify familiar plane and solid objects (e.g., color, position, shape, size, roundness, number of corners). Give and follow directions about location. There are pictures on a table of a ball, a girl, a horse, and a cat. Arrange them according to these directions: 1. Put the picture of the ball above the picture of the horse. 2. Put the picture of the girl on top of the picture of the horse. 3. Put the picture of the cat under the picture of the horse. 2.4 2.5 2.6 2.7 Understand the common language of proximity, position and direction (e.g., near, far, below, above, up, down, behind, in front of, next to, left/right). Know how to duplicate shapes and figures. Understand the concept of lines of symmetry. Use reflections and rotations of geometric figures. Example: The students place a corner of a pattern block in a hinged mirror. Using other pattern blocks, the students will duplicate the design. 2.8 2.9 2.10 2.11 Know how to manipulate and construct two and three dimensional figures. Create drawings or tracings of constructions. Understand how to use different objects to cover the same shape and compare the number used. Compare the area covered by an equal number of objects. STATISTICS, DATA ANALYSIS AND PROBABILITY 1.0 The student organizes, represents, and compares categorical data using simple graphs and charts. PAGE 14 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS The student will: 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Use common attributes to sort objects and data and describe the groups formed. Represent and compare data (e.g., largest, smallest, most often, least often), using pictures, bar graphs, tally charts, and picture graphs. Use Venn diagrams to extend sorting skills to overlapping sets. move from concrete data to representational graphs, charts, etc. Count different objects, tally and groups by tens and ones. Compile and sort collections and explain the results. Understand the vocabulary appropriate to describe collections (all, some, none). 2.0 The student sorts objects and creates and describes patterns involving numbers, shape, size, rhythm, or color. The student will: 2.1 Describe, extend, and explain how to get to the next element in simple repeating patterns (e.g., rhythmic, numeric, color and shape patterns). 3.0 The student understands the concepts of probability. The student will: 3.1 3.2 3.3 Understand the rules of educational games and explain them to others. Understand concepts of chance (e.g., Lotto, number cubes, spinner games). Understand that some events are more likely to occur than others. MATHEMATICAL REASONING 1.0 The student makes decisions about how to set up a problem. The student will: 1.1 1.2 1.3 Select the appropriate approach, materials, and strategies to use to solve problems. Use tools (e.g., manipulatives, sketches) to model problems. Solve problems experientially (count, use materials, act out, discuss). 2.0 The student solves problems and justifies his or her reasoning. The student will: 2.1 Explain the reasoning used and justify the procedures selected orally and in writing. PAGE 15 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 2.2 Make precise calculations and check the validity of the results from the context of the problem. 3.0 The student understands connections between one problem and another. The student will: 3.1 Example: Use variables to investigate concrete situations. Mary is asked to draw a route from her house to the neighborhood grocery store. How might her map differ if she walked, or rode a bike, or was driven by her parent? PAGE 16 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS GRADE 2 By the end of second grade, students understand place value and number relationships as they add and subtract and they use simple concepts of multiplication. They measure quantities with appropriate units. They classify and see relationships among shapes by paying attention to the elements that compose them. They collect and analyze data and verify answers. Throughout this document denotes specific standards from the Mathematics Framework for California Public Schools Throughout this document denotes the most important standards at each grade level All other items are District Standards NUMBER SENSE 1.0 The student understands the relationships among numbers, quantities, and place value in whole numbers up to 1,000. The student will: 1.1 1.2 Count, read, write whole numbers to 1,000 and identify the place value for each digit. Use words, models, and expanded form to represent numbers (to 1,000) (e.g., 45 = 4 tens + 5). Example: Kelly has 308 stickers. How many sets of hundreds, tens, and ones does she have? 1.3 1.4 1.5 Order and compare whole numbers up to 1,000 using the symbols <, =, >. Understand the basic difference between odd and even numbers. Understand place value concepts and patterns. 2.0 The student estimates, calculates, and solves problems involving addition and subtraction of two- and three-digit numbers. The student will: 2.1 2.2 Understand and use the inverse relationship between addition and subtraction to solve problems and check solutions (e.g., an opposite number sentence for 8 + 6 = 14 is 14 - 6 = 8). Find the sum or difference of two whole numbers up to three digits long. PAGE 17 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS Example: 2.3 Use mental arithmetic to find the sum or difference of two 2-digit numbers. In a game, Mysong and Naoki are making addition problems. They make two 2-digit numbers out of the four given numbers 1, 2, 3, and 4. Each number is used exactly once. The winner is the one who makes two numbers whose sum is the largest. Mysong had 43 and 21, while Naoki had 31 and 24. Who won the game? How do you know? Show how you can beat both Mysong and Naoki by making up two numbers with a larger sum than either (Adapted from TIMSS, gr. 4, V-4). (This problem also supports Mathematical Reasoning Standard 1.0.) 2.4 2.5 2.6 2.7 2.8 2.9 Understand strategies to estimate quantities to 100. Understand the concept of regrouping using concrete materials. Understand the concept of regrouping up to 3-digits without concrete materials. Understand equality relationships. Understand how to use calculators to perform addition and subtraction functions after predicting a reasonable answer. Use estimation strategies in computation and problem solving that involve numbers through thousands. 3.0 The student models and solves simple problems involving multiplication and division. The student will: 3.1 Example: Use repeated addition, arrays, counting by multiples to do multiplication. Draw a simple picture of seating 30 people in rows of 10. Show and explain how this is related to multiplication. Do this also for rows of 3, and again for rows of 5. 3.2 Use repeated subtraction, equal sharing, and forming equal groups to do division with remainders. Example: Kim decides to store away his marbles. He knows there are bags that hold up to 10 marbles in each. Kim has 38 marbles, and he tries to spend money on as few bags as he can. How many bags does he have to buy? How many if he has 51 marbles? (Keep in mind that there is no such thing as ―half a bag‖ or ―part of a bag.‖) 3.3 Know the multiplication tables of 2s, 5s, and 10s (to "times 10") and commit to memory. PAGE 18 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 4.0 The student understands that fractions and decimals can refer to parts of a set and to parts of a whole. The student will: 4.1 Example: Recognize and compare unit fractions up from 1/12 to 1/2. True or false? 1. One-fourth of a pie is larger than one-sixth of a pie. 2. 1/4 > 1/3 3. 1/7 < 1/9 4.2 4.3 Recognize fractions of a whole and parts of a group (e.g., 1/4th of a pie, 2/3rds of 15 balls). Know that when all fractional parts are included, such as four-fourths, the result is equal to the whole and to one. 5.0 The student models and solves problems by representing, adding, and subtracting amounts of money. The student will: 5.1 Example: Example: Solve problems using combinations of coins and bills Lee has a wallet with 5 nickels, 9 dimes, and a dollar bill. In how many ways can he pay with correct change for a pen worth $1.15? What about one worth 65 cents? 5.2 Know the decimal notation and the dollar and cents symbols for money. Which of the following shows a correct use of symbols for money? 1. 32 2. 72 3. $1.25 4. 2.57$ The student uses estimation strategies in computation and problem solving that involve numbers that use the ones, tens, hundreds, and thousands places. 6.0 The student will: 6.1 Understand when an estimate is reasonable in measurement (e.g., closest inch). PAGE 19 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS ALGEBRA AND FUNCTIONS 1.0 The student models, represents, and interprets number relationships to create and solve problems involving addition and subtraction. The student will: 1.1 1.2 Example: Use the commutative and associative rules to simplify mental calculations and check results. Use number sentences to model problems involving addition and subtraction. Three classes at your school will see a play together in a large room. Room 1 has 18 students, Room 2 has 34 students, and Room 3 has 19 students. Figure out how many seats you will need. If Room 2 drops out but Room 4 with 29 students joins in, how many seats will you need then? 1.3 1.4 Solve addition and subtraction problems using data from simple charts, picture graphs, and number sentences. Understand missing addends with and without manipulatives. MEASUREMENT AND GEOMETRY 1.0 The student understands that measurement is accomplished by identifying a unit of measure, iterating (repeating) that unit, and comparing it to the item to be measured. The student will: 1.1 Measure the length of objects by iterating (repeating) using non-standard and standard units. Example: Four children measured the width of a room by counting how many paces it took them to cross it. It took Ana 9 paces, Erlane 8, Stephen 10, and Carlos 7. Who had the longest pace? (Adapted from TIMSS, gr. 4, L-8; gr. 8, L-12). 1.2 Use different units to measure the same object and predict whether the measure will be greater or smaller when a different unit is used. Example: Measure the length of your desk with a new crayon and with a new pencil. Which is greater, the number of crayon units or the number of pencil units? PAGE 20 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 1.3 1.4 1.5 1.6 1.7a 1.7b Measure the length of an object to the nearest inch and/or centimeter. Know how to use standard units and tools. Understand measurements of volume and temperature. Select appropriate units of measure. Tell time to the nearest five minutes. Know time relationships (e.g., minutes in an hour, days in a month, weeks in year). Example: Which is a longer period: 3 weeks or 19 days? 27 days or 4 weeks? 1.8 1.9 Know how to determine the duration of time intervals in hours (e.g., 11:00 a.m. to 4:00 p.m.). Understand when an estimate is reasonable in measurements (e.g., closest inch). 2.0 The student knows the elements that compose common figures in the plane and common objects in space. The student will: 2.1 2.2 2.3 2.4 2.5 2.6 Describe and classify plane and solid geometric shapes (e.g., circle, triangle, square, rectangle, sphere, pyramid, cube, rectangular prism) according to the number and shape of faces, edges, and vertices. Put shapes together and take them apart to form other shapes (e.g., two congruent right triangles can form a rectangle). Know how to manipulate and construct models of plane and solid regular figures. Understand the concept of symmetry. Understand reflections of objects. Understand and perform transformations of objects (slides, flips, turns, rotations). STATISTICS, DATA ANALYSIS AND PROBABILITY 1.0 The student collects, records, organizes, displays, and interprets numerical data on bar graphs and other representations. The student will: 1.1 1.2 Record numerical data in systematic ways, keeping track of what/who has been counted. Understand that the same data can be represented in more than one way (e.g., charts with tallies, and bar graphs). PAGE 21 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 1.3 1.4 1.5 1.6 1.7 Know the features of data sets (range and mode). Ask and answer simple questions related to data representations. Understand why objects do not belong to a set. Use vocabulary appropriate to describe sets (all some, none, every, not, many). Develop, record, and communicate classification systems. 2.0 The student understands patterns and how they grow. The student will: 2.1 Understand how to get the next term in linear patterns (e.g., 4, 8, 12; the number of ears on 1 horse, 2 horses, 3 horses, 4 horses). Example: If there are two horses on a farm, how many horseshoes will we need to shoe all the horses? Show, in an organized way, how many horseshoes we will need for 3, 4, 5, 6, 7, 8, 9, and 10 horses. 2.2 Solve problems involving simple patterns (e.g., number, geometric, growth). 3.0 The student explores concepts of chance (spinner and coin activities, number cubes). The student will: 3.1 3.2 Record numerical data in systematic ways, keeping track of what numbers have been spun, rolled, or flipped. Analyze their data in terms of any emerging patterns. MATHEMATICAL REASONING 1.0 The student makes decisions about how to set up a problem. The student will: 1.1 1.2 2.0 Select the appropriate approach, materials, and strategies to use. Use tools such as manipulatives, calculators, or sketches to model problems. The student solves problems and justifies his or her reasoning, both orally and in writing. The student will: 2.1 Use reasoning and justify the procedures selected. PAGE 22 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 2.2 Make precise calculations and check the validity of the results from the context of the problem. 3.0 The student understands connections between one problem and another. The student will: 3.1 Use variables to investigate and model concrete situations. PAGE 23 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS GRADE 3 By the end of third grade, students deepen their understanding of place value and their understanding of and skill with addition, subtraction, multiplication and division of whole numbers. They estimate, measure and describe objects in space. They use patterns to help solve problems. They represent number relationships and conduct simple probability experiments. Throughout this document denotes specific standards from the Mathematics Framework for California Public Schools Throughout this document denotes the most important standards at each grade level All other items are District Standards NUMBER SENSE 1.0 The student understands place value of whole numbers. The student will: 1.1 Example: 2. Count, read, and write whole numbers to 10,000. What is the smallest whole number you can make using the digits 4, 3, 9, and 1? Use each digit exactly once (Adapted from TIMSS gr. 4, T-2). 1.2 1.3 1.4 1.5 Compare and order whole numbers to 10,000. Know the place name and value for each digit in numbers to 10,000. Round off numbers to 10,000 to the nearest ten, hundred and thousand. Use expanded notation to represent numbers (e.g., 3,206 = 3,000 + 200 +6). The student calculates and solves problems involving addition, subtraction, multiplication and division. The student will: 2.1 Example: Add and subtract whole numbers between 0 and 10,000. To prepare for recycling on Monday, Michael collected all the bottles in the house. He found 5 dark green ones, 8 clear ones with liquid still in them, 11 brown ones that used to hold root beer, 2 still with the cap on from his parents’ cooking needs, and 4 more that were over-sized. How many bottles did Michael collect? (This problem also support Mathematical Reasoning Standard 1.1). PAGE 24 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 2.2 2.3 2.4 2.5 Understand multiplication concepts and construct algorithms for multiplication facts to 10 using concrete materials. Know the multiplication table for numbers between 1 and 10. Use the inverse relationship of multiplication and division to compute and check results. Solve simple problems involving multiplication of multi-digit numbers by one-digit numbers (3,671 x 3 = __). Example: Example: A price list in a store states: pen sets, $3; magnets, $4; sticker sets, $6. How much would it cost to buy 5 pen sets, 7 magnets, and 8 sticker sets? 2.6 2.7 Solve division problems in which a multi-digit number is evenly divided by a one-digit number (135 5 = _____ ). Understand the special properties of 0 and 1 in multiplication and division. True or false? 1. 24 x 0 = 24 2. 19 1 = 19 3. 63 x 1 = 63 4. 0 0 = 1 2.8 2.9 Understand how to determine the unit cost when given the total cost and number of units (e.g., five erasers cost 40, how much for one eraser?). Solve problems that require multiple operations and procedures. Example: Example: A tree was planted 54 years before 1961. How old is the tree in 1998? A class of 73 students go on a field trip. The school hires vans, each of which can seat a maximum of 10 students. The school policy is to seat as many students as possible in a van before using the next one. How many vans are needed? 2.10 2.11 Generate and solve word problems using a variety of strategies (e.g., draw a picture, make a list, work backwards, guess and check). Understand different ways an operation can be used (e.g., multiplication as repeated addition, or combining equal groups (e.g., 5 + 5 + 5 = 15, 5 x 3 = 15). Use estimation and mental computation strategies to find sums and differences 2.12 3.0 The student understands the relationship between whole numbers, simple fractions and decimals. PAGE 25 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS The student will: 3.1a Understand the equivalency of fractions represented by drawings or concrete materials. Example: Which is longer, 1/3 of a foot or 5 inches? 2/3 of a foot or 9 inches? 3.1b Add and subtract simple fractions in context. Example: 1/2 of a pizza is the same amount as 2/4 of another pizza that is the same size; show that 3/8 is more than 1/8. 3.2 3.3 Add and subtract simple fractions (e.g., determine that 1/8 + 3/8 is the same as 1/2). Solve problems involving addition, subtraction, multiplication, and division of money amounts using decimal notation. Example: Pedro bought 5 pens, 2 erasers and 2 boxes of crayons. The pens cost 65 cents each, the erasers 25 cents each, and a box of crayons $1.10. The prices include tax, and Pedro paid with a ten-dollar bill. How much change did he get back? 3.4 3.5 Solve problems involving counting back change and converting dollars to cents (e.g., sharing $3.00 between two people). Understand that fractions and decimals are two different representations of the same concept (e.g., 50 cents is 1/2 of a dollar, 75 cents is 3/4 of a dollar). ALGEBRA AND FUNCTIONS 1.0 The student selects appropriate symbols, operations and properties to represent, describe, simplify and solve simple number relationships. The student will: 1.1 1.2 1.3 1.4 Use mathematical expressions, equations, or inequalities to represent relationships of quantities (e.g., write a number sentence that tells how many wheels are on six tricycles). Solve problems involving numeric equations or inequalities (e.g., what number goes in the box, 8 + = 15; what symbol goes in the box, 8 + 4 7 + 6). Select appropriate operational and relational symbols to make an expression true (e.g., 4 __ 3 = 12, what operation symbol goes in the blank?). Express simple unit conversions in symbolic form (e.g., # inches = # feet x 12) PAGE 26 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 1.5 Understand the commutative and associative properties of multiplication (e.g., if 5 x 7 = 35, then what is 7 x 5?, if 5 x 7 x 3 = 105, then what is 7 x 3 x 5?).. 2.0 The student understands simple functional relationships. The student will: 2.1 Solve simple problems involving a functional relationship between two quantities (e.g., find the total cost of multiple items given the per unit cost). Example: John wants to buy a dozen pencils. One store offers pencils at 6 for $1. Another offers them at 4 for 65 cents. Yet another sells pencils at 15 cents each. Where should John purchase his pencils in order to save the most money? 2.2 Extend and recognize a linear pattern by its rules (e.g., the number of legs on a given number of horses can be calculated by counting by 4s or by multiplying the number of horses by 4). Example: Here is the beginning of a pattern of tiles. Assuming that the pattern continues linearly, how many tiles will be in the sixth figure? (Adapted from TIMSS gr. 4, K-6) MEASUREMENT AND GEOMETRY 1.0 The student selects and uses appropriate units and measurement tools to quantify the properties of objects. The student will: 1.1 1.2 Measure time in one minute increments. Select appropriate units (metric and U.S. customary, standard and nonstandard) and tools to estimate and measure length, liquid volume, weight/mass, and temperature. PAGE 27 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 1.3 Estimate or determine the area and volume of solid figures by covering them with squares or by counting the number of cubes that would fill them. Example: Which rectangles is NOT divided into four equal parts? (Adapted from TIMSS fr. 4, K-8) 1.4 1.5 1.6 Find the perimeter of a polygon with integer sides. Perform simple unit conversions within a system of measurement (e.g., centimeters and meters, hours and minutes). Know basic characteristics of the coordinate graphing system. 2.0 The student understands and compares the attributes of plane and solid geometric figures and uses his or her understanding to show relationships and solve problems. The student will: 2.1 2.2 2.3 2.4 2.5 2.6 Use concrete models to determine area and perimeter. Know the defining properties of polygons (including pentagons, hexagons and octagons). Know how to locate lines of symmetry. Know the attributes of triangles (e.g., two equal sides for the isosceles triangle, three equal sides for the equilateral triangle, right angle for the right triangle). Know the attributes of quadrilaterals (e.g., parallel sides for the parallelogram, right angles for the rectangle, equal sides and right angles for the square). Understand the characteristics of angles (e.g., right, acute, obtuse). PAGE 28 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS Example: Which of the following triangles include an angle that is greater than a right angle? 2.7 2.8 Understand the properties of common three-dimensional geometric objects (e.g., cube, rectangular solid, sphere, prism, pyramid, cone, cylinder). Know the common solid objects that are the component parts needed to make a more complex solid object. STATISTICS, DATA ANALYSIS and PROBABILITY 1.0 The student classifies objects and summarizes data. The student will: 1.1 1.2 Example: Use three overlapping attributes to classify objects. Add elements to sets. Students label a 2-way Venn diagram. They find objects to fit the labeled areas of the diagram. 1.3 1.4 1.5 Use appropriate vocabulary to describe sets created (all, some, none, most, both, at least, every, not, many, or). Compose and solve problems that require elimination of possibilities. Record and communicate classification systems used. 2.0 The student conducts simple probability experiments by determining the number of possible outcomes and makes simple predictions. The student will: PAGE 29 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 2.1 Understand that common events can be certain, likely, unlikely, or improbable. Example: Are any of the following certain, likely, or impossible? 1. Take two cubes each with the numbers 1, 2, 3, 4, 5, 6 written on its six faces. Throw them at random, and the sum of the numbers on the top faces is 12. 2. It snows on New Year’s day. 3. A baseball game is played somewhere in this country on any Sunday in July. 4. It is sunny in June. 5. Pick any two one-digit numbers, and their sum is 17. 2.2 Record the possible outcomes for a simple event (e.g., tossing a coin) and systematically keep track of the outcomes when the event is repeated many times. Summarize and display the results of probability experiments in a clear and organized way (e.g., use a bar graph or a line plot). Use the results of probability experiments to predict future events (e.g., use a line plot to predict the temperature forecast for the next day). 2.3 2.4 MATHEMATICAL REASONING 1.0 The student makes decisions about how to approach problems. The student will: 1.1 2.0 1.2 Use a variety of strategies to analyze problems (e.g., identifying relationships, discriminating relevant from irrelevant information, sequencing and prioritizing information, observing patterns). Understand when and how to break a problem into simpler parts. The student uses and communicates (orally and in writing) strategies, skills and concepts in finding solutions. The student will: 2.1 2.2 2.3 2.4 Use estimation to verify the reasonableness of calculated results. Understand strategies used to apply the results from simpler problems to more complex problems. Use a variety of methods (e.g., words, numbers, symbols, charts, graphs, tables, diagrams, models) and tools e.g., calculators) to explain mathematical reasoning. Use appropriate mathematical notation, terms, and clear language to express solutions clearly and logically and to support solutions with evidence in both verbal and symbolic forms. PAGE 30 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 2.5 2.6 Understand the relative advantages of exact and approximate solutions to problems and give answers to a specified degree of accuracy. Make precise calculations and check the validity of the results from the context of the problem. 3.0 The student generalizes from particular problems to other situations. The student will: 3.1 3.2 3.3 Evaluate the reasonableness of the solution in the context of the original situation. Understand methods of deriving solutions and demonstrate conceptual understanding of the derivations by solving similar problems. Generalize from the results obtained and extend them to other circumstances. PAGE 31 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS GRADE 4 By the end of fourth grade, students understand large numbers and addition, subtraction, multiplication and division of whole numbers. They describe and compare simple fractions and decimals. They understand the properties of and the relationships between plane geometric figures. They collect, represent and analyze data to answer questions. Throughout this document denotes specific standards from the Mathematics Framework for California Public Schools Throughout this document denotes the most important standards at each grade level All other items are District Standards NUMBER SENSE 1.0 The student understands place value of whole numbers and decimals to two decimal places, how these relate to simple fractions, and uses concepts of negative numbers. The student will: 1.1a 1.1b 1.2a 1.2b 1.3 1.4 Read and write whole numbers in the millions. Know place value names from thousandths to millions. Order and compare whole numbers and decimals to two decimal places. Use objects to representdecimals (e.g., money, base ten blocks). Round whole numbers through the millions to the nearest ten, hundred, thousand, ten thousand, or hundred thousand. Understand when a rounded solution is called for, and explain why such a solution may be appropriate. Example: Solve each of the following problems and observe the different roles played by the number 37 in each situation: 1. 2. 3. Four children shared 37 dollars equally. How much did each get? Four children shared 37 pennies as equally as possible. How many pennies did each get? Cars need to be rented for 37 children going on a field trip. Each car can take 12 children in addition to the driver. How many cars must be rented? Understand different meanings for fractions (e.g., parts of a whole, parts of a set). PAGE 32 1.5a FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 1.5b 1.6a 1.6b Relate to whole numbers and simple decimals on a number line. Write tenths and hundredths in decimal and fraction notation. Know fraction and decimal equivalents for halves and fourths (e.g., 1/2 = 0.5 or .50; 7/4 = 1 3/4 = 1.75). Example: True or false? 1. 1/4 >2.54 2. 5/2 < 2.6 3. 12/18 = 2/3 (Note the equivalence of fractions.) 4. 4/5 < 13/15 1.7 Understand the equivalent representations of fractions and drawings of parts of a figure. Example: Which number represents the shaded part of the figure? (Adapted from TIMSS gr. 4, M-5) 1. 2. 3. 4. 2.8 0.5 0.2 0.02 1.8 Understand concepts of negative numbers (e.g., on a number line, in counting, in temperature, "owing"). Example: True or false? 1. –9 > -10 2. –31 < -29 1.9 Know the relative position of fractions, mixed numbers, and decimals to two decimal places on the number line. 2.0 The student extends his or her understanding of whole numbers to addition and subtraction of simple decimals. The student will: 2.1 Estimate and compute the sum or difference of whole numbers and positive decimals to two places. Example: Solve 17.91 + 2.18 = ? PAGE 33 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS Example: Solve 55.73 – 48.25 = ? 2.2a 2.2b Round two place decimals to one decimal or the nearest whole number. Use rounding to judge the reasonableness of an answer. 3.0 The student solves problems involving addition, subtraction, multiplication, and division of whole numbers (including the addition and subtraction of negative numbers), and understands the relationships among the operations. The student will: 3.1 3.2 3.3 3.4a 3.4b 3.4c 3.5 3.6 Understand operations and the relationships between them. Use estimation and mental computation strategies to find sums, differences, and products. Understand and use standard algorithms for addition and subtraction of multidigit numbers. Understand and use standard algorithms for multiplying a multi-digit number by a two digit number. Understand and use standard algorithms for dividing a multi-digit number by a one digit number. Use relationships between multiplication and division to simplify computations and to check results. Solve problems involving multiplication of multi-digit numbers by two-digit numbers. Solve problems involving division of multi-digit numbers by one-digit numbers. 4.0 The student knows how to factor small whole numbers. The student will: 4.1 Understand that many whole numbers decompose in different ways (e.g., 12 = 4 x 3 = 2 x 6 = 2 x 2 x 3). Example: List all the distinct prime factors of 264. 4.2 Know that numbers such as 2, 3, 5, 7, 11 do not have any factors except 1 and themselves, and that such numbers are called prime numbers. ALGEBRA AND FUNCTIONS 1.0 The student uses variables, mathematical symbols, and properties to write and simplify expressions and sentences. PAGE 34 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS The student will: 1.1 Use letters, boxes, or other symbols to stand for any number in simple expressions or equations (e.g., demonstrate understanding and use of a concept of a variable). Example: Tanya has read the first 78 pages of a 130-page book. Give the number of the sentence that can be used to find the number of pages Tanya must read to finish the book. (Adapted from TIMSS gr. 4, I-7) 1. 130 + 78 = ______ 2. _____ - 78 = 130 3. 130 – 78 = _____ 4. 130 - _____ = 178 1.2 Interpret and evaluate mathematical expressions that use parentheses. Example: Evaluate the two expressions: (28 – 10) – 8 = _____ and 28 – (10 – 8) = _____. 1.3 Change the format of the equations in the problems, so that certain brackets and parentheses are eliminated. Should look more like this: 24 + 8 6 =? Example: Solve (3 X 12) - Solve 18 + 31 + 5 7 1.4 X 9 = ? Use and interpret formulas (e.g., Area = length times width or A = lw) to answer questions about quantities and their relationships. Example: There are many rules to get from Column A to Column B in the following table. Can you state one rule? (Adapted from TIMSS, gr 4, J-5) Column A 10 15 45 50 Column B 2 3 9 10 1.5 Understand that an equation such as y = 3x + 5 is a prescription for determining a second number when a first number is given; generate a solution table. Y X 8 35 1 PAGE 11 2 ? 3 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 2.0 The student knows how to manipulate equations. The student will: 2.1 2.2 Understand that equals added to equals are equal. Understand that equals multiplied by equals are equal. MEASUREMENT AND GEOMETRY 1.0 The student selects and uses appropriate units and measurement tools to quantify the properties of objects. The student will: 1.1 1.2 1.3 2.0 Use standard measurements (metric and U.S. customary) and tools to estimate and measure length, volume, weight/mass, and temperature. Select and use appropriate units of measurement, according to type and size of unit. Know the relative sizes of units of measure (e.g., inch, foot, and mile). The student understands perimeter and area. The student will: 2.1a 2.1b Use appropriate units (e.g., square centimeter, square meter, square kilometer, square inches, square yard, square mile) to measure the area of rectangular shapes. Determine and compare areas and perimeters of rectangles and non rectangular shapes. Example: Is the area of a 45 X 55 rectangle (in cm2) smaller or bigger than that of a square with the same perimeter? 2.2 Know that rectangles having the same area can have different perimeters. Example: Draw a rectangle whose area is 120 and whose perimeter exceeds 50. Draw another rectangle with the same area whose perimeter exceeds 400. PAGE 36 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS Example: 2.3 Know that rectangles having the same perimeter can have different areas. Draw a rectangle whose perimeter is 40 and whose area is less than 100. 2.4a Understand and use formulas to solve problems involving perimeters and areas of rectangles and squares. Use the area formulas of rectangles and squares to find the areas of more complex figures by dividing them into parts with these basic shapes. 2.4b Example: The length of a rectangle is 6 cm, and its perimeter is 16 cm. What is the area of the rectangle in square centimeters? (Adapted from TIMSS gr. 8, K-5) The student uses two-dimensional coordinate grids to represent points and to graph lines and simple figures. 3.0 The student will: 3.1 3.2 Use graphs to model linear relationships. (e.g., draw the first ten points for the equation y = 3x and connect them using a straight line). Understand that the length of a horizontal line segment equals the difference of the x-coordinates. Example: What is the length of the line segment joining the points (6, -4) and (21, -4)? 3.3 Understand that the length of a vertical line segment equals the difference of the y-coordinates. Example: 4.0 What is the length of the line segment joining the points (121, 3) to (121, 17)? The student understands plane and solid geometric objects and uses this knowledge to show relationships and solve problems. The student will: 4.1 4.2 4.3 4.4 Understand the characteristics of lines (e.g., parallel, perpendicular). Know measurements used to describe circles (e.g., radius, diameter). Understand that figures can be congruent. Understand bilateral and rotational symmetry. Example: Let AB, CD be perpendicular diameters of a circle, as shown. If we reflect across the line segment CD, what happens to A and what happens to B under this reflection? PAGE 37 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS Example: Craig folded a piece of paper in half and cut out a shape along the folded edge. Draw a picture to show what the cutout shape will look like when it is opened up and flattened out (Adapted from TIMSS gr. 4, T-5). 4.5a 4.5b 4.6a 4.6b 4.6c Know the definitions of right angle, acute angles and obtuse angle. Understand that 90, 180, 270, and 360 degrees are, respectively, associated with 1/4, 1/2, 3/4 and full turns. Understand the properties of geometric solids (e.g., prisms, pyramids, etc.) in terms of the number and shape of faces, edges and vertices. Understand two-dimensional representations of three-dimensional objects. Know how to draw patterns for a solid that when folded will make a model of the solid. Know the definitions of different triangles (e.g., equilateral, isosceles, scalene) and identify their features Know the definition of different quadrilaterals (e.g., rhombus, square, rectangle, parallelogram, trapezoid). 4.7 4.8 STATISTICS, DATA ANALYSIS and PROBABILITY 1.0 The student uses a variety of systems to classify objects and summarize data. The student will: 1.1 1.2 2.0 Use a variety of classification systems to classify objects. Record and communicate classification systems. The student organizes, represents, and interprets numerical and categorical data and clearly communicates their findings. The student will: 2.1 Formulate survey questions, systematically collect and represent data (e.g., using number lines, coordinate graphs, tables and charts, and interpret results. PAGE 38 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 2.2 2.3 Understand the measures of central tendency (e.g., mode, median, outliers). Interpret one- and two-variable data graphs to answer questions about a situation. 3.0 The student makes predictions for simple probability situations. The student will: 3.1 Represent all possible outcomes for a simple probability situation in an organized way (e.g., tables, grids, tree diagrams). Example: Nine identical chips numbered 1 through 9 are put in a jar. When a chip is drawn from the jar, what is the probability that is has an even number? (Adapted from TIMSS gr. 8, N-18). 3.2 3.3 Use verbal and numerical forms to express outcomes and probability (e.g., 3 out of 4; 3/4). Determine fairness of games by investigating the probability of winning. MATHEMATICAL REASONING 1.0 The student makes decisions about how to approach problems. The student will: 1.1 1.2 1.3 1.4 Understand a variety of strategies to analyze problems (e.g., identifying relationships, discriminating relevant from irrelevant information, sequencing and prioritizing information, observing patterns). Know when and how to break a problem into simpler parts. Solve analogies. Solve word problems that require reasoning skills. 2.0 The student uses strategies, skills and concepts in finding solutions and to justify reasoning orally and in writing. The student will: 2.1 2.2 2.3 Use estimation to verify the reasonableness of calculated results. Understand how to apply strategies and results from simpler problems to more complex problems. Use a variety of methods (e.g., words, numbers, symbols, charts, graphs, tables, diagrams, models) and tools (i.e., calculators, manipulatives) to explain mathematical reasoning. PAGE 39 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 2.4 2.5a 2.5b 2.6 Use appropriate mathematical notation, terms, and clear language to express a solution clearly and logically and provide supports with evidence in both verbal and symbolic forms. Understand the relative advantages of exact and approximate solutions to problems and give answers to a specified degree of accuracy. Understand the properties of operations (e.g., associative property, commutative property) and the relationships between them. Make precise calculations and check the validity of the results from the context 3.0 The student moves beyond a particular problem by generating problems and generalizing to other situations. The student will: 3.1 3.2 3.3 3.4 Evaluate the reasonableness of the solution in the context of the original situation. Understand how to use a similar problem type to solve a problem. Understand how to generalize from results obtained to extend them to other circumstances. Use multiple approaches to solve problems. PAGE 40 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS GRADE 5 By the end of fifth grade, students increase their facility with the four basic arithmetic operations applied to positive and negative numbers, fractions and decimals. They know and use common measuring units to determine length and area; they know and use formulas to determine the volume of simple geometric figures. Students know the concept of angle measurement and use a protractor and compass in solving problems. They use grids, tables, graphs, and charts to record and analyze data. Throughout this document denotes specific standards from the Mathematics Framework for California Public Schools Throughout this document denotes the most important standards at each grade level All other items are District Standards NUMBER SENSE 1.0 The student computes with very large and very small numbers, positive integers, decimals, and fractions and understands the relationships between decimals, fractions and percents. The student will: 1.1 Example: 1.2 Estimate, round, and manipulate very large (e.g., millions) and very small (e.g., thousandths) numbers. Understand equivalent forms of percents, decimals, and fractions. A test had 48 problems. Joe got 42 correct. 1. What percent were correct? 2. What percent were wrong? 3. If Moe got 93.75% correct, how many problems did he get correct? 1.3 1.4 Understand and compute positive integer powers of non-negative integers. Understand basic number theory concepts (prime numbers, composite numbers, multiples, factors, exponents). Example: List all the factors of 48. List all the factors of 36. List the common factors. PAGE 41 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS Example: Extend the tables shown below: 24 = 16 104 = 10,000 23 = 8 103 = 1,000 22 = 4 102 = 100 21 = ? 101 = ? 20 = ? 100 = ? 1.5 1.6 Use models (e.g., number line, grid, array) to represent positive and negative integers, decimals, fractions and mixed numbers. Construct algorithms and compute with whole numbers and decimals (2-3 digit operations with =, -, x, -). 2.0 The student performs calculations and solves problems involving addition, subtraction, and simple multiplication and division of fractions and decimals. The student will: 2.1 Add, subtract, multiply, and divide with decimals and negative integers and verify the reasonableness of the results. Example: Determine the following numbers: 1. 11 + (-23) 2. (-15) - 128 3. (-27) + (-45) Example: Example: 2.2 Divide with positive decimals multiple digit divisors. Find the quotient: 6 divided by .025 2.3 Solve problems involving the addition, subtraction, and equivalence of fractions and mixed numbers (like and unlike denominators of 20 or less) and express answers in simplest form. Understand the concept of multiplication and division of fractions. 2.4 Given the following three pairs of fractions (3/8 and 1/6, 5 3/4 and 2 1/3, 16 and 12 7/8), find for each pair its: 1. Sum 2. Difference 3. Product 4. Quotient in simplest terms PAGE 42 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 2.5 2.6 Solve problems involving the multiplication and division of fractions. Understand the properties of operations and the use of the parenthesis. ALGEBRA AND FUNCTIONS 1.0 The student uses variables in simple expressions, computes the value of the expression for specific values of the variable, and plots and interprets the results. The student will: 1.1 Use information taken from a graph or equation to answer questions about a problem situation. Example: Joe’s sister Mary is twice as old as he is. Mary is 16. How old is Joe? 1.2a 1.2b Use a letter to represent an unknown number in algebraic expressions. Solve simple algebraic expressions in one variable by substitution. Example: Example: 3x + 2 = 14. What is x ? 1.3 1.4 Understand the distributive property in equations and expressions with variables. Identify and graph ordered pairs in the four quadrants of the coordinate plane. Plot the points (1, 2), (-4, -3), (12, -1), (0, 4), (-4, 0). 1.5 Solve problems involving linear functions with integer values, write the equation, and graph the resulting ordered pairs of integers on a grid. MEASUREMENT AND GEOMETRY 1.0 The student understands and computes volumes and areas of simple objects. The student will: 1.1 Derive and use the formula for the area of a triangle and of a parallelogram by comparing it with the formula for the area of a rectangle. PAGE 43 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS Example: Find the area and perimeter. 1.2 1.3 1.4 1.5 Use two-dimensional patterns to construct cube and rectangular boxes and to compute the surface area for these objects. Understand the concept of volume and use the appropriate units in common measuring systems (cubic centimeter, cubic meter, cubic inches, cubic yard) to compute the volume of rectangular solids. Know appropriate units of measure for, two- and three-dimensional objects (perimeter, area and volume). Understand the relationships among lengths, areas, and volumes of similar solids. 2.0 The student knows the properties of, and relationships between, plane and solid geometric figures. The student will: 2.1 2.2 2.3 2.4 Use appropriate tools (e.g., straight edge, ruler, compass, protractor and drawing software) to measure, identify and draw angles, perpendicular and parallel lines, rectangles and triangles. Know that the sum of the angles of any triangle is 180o and the sum of the angles of any quadrilateral is 3600 and use this information to solve problems. Visualize, draw, and describe two-dimensional views of three-dimensional objects made from rectangular solids. Understand the concept of tessellation (i.e., a repetitive pattern of polygons that fit together with no gaps or holes). PAGE 44 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 3.0 The student selects and uses appropriate units and measurement tools to quantify the properties of objects. The student will: 3.1 3.2 3.3 3.4 3.5 4.0 Use standard measurement units and tools. Select appropriate units of measure. Compare units of measurement. Judge the reasonableness of measurement findings. Understand the accuracy needed for varying tasks. The student will explore concept of scale using coordinate graphing system. The student will: 4.1 Plot on a coordinate graph similar geometric shapes in different sizes with correct proportion. STATISTICS, DATA ANALYSIS and PROBABILITY 1.0 The student displays, analyzes, compares and interprets different data sets, including data sets that are not the same size, and data from the media. The student will: 1.1 1.2 1.3 1.4 1.5 1.6 Know the concepts of mean, median, and mode; compute and compare them in simple examples and understand that they can differ. Use appropriate graphs and representations (e.g., histogram, circle graphs) to organize and display single-variable data in and explain which types of graphs are appropriate for different kinds of data sets. Use fractions and percentages to compare data sets of different size. Understand graphical representations of ordered pairs of data and interpret the meaning of the data in terms of the situation depicted by the graph. Know how to write ordered pairs correctly (e.g., (x, y)). Use sampling experiments to compare results from a number of samples with the entire population. 2.0 The students uses a variety of systems to classify objects and summarize data. The student will: 2.1 2.2 2.3 Record and communicate classification systems. Use appropriate vocabulary to describe sets (intersection, union, subset). Use analogies to solve problems. PAGE 45 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 3.0 The student makes predictions for probability situations. The student will: 3.1 3.2 3.3 Understand how chance and skills are involved in various games. Understand how changing the rules in games can effect the outcome of the game. Estimate the likelihood of future events using a scale of 0% to 100%. MATHEMATICAL REASONING 1.0 The student makes decisions about how to approach problems. The student will: 1.1 1.2 1.3 Use a variety of strategies to analyze problems (e.g., identifying relationships, discriminating relevant from irrelevant information, sequencing and prioritizing, observing patterns. Understand when and how to break a problem into simpler parts. Compose and solve riddles and word problems that require combining and reorganizing clues. 2.0 The student uses strategies, skills and concepts to find solutions and justify reasoning, both orally and in writing. The student will: 2.1 2.2 2.3 2.4 2.5 2.6 Use estimation to verify the reasonableness of calculated results. Apply strategies and results from simpler problems to more complex problems. Use a variety of methods (e.g., words, numbers, symbols, charts, graphs, tables, diagrams, models, and tools (e.g., calculators, manipulatives) to explain mathematical reasoning. Use appropriate mathematical notation, terms, and clear language to express a solution clearly and logically and provide support with evidence in both verbal and symbolic forms. Understand the relative advantages of exact and approximate solutions to problems and give answers to a specified degree of accuracy. Make precise calculations and check the validity of the results from the context of the problem. 3.0 The student moves beyond a particular problem by generalizing to other situations. PAGE 46 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS The student will: 3.1 3.2 3.3 Evaluate the reasonableness of the solution in the context of the original situation. Understand how to use a similar problem type to solve a problem. Understand how to generalize from results obtained to extend them to other circumstances. PAGE 47 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS GRADE 6 By the end of sixth grade, students have mastered the four arithmetic operations with positive and negative numbers, whole numbers, fractions and decimals; they accurately compute and solve problems. They apply their knowledge to statistics and probability. Students understand the concept of and how to calculate the range, mean, median and mode of data sets. They analyze data and sampling processes for possible bias and misleading conclusions, and they use addition and multiplication of fractions routinely to calculate probabilities for compound events. Students conceptually understand and work with ratios and proportions; they compute percentages (e.g., tax, tips, interest). Students know about and the formulas for the circumference and area of a circle. They use letters for numbers in formulas involving geometric shapes and in representing an unknown part of a ratio. They solve 1-step linear equations. Throughout this document denotes specific standards from the Mathematics Framework for California Public Schools Throughout this document denotes the most important standards at each grade level All other items are District Standards NUMBER SENSE 1.0 The student compares and orders integers, fractions, decimals, and mixed numbers. The student will: 1.1 1.2 1.3 1.4 2.0 Estimate using whole numbers and integers. Use number lines to compare and order positive and negative fractions, decimals, and mixed numbers. Understand the equivalency of common fractions, decimals, and percents (e.g., ¼ = .25 = 25%). Understand and write repeating decimals (e.g., .6666....). The student solves problems involving fractions, ratios, proportion, and percentages. The student will: 2.1 2.2a 2.2b 2.3a 2.3b Understand the estimation of percentages. Calculate given percentages of quantities. Solve problems involving discounts at sales, interest earned, and tips. 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 proportion problems. PAGE 48 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 2.3c 2.4 2.5 3.0 Understand cross-multiplication as multiplication of both sides of an equation by a multiplicative inverse. Understand and use ratios in different contexts (e.g., batting averages, miles per hour) and use appropriate notations (a/b, a to b, a:b). Solve problems using ratios, proportions, and percents. The student calculates and solves problems involving addition, subtraction, multiplication and division of rational numbers. The student will: 3.1 3.2 3.3 3.4 Know rules of divisibility for whole numbers (e.g., all even numbers are divisible by 2). Solve problems involving addition, subtraction, multiplication, and division of fractions and explain why a particular operation was used for a given situation. Understand the multiplication and division of fractions (e.g., 5/8 divided by 15/16 = 5/8 x 16/15 = 2/3). Solve addition, subtraction, multiplication, and division problems that use positive and negative numbers and combinations of these operations. Example: The temperature increases 9o F and then drops 17o F. What is the temperature? 3.5 Know how to calculate the least common multiple and greatest common divisor of whole numbers and use them to solve problems with fractions (e.g., to find a common denominator in order to add two fractions or to find the reduced form for a fraction). ALGEBRA AND FUNCTIONS 1.0 The student writes verbal expressions and sentences as algebraic expressions and equations, and evaluates algebraic expressions, solves simple linear equations and graphs and interprets their results. The student will: 1.1 Example: Write and solve one-step linear equations with one variable 6y - 2 = 10. What is y ? 1.2 1.3 Write and evaluate an algebraic expression for a given situation using up to three variables. Use the algebraic order of operations and the commutative, associative and distributive properties to evaluate expressions and provide justifications for each step in the process. Solve problems that require using the correct order of operations manually and by using a scientific calculator. PAGE 49 1.4 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 2.0 The student analyzes and uses tables, graphs and rules to solve problems involving rates and proportions. The student will: 2.1 Example: 2.2 2.3 Know how to convert from one unit of measurement to another (e.g., from feet to miles, from centimeters to inches). Understand that rate is a measure of one quantity per unit value of another quantity (e.g., A student typed an 1,100 word essay in 50 minutes, or 22 words /minute). Solve problems involving rates, average speed, distance and time. If a family takes 4 hours to drive 200 miles on vacation, stops for a 1 hour lunch, then drives 160 more miles in 3 hours, what is their average rate of speed for the entire day? The student investigates geometric patterns and describes them algebraically. 3.0 The student will: 3.1 3.2 Use variables in formulas describing geometric quantities (e.g., P = 2w + 2l, A = 1/2 bh, C = pd). Understand expressions in symbolic form for simple relationships arising from geometry. MEASUREMENT AND GEOMETRY 1.0 The student understands the measurement of plane and solid shapes and use this understanding to solve problems. The student will: 1.1 1.2a 1.2b 1.3 1.4 Use the formulas for perimeter of triangles and quadrilaterals. Understand the concept of a constant number like . Know the formula for the circumference and area of a circle (c = x d, Area = r2). Know common estimates of (3.14; 22/7) and use these values to estimate and calculate the circumference and the area of circles and compare with actual measurements. Use the formulas for the volume of triangular prisms and cylinders (area of base x height) and understand the similarity between these formulas and the formula for the volume of a rectangular solid. 2.0 The student knows the properties of two-dimensional figures. PAGE 50 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS The student will: 2.1 2.2 2.3 Understand the characteristics of angles (e.g., vertical, adjacent, complementary, supplementary). Use the properties of angles (e.g., complimentary and supplementary angles, sum of the angles of a triangle equal 180 degrees) to solve problems involving an unknown angle. Use given information to draw quadrilaterals and triangles (e.g., a quadrilateral having equal sides but no right angles, a right isosceles triangle). 3.0 The student selects and uses appropriate units and measurement tools to quantify the properties of objects. The student will: 3.1 3.2 3.3 Use the appropriate metric and US customary units of length, mass, capacity, time, and temperature. Estimate units of measurement. Use measurement in problem solving situations. STATISTICS, DATA ANALYSIS and PROBABILITY 1.0 The student computes and analyzes statistical measurement for data sets. The student will: 1.1 1.2 1.3 1.4 1.5 Collect and represent data using tables, charts, line and bar graphs. Understand how to calculate the measures of central tendency (e.g., range, mean, median, mode). Understand how additional data added to data sets can effect these computations of measures of central tendency. Understand how the inclusion or exclusion of outliers affect measures of central tendency. Know why a specific measure of central tendency (mean, median, mode) provides the most useful information in a given context. 2.0 The student uses data samples of a population and understands the characteristics and limitations of the samples. The student will: 2.1 2.2 2.3 Understand when it makes sense to use a sample and compare different samples from a population with the data from the entire population. Know different ways of selecting a sample (e.g., convenience sampling, those who respond to a survey, random sampling) and which way makes a sample more representative for a population. Understand how the way a question was asked might have influenced the results obtained, and how the way the results were displayed might have influenced the conclusions reached. PAGE 51 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 2.4 2.5 Understand sampling and how samples may be biased. Identify claims based on statistical data and evaluate the validity of the claims. 3.0 The student determines theoretical and experimental probabilities and uses these to make predictions about events. The student will: 3.1 3.2 3.3a 3.3b 3.4a 3.4b 3.5 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. Use data to estimate the probability for future events (e.g., batting averages or number of accidents per mile driven). Represent probabilities as ratios, proportions, decimals, and percents and check that probabilities computed are reasonable. Know how probability is related to the probability of an event not occurring. Understand that the probability of either of two disjoint events occurring is the sum of the two individual probabilities. Understand that the probability of one event following another, in independent trials, is the product of the two probabilities. Understand the difference between independent and dependent events and how this affects the results for specific probability situations. MATHEMATICAL REASONING 1.0 The student makes decisions about how to approach problems. The student will: 1.1 2.0 1.2 1.3 Use a variety of strategies to analyze problems (e.g., identifying relationships, discriminating relevant from irrelevant information, identifying missing information, sequencing and prioritizing information, observing patterns). Formulate and justify mathematical conjectures based upon a general description of the mathematical question or problem posed. Understand when and how to break a problem into simpler parts. The student uses strategies, skills and concepts in finding solutions. The student will: 2.1 2.2 2.3 Use estimation to verify the reasonableness of calculated results. Use strategies and results from simpler problems to solve more complex problems. Estimate unknown quantities graphically and solve for them using logical reasoning and arithmetic and algebraic techniques. PAGE 52 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 2.4 2.5 2.6 2.7 Use a variety of methods (e.g., words, numbers, symbols, charts, graphs, tables, diagrams, models) and tools e.g., calculators and manipulatives) to explain mathematical reasoning. Express a solution clearly and logically using appropriate mathematical notation, terms, and clear language, and support solutions with evidence, in both verbal and symbolic forms. Understand the relative advantages of exact and approximate solutions to problems and give answers to a specified degree of accuracy. Make precise calculations and check the validity of the results from the context of the problem. 3.0 The student moves beyond a particular problem by generalizing to other situations. The student will: 3.1 3.2 3.3 Evaluate the reasonableness of the solution in the context of the original situation. Use derived solution methods to solve similar problems. Generalize from results obtained and the strategies used in order to extend them to new problem situations. PAGE 53 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS Mathematics Grades 7 - 12 Characteristics of Empowering Mathematics Programs All students fully participate. Students take responsibility for their learning: they actively question, create, and help decide what to do. Teachers are facilitators of learning rather than imparters of information. All students regularly use manipulatives, calculators, and computers. Assessment is integrated into instruction; it focuses on what students understanding and can do rather than on what they don’t know or can’t do. The program is appropriate to the maturity and development of the students as it meets its other goals. The program develops students’ positive disposition towards mathematics in several ways. The program usually introduces computational procedures only where students need them. The program is effective: students learn mathematics. PAGE 54 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS INTRODUCTION TO GRADES 7 THROUGH 12 The standards for grades seven through twelve are organized differently from those for kindergarten through grade six. In this section strands are not used for organizational purposes as they are in the elementary grades because the mathematics studied in grades seven through twelve falls naturally under discipline headings: Pre-Algebra, Algebra, Geometry, and so forth. Many schools teach this material in traditional courses; others teach it in an integrated fashion. To allow schools and teachers flexibility in teaching the material, the standards for grades eight through twelve do not mandate that a particular discipline be initiated and completed in a single grade. The core content of these subjects must be covered; students are expected to achieve the standards however these subjects are sequenced. Standards are provided for Pre-Algebra, Algebra I, Geometry, Algebra II, Trigonometry, Mathematical Analysis, Linear Algebra, Probability and Statistics, Advanced Placement Probability and Statistics, and Calculus. Many of the more advanced subjects are not taught in every middle school or high school. Moreover, schools have different ways of combining the subject matter in these various disciplines. Many combinations of these advanced subjects into courses are possible. For example, a Pre-Algebra course or an Algebra course may be taught over a one-year or a two-year period at the middle school and/or the high school level. What is described in this section are standards for the academic content by discipline; this document does not endorse a particular choice of structure for courses or a particular method of teaching the mathematical content. When students delve deeply into mathematics, they gain not only conceptual understanding of mathematical principles but also knowledge of, and experience with, pure reasoning. One of the most important goals of mathematics is to teach students logical reasoning. The logical reasoning inherent in the study of mathematics allows for applications to a broad range of situations in which answers to practical problems can be found with accuracy. PAGE 55 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS By grade eight, students’ mathematical sensitivity should be sharpened. Students need to start perceiving logical subtleties and appreciate the need for sound mathematical arguments before making conclusions. As students progress in the study of mathematics, they learn to distinguish between inductive and deductive reasoning; understand the meaning of logical implication; test general assertions; realize that one counterexample is enough to show that a general assertion is false; understand conceptually that although a general assertion is true in a few cases, it is not true in all cases; distinguish between something being proven and a mere plausibility argument; and identify logical errors in chains of reasoning. Mathematical reasoning and conceptual understanding are not separate from content; they are intrinsic to the mathematical discipline students master at more advanced levels. Throughout this document denotes specific standards from the Mathematics Framework for California Public Schools Throughout this document denotes the most important standards at each grade level All other items are District Standards PAGE 56 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS Mathematics Course Descriptions Grades 7 - 12 List of Approved Courses Course Code Middle School 5075 5085 5103 5104 2350 2550 2030 2780 2630 2450 2020 2100 2250 2260 2130 2680 2230 2670 7970 2460 2560 Course Title Mathematics 7 Mathematics 8 Pre-Algebra A Pre-Algebra B Algebra I (1 year and 2 year courses) Algebra II Applied Mathematics Calculus Exploring Probability and Statistics Geometry Math Lab Mathematics Exploration for the English Learner (Math ELD) Mathematics with Business Applications Medical Math Pre-Algebra Pre-Calculus Pre-Geometry Statistics Technical Math Transition to Algebra II Transition to College Math High School: Appendix: Advanced Placement Probability and Statistics Linear Algebra Mathematical Analysis Trigonometry PAGE 57 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS Mathematics Course Descriptions Middle School Level PAGE 58 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS MATHEMATICS 7 Repeat for Credit: No Graduation Requirement: N/A Grade: 7 Year Course Course Code: 5075 COURSE DESCRIPTION This course prepares students for Pre-Algebra. They will learn basic fundamental computations, place value, mental math, estimation and rounding, exponential notation, square roots, everyday geometry, and problem solving. PRE-REQUISITE: Mathematics 6 COURSE STANDARDS AND OBJECTIVES The student will: 1. Add, subtract, multiply, and divide whole numbers. 2. 3. 4. 5. 6. Add, subtract, multiply, and divide common fractions and mixed numbers. Add, subtract, multiply, and divide decimals. Understand the relationships among fractions, decimals, and percents. Understand the calculation of percent. Use proportional reasoning to solve mathematical and real-world problems (e.g., constant rate of change, proportions, interest, discounts). Determine which of two items is a better buy, based on unit pricing. Convert from larger to smaller units and from smaller to larger units in the metric system. Understand formulas for finding measures (e.g., perimeter, area, surface area, volume) of simple geometric shapes. Use math skills to determine probability using mathematical/theoretical models (e.g., tree diagrams, area models, counting procedures) 7. 8. 9. 10. PAGE 59 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 11. Solve simple probability problems involving both dependent and independent events. 12. Use a variety of basic problem solving strategies (e.g., work backwards, identify patterns, use similar problem types). Solve one- and two- operation word problems Understand and use basic methods of estimation and rounding. 13. 14. CONTENT, SCOPE, AND SEQUENCE 1. 2. 3. 4. 5. 6. Whole Numbers Decimals Fractions Percent Geometry Probability ASSESSMENT METHODS Tests, quizzes, homework, projects, student presentations, portfolios, group work, teacher observation. SUPPLEMENTARY MATERIALS/EQUIPMENT Calculators, manipulatives, computers PAGE 60 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS MATHEMATICS 8 Repeat for Credit: No Graduation Requirement: N/A Grade: 8 Year Course Course Code: 5085 COURSE DESCRIPTION This course prepares students for high school math. They will learn basic fundamental computations, place value, mental math, estimation and rounding, exponential notation, square roots, everyday geometry, problem solving, linear and co-ordinate graphing, and how to set up and solve simple one and two step equations. PRE-REQUISITE: Mathematics 7 or Pre-Algebra A COURSE STANDARDS AND OBJECTIVES The student will: 1. Add, subtract, multiply, and divide whole numbers. 2. 3. 4. 5. 6. Add, subtract, multiply, and divide common fractions and mixed numbers. Add, subtract, multiply, and divide decimals. Understand the relationships among fractions, decimals, and percents. Understand the calculation of percent. Use proportional reasoning to solve mathematical and real-world problems (e.g., constant rate of change, proportions, ratios, interest). Determine which of two items is a better buy, based on unit pricing. Convert from larger to smaller units and from smaller to larger units in the metric system. Understand formulas for finding measures (e.g., perimeter, area, surface area, volume) of simple geometric shapes. Solve simple ratio and proportion problems. Solve simple probability problems involving both dependent and independent events. PAGE 61 7. 8. 9. 10. 11. FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 12. Use a variety of basic problem solving strategies (e.g., work backwards, identify patterns, use similar problem types). Solve one- and two- operation word problems Understand and use basic methods of estimation and rounding. Translate words into algebraic sentences. Solve simple one and two step equations. Use the coordinate plane to graph linear functions. 13. 14. 15. 16. 17. CONTENT, SCOPE AND SEQUENCE 1. 2. 3. 4. 5. 6. 7. 8. Whole Numbers Decimals Fractions Percent Geometry Probability Equations Graphing ASSESSMENT METHODS Tests, quizzes, homework, projects, student presentations, portfolios, group work, teacher observation. SUPPLEMENTARY MATERIALS/EQUIPMENT Calculators, manipulatives, computers PAGE 62 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS PRE-ALGEBRA A Repeat for Credit: No Graduation Requirements: N/A Grades: 6-8 Year Course Course Code: 5103 COURSE DESCRIPTION This year-long course is the first half of a two-year Pre-Algebra program. It is designed to help students make the transition from elementary to high school mathematics. Areas of emphasis are the further development of basic skills, the extension of problem solving skills, the examination of the various strands of mathematical thinking, the communication of mathematical understanding, the unification of mathematical ideas and relationships and an introduction of pre-algebra skills and concepts. The course is organized around the three unifying ideas for middle school: proportional relationships, relationships, multiples representations, and patterns and generalizations. PRE-REQUISITE: Mathematics 6 or Mathematics 7 COURSE STANDARDS AND OBJECTIVES The student will: 1. Understand the meaning of place value for whole numbers and decimals. 2. 3. 4. 5. Use basic operations (e.g., addition, subtraction, multiplication, division). Use the order of operations. Estimate using whole numbers, fractions, and decimals. Know and apply the arithmetic properties (e.g., distributive property, commutative property, associative property, inverse properties). Know and apply the rules of divisibility. Use prime factorization. Understand and use multiples, the least common multiple, factors, and the greatest common factor. Understand prime and composite numbers. 6. 7. 8. 9. PAGE 63 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 10. Use technology to model and solve problems. (e.g., scientific calculators, computers). Understand and use the appropriate metric and English units of length, mass, capacity, time, and temperature. Convert measurement from one unit to another within a system. Estimate units of measurement. Use measurement in solving real-world problems. Understand the properties of basic geometric shapes. Use geometric tools and methods. Understand the mathematical concepts of congruency, similarity, and symmetry. Find the perimeter, circumference, and area of geometric shapes. Use the basic geometric properties to solve real-world problems. Collect data. Understand the characteristics and measures of central tendency (e.g., mean, median, mode). Represent data using line, bar, circle, and picture graphs. Understand how to determine experimental and simple theoretical probabilities. Read, interpret, and compare graphs. Understand the concept of ratio. Find equivalent ratios. Solve simple proportions. Understand the equivalencies of common fractions, decimals and percents. Estimate percentages. Use ratios, proportions, and percents to solve real-world problems. PAGE 64 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 31. 32. 33. 34. 35. 36. Read, write, and compare integers. Use basic operations with integers. Use integers to solve real-world problems. Use variables to represent unknown quantities. Write and solve simple expressions and equations. Use various problem solving strategies in order to analyze problems and formulate appropriate solution strategies. Use logical reasoning to model and solve problems. Use estimation and logic to verify the reasonableness of solutions. Justify the methods used and solutions. Develop cooperative learning and group skills. 37. 38. 39. 40. CONTENT, SCOPE, AND SEQUENCE 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Basic mathematics skills Measurement Geometry Statistics, probability, and graphing Ratio, proportion, and percent Integers Equations Problem solving Critical thinking Group work and cooperative learning ASSESSMENT METHODS Tests, quizzes, assignments, journals, portfolios, projects, teacher observations, questioning, presentations. SUPPLEMENTARY MATERIALS/EQUIPMENT PAGE 65 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS Appropriate teacher created materials and replacement units, calculators, manipulatives, and computers PAGE 66 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS PRE-ALGEBRA B Repeat for Credit: No Graduation Requirements: N/A Grades: 6-8 Year Course Course Code: 5104 COURSE DESCRIPTION This year-long course is the second half of a two-year Pre-Algebra program. It is designed to help students make the transition from elementary to high school mathematics. Areas of emphasis are the further development of basic skills, the extension of problem solving skills, the examination of the various strands of mathematical thinking, the communication of mathematical understanding, the unification of mathematical ideas and relationships and the development of pre-algebra skills and concepts. The course is organized around the three unifying ideas for middle school: proportional relationships, multiple representations, and patterns and generalizations. PRE-REQUISITE: Pre-Algebra A COURSE STANDARDS AND OBJECTIVES The student will: 1. Use the basic operations with whole numbers, decimals, fractions, integers, and rational numbers. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. Use scientific notation. Use prime factorization. Use exponential notation. Use scientific calculator skills. Select and use the appropriate metric and English units of length. Convert measurement units within a system. Estimate units of measurement. Use measurement to solve real-world problems. Understand the properties of basic geometric solids. Use geometric tools and methods. PAGE 67 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 12. 13. 14. 15. 16. 17. Understand the mathematical concepts of congruency, similarity, and symmetry. Use formulas to calculate the surface area and volume of geometric solids. Use the basic geometric properties to solve real-world problems. Collect data. Analyze data and calculate the mean, median, mode, range. Represent data using line, bar, circle, and picture graphs, histograms, and scatterplots. Calculate and use experimental and theoretical probabilities. Use probabilities to make predictions. Read, interpret, and compare graphs. Understand correlation and cause and effect. Understand the concept of ratio. Find equivalent ratios. Solve simple proportions. Understand the equivalencies of common fractions, decimals and percents. Find a percent of a number. Find what percent one number is of another. Find a number when a percent of it is known. Use ratios, proportions and percents in problem solving situations. Read, write, and compare integers. Use integers to solve real-world problems. Understand the use of variables to represent unknown quantities. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. PAGE 68 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 33. 34. 35. 36. 37. 38. Simplify and solve expressions. Solve simple equations. Solve two-step equations. Solve simple inequalities. Know how to graph linear equations. Use various problem-solving strategies in order to analyze problems and formulate appropriate solution strategies. Use logical reasoning to model and solve problems. Use estimation and logic to verify the reasonableness of solutions. Justify the methods used and solutions. Develop cooperative learning and group skills. 39. 40. 41. 42. CONTENT, SCOPE, AND SEQUENCE 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Basic mathematics skills Measurement Geometry Statistics, probability, and graphing Ratio, proportion, and percent Integers Equations Problem solving Critical thinking Group work and cooperative learning ASSESSMENT METHODS Tests, quizzes, assignments, journals, portfolios, projects, teacher observations, questioning, presentations. SUPPLEMENTARY MATERIALS/EQUIPMENT PAGE 69 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS Appropriate teacher created materials and replacement units ,calculators, manipulatives, computers PAGE 70 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS Mathematics Course Descriptions High School Level PAGE 71 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS Flow Chart Here PAGE 72 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS ALGEBRA I Repeat for Credit: No Graduation Requirements: Mathematics Grades: 7-12 Year Course (10 units) Course Code: 2350 COURSE DESCRIPTION This is an entry level course to the world of mathematics as opposed to arithmetic. In this course students learn how to use variables, solve equations, factor, graph equations, use of real-world problems to motivate and apply theory. PRE-REQUISITE: Pre-Algebra or Pre-Geometry or Applied Mathematics or Technical Math or Exploring Probability and Statistics COURSE STANDARDS AND OBJECTIVES The student will: 1.0 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. Use properties of numbers to demonstrate whether assertions are true or false. 2.0 Understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents. Solve equations and inequalities involving absolute values. Simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x-5) + 4(x-2) = 12. Solve multi-step problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step. 3.0 4.0 5.0 6.0 Graph a linear equation and compute the x- and y-intercepts (e.g., graph 2x + 6y = PAGE 73 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4). 7.0 Verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations by using the point-slope formula. Understand the concepts of parallel lines and perpendicular lines and how those slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point. Solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets. Add, subtract, multiply, and divide monomials and polynomials. Students solve multi-step problems, including word problems, by using these techniques. Apply basic factoring techniques to second- and simple third-degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials. Simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms. Add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques. Solve a quadratic equation by factoring or completing the square. Apply algebraic techniques to solve rate problems, work problems, and percent mixture problems. Understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functions. Determine the domain of independent variables and the range of de-pendent variables defined by a graph, a set of ordered pairs, or a symbolic expression. 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 Determine whether a relation defined by a graph, a set of ordered pairs, or a symbolic expression is a function and justify the conclusion. PAGE 74 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 19.0 Know the quadratic formula and are familiar with its proof by completing the square. Use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations. Graph quadratic functions and know that their roots are the x-intercepts. Use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points. Apply quadratic equations to physical problems, such as the motion of an object under the force of gravity. Use and know simple aspects of a logical argument: 24.1 Explain the difference between inductive and deductive reasoning and identify and provide examples of each. Identify the hypothesis and conclusion in logical deduction. Use counterexamples to show that an assertion is false and recognize that a single counterexample is sufficient to refute an assertion. 20.0 21.0 22.0 23.0 24.0 24.2 24.3 25.0 Use properties of the number system to judge the validity of results, to justify each step of a procedure, and to prove or disprove statements: 25.1 Use properties of numbers to construct simple, valid arguments (direct and indirect) for, or formulate counterexamples to, claimed assertions. Judge the validity of an argument according to whether the properties of the real number system and the order of operations have been applied correctly at each step. Given a specific algebraic statement involving linear, quadratic, or absolute value expressions or equations or inequalities, determine whether the statement is true sometimes, always, or never. 25.2 25.3 CONTENT, SCOPE, AND SEQUENCE 1 Date Organization PAGE 75 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 2. 3. 4 5. 6. 7. 8. 9. 10. 11. 12. Patterns and Calculators Patterns and Graphs Writing and Solving Equations Numerical, Graphical, and Algebraic Ratios Factoring Quadratics Graphing and Systems of Linear Equations Systems of Linear and Nonlinear Equations Pythagorean Theorem Area and sub problems Ratios and Slope Quadratic and Linear Equations TEACHING STRATEGIES Cooperative learning, lecture and discussion, investigations, writing and reading in math, manipulatives, student projects, use of the scientific/graphing calculator, open-ended questions. ASSESSMENT METHODS Tests, quizzes, homework, projects, student presentations, warm-ups, portfolios, group work, teacher observation. SUPPLEMENTARY MATERIALS/EQUIPMENT Calculators and manipulatives PAGE 76 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS ALGEBRA II Repeat for Credit: No Graduation Requirements: Mathematics Grades: 9-12 Year Course (10 units) Course Code: 2550 COURSE DESCRIPTION There is an emphasis on the extended understanding of the concepts presented in Algebra I. The students are introduced to imaginary numbers and work with exponential functions of high degrees. There is more emphasis on problem solving and graphing. PRE-REQUISITE: Algebra I or Geometry or Transition to Algebra II COURSE STANDARDS AND OBJECTIVES The student will: 1.0 Solve equations and inequalities involving absolute value. 2.0 Solve systems of linear equations and inequalities (in two or three variables) by substitution, with graphs, or with matrices. Use operations on polynomials. Factor polynomials representing the difference of squares, perfect square trinomials, and the sum and difference of two cubes. Demonstrate knowledge of how real and complex numbers are related both arithmetically and graphically. Add, subtract, multiply, and divide complex numbers. Add, subtract, multiply, divide, reduce, and evaluate rational expressions with monomial and polynomial denominators and simplify complicated rational expressions, including those with negative exponents in the denominator. Solve and graph quadratic equations by factoring, completing the square, or using the quadratic formula. Students apply these techniques in solving word problems. They also solve quadratic equations in the complex number system. Demonstrate and explain the effect that changing a coefficient has on the graph of PAGE 77 3.0 4.0 5.0 6.0 7.0 8.0 9.0 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS quadratic functions; that is, students can determine how the graph of a parabola changes as a, b, and c vary in the equation y = a(x-b) 2+ c. 10.0 Graph quadratic functions and determine the maxima, minima, and zeros of the function. Prove simple laws of logarithms. 11.1 Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. Judge the validity of an argument according to whether the properties of real numbers, exponents, and logarithms have been applied correctly at each step. 11.0 11.2 12.0 Know the laws of fractional exponents, understand exponential functions, and use these functions in problems involving exponential growth and decay. Use the definition of logarithms to translate between logarithms in any base. Understand and use the properties of logarithms to simplify logarithmic numeric expressions and to identify their approximate values. Determine whether a specific algebraic statement involving rational expressions, radical expressions, or logarithmic or exponential functions is sometimes true, always true, or never true. Demonstrate and explain how the geometry of the graph of a conic section (e.g., asymptotes, eccentricity) depends on the coefficients of the quadratic equation representing it. Given a quadratic equation of the form ax2 + by2 + cx + dy + e = 0, students can use the method for completing the square to put the equation into standard form and can recognize whether the graph of the equation is a circle, parabola, or hyperbola. Students can then graph the equation. Use fundamental counting principles to compute combinations and permutations. Use combinations and permutations to compute probabilities. 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 Know the binomial theorem and use it to expand binomial expressions that are raised to positive integer powers. PAGE 78 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 21.0 Apply the method of mathematical induction to prove general statements about the positive integers. Find the general term and the sums of arithmetic series and of both finite and infinite geometric series. Derive the summation formulas for arithmetic series and for both finite and infinite geometric series. Solve problems involving functional concepts, such as composition, defining the inverse function and performing arithmetic operations on functions. Use properties from number systems to justify steps in combining and simplifying functions. 22.0 23.0 24.0 25.0 CONTENT, SCOPE, AND SEQUENCE 1. 2. 3. 4. 5. 6. 7. 8. Exploring functions, including linear and quadratic Arithmetic and geometric sequences Exponential functions, negative and non-integer, exponents and radicals Parabolas, cubics, exponential and other rational functions and non-functions Logarithms and other inverse functions Polynomials, general systems and complex numbers Probability Introduction to trigonometric functions TEACHING STRATEGIES Cooperative learning, lecture and discussion, investigations, writing and reading in math, student projects, use of the scientific/graphing calculator. ASSESSMENT METHODS Tests, quizzes, homework, warm-ups, teacher observation, projects, student presentations, group work, portfolios. SUPPLEMENTARY MATERIALS/EQUIPMENT Calculators (scientific and graphing) and manipulatives PAGE 79 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS APPLIED MATHEMATICS Repeat for Credit: No Graduation Requirements: Mathematics Grades: 9-12 Year Course ( 10 units) Course Code: 2030 COURSE DESCRIPTION Students will use a problem-solving approach to apply the seven strands of mathematics to real world situations. PRE-REQUISITE: None COURSE STANDARDS AND OBJECTIVES 1.0 The students will have a general understanding of the seven strands of mathematics. The student will: 1.1 1.2 1.3 1.4 Construct a frequency table. Calculate the median, mean, mode. Construct and interpret a bar, line and circle graph. Use variables to represent unknown quantities, write equations and solve them. Set up and solve proportions for a variety of situations. Apply formulas to solve practical problems involving percent. 1.5 1.6 2.0 The students will know how to create a paycheck. The student will: 2.1. Use mathematical skills to determine hourly pay, minimum wage, and salary (e.g., change fractions to decimals, multiply decimals, add decimals, round, computate with fractions). Evaluate weekly timecards. 2.2 PAGE 80 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 2.3 Determine accuracy of paycheck deductions and net income emphasizing finding a rate, comparing numbers and computations with decimals and percents. 3.0 The students will find the best buy. The student will: 3.1 Calculate cost of purchase including sales tax computations involving percentages, decimals and rounding. Understand the techniques of comparison shopping (e.g., finding the unit price, markdowns and sale price, finding a percent, rounding). Determine cost of an automobile including insurance rate factors and cost of maintenance. Understand the elements of responsible consumerism (e.g., recognizing misleading and deceptive practices, pursuit of complaints, proper research and record-keeping, communication skills). 3.2 3.3 3.4 4.0 The students will understand personal banking. The student will: 4.1 Understand the maintenance of savings and checking accounts (e.g., comparison of banking services, reconciling statements using addition and subtraction of decimals, negative numbers, writing whole and decimal numbers). Understand credit (e.g., comparison of interest rates, length of loans, amount of down payment) and the use of formulas, changing fractions to decimals, writing a percent and multiplication of decimals. 4.2 5.0 The students will understand money management. The student will: 5.1 5.2 5.3 6.0 Establish and maintain a (simulated) budget. Understand and make (simulated) investments (e.g., insurance, stocks). Prepare a 1040 tax form. The students will know how to use proper measurement techniques. The student will: PAGE 81 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 6.1 6.2 6.3 Choose appropriate units for measuring objects, rates, and energy. Find measures of length, area, volume, and weight. Use conversion tables and equations to convert units (e.g., Fahrenheit to Celsius). 7.0 The students will understand the foundations of geometry. The student will: 7.1 Understand basic geometric concepts and definitions (e.g., polygon, triangle, quadrilateral, transversal, diagonal, arc, angle, congruence, similarity). Solve problems relating to special polygons (e.g., triangles, parallelograms, rectangles, squares). Use formulas to determine the lateral area, total area, and volume of certain three-dimensional figures (e.g., cubes, rectangular solids, spheres, cones, pyramids, cylinders). Understand the Pythagorean theorem and its converse. 7.2 7.3 7.4 8.0 The students will understand the foundations of statistics and probability. The student will: 8.1 8.2 8.3 Collect and represent data using tables, charts, and line or bar graphs. Compute the range, mean, median, and mode of data sets. Use data to estimate the probability for future events (e.g., batting averages or number of accidents per miles driven). CONTENT, SCOPE, AND SEQUENCE SEMESTER 1 Using date and estimating Reading and making graphs Problem solving using formulas and equations Ratio, Proportion, and Percent Measurement TEACHING STRATEGIES SEMESTER 2 Geometry Earning money in the real world How to manage your money Using money Probability PAGE 82 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS Cooperative learning, lecture and discussion, investigations, reading and writing in math, manipulatives, projects, geometric constructions, use of technology, open-ended questions. ASSESSMENT METHODS A combination of any or all of the following will be used: teacher observation, open-ended questions, oral presentation, projects, portfolios, hybrid testing - summative and formative. SUPPLEMENTARY MATERIALS/EQUIPMENT Teacher generated material, three dimensional geometric models, probability manipulatives such as probability cubes and probability cards, additional manipulatives as needed, and additional texts when appropriate PAGE 83 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS CALCULUS Repeat for Credit: No Graduation Requirements: Mathematics Grades: 12 Year Course (10 units) Course Code: 2780 COURSE DESCRIPTION This course is for students who plan on a college major which requires Calculus. It covers the content of first semester college Calculus, which includes differentiation and integration. It also includes a review of the prerequisite Algebra, Geometry, and Trigonometry courses. PRE-REQUISITE: Pre-Calculus or Statistics COURSE STANDARDS AND OBJECTIVES The student will: 1.0 Use various problem solving strategies in order to analyze problems and formulate appropriate solution strategies. 2.0 3.0 4.0 5.0 Understand, evaluate and use limits. Understand, evaluate and use differentiation. Understand, evaluate and use indefinite and definite integrals. Know how to find maximum and minimum of a function. 5.1 Use maximum and minimum calculation techniques to solve distance, velocity, and acceleration problems. Solve problems about area, volume and distance that involve solving for maximums and minimums. 5.2 6.0 Use calculus to solve more complex problems in content areas covered in previous math courses. Understand the history of mathematics in regards to development of calculus. Use graphing calculator in some calculus problems. Communicate mathematical ideas both orally and in written form. PAGE 84 7.0 8.0 9.0 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 10.0 Know both the formal definition and the graphical interpretation of continuity of a function. Understand and apply of the intermediate value theorem and the extreme value theorem. Know the chain rule, its proof, and its application to the calculation of the derivative of a variety of composite functions. Know how to find the derivatives of parametrically defined functions and use implicit differentiation in a wide variety of problems in physics, chemistry, economics. Understand how to compute derivatives of higher orders. Know and apply Rolle’s theorem, the mean value theorem, and L’Hôpital’s rule. Use differentiation to sketch, by hand, graphs of functions, and identify maxima, minima, inflection points, and intervals in which the function is increasing and decreasing. Know Newton’s method for approximating the zeros of a function. Use differentiation to solve related rate problems in a variety of pure and applied contexts. Know the definition of the definite integral by using Riemann sums and use this definition to approximate integrals. Use the definition of the integral to model problems in physics, economics, and so forth, obtaining results in terms of integrals. Know the proof of the fundamental theorem of calculus and use it to interpret integrals as antiderivatives. Use definite integrals in problems involving area, velocity, acceleration, volume of a solid, area of a surface of revolution, length of a curve, and work. Know the definitions and properties of inverse trigonometric functions and the expression of these functions as indefinite integrals. Understand the algorithms involved in Simpson's rule and Newton's method. 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 PAGE 85 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 25.0 26.0 Understand improper integrals as limits of definite integrals. Understand the definitions of convergence and divergence of sequences and series of real numbers. Understand and compute the radius (interval) of the convergence of power series. Differentiate and integrate the terms of a power series in order to form new series from known ones. Calculate Taylor polynomials and Taylor series of basic functions, including the remainder term. Know the techniques of solution of selected elementary differential equations and their applications to a wide variety of situations, including growth-and-decay problems. 27.0 28.0 29.0 30.0 CONTENT, SCOPE, AND SEQUENCE 1. 2. 3. 4. 5. 6. 7. Review of previous math courses Limits and their properties Differentiation Application of differentiation Integration Inverse functions Applications of integration TEACHING STRATEGIES Cooperative learning, lecture and discussion, writing and reading in math, use of the scientific/graphing calculators, projects. ASSESSMENT METHODS Tests, quizzes, homework, project, student presentations, portfolios, group work, participation, final exam. SUPPLEMENTARY MATERIALS/EQUIPMENT Graphing calculators and manipulatives PAGE 86 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS EXPLORING PROBABILITY AND STATISTICS Repeat for Credit: No Graduation Requirement: Mathematics Grades: 10-12 Year Course (10 units) Course Code: 2630 COURSE DESCRIPTION This is a non-college preparatory statistics course. Current events will be a basis for projects and class discussion with an emphasis on finding, evaluating, and understanding statistics and the relevance to our daily lives. Math topics from previous courses will be reviewed as needed. PRE-REQUISITE: Pre-Algebra or Algebra I COURSE STANDARDS AND OBJECTIVES 1.0 The student critically evaluates reports from common media sources, including newspapers, magazines, television, and the Internet. The student will: 1.1 1.2 1.3 1.4 1.5 1.6 2.0 Understand vocabulary of statistics. Understand common methods of statistical misrepresentation. Recognize unsupported claims and areas of potential bias. Know what statistics should be included in a good report. Understand reported research results. Understand reported polling results. The student collects and analyzes data. The student will: 2.1 2.2 2.3 2.4 Understand random sampling procedures. Know methods to minimize bias collection and analysis procedures. Understand measures of central tendency (e.g., mean, median, and mode). Understand measures of dispersion (e.g., range, variance, standard deviation). PAGE 87 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 2.5 2.6 3.0 Understand various graphs and visual representations. Understand z scores, z and t tests, and confidence intervals. The student creates and analyzes graphs and other visual representations of data. The student will: 3.1 3.2 3.3 3.4 3.5 4.0 Understand bar and pie graphs. Understand histograms. Understand stem and leaf plots. Understand scatter plots. Understand regression lines. The student uses logical argumentation, presents and defends opinions and/or positions based on reasoned analysis of data. The student will: 4.1 4.2 4.3 5.0 Understand and write beginning proofs. Use ideas of mathematical proof to structure written reports. Use calculated statistical quantities to support assertions. The student explores probability with hands on activities and projects. The student will: 5.1 5.2 5.3 5.4 5.5 Calculate probability for independent events. Calculate probability for dependent events. Calculate probability for conditional events. Understand introductory counting theory (e.g., combinations, permutations). Understand probability distributions, especially as related to hypothesis testing in designed experiments. PAGE 88 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 6.0 The students design and implement surveys, observational studies, and experiments, using acceptable statistical methodology and interpret their results. The student will: 6.1 6.2 6.3 6.4 Understand random sampling procedures. Know the sample size needed for a given degree of accuracy. Design studies based on random sampling procedures. Know the difference between surveys, observational studies, and experiments. Analyze their own studies using mean, median, and mode. Analyze their own studies using range, variance, and standard deviation. Analyze their own studies for statistical significance using hypothesis testing. 6.5 6.6 6.7 7.0 The students use technology as a tool for aiding in collecting and analyzing data. The student will: 7.1 Use computers with spreadsheets, statistical computing packages, and Internet Use scientific and graphing calculators. 7.2 CONTENT, SCOPE, AND SEQUENCE 1. 2. 3. 4. 5. 6. 7. 8. Definition of statistics and critical observation of its uses and application. Description of sample data and graphical displays of data. Measures of central tendency including mean, median, and mode. Explore measures of dispersion including range, variance, and standard deviation. Probability. Introduction to data analysis. Introduction to correlation and regression. Projects integrating multiple statistical topics throughout course. ASSESSMENT METHODS Tests, quizzes, homework, projects, student presentations, warm ups, group work, portfolios, teacher observation. SUPPLEMENTARY MATERIALS/EQUIPMENT PAGE 89 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS Calculators, manipulatives, computers PAGE 90 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS GEOMETRY Repeat for Credit: No Graduation Requirements: Mathematics Grades: 8-12 Year Course (10 units) Course Code: 2450 COURSE DESCRIPTION This course, which has a prerequisite of Algebra I, is a study of geometric concepts as related to plane and solid figures. It encompasses a study of lines, triangles, polygons, circles, spheres, prisms, coordinates, inductive and deductive reasoning. There is an emphasis on students discovering the theorems of geometry through investigations. PRE-REQUISITE: Algebra I or Algebra II COURSE STANDARDS AND OBJECTIVES The student will: 1.0 Know how to identify and give examples of undefined terms, axioms, theorems, and inductive and deductive reasoning. Know how to write geometric proofs, including proofs by contradiction. Know how to construct and judge the validity of a logical argument and give counter examples to disprove a statement. Use geometric modeling to represent and solve real world problems. Understand the properties of figures and generalize from observation. Make and test geometric conjectures and formulate counter examples. Construct arguments by using inductive reasoning to make discoveries and deductive reasoning to logically verify their discoveries. Use the compass and straight edge to make basic constructions to demonstrate skills in visualization and pictorial representation. Use algebraic methods to solve geometric problems. PAGE 91 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 10.0 11.0 12.0 13.0 Understand the application of geometry to the physical world around them. Understand oral and written expressions of mathematical ideas. Prove basic theorems involving congruence and similarity. Prove that triangles are congruent or similar, using the concept of corresponding parts of congruent triangles. Know and use the triangle inequality theorem. Prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles. Know, derive, and solve problems involving the perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures. Compute the volumes and surface areas of prisms, pyramids, cylinders, cones, cylinders, and spheres. Compute areas of polygons, including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms, and trapezoids. Know how changes in dimensions affect the perimeter, area, and volume of common geometric figures and solids. Use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems. Prove relationships between angles in polygons by using properties of complementary, supplementary, vertical, and exterior angles. Prove the Pythagorean theorem. Use the Pythagorean theorem to determine distance and find missing lengths of sides of right triangles. Perform basic constructions with a straightedge and compass, such as angle bisectors, perpendicular bisectors, and the line parallel to a given line through a point off the line. Prove theorems by using coordinate geometry, including the midpoint of a line segment, the distance formula, and various forms of equations of lines and circles. PAGE 92 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 26.0 Know the definitions of the basic trigonometric functions defined by the angles of a right triangle. Know and use elementary relationships between trigonometric functions (e.g., tan(x) = sin(x)/cos(x), (sin(x))2 + (cos(x)) 2 = 1). Use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side. Know and use angle and side relationships in problems with special right triangles, such as 30/, 60/, and 90/ triangles and 45/, 45/, and 90/ triangles. Prove and solve problems regarding relationships among chords, secants, tangents, inscribed angles, and inscribed and circumscribed polygons of circles. Know the effect of rigid motions on figures in the coordinate plane and space, including rotations, translations, and reflections. 27.0 28.0 29.0 30.0 31.0 CONTENT, SCOPE, AND SEQUENCE 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. Geometric Art Inductive Reasoning Undefined Terms and Basic Vocabulary Geometric Constructions Polygons and Related Segments Congruence Circles Area Pythagorean Theorem Volume Similarity Trigonometry Geometric Proofs TEACHING STRATEGIES Cooperative learning, lecture and discussion, investigations, writing in math, manipulatives, projects, geometric constructions, use of technology, open-ended questions. ASSESSMENT METHODS PAGE 93 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS Homework, tests, quizzes, warm-ups, assignments, notebooks, portfolios, student presentations, projects, group work, teacher observation. SUPPLEMENTARY MATERIALS/EQUIPMENT Student geometry tool kit (compass, ruler, straight edge, protractors, calculator, graph paper), manipulatives, three dimensional models, graphing calculator PAGE 94 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS MATH LAB Repeat for Credit: No Graduation Requirements: Mathematics Grades: 9-12 Year Course (10 units) Course Code: 2020 COURSE DESCRIPTION This course reviews basic math proficiency skills needed for graduation. It will also provide the student with opportunities to learn practical uses of those skills already mastered. PRE-REQUISITE: None COURSE STANDARDS AND OBJECTIVES The student will: 1. Understand place value of each digit in a numeral from million to millionths. 2. 3. 4. Add, subtract, multiply, and divide whole numbers. Convert fractions to the lowest term equivalent. Convert an improper fraction to a mixed number and a mixed number to an improper fraction. Add, subtract, multiply, and divide fractions and/or mixed numbers. Understand the equivalencies of fractions, decimals, and percents. Convert a decimal to a fraction and/or mixed number. Add, subtract, multiply, and divide decimals. Understand how to calculate an average. Solve simple equations (e.g., 3 x N=12). Use ratio and proportions to solve problems. Use formulae for area and perimeter. Use metric units to calculate the area and volume of rectangular objects. PAGE 95 5. 6. 7. 8. 9. 10. 11. 12. 13. FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 14. 15. 16. 17. Understand how to find percent. Solve simple word problems. Understand rounding numbers. Use common units to make simple measurements. CONTENT, SCOPE, AND SEQUENCE Semester 1 Place value and rounding Whole numbers Fractions Decimals Semester 2 Percent Applying percent Ratio to proportion Applications TEACHING STRATEGIES Cooperative learning, lecture and discussion, investigations, reading and writing in math, manipulatives, projects, geometric constructions, use of technology, open-ended questions. ASSESSMENT METHODS Tests, quizzes, homework, project, student presentations, portfolios, group work, participation, final exam SUPPLEMENTARY MATERIALS/EQUIPMENT Calculators (scientific and graphing) and manipulatives PAGE 96 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS MATHEMATICS EXPLORATION FOR THE ENGLISH LEARNER (MATH ELD) Repeat for Credit: No Graduation Requirements: Mathematics Grades: 9-12 Year Course (10 units) Course Code: 2121 COURSE DESCRIPTION This course is designed for the English learner student to transition to the American mathematics system and be able to successfully function in the next appropriate level mathematics class including college prep classes. Students will develop literacy in English, core curricular math skills, technical math vocabulary, and understanding of American mathematical structures. The focus will be on logic and problem solving strategies that address the mathematics strands outlined in the California State Framework. PRE-REQUISITE: None COURSE STANDARDS AND OBJECTIVES 1.0 The students will develop literacy by listening to English using mathematics. The student will: 1.1 Understand and follow simple directions using the vocabulary of math (e.g., point to, show me, select). Understand basic characteristics for each area of mathematics (e.g., shape, size, amount, relational). Understand how to categorize and sequence objects. 1.2 1.3 2.0 The students will develop literacy in speaking English in relation to core curricular math skills. The student will: 2.1 2.2 2.3 3.0 Use short oral responses to answer simple questions. Understand different types of numeration (e.g., counting, place value) Identify and name numbers, symbols, relational terminology, and operations. The students will develop literacy in reading English technical math PAGE 97 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS vocabulary. The student will: 3.1 Know the meaning of single words, phrases, and simple mathematical sentences. Use context clues to identify key words. Understand and interpret basic math vocabulary. Know high frequency math vocabulary used in English. 3.2 3.3 3.4 4.0 The students will understand English by writing American mathematical structures. The student will: 4.1 4.2 4.3 4.4 4.5 Write in primary language if ability exists. Translate from primary language into English. Solve short math problems. Make and label simple tables, charts, and graphs. Understand equivalent forms to represent numbers (e.g., integers, fractions, decimals, percents). Use appropriate written mathematical vocabulary. 4.6 CONTENT, SCOPE, AND SEQUENCE Semester 1 Math as Communication Basic Geometry Number and Number Relationships Number Systems and Number Theory Mathematical Connections Semester 2 Problem Solving Measurement Patterns and Functions Statistics TEACHING STRATEGIES PAGE 98 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS Cooperative learning, lecture and discussion, investigations, reading and writing in math, manipulatives, projects, geometric constructions, use of technology, open-ended questions. ASSESSMENT METHODS A combination of any or all of the following will be used: teacher observation, open-ended questions, oral presentations, projects, portfolios, hybrid testing - summative and formative. SUPPLEMENTARY MATERIALS/EQUIPMENT Teacher generated materials, three dimensional geometric models, probability manipulatives such as probability cubes and probability cards, additional manipulatives as needed, and additional texts when appropriate PAGE 99 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS MATHEMATICS WITH BUSINESS APPLICATIONS Repeat for Credit: No Graduation Requirement: Mathematics Grades: 10-12 Year Course (10 units) Course Code: 2250 COURSE DESCRIPTION This is a non-college preparatory mathematics course with a focus on business applications. Emphasis will be on real world application of computational skills to solve business and consumer problems. Students requiring additional work to improve their math skills will have an opportunity to do so while solving real life problems. Math topics from previous courses will be reviewed as necessary. PRE-REQUISITE: Pre-Algebra or Algebra I or Geometry COURSE STANDARDS AND OBJECTIVES 1.0 The student applies basic math skills in computations for employment, personal business, and computer applications. The student will: 1.1 Calculate the amount of gross pay given pay rates, hours worked, and/or commission schedule Analyze earnings statements (e.g., verify authorized deductions, taxes, net pay) and find the percentage component of each deduction. Calculate the amount of interest earned on a savings or checking account using both algebraic formulas and interest tables. Calculate the final cost of consumer purchases after considering sale price, discounts, and taxes payable. Calculate the amount of interest due on credit purchases and consumer loans using both algebraic formulas and interest tables. Calculate the break-even point for a business product using algebraic formulas. Using algebraic formulas to determine the percentage of defective PAGE 100 1.2 1.3 1.4 1.5 1.6 1.7 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS components in a set. 1.8 Compute the monthly payment, total amount paid, and total interest charged on mortgage loans. 2.0 The student analyzes and solve business related problems using several mathematical problem solving skills. The student will: 2.1 2.2 2.3 2.4 2.5 2.6 3.0 Construct tables of information. Recognize a wide variety of patterns. Use guess and check to solve problems. Write equations to model problem situations. Make a Venn diagram to model and solve problem situations. Use drawings to model problem situations. The student creates and analyzes graphs and other presentations of business and personal data. The student will: 3.1 3.2 3.3 3.4 3.5 3.6 4.0 Create and analyze line graphs. Create and analyze bar and pie graphs. Create and analyze stem and leaf plots. Create and analyze histograms. Create and analyze scatter plots. Create and analyze regression lines. The student critically evaluates consumer advertisements and business reports from common media sources (e.g., newspapers, magazines, television, the Internet). The student will: 4.1 Understand the vocabulary used in consumer advertisements. PAGE 101 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 4.2 4.3 4.4 4.5 Understand standard business terms and their uses. Recognize unsupported claims and false advertising. Understand and recognize pressure sales techniques. Know which government agencies monitor consumer advertising and business product safety. Understand the factors involved in making better consumer decisions. Understand the factors involved in evaluating businesses as potential employers. 4.6 4.7 5.0 The student develops a realistic personal finance plan using material from textbooks, projects, and hands on activities. The student will: 5.1 5.2 5.3 5.4 5.5 5.6 6.0 Establish financial goals and objectives (e.g., make a budget). Understand how income affects goals. Know how to manage income and credit. Know how to acquire and protect assets. Understand how savings can help achieve financial goals. Know how to take control of achieving goals using a financial plan. The student explores mathematics in business with hands on activities and projects. The student will: 6.1 Understand how to review and critically analyze business reports from local businesses. Know what math skills employers expect from job applicants. 6.2 CONTENT, SCOPE, AND SEQUENCE 1. Basic mathematics review PAGE 102 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 2. 3. 4. 5. 6. 7. 8. 9. Mathematics of a paycheck: gross income, net income, taxes Interest rates: checking and savings account Purchases: cash or credit cards Loans Automobile transportation: what are the real costs? After high school: housing costs, insurance, investments The world of work: employment, production, and employee benefits Marketing and consumerism ASSESSMENT METHODS Tests, quizzes, written reports, homework, group work, portfolios, student presentations, teacher observation, projects, homework. SUPPLEMENTARY MATERIALS/EQUIPMENT Calculators, computers, manipulatives, newspapers, magazines, and the Internet PAGE 103 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS MEDICAL MATH Repeat for Credit: No Graduation Requirement: Third Year Mathematics/ Electives Grades: 10-12 Year Course (10 units) Course Code: 2260 COURSE DESCRIPTION This course prepares students for post high school technical training in the health occupation field. They will learn fractions, percents, metric measurement, ratios and proportions, graphs, accounting techniques, and medical dosages as they apply to the health occupations field. PRE-REQUISITE: None COURSE STANDARDS AND OBJECTIVES The student will: 1. Add, subtract, multiply, and divide common fractions. 2. 3. 4. 5. 6. 7. 8. 9. Add, subtract, multiply, and divide decimal fractions using multiple operations. Understand basic principles of percentages, interest, and discounts. Know the basic principles of mass and weight measurement. Know the basic principles of volume and liquid measurement. Know the basic principles of Celsius and Fahrenheit measurement. Solve problems of ratio and proportion. Know and use the measurement instruments used in health occupations. Use conversion charts used in health occupations (temperature, pulse, respiration, height, weight). Know the accounting techniques and strategies used by today's health care professionals. Identify equipment used in the administration of medications. Identify various syringes by total volume, use, and calibration. PAGE 104 10. 11. 12. FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 13. 14. 15. 16. 17. 18. 19. 20. Read a given volume of medication contained in a syringe. Identify commonly used medications. Understand the difference between dosage strength and supply dosage. Calculate oral drug dosages using proportion. Determine number of tablets needed to deliver a prescribed dose. Determine volume of liquid needed to deliver a prescribed dose. Calculate pediatric dosages based on recommended usual doses using a proportion. Express the flow rate of IV fluids in milliliters per hour based on the physician's order. Understand how to convert a flow rate expressed in milliliters per hour to in drops per minute. 21. CONTENT, SCOPE, AND SEQUENCE 1. 2. 3. 4. 5. 6. 7. 8. 9. 10 Common Fractions Decimal Fractions Percent Interest and Averages Metric and Other Measures Ratio and Proportions Measurement Instruments Graphs and Charts Accounting and Business Weights and Measures Dosage Calculations ASSESSMENT METHODS Tests, quizzes, projects, student presentations, portfolios, group work, teacher observation. SUPPLEMENTARY MATERIALS/EQUIPMENT Calculators, manipulatives, computers PAGE 105 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS PRE-ALGEBRA Repeat for Credit: No Graduation Requirements: Mathematics Grades: 9-12 Year Course (10 units) Course Code: 2130 COURSE DESCRIPTION This course is designed for the student who is not ready for the abstract world of Algebra I. Emphasis is placed on a problem-solving approach. logic and language geometry functions discrete mathematics measurement number algebra statistics and probability PRE-REQUISITE: None COURSE STANDARDS AND OBJECTIVES NUMBER SENSE 1.0 Students know the properties of, and compute with, rational numbers expressed in a variety of forms: The student will: 1.1 Read, write, and compare rational numbers in scientific notation (positive and negative powers of 10) with approximate numbers using scientific notation. Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to wholenumber powers. Convert fractions to decimals and percents and use these representations in estimations, computations, and applications. Differentiate between rational and irrational numbers. 1.2 1.3 1.4 1.5 Know that every rational number is either a terminating or repeating PAGE 106 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS decimal and be able to convert terminating decimals into reduced fractions. 1.6 1.7 Calculate the percentage of increases and decreases of a quantity. Solve problems that involve discounts, markups, commissions, and profit and compute simple and compound interest. 2.0 Students use exponents, powers, and roots and use exponents in working with fractions: The student will: 2.1 Understand negative whole-number exponents. Multiply and divide expressions involving exponents with a common base. Add and subtract fractions by using factoring to find common denominators. Multiply, divide, and simplify rational numbers by using exponent rules. Use the inverse relationship between raising to a power and extracting the root of a perfect square integer; for an integer that is not square, determine without a calculator the two integers between which its square root lies and explain why. Understand the meaning of the absolute value of a number; interpret the absolute value as the distance of the number from zero on a number line; and determine the absolute value of real numbers. 2.2 2.3 2.4 2.5 ALGEBRA AND FUNCTIONS 1.0 Students express quantitative relationships by using algebraic terminology, expressions, equations, inequalities, and graphs: The student will: 1.1 Use variables and appropriate operations to write an expression, an equation, a inequality, or a system of equations or inequalities that represents a verbal description (e.g., three less than a number, half as large as area A). Use the correct order of operations to evaluate algebraic expressions such as 3(2x + 5)2 Simplify numerical expressions by applying properties of rational numbers (e.g., identity, inverse, distributive, associative, commutative) and justify PAGE 107 1.2 1.3 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS the process used. 1.4 Use algebraic terminology (e.g., variable, equation, term, coefficient, inequality, expression, constant) correctly. Represent quantitative relationships graphically and interpret the meaning of a specific part of a graph in the situation represented by the graph. 1.5 2.0 Students interpret and evaluate expressions involving integer powers and simple roots: The student will: 2.1 Interpret positive whole-number powers as repeated multiplication and negative whole-number powers as repeated division or multiplication by the multiplicative inverse. Simplify and evaluate expressions that include exponents. Multiply and divide monomials; extend the process of taking powers and extracting roots to monomials when the latter results in a monomial with an integer exponent. 2.2 3.0 Students graph and interpret linear and some nonlinear functions: Graph functions of the form y = nx2 and y = nx3 and use in solving problems. Plot the values from the volumes of three-dimensional shapes for various values of the edge lengths (e.g., cubes with varying edge lengths or a triangle prism with a fixed height and an equilateral triangle base of varying lengths). Graph linear functions, noting that the vertical change (change in y-value) per unit of horizontal change (change in x-value) is always the same and know that the ratio (―rise over run‖) is called the slope of a graph. Plot the values of quantities whose ratios are always the same (e.g., cost to the number of an item, feet to inches, circumference to diameter of a circle). Fit a line to the plot and understand that the slope of the line equals the quantities. The student will: 3.1 3.2 3.3 3.4 4.0 Students solve simple linear equations and inequalities over the rational numbers: PAGE 108 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS The student will: 4.1 Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results. Solve multi-step problems involving rate, average speed, distance, and time or a direct variation. 4.2 MEASUREMENT AND GEOMETRY 1.0 Students choose appropriate units of measure and use ratios to convert within and between measurement systems to solve problems: The student will: 1.1 Compare weights, capacities, geometric measures, times, and temperatures within and between measurement systems (e.g., miles per hour and feet per second, cubic inches to cubic centimeters). Construct and read drawings and models made to scale. Use measures expressed as rates (e.g., speed, density) and measures expressed as products (e.g., person-days) to solve problems; check the units of the solutions; and use dimensional analysis to check the reasonableness of the answer. 1.2 1.3 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. The student will: 2.1 Use formulas routinely for finding the perimeter and area of basic twodimensional figures and the surface area and volume of basic threedimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders. Estimate and compute the area of more complex or irregular two- and three-dimensional figures by breaking the figures down into more basic geometric objects. 2.2 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 PAGE 109 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS and the volume is multiplied by the cube of the scale factor. 2.4 Relate the changes in measurement with a change of scale to the units used (e.g., square inches, cubic feet) and to conversions between units (1 square foot = 144 square inches or [1 ft2] = [144 in2 ], 1 cubic inch is approximately 16.38 cubic centimeters or [1in3 ] = [16.38 cm3 ]). 3.0 Students know the Pythagorean theorem and deepen their understanding of plane and solid geometric shapes by constructing figures that meet given conditions and by identifying attributes of figures: The student will: 3.1 Identify and construct basic elements of geometric figures (e.g., altitudes, mid-points, diagonals, angle bisectors, and perpendicular bisectors; central angles, radii, diameters, and chords of circles) by using a compass and straightedge. Understand and use coordinate graphs to plot simple figures, determine lengths and areas related to them, and determine their image under translations and reflections. Know and understand the Pythagorean theorem and its converse and use it to find the length of the missing side of a right triangle and the lengths of other line segments and, in some situations, empirically verify the Pythagorean theorem by direct measurement. Demonstrate an understanding of conditions that indicate two geometrical figures are congruent and what congruence means about the relationships between the sides and angles of the two figures. Construct two-dimensional patterns for three-dimensional models, such as cylinders, prisms, and cones. Identify elements of three-dimensional geometric objects (e.g., diagonals of rectangular solids) and describe how two or more objects are related in space (e.g., skew lines, the possible ways three planes might intersect). 3.2 3.3 3.4 3.5 3.6 STATISTICS, DATA ANALYSIS, AND PROBABILITY 1.0 Students collect, organize, and represent data sets that have one or more variables and identify relationships among variables within a data set by hand and through the use of an electronic spreadsheet software program: PAGE 110 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS The student will: 1.1 Know various forms of display for data sets, including a stem-and-leaf plot or box-and-whisker plot; use the forms to display a single set of data or to compare two sets of data. Represent two numerical variables on a scatter plot and informally describe how the data points are distributed and any apparent relationship that exists between the two variables (e.g., between time spent on homework and grade level). Understand the meaning of, and be able to compute, the minimum, the lower quartile, the median, the upper quartile, and the maximum of a data set. 1.2 1.3 MATHEMATICAL REASONING 1.0 Students make decisions about how to approach problems: The student will: 1.1 Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, identifying missing information, sequencing and prioritizing information, and observing patterns. Formulate and justify mathematical conjectures based on a general description of the mathematical question or problem posed. Determine when and how to break a problem into simpler parts. 1.2 1.3 2.0 Students use strategies, skills, and concepts in finding solutions: The student will: 2.1 2.2 Use estimation to verify the reasonableness of calculated results. 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. Make and test conjectures by using both inductive and deductive reasoning. 2.4 PAGE 111 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 2.5 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. 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. Indicate the relative advantages of exact and approximate solutions to problems and give answers to a specified degree of accuracy. Make precise calculations and check the validity of the results from the context of the problem. 2.6 2.7 2.8 3.0 Students determine a solution is complete and move beyond a particular problem by generalizing to other situations: The student will: 3.1 Evaluate the reasonableness of the solution in the context of the original situation. Note the method of deriving the solution and demonstrate a conceptual under-standing of the derivation by solving similar problems. Develop generalizations of the results obtained and the strategies used and apply them to new problem situations. 3.2 3.3 CONTENT, SCOPE, AND SEQUENCE 1. Basic Mathematics Skills 2. Measurement 3. Ratios, Proportions and Percentages 4. Discrete Mathematics 5. Algebraic Equations 6. Statistics, Probability and Graphing 7. Plane Geometry 8. Critical Thinking and Problem Solving Skills TEACHING STRATEGIES Cooperative learning, lecture and discussion, investigations, writing in math, manipulatives, student projects, use of technology, problem of the week, open-ended questions. ASSESSMENT METHODS PAGE 112 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS Tests, quizzes, homework, projects, student presentations, warm-ups, portfolios, group work, teacher observation. SUPPLEMENTARY MATERIALS Calculators, manipulatives, etc. PAGE 113 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS PRE-CALCULUS (TRIGONOMETRY & ANALYTIC GEOMETRY) Repeat for Credit: No Graduation Requirements: Mathematics Grades: 10-12 Year Course (10 units) Course Code: 2680 COURSE DESCRIPTION This course is designed for the college or university prep student who has demonstrated a high degree of skill and understanding in mathematics. This course emphasizes properties of right triangle, solving triangles, trigonometric functions, relations, inverses, laws and conics. PRE-REQUISITE: Algebra II and Geometry or Transition to College Math or Statistics COURSE STANDARDS AND OBJECTIVES The student will: 1. Use various problem solving strategies in order to analyze problems and formulate appropriate solution strategies. 2. Understand how to express, interpret and graph a variety of functions (e.g., quadratic, logarithmic, exponential, trigonometric. Solve equations, inequalities and systems of equations (e.g., linear, quadratic, trigonometric, exponential, logarithmic and rational) using a variety of techniques. Understand the basic trigonometric functions (i.e., sine, cosine, tangent) and use them when solving problems. Solve systems of multi-variable equations and inequalities. Understand basic applications and operations on matrices (e.g., inverse, determinants). Understand the properties of sequences and series and solve problems. Use theoretical and experimental methods to solve probability problems. Communicate mathematical ideas both orally and in written form. 3. 4. 5. 6. 7a. 7b. 8. PAGE 114 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS CONTENT, SCOPE, AND SEQUENCE 1. Review of algebra 2. Functions and graphs 3. Polynomial and rational functions 4. Exponential and logarithmic functions 5. Trigonometry 6. Analytic trigonometry 7. Applications of trigonometry 8. Systems of equations and inequalities 9. Matrices and determinants 10. Sequences, series, and probability 11. Analytic geometry: conics, polar coordinates TEACHING STRATEGIES Cooperative learning, lecture and discussion, investigations, reading and writing in math, manipulatives, projects, geometric constructions, use of technology, open-ended questions. ASSESSMENT METHODS Tests, quizzes, homework, projects, student presentations, warm-ups, portfolios, group work, teacher observation. SUPPLEMENTARY MATERIALS/EQUIPMENT Calculators, manipulatives PAGE 115 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS PRE-GEOMETRY Repeat for Credit: No Graduation Requirements: Mathematics Grades: 9-12 Year Course (10 units) Course Code: 2230 COURSE DESCRIPTION This is a second year course that continues the strands studied in Pre-Algebra, with an emphasis on the geometry strand. Strands studies are: algebra number measurement volume and area ratios spatial figures similarity scale congruence Cartesian plane polygons and circles trigonometry PRE-REQUISITE: Pre-Algebra or Algebra I COURSE STANDARDS AND OBJECTIVES Students will: 1. Measure and construct line segments, rays, and angles. 2. Use a protractor and/or compass to measure and/or construct right angles, triangles, and circles. use inductive and/or deductive reasoning to reach a conclusion based on a series of examples and/or observations. Solve linear equations and understand their relationship to the graphs of functions. Use proportional reasoning to solve real-world problems (e.g., ratio, proportion, percent, similarity, right triangles). Understand definitions and vocabulary of Geometry such as (but not limited to) line, point, plane, line segment, intersection, locus, postulate, theorem, etc. Use the coordinate system to plot, graph, measure, and construct geometric shapes. Use scientific calculators to solve problems. PAGE 116 3. 4. 5. 6. 7. 8. FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 9. 10. 11. 12. 13. Understand angle relationships formed by parallel lines and transversals. Be able to communicate geometric relationships both orally and in written form. Understand and analyze tessellations. Prove triangle congruence and similarity based on the postulates and theorems. Determine area, perimeter, and circumference of polygons and circles using postulates and theorems. Understand and recognize various space figures such as spheres, cubes, pyramids, and prisms. Understand and use trigonometric ratios to solve problems. 14. 15. CONTENT, SCOPE, AND SEQUENCE 1. Data Organization 2. Distance and Measurement 3. Angles and Triangles 4. Parallel Lines and Quadrilaterals 5. Similarity and Scale Change 6. Pythagorean Theorem 7. Perimeter, Area, and Volume 8. Circles 9. Process of Geometric Proofs 10. Space Figures 11. Coordinate Geometry 12. Trigonometry TEACHING STRATEGIES Cooperative learning, lecture and discussion, investigations, writing in math, manipulatives, student projects, use of technology, teacher modeled constructions and drawings. ASSESSMENT METHODS PAGE 117 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS Tests, quizzes, homework, projects, student presentations, warm-ups, problem of the week, portfolios, group work, teacher observations. SUPPLEMENTARY MATERIALS/EQUIPMENT Calculators, protractors, compasses and straightedges, manipulatives, overhead projector. PAGE 118 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS STATISTICS Repeat for Credit: No Graduation Requirement: Mathematics Grades: 11-12 Year Course (10 units) Course Code: 2670 COURSE DESCRIPTION This course is an introduction to the vocabulary and techniques involved in collecting and analyzing data in the fields of business, social sciences, natural sciences, behavioral sciences, and agriculture. Topics in probability are included. Students will use calculators and the computer. Activities and projects give students experience using their skills. PRE-REQUISITE: Completion of Algebra II with a C or better. Pre-Calculus or Transition to College Math or Calculus (may be taken concurrently with Pre-Calculus or Calculus) COURSE STANDARDS AND OBJECTIVES 1.0 The students will develop an understanding of descriptive statistics and its uses. The student will: 1.1 1.2 Construct a frequency distribution and a histogram. Interpret and draw circle graphs, bar graphs, pictographs, and stem-and-leaf diagrams. Understand misleading presentations of data. Understand and calculate measures of central tendency (mean, median, mode) and measures of variation (range, variance, standard deviation). 1.3 1.4 2.0 The students will develop an understanding of probability, especially in its relation to statistics. The student will: 2.1 2.2 Use counting methods (e.g., multiplication rule, permutations and combinations) to count the number of outcomes in a sample space or an event. Understand the concepts of independent and dependent events and how they PAGE 119 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS are related to compound events and conditional probability. 2.3 2.4 Understand Bayes' Formula for determining probability. Understand the concept of discrete probability distributions and compute the mean and standard deviation of the distributions. Understand the normal distribution and its applications (e.g., quality control, testing, scheduling). 2.5 3.0 The students will understand the methods and uses of inferential statistics. The student will: 3.1 3.2 Understand what a random sample is and how to obtain one. Compute the standard error of the mean and to use the Central Limit Theorem to make predictions about and calculate probabilities for the sample means. Understand sampling distributions and confidence intervals. Know how large a sample is needed to have the mean fit within a maximum allowable error. Know how to test hypotheses concerning means, differences between means, and proportions for large and small samples. Determine whether two variables are related by use of linear correlation and regression. Use chi-square tests for testing the equality of proportions, independent of data on a contingency table, and goodness of fit to a mathematical expectation. 3.3 3.4 3.5 3.6 3.7 PAGE 120 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS CONTENT, SCOPE, AND SEQUENCE 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. Definition of statistics and observation of its uses and applications Description of sample data and graphical displays of data Measures of central tendency: mean, median and measures of dispersion: standard deviation Histograms and frequency curves Probability, combinations, permutations Binomial distribution Normal distributions Confidence intervals Hypothesis testing Sampling and the Central Limit Theorem Linear correlation and regression Chi-Square distributions ASSESSMENT METHODS Completion of assignments and projects, group participation, teacher designed tests and quizzes, and class participation. SUPPLEMENTARY MATERIALS/EQUIPMENT Calculators and manipulatives PAGE 121 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS TECHNICAL MATH Repeat for Credit: No Graduation Requirement: Math/Vocational Education Grades: 11-12 Year Course (10 units) Course Code: 7970 COURSE DESCRIPTION This course provides students with a basic understanding of math concepts as they apply to industrial technology. Units of study will include measuring tools, geometry, trigonometry, surveying, cost estimating, ohms law, pulley and gear ratios, and personal finance. PRE-REQUISITE: Pre-Geometry or Algebra I COURSE STANDARDS AND OBJECTIVES 1.1 The students will understand the basics of measurement. The student will: 1.1 1.2 1.3 1.4 2.0 Use common measuring tools used in industry. Use properly conventional and metric units of measure. Use conversion charts. Perform calculations using various units of measure. The students will understand the fundamental of perimeter, area, and volume. The student will: 2.1 2.2 2.3 Know and use proper formulas to calculate perimeter, area, and volume. Know the components used in a basic hydraulic system. Use proper formulas to calculate pressure, flow, and volume problems for a hydraulic system. 3.0 The students will understand the basics of geometry. PAGE 122 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS The student will: 3.1 3.2 3.3 3.4 3.5 3.6 4.0 Know the names of geometric shapes. Use layout tools to layout geometric shapes. Develop patterns and stretch-outs for sheet metal. Develop transitions for sheet metal. Identify sheet metal by gage and type. Develop cost estimates for sheet metal projects. The students will understand the fundamentals of applied trigonometry. The student will: 4.1 4.2 4.3 Know the properties of right triangle and their use in construction. Use a right triangle to lay out a building foundation Know common terms used in roof construction, such as run, rise, span, and overhang. Use a framing square to lay out a rafter and a stair. Understand how a right triangle is used in vector addition in RL circuits. 4.4 4.5 5.0 The students will know the principles of surveying. The student will: 5.1 5.2 5.3 5.4 5.5 5.6 5.7 Use the pacing methods to estimate distances. Know how to tape off a distance. Measure land areas and calculate square footage and acreage. Know how to set up a building level. Know how to perform differential leveling problems. Know how to lay out and level a foundation. Know how to identify land parcels using the U.S. public land surveys. PAGE 123 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 5.8 5.9 5.10 6.0 Read maps and identify townships and sections. Know how to run a compass course using azimuths and bearings. Know careers that require knowledge of surveying. Students will be introduced to cost estimating. The student will: 6.1 6.2 6.3 6.4 Know common construction materials and methods by which they are priced. Read construction blueprints and identify specified materials. Know common building components. Estimate accurately the cost of projects constructed from lumber, metal, and concrete. 7.0 The students will understand basic relationships of electricity. The student will: 7.1 7.2 7.3 7.4 8.0 Know the components required for a basic electrical circuit. Understand series, parallel, and combination circuits. Know and use OHMs law formulas. Know and use power formulas. The students will understand the principles of mechanical power transmission systems. The student will: 8.1 8.2 Understand common methods of mechanical power transmission. Understand first, second, and third class levers and calculate their mechanical advantage. Know types of gears. 8.3 9.0 8.4 Calculate gear ratios. The students will know the basics of personal finance. The student will: PAGE 124 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11 9.12 Know terms used on paycheck stubs. Calculate gross and net pay. Know how to complete a state and federal tax form. Propose a realistic spending plan. Identify and calculate transportation costs. Know various types of retirement investment plans. Write a bank deposit properly. Know how to write a check properly. Know how to balance a monthly check statement. Understand the advantages and disadvantages of the use of credit. Know the importance of auto, health, and life insurance. Understand common insurance terms. CONTENT, SCOPE, AND SEQUENCE 1. Measuring a. units of measure b. measuring tool identification and use c. reading tools of measurement (1) U.S. conventional rules, scales to include 1/8", 1/16", 1/32", 1/64" (addition, subtraction, multiplication, division of fractions with these common denominators) (2) S.I. metric rules (3) triangular drafting scales (4) mechanical engineer's scales (5) micrometers, inch and metric (6) dial and vernier calipers (7) torque wrenches (8) miscellaneous gages, thread pitch, drill, radius, feeler, pressure, sheet metal (9) electrical meters Perimeter, area, volume a. review of common geometric shapes PAGE 125 2. FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS b. c. d. 3. calculation of perimeter, area, volume cylinder displacement hydraulics, pressure, flow, and volume Basic Geometry a. review of geometric shapes b. layout of geometric shapes c. pattern and stretch out development d. transition development e. sheet metal material identification and cost estimation Applied Trigonometry a. right triangles b. stair and rafter layout c. vector addition, RL circuits Surveying a. distance measuring (1) pacing (2) taping/chaining (3) acreage calculations b. introduction to leveling (1) safety and setup of building level (2) differential leveling (3) foundation layout and leveling c. U.S. public land surveys (1) subdivision of township and sections (2) map reading d. compass reading (1) azimuths and bearings (2) running a compass course e. careers in surveying Introduction to cost estimating a. construction materials identification and methods of pricing (1) lumber, board feet, and linear measure (2) plyboard, wallboard, and sheeting materials (3) concrete and masonry materials (4) paint and wallpaper b. basic construction blueprint reading c. building estimate d. metal identification (1) shapes, sizes, and weights PAGE 126 4. 5. 6. FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS (2) (3) 7. cost calculation blueprint reading and cost estimation Basic relationships of electricity a. ohm's law formulas b. power formulas c. series circuits, parallel circuits, combination circuits Mechanical power transmission systems a. inclined plane and mechanical advantage b. first, second, third class levers and their mechanical advantages c. pulleys and mechanical advantage d. gears (1) types of gears (2) gear ratios e. chain and sprocket systems Personal finance a. reading paycheck stubs (1) terms (2) calculating gross and net pay b. income tax preparation (1) W-2 form (2) completing a tax return form c. personal budget preparation and savings and investments (1) personal living expenses (2) transportation costs (3) retirement planning d. banking (1) making bank deposits (2) writing checks (3) using the check register (4) using a savings account (5) use of credit and interest rates e. insurance (1) auto (2) health (3) life (4) workman's compensation (5) liability insurance 8. 9. ASSESSMENT METHODS PAGE 127 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS Teacher observation, lab assignments, class assignments, written exams, oral exams SUPPLEMENTARY MATERIALS/EQUIPMENT Measuring tools such as micrometers, calipers and gages, torque wrenches, drafting equipment, electrical meters, building level and surveying tapes PAGE 128 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS TRANSITION TO ALGEBRA II Repeat for Credit: No Graduation Requirements: Mathematics Grades: 10-12 Year Course (10 units) Course Code: 2460 COURSE DESCRIPTION This course is designed for students who have passed Algebra I and geometry, but do not feel confident about the next level of mathematics, which is Algebra II. This course places an emphasis on cooperative learning and problem solving. Topics covered are: probability and statistics, functions, inverse functions, linear programming, trigonometry, series and sequence of numbers, problem solving, and a review of Algebra concepts. PRE-REQUISITE: Geometry COURSE STANDARDS AND OBJECTIVES Students will: 1. Use a variety of strategies for solving problems (e.g., graphs, equivalent representations, working backwards). 2. Understand methods to solve problems which involve sequences and series (e.g., looking for a pattern, algebraic manipulation) Understand counting procedures and reasoning (e.g., combinations, permutation, probability). Understand inverse functions and their graphs. Use trigonometric functions to solve real-world problems. Understand the relationship between equations and their graphs. Express mathematical ideas both orally and in written form. 3. 4. 5. 6. 7. PAGE 129 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS CONTENT, SCOPE, AND SEQUENCE 1. 2. 3. 4. 5. 6. 7. 8. Problem solving Sequences and series Counting and probability Algebra review with graphing calculator Linear programming Functions Inverse functions Trigonometry TEACHING STRATEGIES Cooperative learning, lecture and discussion, investigations, problem of the week, writing in math, scientific/graphing calculators, open-ended questions. ASSESSMENT METHODS Tests, quizzes, homework, projects, student presentations, warm-ups, portfolios, group work, teacher observation. SUPPLEMENTARY MATERIALS/EQUIPMENT Algebra Review Warm-Ups, Algebra With Pizzazz, calculators (both scientific and graphing), manipulative. PAGE 130 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS TRANSITION TO COLLEGE MATH Repeat for Credit: No Graduation Requirements: Mathematics Grades: 10-12 Year Course (10 units) Course Code: 2560 COURSE DESCRIPTION This course is designed for students who have completed Algebra II but do not feel confident about taking trigonometry. On completion of this course students should be better prepared to pass college screening examinations and will have more options open in college majors which require mathematics. Topics covered are: probability and statistics, functions, inverse functions, linear programming, trigonometry, series and sequence of numbers, Algebra II review, and a special emphasis on problem solving and cooperative groups. PRE-REQUISITE: Algebra II COURSE STANDARDS AND OBJECTIVES Students will: 1. Use a variety of strategies for solving problems (e.g., graphs, equivalent representations, working backwards). 2. Understand methods to solve problems which involve sequences and series (e.g., looking for a pattern, algebraic manipulation) Understand counting procedures and reasoning (e.g., combinations, permutation, probability). Understand how to break a complex problem into simpler parts or use a similar problem type to solve a problem. Understand inverse functions and their graphs. Use trigonometric functions to solve real-world problems. Understand the relationship between equations and their graphs. 3. 4. 5. 6. 7. 8. Solve multi-step problems. CONTENT, SCOPE, AND SEQUENCE PAGE 131 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 1. 2. 3. 4. 5. 6. 7. 8. 9. Problem solving Sequence and series Counting and probability Linear programming Functions Inverse functions Trigonometry Conic sections Advanced problem solving TEACHING STRATEGIES Cooperative learning, lecture and discussion, investigations, problem of the week, writing in math, scientific/graphing. ASSESSMENT METHODS Tests, quizzes, homework, projects, student presentations, warm-ups, portfolios, group work, teacher observation. SUPPLEMENTARY MATERIALS/EQUIPMENT Algebra Review Warm-Ups, Algebra with Pizzazz, calculators (both scientific and graphinga0, manipulatives. PAGE 132 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS Appendix State Content Standards Advanced Placement Probability and Statistics (included in Statistics) Linear Algebra (included in Algebra I, Algebra II, and Pre-Calculus) Mathematical Analysis (included in Geometry and Pre-Calculus) Trigonometry (included in Geometry and Pre-Calculus) PAGE 133 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS STATE CONTENT STANDARDS ADVANCED PLACEMENT PROBABILITY AND STATISTICS This discipline is a technical and in-depth extension of probability and statistics. In particular, mastery of academic content for advanced placement gives students the background to succeed in the Advanced Placement examination in the subject. 1.0 Students solve probability problems with finite sample spaces by using the rules for addition, multiplication, and complementation for probability distributions and understand the simplifications that arise with independent events. 2.0 Students know the definition of conditional probability and use it to solve for probabilities in finite sample spaces. 3.0 Students demonstrate an understanding of the notion of discrete random variables by using this concept to solve for the probabilities of outcomes, such as the probability of the occurrence of five or fewer heads in 14 coin tosses. 4.0 Students understand the notion of a continuous random variable and can interpret the probability of an outcome as the area of a region under the graph of the probability density function associated with the random variable. 5.0 Students know the definition of the mean of a discrete random variable and can determine the mean for a particular discrete random variable. 6.0 Students know the definition of the variance of a discrete random variable and can determine the variance for a particular discrete random variable. 7.0 Students demonstrate an understanding of the standard distributions (normal, binomial, and exponential) and can use the distributions to solve for events in problems in which the distribution belongs to those families. 8.0 Students determine the mean and the standard deviation of a normally distributed random variable. 9.0 Students know the central limit theorem and can use it to obtain approximations for probabilities in problems of finite sample spaces in which the probabilities are distributed binomially. 10.0 Students know the definitions of the mean, median, and mode of distribution of data and can compute each of them in particular situations. 11.0 Students compute the variance and the standard deviation of a distribution of data. PAGE 134 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 12.0 Students find the line of best fit to a given distribution of data by using least squares regression. 13.0 Students know what the correlation coefficient of two variables means and are familiar with the coefficient’s properties. 14.0 Students organize and describe distributions of data by using a number of different methods, including frequency tables, histograms, standard line graphs and bar graphs, stem-and-leaf displays, scatterplots, and box-and-whisker plots. 15.0 Students are familiar with the notions of a statistic of a distribution of values, of the sampling distribution of a statistic, and of the variability of a statistic. 16.0 Students know basic facts concerning the relation between the mean and the standard deviation of a sampling distribution and the mean and the standard deviation of the population distribution. 17.0 Students determine confidence intervals for a simple random sample from a normal distribution of data and determine the sample size required for a desired margin of error. 18.0 Students determine the P-value for a statistic for a simple random sample from a normal distribution. 19.0 Students are familiar with the chi-square distribution and chi-square test and understand their uses. PAGE 135 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS STATE CONTENT STANDARDS LINEAR ALGEBRA The general goal in this discipline is for students to learn the techniques of matrix manipulation so that they can solve systems of linear equations in any number of variables. Linear algebra is most often combined with another subject, such as trigonometry, mathematical analysis, or pre-calculus. 1.0 Students solve linear equations in any number of variables by using Gauss-Jordan elimination. 2.0 Students interpret linear systems as coefficient matrices and the Gauss-Jordan method as row operations on the coefficient matrix. 3.0 Students reduce rectangular matrices to row echelon form. 4.0 Students perform addition on matrices and vectors. 5.0 Students perform matrix multiplication and multiply vectors by matrices and by scalars. 6.0 Students demonstrate an understanding that linear systems are inconsistent (have no solutions), have exactly one solution, or have infinitely many solutions. 7.0 Students demonstrate an understanding of the geometric interpretation of vectors and vector addition (by means of parallelograms) in the plane and in three-dimensional space. 8.0 Students interpret geometrically the solution sets of systems of equations. For example, the solution set of a single linear equation in two variables is interpreted as a line in the plane, and the solution set of a two-by-two system is interpreted as the intersection of a pair of lines in the plane. 9.0 Students demonstrate an understanding of the notion of the inverse to a square matrix and apply that concept to solve systems of linear equations. 10.0 Students compute the determinants of 2 x 2 and 3 x 3 matrices and are familiar with their geometric interpretations as the area and volume of the parallelepipeds spanned by the images under the matrices of the standard basis vectors in two-dimensional and three-dimensional spaces. 11.0 Students know that a square matrix is invertible if, and only if, its determinant is PAGE 136 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS nonzero. They can compute the inverse to 2 x 2 and 3 x 3 matrices using row reduction methods or Cramer’s rule. 12.0 Students compute the scalar (dot) product of two vectors in n-dimensional space and know that perpendicular vectors have zero dot product. PAGE 137 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS STATE CONTENT STANDARDS MATHEMATICAL ANALYSIS This discipline combines many of the trigonometric, geometric, and algebraic techniques needed to prepare students for the study of calculus and strengthens their conceptual understanding of problems and mathematical reasoning in solving problems. These standards take a functional point of view toward those topics. The most significant new concept is that of limits. Mathematical analysis is often combined with a course in trigonometry or perhaps with one in linear algebra to make a yearlong pre-calculus course. 1.0 Students are familiar with, and can apply, polar coordinates and vectors in the plane. In particular, they can translate between polar and rectangular coordinates and can interpret polar coordinates and vectors graphically. 2.0 Students are adept at the arithmetic of complex numbers. They can use the trigonometric form of complex numbers and understand that a function of a complex variable can be viewed as a function of two real variables. They know the proof of DeMoivre’s theorem. 3.0 Students can give proofs of various formulas by using the technique of mathematical induction. 4.0 Students know the statement of, and can apply, the fundamental theorem of algebra. 5.0 Students are familiar with conic sections, both analytically and geometrically: 5.1 Students can take a quadratic equation in two variables; put it in standard form by completing the square and using rotations and translations, if necessary; determine what type of conic section the equation represents; and determine its geometric components (foci, asymptotes, and so forth). Students can take a geometric description of a conic section—for example, the locus of points whose sum of its distances from (1, 0) and (-1, 0) is 6—and derive a quadratic equation representing it. 5.2 6.0 Students find the roots and poles of a rational function and can graph the function and locate its asymptotes. 7.0 Students demonstrate an understanding of functions and equations defined parametrically and can graph them. PAGE 138 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 8.0 Students are familiar with the notion of the limit of a sequence and the limit of a function as the independent variable approaches a number or infinity. They determine whether certain sequences converge or diverge. PAGE 139 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS STATE CONTENT STANDARDS TRIGONOMETRY Trigonometry uses the techniques that students have previously learned from the study of algebra and geometry. The trigonometric functions studied are defined geo-metrically rather than in terms of algebraic equations. Facility with these functions as well as the ability to prove basic identities regarding them is especially important for students intending to study calculus, more advanced mathematics, physics and other sciences, and engineering in college. 1.0 Students understand the notion of angle and how to measure it, in both degrees and radians. They can convert between degrees and radians. 2.0 Students know the definition of sine and cosine as y- and x-coordinates of points on the unit circle and are familiar with the graphs of the sine and cosine functions. 3.0 Students know the identity cos2 (x) + sin2 (x) = 1: 3.1 Students prove that this identity is equivalent to the Pythagorean theorem (i.e., students can prove this identity by using the Pythagorean theorem and, conversely, they can prove the Pythagorean theorem as a consequence of this identity). Students prove other trigonometric identities and simplify others by using the identity cos2 (x) + sin2 (x) = 1. For example, students use this identity to prove that sec2 (x) =tan2 (x) + 1. 3.2 4.0 Students graph functions of the form f(t) = A sin (Bt + C) or f(t) = A cos (Bt + C) and interpret A, B, and C in terms of amplitude, frequency, period, and phase shift. 5.0 Students know the definitions of the tangent and cotangent functions and can graph them. 6.0 Students know the definitions of the secant and cosecant functions and can graph them. 7.0 Students know that the tangent of the angle that a line makes with the x-axis is equal to the slope of the line. 8.0 Students know the definitions of the inverse trigonometric functions and can graph the functions. 9.0 Students compute, by hand, the values of the trigonometric functions and the inverse trigonometric functions at various standard points. PAGE 140 FAIRFIELD-SUISUN UNIFIED SCHOOL DISTRICT GRADES K-12 MATHEMATICS 10.0 Students demonstrate an understanding of the addition formulas for sines and cosines and their proofs and can use those formulas to prove and/or simplify other trigonometric identities. 11.0 Students demonstrate an understanding of half-angle and double-angle formulas for sines and cosines and can use those formulas to prove and/or simplify other trigonometric identities. 12.0 Students use trigonometry to determine unknown sides or angles in right triangles. 13.0 Students know the law of sines and the law of cosines and apply those laws to solve problems. 14.0 Students determine the area of a triangle, given one angle and the two adjacent sides. 15.0 Students are familiar with polar coordinates. In particular, they can determine polar coordinates of a point given in rectangular coordinates and vice versa. 16.0 Students represent equations given in rectangular coordinates in terms of polar coordinates. 17.0 Students are familiar with complex numbers. They can represent a complex number in polar form and know how to multiply complex numbers in their polar form. 18.0 Students know DeMoivre’s theorem and can give nth roots of a complex number given in polar form. 19.0 Students are adept at using trigonometry in a variety of applications and word problems. PAGE 141

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