Mathematics

Overview

MATHEMATICS PHILOSOPHY

The Mathematics program in Madison prepares students to be mathematically literate, understand major mathematics concepts, possess computational facility and have the ability to apply these understandings to situations in daily life. Our goal is for students to make informed decisions about the world around them and to have the interest and confidence to meet an increasingly quantitative, data-rich global society’s needs and challenges.

Teachers carefully choose complex mathematical tasks in which to engage students. Students draw on knowledge from a wide variety of mathematical topics, sometimes approaching the same problem from different mathematical per­spectives or representing the mathematics in different ways, until they find methods that enable them to make progress. Alone or in groups and with appropriate access to technology, they work productively and reflectively, with the guidance of their teachers. Students communicate, orally and in writing, their ideas and results to justify their argument and seek feedback from their audience.

DELIVERY METHOD / VALUES

To teach math effectively teachers of mathematics need strong content knowledge combined with good instructional skills and sound pedagogy. Teachers create inquiries, choose problems, and explore multiple solution pathways that will foster student engagement. Teachers regularly assess understanding by asking the student to justify, in a way that is appropriate to the student’s mathematical maturity, why a particular mathematical statement is true, or where a mathematical rule comes from. Teachers take a balanced approach to mathematics education, which places equal importance on conceptual understanding, computational and procedural fluency and problem solving through a variety of strategies and tools.

STANDARDS

In the creation of the Madison Public Schools mathematics curriculum, Madison’s 21st Century Capacities served as the foundation and Common Core State Standards (CCSS) were carefully considered for developmental and conceptual appropriateness for each course.

The curriculum communicates high standards and expectations for students to routinely apply the CCSS Mathematical Practices:

    • Make sense of problems and persevere in solving them
    • Reason abstractly and quantitatively
    • Construct viable arguments and critique the reasoning of others
    • Model with mathematics
    • Use appropriate tools strategically
    • Attend to precision
    • Look for and make use of structure

LOCAL VALUES IN CURRICULUM DESIGN

These priorities were used to develop K-12 Transfer Goals that are most frequently revisited and assessed in curricular units throughout a student’s academic career:

Mathematics Transfer Goals

Students will be able to…

  1. model relationships among quantities.
  2. manipulate equations/expressions or objects to create order and establish relationships.
  3. represent and interpret patterns in numbers, data and objects.
  4. draw conclusions about graphs, shapes, equations, or objects.
  5. demonstrate fluency with math facts, computation and concepts.
  6. make sense of a problem, initiate a plan, execute it, and evaluate the reasonableness of the solution.
  7. use appropriate tools to make reaching solutions more efficient, accessible and accurate.
  8. apply familiar mathematical concepts to a new problem or apply a new concept to rework a familiar problem.
  9. evaluate the accuracy and efficiency of a given solution.
  10. justify reasoning using clear and appropriate mathematical language.

The capacities are embedded and assessed in each unit design. Every content area is unique, and some have clusters by the very nature of their discipline.


Guiding Documents

Kindergarten

Kindergarten – MATH Curriculum Overview 2016-2017

UnitDescription

Unit A

Working Through Five

In the first unit, we are establishing our math workshop environment for the year.Routines of Think-Pair-Share and choral counting develop the number word sequence to 20.Three major concepts make it possible for students to answer the question, “How many?” To do so, students must be able to apply the number word sequence, and possess both one-to-one correspondence and cardinality. Arranging objects in an orderly way before counting fosters students’ ability to recognize the quantity without counting.. The five-frame, ten-frame, and finger patterns are key models featured in this unit to help students subitize quantities from 0 to 10. This unit sets a solid foundation in learning combinations to 10.In Number Talks and Daily Workouts, students will also learn to recognize and name two-dimensional shapes in various contexts, practice numeral recognition and counting forward and backward to 10, and begin learning combinations of 5 through activities surrounding five-frames.

21st Century Capacities: Product Creation

Unit B

Five and Ten, Do It Again!

Students continue to develop the concepts addressed in the first unit related to counting, instant recognition of numbers, number sequence, one-to-one correspondence, and cardinality.The question of “how many?” begins to shift to “which is more and which is less?” Student activities focus on promoting flexible ways ofrepresenting and recognizing quantities, not memorizing combinations.

In this unit, students visually represent numbers by using five frames, ten frames, number racks, standard finger patterns, and tallies.Theyfind and recognize combinations of numbers that make 5, recognize and compare quantities within 10, andbegin simple addition and subtraction.

21st Century Capacities: Synthesizing

Unit C

How Much? How Many?

This unit introduces counting through the use of the number line and length measurement. The concept of equality, which was formally introduced in the previous unit, is developed further in this unit when students measure length and work with money. Students are introduced to the number line model through hands-on activities that help them interpret the structure of the number line. Students investigate the number line model as a tool that can be used to order and compare numbers less than 20.The number line is used learn how to solve addition and subtraction problems within 10.They investigate terms such as length, longer and shorter.In addition, they are exposed to the value of a penny and a nickel.

21st Century Capacities: Analyzing

Unit D

Shaping Up

Students begin to examine, identify, compare, and sort two-dimensional and three-dimensional shapes.They explore largely through play,how to describe the world around them using geometry terms.Characteristics of shapes are realized through careful analysis as students notice how some are helpful in defining the geometry of a shape, while others are not.They will construct and deconstruct a variety of shapes in order to build both realistic and imagined objects.

21st Century Capacities: Analyzing, Product Creation

Unit E

Ten and Then Some

In this unit, students will use a variety of materials to represent mathematical situations.Students will read, write, and compare numerals with one-to-one correspondence and cardinality.They will also relate comparing numbers to comparing the weight of two objects. Students will break numbers into their component parts based on place value in order to recognize numbers 11-20 as “ten and some more”.They also compare numbers to 20 using greater than and less than.

21st Century Capacities:Analyzing, Presentation

Unit F

Problems All Around Us

In the final unit, students strengthen their understanding of quantity, number combinations and written notation to 20. They spend more time developing fluency with addition and subtraction to 5 and continue to develop strategies for adding and subtracting to 10. A deeper understanding of subtraction is developed as they begin to see subtraction as both taking away and comparing.Students learn to identify and solve problems by applying known facts or using materials to model and then solve problems.

21st Century Capacities:Analyzing, Problem Identification

Grade 1

Grade 1 – MATH Curriculum Overview 2016-2017

UnitDescription

Unit A

Numbers All Around Us

To begin the year, students will establish routines for the math workshop and Number Corner environment. Students use Work Places as regular opportunities to socially engage in mathematical learning while sharing strategies with fellow students. Small guided math groups are facilitated during this time to help students consolidate or extend their learning.

The first unit is designed to help students develop a sense of numbers and their relationships to one another through looking at several key counting and number concepts. The unit begins with organizing and counting objects moving to counting forward and backward and grouping and counting in 2s, 5s, and 10s. Subitizing, the ability to know a quantity without counting each individual part of a set, is developed through the use of several models such as number racks, ten frames, tally marks, graphs, and number lines. By understanding the structure of the models, students can begin to see numbers in parts and groups. Being able to subitize is a key step in developing strategies to add and subtract. In Kindergarten, students worked in depth on this skill so this unit will review and reinforce how to “see numbers”. As first graders, students will begin to develop flexibility with numbers in problem contexts involving combining and separating numbers. By the end of the unit, students understand how to use, visualize and create models such as number racks and ten frames to solve a problem that they have analyzed in order to find various solutions.

21st Century Capacities: Product Creation, Analyzing

Unit B

Equal: To Be or Not to Be?

In this unit, students will continue modeling with mathematical tools to build confidence using efficient and effective strategies to add and subtract single-digit numbers. They will continue to develop flexibility with numbers through visualizing combinations of 5 and 10. They develop mastery with number facts up to 10 and the use of strategies to model number families to 20. While students have been using the equals sign before, this is the first time they learn that two expressions are of equal value (rather than just the symbol that means “the answer”). They explore finding missing addends and subtrahends (the number being subtracted). Students identify, select, and implement efficient strategies when problem solving.

21st Century Capacities: Synthesizing, Problem Identification

Unit C

Leap Frogging

This unit revolves around the number line, helping students visualize number relationships in order to count and calculate. Closed and open number lines are used both as models of our number system, as well as models for beginning operations with addition and subtraction. Numbers lines with both large scales (skip-count by 10s or 50s) and small scales (skip count by 1s or 5s) ranging to 120 are introduced. Students learn that addition and subtraction problems can be solved in different ways, each of which anchors on fundamental understandings of number. As students become confident with the placement of numbers on the number line, they begin to use the number line to solve story problems. Students use their understanding of number placement to compare two numbers with application to real world scenarios as they measure, compare, and order measurements of penguins, finding differences and writing inequality statements.

21st Century Capacities: Synthesizing , Product Creation

Unit D

One of These Shapes Is Not Like the Others

In this unit, students build upon their Kindergarten understanding to examine, identify, compare, and sort two-dimensional and three-dimensional shapes. They explore how to describe the world around them using geometry terms. Characteristics of shapes are realized through careful analysis as students notice how some are helpful in defining the geometry of a shape, while others are not. They will construct and deconstruct a variety of shapes in order to build both realistic and imagined objects and develop an understanding of how shapes can be divided into equal parts.

21st Century Capacities: Collective Intelligence;Imagining

Unit E

The Problem with Penguins

In this unit, first graders will continue to develop fluency with addition and subtraction within 10 and use strategies within 20. They usetools to model, solve, and create story problems of all types.Through careful analysis, they will begin to recognize patterns within problem types within an engaging context of penguins.

21st Century Capacities:Analyze, Imagining

Unit F

To 100 and Beyond!

The focus of this unit is on place value, deepening understanding of numbers to 120.Students will estimate, count, compare, add, and subtract two-digit numbers using models including the number line and sticks & bundles.Computation strategies, such as making “jumps” of 2s, 5s, and 10s on pathways develop students’ problem solving ability.The use of coins is incorporated to further explore place value at the end of the unit.

21st Century Capacities:Synthesizing

Grade 2

Grade 2 - Math Curriculum Overview 2016-2017

UnitDescription

Unit A

Figure the Facts

To begin the year, students will establish routines within the math workshop and Number Corner environment. Students use Work Places as regular opportunities to socially engage in mathematical learning while sharing strategies with fellow students. Small guided math groups are facilitated during this time to help students consolidate or extend their learning.

In this first unit, students develop confidence and fluency with number relationships, operations, and facts in the range of 0 to 20. This operational sense depends heavily on a solid number foundation developed in earlier grades. The goal of this unit is to help students develop solid understandings of addition and subtraction and some of the ways in which these two operations complement each other, which will lead to the development of confidence and fluency with the number facts as they appear in real-world contexts. Fact retrieval is based on models and the use of strategies as opposed to rote memorization and recall. They can create a variety of combinations of 20 and justify their solutions using models, pictures and words.

21st Century Capacities: Product Creation

Unit B

Climbing the Beanstalk

In the second unit, students build upon their operational sense with number relationships to 20 developed in the first unit as they explore base ten concepts and models within 1,000. Students focus on the first three place value units: ones, tens, and hundreds.

Students use models for grouping including tallying with bundled objects, counters, base ten area pieces, and the number line.

They solve word problems involving addition and subtraction within 100 using splitting strategies and the open number line. In addition, students recognize that subtraction is finding the distance between 2 points on a number line.

21st Century Capacities: Product Creation, Synthesizing

Unit C

Sizing It Up

The focus of this unit shifts from earlier work with addition, subtraction and place value concepts to those concerning measurement. Students will discover the need for a standard unit of measurement as their attempts to measure without one become widely varied and confusing. Students are encouraged in a playful approach with inchworms, footworms, and yardworms to recognize connections and relationships between units of measure. They explore the effect that the size of the unit has on the corresponding measurement. This understanding lends itself to informal pictorial experience with ratios and proportional reasoning, laying groundwork for the multiplicative thinking required in third grade. With this understanding comes greater ability to justify a most appropriate tool and/or unit to use when measuring objects of various sizes. Because of this, students will also become more adept at making unit conversions.

21st Century Capacities: Synthesizing, Analyzing

Unit D

How ... is 1000?

In this unit, students develop a deeper understanding of place value with numbers to 1,000. This understanding builds upon concepts and models students refined for adding and subtracting within 100 in previous units. Students compose and decompose numbers based on place value using multiple models and representations including sticks, cubes, paper clips and coins in order to understand sets of 10 and 100 as single entities. Students develop a greater place value understanding as they realize that any number can be decomposed based on place value groupings. Students see that multi-digit numbers are formed by following the same counting pattern present in single digit counting.

21st Century Capacities: Analyzing, Product Creation

Unit E

Name It, Make It, Shape It, Break It, Build It , Move It and Compare It

This unit will build upon the two-dimensional concepts students learned in first grade as they investigate two-dimensional shapes, fractions (halves and fourths), congruence and symmetry using a variety of tools and models. Students will identify, describe, construct, draw, compare, contrast, and sort various types of triangles and quadrilaterals, as well as other shapes.

21st Century Capacities: Analyzing, Presentation

Unit F

The Ants Go Marching 10 by 10!

This unit incorporates concepts of multi-digit addition and subtraction within story problem contexts. Students will spend time working together to solve and create story problems involving adding and subtracting 3-digit numbers within real-world applications such as a toy store and party planning. Emphasis is placed on student-invented and generated strategies, such as concrete models, drawings, and strategies based on place value through 1,000.

21st Century Capacities:Synthesizing, Collective Intelligence

Grade 3

GRADE 3 – Mathematics Curriculum Overview 2016-2017

UnitDescription

Unit 1

Launching Addition and Subtraction Patterns

Unit 1 focuses on patterns in addition and subtraction facts, the pattern of adding 10s, and problem solving. The first lessons set the tone for the math workshop and establish expectations for working cooperatively on learning tasks. Students revisit the addition and subtraction strategies for facts to 20, which they learned in second grade. Later in the unit, the students apply the addition and subtraction strategies they have learned to add and subtract multi-digit numbers efficiently on the open number line. They also practice place value splitting with addition.

21st Century Capacities: Collective Intelligence, Analyzing

Unit 2

Multiplication: An Array of Strategies

In Unit 2, students begin to develop a conceptual understanding of multiplication. Investigations begin with contexts and problems that invite students to think about equal groups and multiplicative comparisons.Students are introduced to one-on-one counting, skip-counting and repeated addition as they develop a basic understanding of multiplication. They make use of a variety of models for multiplication, including equal groups, arrays, the number line, and ratio tables to develop strategies for multiplication.They apply what they have learned by solving problems that involve graphs and story problems with multiple steps.

21st Century Capacities: Analyzing, Product Creation

Unit 3

Multi-Digit Addition and Subtraction

This unit reviews and extends upon patterns within place value and larger addition and subtraction situations. The first lessons introduce the concept of rounding to the nearest ten and/or hundred, which is then used as a strategy to estimate and partition three-digit numbers in order to add and subtract efficiently. Students expand their repertoire of addition and subtraction strategies learned in second grade. In this unit, students will gain experiences and strategies for making sense of problems and communicating effectively about the accuracy and efficiency of various solutions.

21st Century Capacities:Problem Identification

Unit 4F

Fractions

In this unit, students will begin by building, comparing, and investigating relationships between unit and common fractions using several models including parts of a whole, parts of a set, and number line models. Using models such as an egg carton, ruler, 12 foot section of adding machine tape, and circle graph, students see real world applications for fractions.The unit then connects this to related work with data collection, representation, and interpretation.

21st Century Capacities: Synthesizing

Unit 4M

For Good Measure

This unit focuses on measurement concepts and skills. Students tell time to the minute and solve elapsed time problems. Then they explore why and how we measure concepts such as biggest, tallest, and fastest. Students estimate, measure, and compare the masses of different objects and work with volume to solve measurement-related story problems. The unit builds upon the strategies to add and subtract 3-digit numbers that were introduced in Unit 3.

21st Century Capacities: Synthesizing

Unit 5

Multiplication, Division & Area

Unit 5 returns to the study of multiplication, especially as it relates to division. Students again build arrays, but use them to model and solve division as well as multiplication problems. Story problems play a major role in the unit, helping students to connect their everyday experiences with division to more formal mathematical concepts. As they solve and pose story problems, students encounter different interpretations of division—area, sharing and grouping—and have numerous opportunities to build understandings of these different models and meanings. The connection between multiplication and division is also drawn through work that revolves around fact families.Toward the end of the unit, area isintroduced, a topic that will be revisited in Unit 6.

21st Century Capacities: Analyzing

Unit 6

Quadruple the Fun

In Unit 6, students analyze polygons in various contexts including in relationship to fractions, area and perimeter. They develop increasingly precise ways to describe, classify, and make generalizations about two-dimensional shapes, particularly quadrilaterals. Models such as tangrams, toothpicks, colored tiles, linear units, and geoboards help build an understanding that shared characteristics can define a larger category. Polygons are also measured in terms of perimeter and area. In addition, quadrilaterals are partitioned into parts with equal areas and the area of each equal part is expressed as a unit fraction of the whole.

21st Century Capacities: Analyzing, Design

Grade 4

GRADE 4 – Mathematics Curriculum Overview 2016-2017

UnitDescription

Unit 1

Multiplicative Thinking

This unit focuses on developing concepts related to multiplication and division through models (open number line, tile arrays, area model and the ratio table), strategies for multiplication facts and multiplicative comparisons. Students continue to transition from additive to multiplicative thinking, a process begun in third grade, by studying multiplicative comparisons presented in story problems involving both multiplication and division. The first lessons set the tone for the year with community building andestablish expectations for working cooperatively on learning tasks within the math workshop. Students set goals for their learning related to multiplication fact fluency which they will reflect upon throughout the year.

21st Century Capacities: Collective Intelligence, Reflection

Unit 2

Multiplication, Division and Strategies Oh My!

This unit focuses on an applied and visual approach to multi-digit multiplication and early division with remainders. Students deepen their understandings of multiplication and division continuing on the journey to multiplicative reasoning developed in unit 1. They apply number sense to developing useful models such as the ratio table and the array or area model and mental strategies such as doubling and halving for multiplying and dividing with an increasing degree of efficiency. They also continue to develop proficiency with basic multiplication and division facts. As they are solving various problems, students justify their reasoning using clear models and mathematical language as they create products.

21st Century Capacities: Synthesizing, Product Creation

Unit 3

Full of Wholes

In this unit, students use concrete manipulatives and visual models to explore unit fractions, common fractions, mixed numbers, equivalent fractions, and decimals as well as addition and subtraction of fractions. Students begin to understand how two fractions with unlike numerators and unlike denominators can be equal and they develop methods for generating and recognizing equivalent fractions. The connection between unit fractions and common fractions leads toward multiplying fractions by whole numbers. Fraction works extends into decimals by considering the equivalence of tenths and hundredths. Students must understand that comparisons of fractions or decimals are valid only when the two fractions or decimals refer to the same whole.

21st Century Capacities:Analyzing

Unit 4

It All Adds Up To This

Unit 4 focuses on place value to 1,000,000 and multi-digit addition strategies. In this unit, a strand of numeric exploration and investigation that was launched in Grade 1 and developed throughout Grades 2 and 3 comes to a logical conclusion as students are introduced to the standard, or traditional, algorithms for multi-digit addition and subtraction. They also continue to explore other strategies for addition and subtraction determining which strategy is most efficient to solve a given problem.Later in the unit, students apply their knowledge of place value and multi-digit computation to solve problems involving length, time, volume, mass and weight.

21st Century Capacities: Synthesizing, Presentation

Unit 5

Geometry and Measurement: We Believe in Angles

The work within this unit gives students opportunities to compare, analyze, classify, and measure polygons and angles.They develop understanding of numerous properties of shapes, including symmetry, congruence, parallel and perpendicular sides.Determining measurements such as perimeter, area, and angle measurement are introduced. The purpose of this unit is to deepen their thinking from visualization and analysis stages to that of informal deduction, or “if-then” reasoning.

21st Century Capacities: Analyzing

Unit 6

A Fraction of This, A Multiple of That

The instruction in Unit 6 picks up where Unit 2 left off, further developing the skills and concepts associated with multi-digit multiplication and division. Students discover that the models they have been using and strategies they have developed for multi-digit multiplication work equally well for division. They learn to divide numbers into the thousands by 1-digit numbers, using strategies based on the relationship between multiplication and division, as well as on place value, and the properties of operations. Students also explore the relationship between division and fractions and decimals.

21st Century Capacities: Decision Making, Collective Intelligence

Grade 5

Grade 5 - Mathematics Curriculum Overview 2016-2017

UnitDescription

Unit 1

Expressions, Equations and Volume

In this unit, students use the study of volume to review and extend a host of skills and concepts related to multiplication. Students investigate a scenario in which they find different ways to arrange 24 cubes into a rectangular prism. These prompts a deep look at the associative and commutative properties of multiplication as students use expressions with parenthesis to represent different rectangular prisms. Students develop major multi-digit multiplications strategies to solve real world and mathematical problems in elegant and efficient ways. The link between multiplication and division is revisited through the lens of the area model and extended into dividing 3-digit and 2-digit numbers. In addition, as a precursor to unit 2, students will be introduced to the money model with fractions while solving problems of the day.

21st Century Capacities: Analyzing

Unit 2

Adding and Subtracting Fractions

In this unit, students add and subtract fractions with unlike denominators, using a variety of strategies to find common denominators. Money and clocks serve to help students develop intuitions about finding common denominators in order to compare, add, and subtract fractions. Students are introduced to the use of ratio tables to rewrite fractions with common denominators. They extend these strategies and models to solving a variety of story problems, and make generalizations about finding common denominators. Students then gain more explicit experience with greatest common factors and least common multiples as they find common denominators and learn to simplify fractions.

21st Century Capacities: Synthesizing

Unit 3

Place Value and Decimals

In this unit, students study skills and concepts related to place value, from reading, writing and comparing decimals to rounding and examining decimal the decimal patterns of multiplying and dividing numbers by 10. Students use their place value understandings of whole numbers and decimals to add and subtract decimals to the hundredths.

21st Century Capacities:Synthesizing

Unit 4

Multiplying and Dividing Whole Numbers and Decimals

In this unit, students return to the study of multiplication and division strategies, including the standard multiplication algorithm. In the first two modules, students investigate a number of strategies that capitalize on their estimation and mental math skills and help them to continue to develop strong number sense. These include strategies that leverage the relationships between multiplication and division; the fact that 5 is half of 10; the relationships between fractions, decimals, and whole numbers and the process of doubling and halving. In Module 3 the teacher formally introduces the standard multiplication algorithm after reviewing the area model and partial products. Module 4 reinforces the connection between multiplication and division, using the area model and ratio tables to help students develop a degree of comfort with long division.

21st Century Capacities: Analyzing

Unit 5

Multiplying and Dividing Whole Numbers and Fractions

In Unit 5, students extend their understandings of multiplication and division to working with fractions. During the first module, students review and extend skills and concepts first introduced in Grade 4 to solidify their understandings of whole-number-by-fraction multiplication. In Modules 2 and 3, they use rectangular arrays to model and solve fraction-by-fraction multiplication problems. Module 4 features an introduction to division of whole numbers by unit fractions, and unit fractions by whole numbers. There is a strong emphasis throughout the unit on sense-making and understanding, as students tackle material that is conceptually challenging.

21st Century Capacities: Synthesizing

Unit 6

Graphing-Geometry-Volume

In this unit, students are formally introduced to several new geometric concepts, including coordinate graphing and the use of hierarchies to classify two-dimensional shapes by their properties and working with protractors for angle measure and angle drawing. Students also review volume, working from counting the cubes that will fit into a box to measuring prisms in continuous units and using standard formulas (V = l x w x h and V = b x h) to find their volumes.

21st Century Capacities: Synthesizing

Unit 7

Division and Decimals

In this unit, students continue their study of division, including its relationship to multiplication. First, students work through class activities to find partial quotients as they divide 3 and 4 digit dividends by 2-digit divisors. Next, the focus is on sharing and grouping interpretations of division, providing opportunities to review the skills and concepts associated with dividing unit fractions by whole numbers and vice versa. Students also solve and discuss a wide variety of division story problems, including contexts that require decisions about how to handle remainders. Finally, students review and extend their thinking about the effects of multiplying and dividing powers of ten, as well as multiplying and dividing decimal numbers.

21st Century Capacities: Synthesizing

Grade 6

Grade 6 - Mathematics Curriculum 2016-2017

UnitDescription

Unit 1

“Are We Related?” Ratios, Proportions, Rates, Percents

In this unit, students solve problems that involve ratio - a comparison of two quantities. Students will learn to write, identify, and use ratios and rates to compare given quantities including those involved in converting measurements. Students will use the relationships between fractions, decimals, percents, and ratios to explore how to calculate the percent of a number. While grade 5 students have not formally worked with ratios or worked with proportions, students have experience with: ratio tables, converting among different-sized standard measurement units within a given measurement system, and using these conversions in solving multi-step, real world problems and renaming fractions.

Students move between different representations of the same quantity (or relationship) in order to:

●solve problems and make predictions in scenarios that involve ratios.

●convert within the customary and metric systems.

●find unit rates and compare situations using unit rates.

●fluently convert among fractions, decimals, and percents in both receiving and sharing information.

In the next unit students will continue working with fractions and decimals to connect ratio and rate to whole number multiplication and division. They will be extending the notion of a number to the system of rational numbers.

21st Century Capacities: Synthesizing , Analyzing

Unit 2

“Parts of a Whole” Fractions and Decimals

Students will extend their previous understanding of decimals and multiplications and division to multiply and divide decimals using the standard algorithm. In grade 5 students multiplied a fraction by a whole number. They will now extend this concept to multiply and divide fractions and mixed numbers. While working with fractions and decimals students will estimate solutions to problems to be able to evaluate the accuracy of their final answers. Number lines, equivalent fractions, and benchmarks will allow students to draw conclusions for accurate estimates.

21st Century Capacities: Analyzing, Synthesizing

Unit 3

“Fact or Fiction?Statistics

Building on, and reinforcing their understanding of number, students begin to develop their ability to think statistically. First, they learn what makes a good statistical question. Students recognize that different ways to measure center yield different values. Students recognize that a measure of variability can also be useful for summarizing data because two very different sets of data can have the same mean and median yet be distinguished by their variability. Students learn to describe and summarize numerical data sets, identifying clusters, peaks, gaps, and symmetry, considering the context in which the data were collected. Although the students will be creating data displays, throughout the unit, the emphasis should be on the student reading, understanding and critically reflecting on displayed data.

21st Century Capacities: Presentation, Synthesizing

Unit 4

“Constructing and Deconstructing”

In this unit students decompose shapes to familiar parts to find area and volume. Students extend their understanding of area to three dimensional objects to find surface area. The students will use all four quadrants of the coordinate plane to solve problems about polygons. This is the students’ first formal work with negative numbers.

21st Century Capacities: Synthesizing, Presentation

Unit 5

“From Concrete to Abstract”: Expressions and Equations, Integers

In this unit students apply and extend previous understandings of arithmetic to algebraic expressions, equations and inequalities. They will model using algebraic equations and inequalities and solve simple one step algebraic equations. They will represent and analyze quantitative relationships between dependent and independent variables. Finally, they will apply and extend previous understandings of numbers to integers to order to compare and order and gain insight into the magnitude of an integer. Students will use integers to work within all four quadrants of the coordinate plane.

21st Century Capacities: Analyzing


Grade 7

Grade 7 Mathematics

Grade 7 - Mathematics Curriculum Overview 2016-2017

UnitDescription

Unit A

Building Blocks

This first unit gives students the essential tools they will use throughout their work in math including:

●extending the number system to include integers

●the concept of a variable and how it can be manipulated

●solving algebraic equations and giving a clear argument to justify a solution.

During this unit, students will review working with decimal operations so they are ready to work with rational numbers that include negatives in Unit C.

21st Century Capacities: Presentation, Analyzing

Unit B

2D and 3D Geometry

During this Geometry unit students move from finding area and perimeter of two dimensional shapes to finding the surface area and volume of three dimensional shapes. Although finding the area of some of the shapes is a review, finding the area and circumference of circles is new for the students. Students should not be given formulas for finding surface area but instead should find surface area based on the nets of these shapes. When working with volume of prisms the emphasis should be on thinking of volume as the area of the base times the height of the prism to generalize for any prism. The concepts of area and volume are used in applications throughout the unit.

21st Century Capacities: Analyzing, Synthesizing

Unit C

Rational Numbers

In this unit students will extend their knowledge about rational numbers by extending the concept to negative numbers. Students will use number lines to add and subtract rational numbers. Students will be encouraged throughout the unit to think about whether their answer will be positive or negative before they begin to compute solutions. Students will be encouraged to use tools to overcome obstacles to solve problems. Students will be encouraged to persevere as they learn by using tools and strategies to create solutions. Students will be encouraged to persevere in solving problems by using tools and strategies they have at their disposal to solve problems. Students will learn what to do when they are stuck.

21st Century Capacities: Perseverance, Analyzing

Unit D

Ratios, Proportions and Percents

Students extend their understanding of ratios and develop understanding of proportionality to solve single- and multi-step problems. Students use their understanding of ratios and proportionality to solve a wide variety of percent problems, including those involving discounts, interest, taxes, tips, and percent increase or decrease. Students solve problems about scale drawings by relating corresponding lengths between the objects or by using the fact that relationships of lengths within an object are preserved in similar objects. They determine if a relationship is proportional. In Pre-Algebra students will graph proportional relationships and understand the unit rate informally as a measure of the steepness of the related line, called the slope.

21st Century Capacities: Analyzing, Synthesizing

Unit E

Probability and Statistics

Students use organized lists and trees to write sample spaces about events and determine probabilities associated with the events including simple probabilities and conditional probabilities involving ‘and’ and ‘or’. As students use tables to find probabilities they may begin to find shortcuts for finding probabilities. Students also develop and use simulations to model compound events. Given a sample of a population students will make predictions about the entire population. Building on their work describing the center of spread of data in grade 6, students will now compare the center and spread of two sets of data. Throughout the unit students are made aware that presentation of data can affect interpretation of information.

21st Century Capacities: Analyzing, Presentation

Unit F

Geometry: All the Angles

This short Geometry unit focuses on angle measurements. Students review how to use a protractor and use the protractor to examine angles including those in a triangle. Students revisit solving equations by solving for an unknown in Geometric diagrams involving angles.

21st Century Capacities: Analyzing

Grade 7 - Pre-Algebra

Grade 7 Pre-Algebra Mathematics Curriculum Overview 2016-2017

UnitDescription

Unit A

Tools of the Trade

This first unit gives students the essential tools they will use throughout their work in math and science including:

●extending the number system to include integers

●the concept of a variable and how it can be manipulated

●solving algebraic equations and inequalities and giving a clear argument to justify a solution.

During this unit, students will review working with fractions and decimals so they are ready to work with rational numbers that include negatives in Unit B.

21st Century Capacities: Product Creation, Analyzing

Unit B

Parts of Whole, Factors and Rational Numbers

Students will build on their knowledge of rational numbers by extending their work into rational numbers with negative values. The expectation is that students are fluent with positive rational numbers (decimals and fractions) before starting this unit. The theme of moving between different representation s of numbers continues into the second part of the unit as students work with monomials. Their work with scientific notation gives the skill of simplifying monomials an application.

21st Century Capacities: Synthesizing

Unit C

Geometry

Students use ideas about distance and angles, how they behave under translations, rotations, reflections, and dilations. In later math courses, these transformations will be made on functions. Students continue their work with area from Grade 6 to solve problems involving the area and circumference of a circle and surface area of three-dimensional objects. They reason about relationships among two-dimensional figures and among angles formed by intersecting lines using informal geometric constructions. Students will use equations and inequalities, including those with a variable on both sides, to solve geometric problems. Students work with three-dimensional figures, relating them to two-dimensional figures by examining cross sections and nets. They solve real-world and mathematical problems involving area, surface area, and volume of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes and right prisms, cones, cylinders, and spheres.

21st Century Capacities:Synthesizing

Unit D

Ratio, Proportion, Percent

Students extend their understanding of ratios and develop understanding of proportionality to solve single- and multi-step problems. Students use their understanding of ratios and proportionality to solve a wide variety of percent problems, including those involving discounts, interest, taxes, tips, and percent increase or decrease. Students solve problems about scale drawings by relating corresponding lengths between the objects or by using the fact that relationships of lengths within an object are preserved in similar objects. They distinguish proportional relationships from other relationships. In the last unit of this course students will graph proportional relationships and understand the unit rate informally as a measure of the steepness of the related line, called the slope.

21st Century Capacities: Analyzing, Synthesizing

Unit E

Statistics and Probability

Students build on their previous work with single data distributions to compare two data distributions and address questions about differences between populations. They begin informal work with random sampling to generate data sets and learn about the importance of representative samples for drawing inferences.

21st Century Capacities: Analyzing, Presentation

Unit F

Visualizing Solutions to Equations

In this unit students grasp the concept of a function as a rule that assigns to each input exactly one output. They understand that functions describe situations where one quantity determines another. They can translate among representations and partial representations of functions (noting that tabular and graphical representations may be partial representations), and they describe how aspects of the function are reflected in the different representations.

Students use linear equations and systems of linear equations to represent, analyze, and solve a variety of problems. Students recognize equations for proportions (y/x = m or y = mx) as special linear equations (y = mx + b), understanding that the constant of proportionality (m) is the slope, and the graphs are lines through the origin. They understand that the slope (m) of a line is a constant rate of change, so that if the input or x-coordinate changes by an amount A, the output or y-coordinate changes by the amount mA. Students also use a linear equation to describe the association between two quantities in bivariate data (such as arm span vs. height for students in a classroom). Interpreting the model in the context of the data requires students to express a relationship between the two quantities in question and to interpret components of the relationship (such as slope and y-intercept) in terms of the situation.

21st Century Capacities: Synthesizing

Grade 8

Grade 8 Pre-Algebra

Grade 8 Pre-Algebra Mathematics Curriculum Overview 2016-2017

UnitDescription

Unit A

Graphing

In this unit students will work with graphing on the coordinate plane in three different ways. First they will use graphing to represent equations with two variables. Then they will create best fit lines for scatter plots to help describe relationships and make predictions. Finally, students will graph transformations on the coordinate plane and note the differences and similarities between the pre and post images. The concept of transformations will help them graph functions in later math courses.

21st Century Capacities: Analyzing

Unit B

Equations

This unit is a very important foundation for later math. Students learn how to solve equations and inequalities. It is important for them to be able to justify their solution both to check their reasoning and to allow others to understand their argument. Students will learn how to model using algebraic equations and inequalities. At this point, the problems students are working with can often be solved with arithmetic. Students should be encouraged to build on that understanding to create an algebraic model.

21st Century Capacities: Product Creation, Synthesizing

Unit C

Ratio, Percent, Proportion

Students master the concept of rate of change and firmly establish the relationship between a graph, a table, an equation and a verbal description of a function that has a constant rate of change. In this unit students work with percents with the goal that students become fluent working with percents to get approximate answers mentally and exact answers.

21st Century Capacities:Synthesizing, Analyzing

Unit D

Geometry

In Grade 7, students used facts about supplementary, complementary, vertical, and adjacent angles to find the measures of unknown angles. This unit extends that knowledge to angle relationships that are formed when two parallel lines are cut by a transversal and why the exterior angles of a triangle is the sum of the two remote interior angles of the triangle. Students also learn and use the Pythagorean Theorem and are shown an informal proof of the theorem to build understanding. Finally, students work with three dimensional shapes and use volume formulas to solve problems in context.

21st Century Capacities: Synthesizing, Product Creation

Unit E

Factors and Monomials

Students understand the structure of exponents by expanding multiplication and division of expressions and raising a power to a power and then simplifying those expressions using concepts from multiplication of whole numbers and simplifying fractions. Properties of exponents are extended by raising integers and monomials to a negative exponent. Students use the properties of exponents they developed with positive exponents and accept them as true for all integer exponents and are shown the value of learning those properties.

Students’ understanding of integer exponents is expanded to scientific notation. Students learn that positive powers of ten are large numbers and negative powers of 10 are very small numbers. Students will express large and small numbers in the form of a single digit times a power of 10 and express how many times as much one of these numbers is compared to another. Lessons will demonstrates the need for such a notation and then how to compare and compute with numbers in scientific notation. Also, in this unit, students will use what they know about exponential notation, properties of exponents, and scientific notation to interpret results that have been generated by technology. By the end of the unit, students are able to compare and perform operations on numbers given in both decimal and scientific notation.

21st Century Capacities: Synthesizing

Unit F

Graphing Lines

Students grasp the concept of a function as a rule that assigns to each input exactly one output. They understand that functions describe situations where one quantity determines another. They can translate among representations and partial representations of functions (noting that tabular and graphical representations may be partial representations), and they describe how aspects of the function are reflected in the different representations.

Students use linear equations and systems of linear equations to represent, analyze, and solve a variety of problems. Students recognize equations for proportions (y/x = m or y = mx) as special linear equations (y = mx + b), understanding that the constant of proportionality (m) is the slope, and the graphs are lines through the origin. They understand that the slope (m) of a line is a constant rate of change, so that if the input or x-coordinate changes by an amount A, the output or y-coordinate changes by the amount mA. Students also use a linear equation to describe the association between two quantities in bivariate data (such as arm span vs. height for students in a classroom). Students interpret components of the relationship (such as slope and y-intercept) in terms of the situation.

Students solve systems of two linear equations in two variables and relate the systems to pairs of lines in the plane; these intersect, are parallel, or are the same line. Students use linear equations, systems of linear equations, linear functions, and their understanding of slope of a line to analyze situations and solve problems.

21st Century Capacities: Synthesizing, Product Creation

Grade 8 Pre-Algebra Level B

Grade 8 Pre-Algebra Level B Mathematics Curriculum Overview 2016-2017

UnitDescription

Unit A

Graphing

In this unit students will work with graphing on the coordinate plane in three different ways. First they will use graphing to represent equations with two variables. Then they will create best fit lines for scatter plots to help describe relationships and make predictions. Finally, students will graph transformations on the coordinate plane and note the differences and similarities between the pre and post images. The concept of transformations will help them graph functions in later math courses.

21st Century Capacities: Analyzing

Unit B

Equations

This unit is a very important foundation for later math. Students learn how to solve equations and inequalities. It is important for them to be able to justify their solution both to check their reasoning and to allow others to understand their argument. Students will learn how to model using algebraic equations and inequalities. At this point, the problems students are working with can often be solved with arithmetic. Students should be encouraged to build on that understanding to create an algebraic model.

It is helpful to provide students with model examples of each equation type that they can refer back to and match with the particular problem they are given. Listing steps for each equation type and allowing students access to these steps also helps students when solving more complex equations.

21st Century Capacities: Product Creation, Synthesizing

Unit C

Ratio, Percent, Proportion

Students again revisit the concept of rate of change and firmly establish the relationship between a graph, a table, an equation and a verbal description of a function that has a constant rate of change. In this unit students work with percents with the goal that students become fluent working with percents to get approximate answers mentally and exact answers.

21st Century Capacities: Synthesizing, Analyzing

Unit D

Geometry

In Grade 7, students used facts about supplementary, complementary, vertical, and adjacent angles to find the measures of unknown angles. This unit extends that knowledge to angle relationships that are formed when two parallel lines are cut by a transversal and why the exterior angles of a triangle is the sum of the two remote interior angles of the triangle. Students also learn and use the Pythagorean Theorem and are shown an informal proof of the theorem to build understanding. Finally, students work with three dimensional shapes and use volume formulas to solve problems in context.

21st Century Capacities: Synthesizing, Product Creation

Unit E

Factors and Monomials

Students understand the structure of exponents by expanding multiplication and division of expressions and raising a power to a power and then simplifying those expressions using concepts from multiplication of whole numbers and simplifying fractions. Properties of exponents are extended by raising integers and monomials to a negative exponent. Students use the properties of exponents they developed with positive exponents and accept them as true for all integer exponents and are shown the value of learning those properties.

Students’ understanding of integer exponents is expanded to scientific notation. Students learn that positive powers of ten are large numbers and negative powers of 10 are very small numbers. Students will express large and small numbers in the form of a single digit times a power of 10 and express how many times as much one of these numbers is compared to another. Lessons will demonstrates the need for such a notation and then how to compare and compute with numbers in scientific notation. Also, in this unit, students will use what they know about exponential notation, properties of exponents, and scientific notation to interpret results that have been generated by technology. By the end of the unit, students are able to compare and perform operations on numbers given in both decimal and scientific notation.

21st Century Capacities: Synthesizing

Unit F

Graphing Lines

Students grasp the concept of a function as a rule that assigns to each input exactly one output. They understand that functions describe situations where one quantity determines another. They can translate among representations and partial representations of functions (noting that tabular and graphical representations may be partial representations), and they describe how aspects of the function are reflected in the different representations.

Students use linear equations and systems of linear equations to represent, analyze, and solve a variety of problems. Students recognize equations for proportions (y/x = m or y = mx) as special linear equations (y = mx + b), understanding that the constant of proportionality (m) is the slope, and the graphs are lines through the origin. They understand that the slope (m) of a line is a constant rate of change, so that if the input or x-coordinate changes by an amount A, the output or y-coordinate changes by the amount mA. Students also use a linear equation to describe the association between two quantities in bivariate data (such as arm span vs. height for students in a classroom). Students interpret components of the relationship (such as slope and y-intercept) in terms of the situation.

Students solve systems of two linear equations in two variables and relate the systems to pairs of lines in the plane; these intersect, are parallel, or are the same line. Students use linear equations, systems of linear equations, linear functions, and their understanding of slope of a line to analyze situations and solve problems.

21st Century Capacities: Synthesizing, Product Creation

Algebra I Level 2

UnitDescription

Unit A

“Cracking Codes” Patterns and Repetition in our World

Student will be engaged in learning mathematical skills within the context of interesting problems that connect to real world issues. Students will be expected to use the 21st Century Skill: Analyzing to independently use their learning in new situations. They will learn to interpret patterns in the real world and use mathematics to evaluate complex situations, apply properties of patterns to inform decisions by analyzing information, and design and create representations of data.

Students coming from Pre-Algebra should be proficient in combining like terms, order of operations, the rules of integer operations, graphing on a coordinate plane, finding rate of change and the meaning of the intercepts.

The students begin with hands-on activities building concrete models which helps them generalize to tables, graphs and equations. They will recognize and extend addition, multiplication and other patterns including fractions and decimals. They are asked to write explicit and recursive rules for the patterns. Later students will explore rules for arithmetic and geometric sequences that will help them find values and terms that extend the usefulness of patterns. They will explore the rules that will reach a higher understanding of the patterns to explain real world phenomenon. Students will develop strategies to select the most efficient method to solve problems involving arithmetic or geometric sequences.

Challenges for students can be: 1) distinguishing between arithmetic and geometric sequences, 2) accurately representing data on a graph, 3) independently transferring skills to new situations.

In the next unit students will extend pattern rules to function relationships. The students will use technology to generate and display patterns.

21st Century Capacities: Analyzing

Unit B

Relationships (Equations, Inequalities and Functions)

Students begin the unit with a quick review of solving equations and inequalities. Focus should be on solving more difficult equations (ie. equations with a variable on both sides) and inequalities (those with negative numbers and with the variable on the right hand side of the inequality). Students are asked to justify their work using math and words. Asking for step by step explanations in words will help them in justifying steps in proof writing in Geometry. The concept of solving equations is expanded to solving literal equations, solving compound equations and solving absolute value equations and inequalities.

Students learn function notation during the second part of this unit. The goal is for them to be able to move fluently between representations of a function.

21st Century Capacities: Analyzing, Product Creation

Unit C

What’s In A Line? - Elements of Linear Equations

In this unit students will learn how to model with, interpret and graph linear functions. The ability to fluidly move between different representations of linear relationships is a skill that students will continue to use and to build upon in Algebra and later math courses. Students will use technology to experiment with changing the parameters of a linear equation and noting how those changes affect the graph of the relationship.

21st Century Capacities:Analyzing, Synthesizing, Product Creation

Unit D

Describing Data - Identifying Trends and Making Decisions

Describing Data extends linear thinking to statistical modeling. First, students develop measures of central tendency by studying dispersion through the 5-number summary and the corresponding box and whisker graph. Students then make frequency tables and histograms that shape discussions about skewness.

Next, students compare two quantities in scatterplots and add context to Unit C concepts of slope and line of best fit. Students model linear relationships both manually with trend-lines and digitally with graphing calculators or software. Students use models to make predictions both inside and outside of the known range and understand limitations of those predictions. Students describe strength of fit using correlation coefficients, which strengthen understandings of slope from Unit C. Students are challenged to explain the difference between correlation and causation. Students explain the impact of an outlier on linear models.

Students expand their notions of linear models to piecewise functions. This is a prelude to other nonlinear modeling, including exponential and quadratic models which will resurface later in the course.

21st Century Capacities: Analyzing, Synthesizing

Unit E

Linear Systems: Points In Common

In this unit students will use previously learned skills in graphing equations and extend those to graph systems of equations and graph inequalities and graph systems of inequalities. Students will model using systems of equations or inequalities. Students will also solve systems of equations using substitution or elimination. Students will be encouraged to analyze a system before solving it to determine the most efficient method to use to solve the system.

21st Century Capacities: Synthesizing, Product Creation

Unit F

Beyond Straight Lines - Quadratic and Absolute Value Functions

In this unit students work with quadratic expressions, quadratic equations, radicals and rational expressions to see how changing the form of an expression or equation can give the item a clearer meaning and can make it easier to work with. By the end of the unit students should be able to fluently solve quadratic equations. They should be able to fluently identify transformations made to the parent function so they are able to visualize the graph to make estimations and to check to see if their solution makes sense. The unit ends with students using their new factoring skills to simplify rational expressions and equations into manageable problems.

21st Century Capacities: Analyzing, Product Creation

Unit G

“Growth and Decay” - Understanding Exponential Functions

This unit builds on concepts of a function and patterns of change as students work with interesting and significant relationships that are exponential in nature. Students study rules of exponents and develop meaning for negative and rational exponents. Then they will apply those rules to exponential functions. Students will transform functions as they did with linear, quadratic, and absolute value models. When comparing an exponential model with a linear model, the question is not if the exponential model will generate very large or very small inputs, but rather when. Students will gain an appreciation for the power of mathematics in identifying and addressing solutions and making predictions and decisions about significant real world problems.

21st Century Skills: Product Creations, Synthesizing

Grade 9-12

Algebra I Level 2

UnitDescription

Unit A

“Cracking Codes” Patterns and Repetition in our World

Student will be engaged in learning mathematical skills within the context of interesting problems that connect to real world issues. Students will be expected to use the 21st Century Skill: Analyzing to independently use their learning in new situations. They will learn to interpret patterns in the real world and use mathematics to evaluate complex situations, apply properties of patterns to inform decisions by analyzing information, and design and create representations of data.

Students coming from Pre-Algebra should be proficient in combining like terms, order of operations, the rules of integer operations, graphing on a coordinate plane, finding rate of change and the meaning of the intercepts.

The students begin with hands-on activities building concrete models which helps them generalize to tables, graphs and equations. They will recognize and extend addition, multiplication and other patterns including fractions and decimals. They are asked to write explicit and recursive rules for the patterns. Later students will explore rules for arithmetic and geometric sequences that will help them find values and terms that extend the usefulness of patterns. They will explore the rules that will reach a higher understanding of the patterns to explain real world phenomenon. Students will develop strategies to select the most efficient method to solve problems involving arithmetic or geometric sequences.

Challenges for students can be: 1) distinguishing between arithmetic and geometric sequences, 2) accurately representing data on a graph, 3) independently transferring skills to new situations.

In the next unit students will extend pattern rules to function relationships. The students will use technology to generate and display patterns.

21st Century Capacities: Analyzing

Unit B

Relationships (Equations, Inequalities and Functions)

Students begin the unit with a quick review of solving equations and inequalities. Focus should be on solving more difficult equations (ie. equations with a variable on both sides) and inequalities (those with negative numbers and with the variable on the right hand side of the inequality). Students are asked to justify their work using math and words. Asking for step by step explanations in words will help them in justifying steps in proof writing in Geometry. The concept of solving equations is expanded to solving literal equations, solving compound equations and solving absolute value equations and inequalities.

Students learn function notation during the second part of this unit. The goal is for them to be able to move fluently between representations of a function.

21st Century Capacities: Analyzing, Product Creation

Unit C

What’s In A Line? - Elements of Linear Equations

In this unit students will learn how to model with, interpret and graph linear functions. The ability to fluidly move between different representations of linear relationships is a skill that students will continue to use and to build upon in Algebra and later math courses. Students will use technology to experiment with changing the parameters of a linear equation and noting how those changes affect the graph of the relationship.

21st Century Capacities:Analyzing, Synthesizing, Product Creation

Unit D

Describing Data - Identifying Trends and Making Decisions

Describing Data extends linear thinking to statistical modeling. First, students develop measures of central tendency by studying dispersion through the 5-number summary and the corresponding box and whisker graph. Students then make frequency tables and histograms that shape discussions about skewness.

Next, students compare two quantities in scatterplots and add context to Unit C concepts of slope and line of best fit. Students model linear relationships both manually with trend-lines and digitally with graphing calculators or software. Students use models to make predictions both inside and outside of the known range and understand limitations of those predictions. Students describe strength of fit using correlation coefficients, which strengthen understandings of slope from Unit C. Students are challenged to explain the difference between correlation and causation. Students explain the impact of an outlier on linear models.

Students expand their notions of linear models to piecewise functions. This is a prelude to other nonlinear modeling, including exponential and quadratic models which will resurface later in the course.

21st Century Capacities: Analyzing, Synthesizing

Unit E

Linear Systems: Points In Common

In this unit students will use previously learned skills in graphing equations and extend those to graph systems of equations and graph inequalities and graph systems of inequalities. Students will model using systems of equations or inequalities. Students will also solve systems of equations using substitution or elimination. Students will be encouraged to analyze a system before solving it to determine the most efficient method to use to solve the system.

21st Century Capacities: Synthesizing, Product Creation

Unit F

Beyond Straight Lines - Quadratic and Absolute Value Functions

In this unit students work with quadratic expressions, quadratic equations, radicals and rational expressions to see how changing the form of an expression or equation can give the item a clearer meaning and can make it easier to work with. By the end of the unit students should be able to fluently solve quadratic equations. They should be able to fluently identify transformations made to the parent function so they are able to visualize the graph to make estimations and to check to see if their solution makes sense. The unit ends with students using their new factoring skills to simplify rational expressions and equations into manageable problems.

21st Century Capacities: Analyzing, Product Creation

Unit G

“Growth and Decay” - Understanding Exponential Functions

This unit builds on concepts of a function and patterns of change as students work with interesting and significant relationships that are exponential in nature. Students study rules of exponents and develop meaning for negative and rational exponents. Then they will apply those rules to exponential functions. Students will transform functions as they did with linear, quadratic, and absolute value models. When comparing an exponential model with a linear model, the question is not if the exponential model will generate very large or very small inputs, but rather when. Students will gain an appreciation for the power of mathematics in identifying and addressing solutions and making predictions and decisions about significant real world problems.

21st Century Skills: Product Creations, Synthesizing

Algebra I Level 3

UnitDescription

Unit A

“Cracking Codes” Patterns and Repetition in our World

Student will be engage in learning mathematical skills within the context of interesting problems that connect to real world issues. Students will be expected to use the 21st Century Skill: Analyzing to independently use their learning in new situations. They will learn to interpret patterns in the real world and use mathematics to evaluate complex situations, apply properties of patterns to inform decisions by analyzing information, and design and create representations of data.

Students coming from Pre-Algebra should be proficient in combining like terms, order of operations, the rules of integer operations, graphing on a coordinate plane, finding rate of change and the meaning of the intercepts.

The students begin with hands-on activities building concrete models which helps them generalize to tables, graphs and equations. They will recognize and extend addition, multiplication and other patterns including fractions and decimals. They are asked to write explicit and recursive rules for the patterns. Later students will explore rules for arithmetic and geometric sequences that will help them find values and terms that extend the usefulness of patterns. They will explore the rules that will reach a higher understanding of the patterns to explain real world phenomenon. Students will develop strategies to select the most efficient method to solve problems involving arithmetic or geometric sequences.

Challenges for students can be: 1) distinguishing between arithmetic and geometric sequences, 2) accurately representing data on a graph, 3) independently transferring skills to new situations

In the next unit students will extend pattern rules to function relationships. The students will use technology to generate and display patterns.

21st Century Capacities: Analyzing

Unit B

Relationships (Equations, Inequalities and Functions)

In this unit students solve and model with equations and inequalities. Students are asked to justify their work using math and words. Asking for step by step explanations in words will help them solidify their understanding and later will help in justifying steps in proof writing in Geometry. The concept of solving equations is expanded to solving literal equations, solving compound equations and inequalities.

During the second part of this unit, students are introduced to the concept of a function. Focus includes identification of relationships that are not functions, defining the domain and range of a function and distinguishing between linear and non-linear functions. Students then go on to practice applying functions through various contextual problems. Students collect data, make a table and graph data then identify the type of function. Students learn function notation during this unit. The goal is for them to be able to move fluently between representations of a function.

21st Century Capacities: Analyzing

Unit C

“What’s In A Line?” - Elements of Linear Equations

In this unit students will learn how to model with, interpret and graph linear functions. The ability to fluidly move between different representations of linear relationships is a skill that students will continue to use and to build upon in Algebra and later math courses. Students will use technology to experiment with changing the parameters of a linear equation and noting how those changes affect the graph of the relationship.

21st Century Capacities: Synthesizing, Analyzing

Unit D

Describing Data - Identifying Trends and Making Decisions

Describing Data extends linear thinking to statistical modeling. First, students develop measures of central tendency by studying dispersion through the 5-number summary and the corresponding box and whisker graph.

Next, students compare two quantities in scatterplots and add context to Unit C concepts of slope and line of best fit. Students model linear relationships both manually with trend lines and digitally with graphing calculators or software. Students use models to make predictions both inside and outside of the known range and understand limitations of those predictions. Students describe strength of fit using correlation coefficients, which strengthen understandings of slope from Unit C. Students are challenged to explain the difference between correlation and causation. Students explain the impact of an outlier on linear models.

This is a prelude to other nonlinear modeling, including quadratic models which will resurface later in the course.

21st Century Capacities: Synthesizing, Analyzing

Unit E

Linear Systems: Points In Common

In this unit students will use previously learned skills in graphing equations and apply them in order to graph systems of equations. Students will also be encouraged to determine the systems of equations that can be determined from different application problems. Interpretation of solutions found, number of solutions found and their meaning in the context of the applied problem.

Students will also solve systems of equations using substitution or elimination. Students should be encouraged to analyze a system before solving it to determine the most efficient method to use to solve the system.

21st Century Capacities: Synthesizing, Product Creation

Unit F

“Beyond Straight Lines” - Quadratic Functions

In this unit, students work with quadratic expressions, quadratic equations and rational expressions to see how changing the form of an expression or equation can give the item a clearer meaning and can make it easier to work with. By the end of the unit students should be able to fluently factor and solve quadratic equations.

21st Century Capacities: Analyzing, Product Creation

Algebra II Level 1

UnitDescription

Unit 1

Equations and Inequalities

This brief unit is a quick refresher of fundamental Algebra I topics including factoring polynomials, simplifying rational expressions, solving single variable equations and inequalities. Students will use these skills throughout the entire course.

21st Century Capacities: Analyzing, Presentation

Unit 2

Relations and Functions

In this unit we move from working with single variables to multiple variables in equations. Functions and function notation will be the focus of this unit and every unit after this unit. Students will understand the concept of function, function notation, types of functions, transformations of functions, operations on functions, inverse functions and graphing functions. Students will be able to identify the domain and range of a function. Students should be able to work with functions in multiple representations: algebraic, graph and table of values.

21st Century Capacities: Analyzing, Presentation

Unit 3

Quadratic Equations and Complex Numbers

The goal of this unit is for students to become fluent in interpreting, solving, and graphing quadratic functions with rational and irrational solutions as well as complex roots. The connection between completing the square and equations of circles is made. Students model using quadratic functions.

21st Century Capacities: Analyzing

Unit 4

Polynomial Functions

We move from quadratics to a study of polynomials and the relationship between the degree, the number of terms and the zeros. Multiplicity of zeros will be investigated and students will discover the relationship between the number of zeros the graph. The Rational Roots Theorem, Remainder Theorem and Factor Theorem will also be investigated in this unit.

21st Century Capacities: Analyzing

Unit 5

Rational Expressions and Functions

In this unit students will extend their understanding of polynomials functions and their graphs to rational functions and their graphs. Students are encouraged to connect operations on rational expressions to operations on fractions learned in earlier math courses. Polynomial and rational inequalities are also explored in this unit.

21st Century Capacities: Analyzing

Unit 6

Exponential and Log Functions

This unit is the study of exponential and logarithmic functions. Understanding the inverse relationship between exponential and logarithmic functions is important. The properties and rules of logarithms will be related to exponential rules and then used in application problems including Newton’s Law of Cooling, compound interest and exponential growth and decay.

21st Century Capacities: Analyzing

Unit 7

Trigonometric Functions

This is the first time that many students will see any trigonometry beyond SOHCAHTOA, for example, radian measure, law of sines and law of cosines, the reciprocal trigonometric functions, trigonometric graphs. We chose specifically to emphasize the Unit Circle, and to simply use “circle definitions” for problems like sin(240). Students should come away from this unit feeling like many trigonometric topics can simply be done with x, y, and r (circle definitions). There are many variations of this type of problem, but students should feel the unity among them. Students need to know the basic side patterns for special right triangles along with how to draw angles in standard position.

21st Century Capacities: Analyzing, Collective Intelligence

Algebra II Level 2

UnitDescription

Unit 1

Equations and Inequalities

This short unit focuses on prior mathematical knowledge of solving multi-step equations and inequalities. Students are expected to apply their algebraic knowledge and understanding through the application to real-world problems.

21st Century Capacities: Analyzing, Collective Intelligence

Unit 2

Relations and Functions

Students will appreciate the importance of functions and their domains and will use input/output language throughout unit. A significant part of this unit is transformations on parent functions, having students understand how parameters affecting the inputs differ from the parameters affecting the outputs. Graph analysis is introduced but somewhat limited in scope. Students will also explore systems of linear equations, systems of inequalities, and linear programming to see real world applications.

21st Century Capacities: Analyzing, Presentation

Unit 3

Quadratic Equations and Complex Numbers

Students will understand what a radical is and how to simplify and combine in order to solve quadratics that are not factorable. Students will learn a variety of ways to solve quadratic equations and then will be challenged to choose the most efficient method for solving a given equation. Students can visualize the solutions of quadratic equations through graphing (e.g., min, max, transformations, complex roots). Completing the squares is used as an introduction to the equation of circles to further understanding of transformations. Students will demonstrate their efficiency through the solving of application problems.

21st Century Capacities: Collective Intelligence

Unit 4

Polynomial Functions

Students will perform operations on polynomials, adding, subtracting, multiplying and dividing. They will be able to graph polynomial functions by factoring to find the zeroes and understanding end behavior and multiplicities.

21st Century Capacities: Analyzing, Collective Intelligence

Unit 5

Rational Expressions and Functions

Students will perform operations on rational expressions: simplifying, adding, subtracting, multiplying and dividing. They will also solve rational equations. They will be able to graph rational functions by finding the vertical and horizontal asymptotes, intercepts, and testing points.

21st Century Capacities: Analyzing, Collective Intelligence

Unit 6

Exponential and Log Functions

Students will understand exponential functions and their graphs. Students will be introduced to logarithms as the inverse of exponential functions. They will use the properties of logarithms to solve both exponential and logarithmic equations. Real world application problems will be introduced.

21st Century Capacities: Analyzing, Synthesizing

Unit 7

Trigonometric Functions

This is the first time that many students will see any trigonometry beyond SOHCAHTOA, for example, radian measure, reciprocal trig functions, trig graphs. We introduce trig functions of all angles using a circle and reference angles and then move on to use the Unit Circle as a special case. Students learn about the basic characteristics of sine and cosine graphs and then learn about transformations of these functions.

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Algebra II Level 3

UnitDescription

Unit A

Rebuilding Algebra Skills

This brief unit is a review of Pre-Algebra and Algebra 1 topics that are integral for succeeding in Algebra 2. Care should be given to teach each topic in a way that illuminates the reasons behind the methodology. For example, students should understand why, when terms are on the same side of an equations, they are combined, but when on opposite sides of the equation, we need to add the opposite to eliminate a term from one side, or why absolute value equations may have two solutions, one, or no solutions at all.

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Unit B

Equations on the Coordinate Plane

The purpose of this unit is to use math to analyze situations in which the rate of change is constant and to model those situations using linear equations. Students should make a connection between tabular, algebraic, and graphic representations of relations. In later units students will use the concepts and skills from this unit to work with quadratic and exponential functions.

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Unit C

Quadratic Equations and Parabolas

The purpose of this unit is to move beyond linear functions and to learn strategies to solve quadratic equations. Students should understand that the power of 2 creates a specific shaped graph (parabola). Students should also learn the importance of the complex number system, and should be taught about the history of complex numbers not being all that different from the history of negative numbers.

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Unit D

Functions

The focus of Unit D is for students to learn what is a mathematical function and its importance in problem solving. Students will also explore and learn to use the concept of function notation. Even though function notation is awkward to learn and seems more cumbersome, it is a great tool that allows mathematicians to communicate more clearly. Students will learn to work flexibly between all representations of a relation or function (table, list, equation, graph, and mapping diagram).

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Unit E

Trigonometry

Students will learn the basics of right triangle trigonometry, and will be able to apply trig ratios to solve word problems. Students will learn how to measure angles using radians, how to sketch angles in standard position, etc. The goal of this unit is to expose students to enough trigonometry for them to understand its value in the real world and to be successful in higher math.

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Unit F

Exponential and Logarithmic Functions

The purpose of this unit to expose students to ways of manipulating expressions using exponents. Students are expected to have a conceptual understanding of the rules around exponents and logarithms. They should explore the logic behind the development of negative exponents, zero as an exponent, and rational exponents. These should not just be taught as rules.

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Geometry Level 1

UnitDescription

Unit A

A Introduction to Geometry

This unit introduces students to the majority of terminology used in Geometry. Transformations, logical thinking and proofs are all part of the unit and will be referred to throughout the course. Students will be able to complete a two-column geometric proof by the end of the unit. Geometric software, along with compass and straightedge, will be used for constructions.

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Unit B

Congruent Triangles

This unit focuses on triangle classifications and proving triangles congruent. Proof is a very important concept throughout the unit. Students should become fluent in completing proofs by the end of this unit by seeing the patterns and structure within proofs.

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Unit C

Lines in a Plane

In this unit students develop proofs to fairly complex problems. Along with two column proofs students are encouraged to give verbal and/or paragraph arguments always with the idea of a clear, logical argument with mathematical justification as a priority.

A major focus in this unit is on quadrilaterals. Properties of quadrilaterals are introduced through various discovery activities in order to build a quadrilateral tree and to be able to classify the special quadrilaterals. Parallelograms are explored in further detail as students learn about sufficient conditions for parallelograms. Coordinate plane geometry is used to classify quadrilaterals.

Finally, the students mover beyond two dimensional shapes and study lines and planes in three dimensional space.

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Unit D

Polygons

We move from quadrilaterals to this unit which explores triangles and polygons and the measures of their interior and exterior angles, including “regular” polygons. Students are encouraged to see diagrams and shapes as compositions of smaller, often repeated, shapes. Students will learn the concept of “similar” polygons and the ratios of their corresponding sides, perimeters and areas.

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Unit E

Right Triangles

This unit is an exploration of families of right triangles, the Pythagorean theorem, and right triangle trigonometry. It includes the 30-60-90 and 45-45-90 right triangles and the relationship between the lengths of their sides. Word problems focus on angles of elevation and angles of depression.

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Unit F

Circles

During this unit students use many concepts learned throughout the course to solve problems involving circles. Segments and angles associated with circles are examined. Problems on the coordinate plane again bridge Algebra and Geometry skills and concepts.

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Unit G

Area, Surface Area, Volume

This short unit on area, surface area and volume gives students an opportunity to apply the Geometry they have learned throughout the year. The goal for students is to understand the formulas involved through deriving the formulas rather than simply memorize the formulas.

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Unit H

Advanced Coordinate Geometry

In this final unit, students link what they have learned in Algebra I about graphing equations to the concepts they have learned throughout this Geometry course. The work in this unit will create a smooth bridge to the work done in Algebra II.

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Geometry Level 2

UnitDescription

Unit A

A Introduction to Geometry

This unit introduces students to the majority of terminology and core concepts used in Geometry. Constructions, transformations, logical thinking and proofs are all part of the unit and will be referred to throughout the course. Students will be able to complete a two-column geometric proof by the end of the unit. Geometric software, along with compass and straightedge, may be used for constructions.

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Unit B

Congruent Triangles

This unit focuses on triangle classifications and proving triangles congruent. Proof is a fundamental concept throughout the unit. Proving and using congruent triangles will be used throughout the course.

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Unit C

Lines in a Plane

In this unit we de-emphasize two column proofs and concentrate more heavily on diagram type problems and the notion that there are other ways to make a mathematical proof including verbal and paragraph arguments but the idea of a clear, logical argument with mathematical justification for each step remains constant. .

A major focus in this unit is on quadrilaterals. Properties of quadrilaterals are introduced through various discovery activities in order to build a quadrilateral tree and to be able to classify the special quadrilaterals. Parallelograms are explored in further detail as students learn about sufficient conditions for parallelograms. Coordinate plane geometry is incorporated frequently to bridge Algebra and Geometry concepts.

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Unit D

Polygons

We move from quadrilaterals to this unit which explores triangles and polygons and the measures of their interior and exterior angles, including “regular” polygons. Students are encouraged to see diagrams and shapes as compositions of smaller, often repeated, shapes. Students will learn the concept of “similar” polygons and the ratios of their corresponding sides, perimeters and areas. Formal proofs are not done in this unit.

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Unit E

Right Triangles

This unit is an exploration of families of right triangles, the Pythagorean theorem, and right triangle trigonometry. It includes the 30-60-90 and 45-45-90 right triangles and the relationship between the lengths of their sides. Word problems focus on angles of elevation and angles of depression.

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Unit F

Circles

During this unit students use many concepts learned throughout the course to solve problems involving circles. Segments and angles associated with circles are examined. Problems on the coordinate plane again bridge Algebra and Geometry. Proofs are not done in this unit.

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Unit G

Area, Surface Area, Volume

This short unit on area, surface area and volume gives students an opportunity to apply the Geometry they have learned throughout the year. Formulas are provided to students so they focus can be on application and complex thinking rather than recall of formulas. Students solve a variety of application problems that involve surface area, area and/or volume.

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Geometry Level 3

UnitDescription

Unit A

A Introduction to Geometry

This unit introduces students to the majority of terminology used in Geometry. Constructions, transformations, logical thinking and proofs are all part of the unit and will be referred to throughout the course. Students will be able to complete a two-column geometric proof by the end of the unit. Geometric software, along with compass and straightedge, will be used for constructions and transformations.

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Unit B

Triangles

This unit focuses on triangle classifications and proving triangles congruent. Properties of triangles are applied to proofs so that students have experienced with the proof process. Proofs can be differentiated to students as they develop skill in the process by using word banks, missing statements or reasons, or cut-up proofs where student must re-order steps to establish sequence. Students will be extended to create 5-10 step proofs without assistance by the end of the unit. Segments that can be drawn in a triangle and their properties are explored.

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Unit C

Similarity

This unit extends students’ understanding of relationships between triangles (and shapes in general). They will learn what it means for shapes to be similar (congruent angles and proportional sides) and solve for sides of similar triangles. Students measuring the height of unknown objects using similarity. The second half of the unit focuses on right triangles, using the idea of similarity to introduce the concept of Trigonometry.

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Unit D

Polygons and Quadrilaterals

In this unit, students learn first about polygons and then focus on quadrilaterals. Properties of quadrilaterals are introduced through various discovery activities in order to build a quadrilateral tree and to be able to classify the special quadrilaterals. Parallelograms are explored in further detail as students learn about sufficient conditions for parallelograms. Areas of quadrilaterals are examined, and coordinate plane geometry is used to classify quadrilaterals.

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Unit E

Circles

This unit provides students a thorough study of circles. Students learn about the segments (in particular tangents) and angles (central and inscribed) associated with circles. Equations of circles on the coordinate plane are taught. The formula for circle circumference is reviewed, and students explore an informal proof of the area of a circle. Arc length and sector area are introduced.

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Unit F

Area, Surface Area, Volume

This unit explores the 3D world of surface area and volume. Students will discover the formulas for prisms, cylinders, cones, pyramids, and spheres. They will then use these concepts to determine the surface area and volume of composite figures. 2D cross-sections of 3D objects will be investigated with online applets. Finally, the effects of dilating dimensions on surface area and volume will be introduced.

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Unit G

Probability

This brief unit introduces students to probability. It begins by arranging information from sets into Venn Diagrams. Then the Venn Diagrams are used to determine probabilities. Next, some geometric probabilities related to length and area are explored. Finally, the fundamental counting principle is covered and applied to permutations and combinations.

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Integrated Algebra & Geometry

UnitDescription

Unit A

Tools for Algebra

The unit starts with an investigation of unit analysis and formula application. Then the unit moves on to a review of fraction operations and estimation techniques with instruction on the use of calculators in word problems. Similar practices are applied to decimal operations, estimation and problem solving. Fraction, decimal and percent conversions are reviewed for application in problem solving based on direct variation, similar polygons and percents.

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Unit B

Operations on Signed Numbers

Students will apply negative numbers to explain real world events. Students integrate problem solving and skill building that extends from positive numbers in Unit A to properties of integers and rational numbers in Unit B. Students develop a portfolio and track stocks to see the impact of positive and negative growth on assets. Students will examine short versus long term asset choices to determine which are best for their college and retirement savings.

21st Century Capacities: Decision Making

Unit C

Exponents and Roots

Students will explore the meaning of exponents and the rules for multiplying and dividing numbers in exponential form. Operations are extended to include negative exponents. A review of scientific notation with both negative and positive powers of 10 is included, with references to content in Integrated Science. Students solve expressions involving several steps with numbers in exponent form. Students investigate the square roots and explore their connection to rational exponents. Students apply square roots to solve missing lengths of right triangle using the pythagorean theorem. Students are challenged to apply principles of exponents and square roots to approximate the volume of a lean-to shelter.

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Unit D

Exploring Functions

This unit explores functions, which are expressed as expressions, tables and graphs. Students review the properties of equality and inequality so that they are able to manipulate equations and inequalities to solve for missing inputs and outputs. Students review the coordinate plane, graphing points, and linear functions.

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Introduction to Calculus Level 2

UnitDescription

Unit A

Rebuilding the Prerequisite Skills for Calculus

This course is not meant to take the place of AB Calculus or Calculus I but meant to be an introduction to the basic conceptual foundations of differentiation and integration and the algebraic applications used in formulaic differentiation and integration. Major concepts taught are differentiation (Chain rule and implicit application, curve sketching and related rates) and integration (substitution, application, simple initial value problems).

Even though this is an introductory course, students will be required to use specific, correct notation in all written work. Limit notation, parentheses, etc, if left out, completely change the meaning of the written expression.

The initial unit serves as a summarized review of the concepts taught in PreCalculus which are a prerequisite to the study of differentiation and integration in calculus. Students may spend up to 25% of the course time in this unit. Students are encouraged to work in groups to help each other as needed to strengthen skills and understanding.

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Unit B

Limits and Continuity

Students are introduced to the concept of a limit. Different representations will provide students with a deep understanding of limits. Simple computations of limits are introduced. Students will understand continuity of its’ implications.

For all written exercises, students must give a specific justification for any answer. Notation used must be correct. For example:

Unit B Example

Prerequisite skills to review:

●toolkit graphs

●rational, absolute value functions

●radical functions

●domain restrictions

●factoring polynomials

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Unit C

Derivatives

Students are introduced to the derivative as the slope of the line tangent to a function at any point. Students will practice finding derivative first using the limit definition and then using the rules and algebraic computation and the chain rule. Student will study implicit differentiation and apply this concept to related rate problems.

By then end of this unit students should be able to determine the equation of the tangent to a graph using explicit or implicit derivatives. Students should also be able to distinguish between position, velocity and acceleration and how they relate to each other in terms of being derivatives of each other.

There are many places to be careful and apply numerous rules at the same time. If extra time is necessary, take that time. Students must know derivative before they can do integral.

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Unit D

Using Calculus to Sketch Curves

Students will use methods of calculus to determine critical points, points of local and absolute extrema, intervals of increasing and/or decreasing, points of inflection, and intervals of concavity. Together with material already covered (such as x and y intercept(s), vertical and/or horizontal, and/or slant asymptotes, etc) they will draw clear sketches of graphs without using the graphing calculator. In addition, linear approximations and L’Hopital’s Rule are covered. Students are given time throughout the unit to work with peers to solve problems.

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Unit E

Integration

Students are introduced to the concept of an antiderivative using the concepts learned in Unit C. Using Riemann sums and sigma notation, students will see the connection of an integral to the area under a curve. Students will explore definite and indefinite integrals. Students will learn the integration technique of substitution.

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Introduction to Computer Science Level 1 & 2

UnitDescription

Unit 1

Introduction to Computer Programming with Visual Basic

Throughout this course students will develop algorithms and apply logic to use a computer to solve and model a real world problem.

Throughout this course students learn how to use logic and sets of instructions to have a computer accomplish a task.

In this unit, students will gain a general understanding of what a computer program is, how it works, and how to write one using a language such as Visual Basic and an Integrated Development Language such as Microsoft Visual Studio. Students will understand the flow of a program, how to respond to user-generated events, how to add user interface elements to a form, and how to save and run a program. Students will become acquainted with much of the terminology as well as the technology that is used throughout the course.

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Unit 2

Working with Variables, Constants and Calculations

In this unit, students will learn how a computer stores and manipulates various types of data including numeric and textual information. Students will learn how to perform basic arithmetic calculations such as adding, subtracting, multiplying, and dividing, as well as how to write code to count and accumulate values.

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Unit 3

Conditional Logic and Decision Making

Students will develop the ability to read, write, and use conditional statements to model the decision-making process used in the real world. Students will build on their ability to use variables and calculations by applying conditional logic to their use. Students will first learn how to create simple, single variable conditional statements, and will eventually learn how to model more complex decision making with compound conditional statements and singly nested conditional statements.

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Unit 4

Functions and Subroutines

In this unit, students will learn how we can break large tasks into smaller, reusable units of work called subroutines and functions. Students will learn the value of subroutines and functions, how to write them, and how to pass arguments to them. Students will continue exploring how to use built-in functions to efficiently code solutions to problems.

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Unit 5

Iteration and Computer Simulation

In this unit, students will learn how to solve problems that require looping, also known as iterations. Students will learn several different ways to structure loops, and how iteration can be a valuable problem solving technique. Students will also gain experience modeling real world events through computer simulations that are implemented using loops.

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Unit 6

Working with Strings

This unit marks a shift in the course from a focus on learning the building blocks of computer programming to using those building blocks to solve problems. In this unit, students are exposed to many of the built-in methods of the String class, and then use these methods to solve challenges involving strings that require many of the skills they’ve learned in prior units.

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Unit 7

Lists, Arrays and Problem Solving

This unit continues the theme of using a computer program as a problem solving tool. Students will learn how to use arrays and lists to represent real-world objects and how to manipulate those lists to arrive at solutions. A general four-step approach to problem solving will be explored, and students will have an opportunity to practice the approach on a series of challenging exercises.

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Unit 8

Object Oriented Concepts and Culminating Activity

In this final unit, students will gain an appreciation for object oriented programming concepts including inheritance, encapsulation, and polymorphism. Students will also have the opportunity to apply the knowledge they have learned throughout the course in a culminating programming activity.

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Pre-Calculus Level 1

Unit Description

Unit A

Trigonometry

This unit covers all the basics of trigonometry, from radian measure to right triangle and unit circle definitions to graphing. These fundamentals will be built on in further units, so it is important students understand these concepts thoroughly, without relying on the calculator until the applications are taught.

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Unit B

Analytic Trigonometry

This unit extends the topics covered in unit A. It begins with simplifying expressions so that students understand how proving trigonometric identities depends on showing one side of an equation simplifies to the other without manipulating both sides simultaneously. Students also learn how to solve trigonometric equations, in particular when the angle has been multiplied by a factor within the trigonometric function. The graphing calculator is introduced here to show how the solutions may be verified. The second half of the unit covers many formulas that extend the number of angles for which exact values of the trigonometric functions may be found.

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Unit C

Vectors

This two part unit explores applications of trignometry, most importantly vectors.

In part 1 the Laws of Sines and Cosines are derived, allowing us to solve for sides and angles in oblique (non-right) triangles. Students need to be aware that in the SSA case, there could be no, one, or two possible triangles and why this happens. Area of triangles is also covered at this time for the SAS and SSS cases (Heron’s Formula). After oblique triangles, most of the unit is spent on vectors, quantities with both magnitude and direction (velocity, force, etc). Students will learn how to express them in component form as well as a magnitude and direction angle. They will learn how to perform several operations on vectors, including the dot product. This operation is used to find angles between vectors and projections of vectors. Many applications of vectors are discussed, including plane and wind problems, force balancing, weights on ramps, and work. The unit finishes with new topic, complex numbers. By converting from a + bi form to a trigonometric form, some calculations (raising to powers and finding roots) can be done much quicker by using DeMoivre’s Theorem.

Part 2 extends the topic of vectors to 3-dimensional space. The unit begins by discussing the 3D coordinate system, including how to plot points, find distance between points, midpoints and equations of spheres. Vectors are useful in 3D to determine if points are collinear. Angles between vectors is revisited, along with the dot product. A new operation is taught, the cross product, which is only possible in 3D space. Students will learn that this operation creates a new vectors which is normal (perpendicular) to the plane containing the two original vectors and can be used to find area and volume of parallelogram-type figures. Finally, vectors are used to determine equations of lines and planes in 3D space and determining the distance between a point and a plane.

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Unit D

Systems of Equations

This short unit reviews systems of equations, with a new perspective of the graphing calculator. The traditional methods of substitution and elimination are covered, but with more complicated systems than just linear equations. The graphing calculator is used to confirm exact answers found algebraically and also used to solve equations that would be very difficult to solve by hand (ex: natural logs). No solution and infinite solutions results are discussed in the context of the intersections of the graphs of the equations. Linear systems of three variables are at first solved by hand through repeated elimination and back substitution, but these methods are quickly replaced with matrices on the graphing calculator. A brief overview of matrices is given, but most of the focus is on solving multivariable linear systems. Finally, systems are used for a new concept, partial fraction decomposition.

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Unit E

Sequences, Series, and Probability

This unit is all about recognizing patterns. Formulas for nth terms and summations of arithmetic and geometric sequences are derived. The concept of induction is introduced as a way to prove other formulas for series, divisibility, and inequalities. Then the binomial theorem is taught as an application of another pattern. Finally, combinations and permutations are covered and these counting principles are used to find probabilities.

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Unit F

Topics in Analytic Geometry

This large unit covers a variety of topics in Analytic Geometry (mostly graphing related). The concept of slope is analyzed with respect to the angles lines make with axes and other lines. A major portion of the unit focuses on the conic sections: parabolas, ellipses, and hyperbolas. These equations/graphs are introduced as loci of points and their various applications to the real world are explored, in particular their reflective properties. The rotation of their axes is not emphasized. Two new types of graphing are studied: parametric equations and polar equations which connect back to the conic sections.

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Unit G

Introduction to Calculus

This short unit introduces students to the concepts of limits and derivatives. Various techniques for determining limits are explored: graphically, numerically (table), comparing the one-sided limits, and algebraically or direct substitution, if possible. Limits are then applied to the slope of secant lines to determine the slope of a curve at a single point (slope of the tangent line) using the limit definition of a derivative. Infinite limits and limits of summations are discussed and if there is time, used to determine the area under a curve.

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Pre-Calculus Level 2

UnitDescription

Unit A

Graphs and Equations

In this unit, students review graphing and properties of linear equations. Technology such as graphing calculators will be used to model the linear relationship between two variables. Students will also review equation solving techniques which will be used throughout the remainder of the course.

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Unit B

Functions and their Graphs

In this unit, students will be re-familiarized with the definition of functions, function notation, and operations on functions. Students will learn how to express the relationship between two variables as a function, how to graph these relationships on a graphing calculator and how to determine the optimal values of a function in an appropriate domain. Function families that will be covered in this unit include polynomials, rationals and radicals. A generalized transformation rule will be introduced and will be used for the remainder of the year.

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Unit C

Exponential and Logarithmic Functions

This unit covers the basic properties of exponential and logarithmic functions. Students will be graphing and solving exponential and logarithmic functions. Students will also model “real world” situations with exponential and logarithmic functions.

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Unit D

Trigonometry

This unit covers all the basics of trigonometry, from radian measure to right triangle and unit circle definitions to graphing. These fundamentals will be built upon in further units, so it is important students understand these concepts thoroughly, without relying on the calculator.

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Unit E

Applications of Trigonometric Functions

This unit will cover solving right and oblique triangles using Trigonometry. Right triangles will be solved using right triangle trigonometry and oblique triangles will be solved using the Law of Sines or the Law of Cosines. Students will apply these concepts to “real world” problems. Area will also be covered in this unit.

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Pre-College Algebra & Trigonometry

UnitDescription

Unit 1

Rebuilding the Foundations

Many students who take this course have used memorization in previous courses. This method of learning math is not sustainable and has led to frustration and only partial understanding. In this course, students will be encouraged to reach for complete conceptual understanding of every topic, and discouraged from memorizing all but a very few small things like definitions. Learning math this way is more rewarding, enjoyable experience, and students will feel empowered to continue on in mathematics.

The goal of this mini-unit is to rewire student thinking about concepts that are fundamental to Algebra. We will look at the familiar concepts of order of operations, fractions, mathematical properties, operations on polynomials, and graphing from a new perspective that will enable students to transfer their knowledge to more abstract applications in the future. Example: fractions will be taught by prime factorization so that students can transfer their knowledge to the simplification of rational expressions.

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Unit 2

Equations and Lines

In this unit students will understand that the way we solve equations is based on an understanding of the order of operations. There are certain equations that require more than just an algebraic step (absolute value requires a leap of logic). Literal equations are a great opportunity for students to hone their equation solving skills which will be beneficial in other classes, like science.

For writing equations of lines, students should get to the point where they no longer think of every problem as a unique case, but as all being basically the same, just slight variations. Students should appreciate the usefulness of all forms of lines, not just y = mx + b, and should be allowed to leave answers in any form. Students should leave the unit displaying confidence in their understanding of how slope can be interpreted in the real world.

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Unit 3

Functions and Transformations

Students will understand the importance of functions - they give us the ability to predict because they have only one output given an input. There is an emphasis on input/output language throughout unit. Piecewise, compositions, and inverses are explored. A significant part of this unit is about transformations, having students understand how parameters affecting the inputs differ from the parameters affecting the outputs. The ABCD method of transforming functions (with point mapping) will really help for when many different parameters are used at once. Graph analysis is introduced but somewhat limited in scope.

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Unit 4

Polynomials

Students will be extremely proficient and confident at factoring in order to be able to solve polynomials later in this unit and to facilitate ease with Unit 5: simplifying rational expressions and solving rational equations. Complex numbers are briefly touched on, as is the quadratic formula, but only as tools for solving higher degree polynomial equations. Students should know by the end of this unit that the number of solutions to a polynomial equation is the same as the degree of the polynomial (The Fundamental Theorem of Algebra). Students will graph polynomials with attention to intercepts and end behavior. Quadratics are considered an optional topic.

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Unit 5

Rational Functions

The main goals of this unit include improving students’ fluency with rational expressions - understanding difference between expressions and equations and the strategies/approaches we can use to simplify each. Students will build confidence to handle more complex math problems they will encounter in future math classes. Students will make connections between equations and graphs in terms of asymptotes and domain and limits.

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Unit 6

Radicals, Exponents, and Logarithms

This unit will help students build fluency with radicals and rational exponents. They should know why negative exponents mean “divide” and rational exponents are equivalent to radical notation. We want students to know that a logarithm is used to solve for a variable in the exponent. Logarithms are a way to solve difficult (near impossible) problems in a fast, easy way. Use the language for rational exponents: “8^(⅔) means: 2 of the 3 ‘identical factors’ that multiply to 8.”

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Unit 7

Trigon

This is the first time that many students will see any trigonometry beyond SOHCAHTOA, for example, radian measure, law of sines and law of cosines, the reciprocal trig functions, trig graphs. We chose specifically to deemphasize Unit Circle, and to simply use “circle definitions” for problems like sin(240). Students should come away from this unit feeling like many trig topics can simply be done with x, y, and r (circle definitions). There are many variations of this type of problem, but students should feel the unity among them. Students need to know the basic side patterns for special right triangles along with how to draw angles in standard position.

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Statistics Levels 2 & 3

UnitDescription

Unit A

Introduction to Single Variable Statistics

Unit A begins with an overview of statistics and how they impact our lives. Students will examine univariate and bivariate data and make sense out of it using statistical methods and displays. Graphing calculators are used throughout the course.

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Unit B

Research Design

This unit explores the process of collecting and interpreting data. Students investigate sampling as a method of understanding information about populations. It includes discussion of uncertainty in samples and how the margin of error narrows as sample size grows. Students review articles with sampling and review the validity of the statistical processes used to obtain data. Experimentation is introduced. Students learn about the basic principles of experiment design, including: explanatory versus response variables, the definition of statistical significance, adjusting for confounding variables, and double blind experiments. Students explore the ethical complexities of experimentation in a review of the movie Miss Evers’ Boys, which is a historical account of the controversial Tuskegee Syphilis Study.

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Unit C

Chance

In this unit, we discuss the basic ideas and methods of probability. Our goal is not just to help students answer questions like “What’s the probability that you get no heads if you toss a fair coin 5 times?” We aim to show students the role that probability plays in statistical inference. Contrast the previous question with this one: “Suppose you toss a coin five times and get no heads. Is the coin fair?” That’s a statistics question, but you need to understand probability to answer it.

Probability is about much more than coins, dice, and cards. It’s about making decisions in the face of uncertainty. People use probability to assess the results of drug tests, to determine the strength of certain kinds of evidence in a court case, to set insurance premiums, to choose an investment strategy, and to weigh the risks and benefits of medical treatment options. Of course, probability also plays an integral role in games of chance, from state and national lotteries to casino favorites like slot machines, craps, roulette, and Texas Hold ‘Em. In Unit C, we try to strike a balance between applications involving games of chance (which motivated the study of probability in the first place) and interesting uses of probability in everyday life.

21st Century Capacities: Analyzing

Unit D

Inference

Unit D deals with the reasoning of statistical inference. It presents methods for estimating and testing claims about a population proportion. Discussion about confidence intervals builds on foundations laid in previous learning about normal distributions and sampling. Extensions are made to testing for an association between two categorical variables, and estimating and testing claims about a population mean.

21st Century Capacities: Analyzing, Synthesizing