K-12 MatheMatics curriculuM · K-12 MatheMatics DepartMent 2012-2013 Mission stateMent The aim of the Mathematics Department is to expose students to the benefits and enjoyment of

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K-12MatheMaticscurriculuM

2012-2013

New Ulm Public SchoolsAccepted by the Board of Education 4/25/13

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7-12 MatheMatics DepartMent course offerings

Class Grade Prerequisite lenGth required Credit(s)Math 7A 7 None Semester YesMath 7B 7 None Semester YesHonors Math 7A 7 Meet MCA Score Requirement Semester *alternateHonors Math 7B 7 Honors Math 7A Semester *alternate

Algebra IA 8 Math 7 or Honors Math 7A/B Semester YesAlgebra IB 8 Math 7 or Honors Math 7A/B Semester YesHonors Algebra IA 8 Math 7 or Honors Math 7A/B Semester *alternateHonors Algebra IB 8 Math 7 or Honors Math 7A/B Semester *alternate

Geometry A 9-12 Algebra IA/B or Honors Algebra IA/B Semester Yes 1Geometry B 9-12 Geometry A or Honors Geometry A Semester Yes 1Honors Geometry A 9-12 Algebra IA/B or Honors Algebra IA/B Semester *alternate 1Honors Geometry B 9-12 Honors Geometry A Semester *alternate 1

Algebra IIA 9-12 Algebra IA/B or Honors Algebra IA/B Semester Yes 1 Algebra IIB 9-12 Algebra IIA or Honors Algebra IIA Semester Yes 1Honors Algebra IIA 9-12 Algebra IA/B or Honors Algebra IA/B Semester *alternate 1Honors Algebra IIB 9-12 Honors Algebra IIA Semester *alternate 1

Probability & Statistics 9-12 Algebra IA/B or Honors Algebra IA/B Semester Yes 1

Pre-Calculus A 10-12 Geometry & Algebra II (or Honors) Semester Elective 1Pre-Calculus B 10-12 Pre-Calculus A Semester Elective 1Calculus A 11-12 Pre-Calculus A/B or Trig & Special Functions Semester Elective 1Calculus B 11-12 Calculus A Semester Elective 1

Algebra IIC 10-12 Algebra IIA/B Semester Elective 1

GRAD Math 12 Have Not Met GRAD Requirements Semester Administrative 1 Placement Only

ColleGe Courses

College Trigonometry 11-12 Pre-Calculus A Semester Elective 1& Special Functions

College Calculus 11-12 Pre-Calculus A/B or Trig & Special Functions Year Elective 2

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new ulM public schools

K-12 MatheMatics DepartMent2012-2013

Mission stateMent

The aim of the Mathematics Department is to expose students to the benefits and enjoyment of mathematics by providing progressive, high quality instruction for all students, at all levels. The de-partment seeks to prepare students for everyday life and advanced study of mathematics by teaching mastery of skills/standards essential to meet this goal.

exit outcoMes/essential learner outcoMes (elo)The Minnesota Academic Standards in Mathematics set the expectations for achievement in mathe-matics for K-12 students in Minnesota. This document is grounded in the belief that all students can and should be mathematically proficient. All students should learn important mathematical concepts, skills, and relationships with understanding. The standards and benchmarks presented here describe a connected body of mathematical knowledge that is acquired through the processes of problem solv-ing, reasoning and proof, communication, connections, and representation. The standards are placed at the grade level where mastery is expected with the recognition that intentional experiences at ear-lier grades are required to facilitate learning and mastery for other grade levels.

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K-12 assessMent of strengths anD liMitations

Strengths of the department:K-3 1. The current math curriculum, HSP Math, is available online for both students and parents to reference at home. 2. The current math curriculum is aligned to the 2007 State Standard that provides clear and consistent expecta-

tions for all students. 3. Math terminology, language, and vocabulary is consistent throughout the K-3 curriculum. 4. There are consistent expectations through the Essential Learner Outcomes with both formative and summative

assessments that ensure high expectations for all students. 5. Students are engaged interactively through the use of technology and a variety of manipulatives including base

10 blocks, pattern blocks, and connecting cubes. 6. Students who meet the MCA proficiency level are above the state average for grade 3.

4-6 1. The current math curriculum is aligned to the 2007 State Standard that provides clear and consistent expecta-

tions for all students. 2. There are consistent expectations through the Essential Learner Outcomes with both formative and summative

evaluations that ensure high expectations for all students. 3. A range of assessments are in place to identify students who need remediation and intervention as well as for

those students who have mastered the standard and benchmarks. 4. The iTools on Think Central are valuable teaching tools and contain effective resources for struggling students. 5. Departmentalization has allowed for a more specialized approach to teaching the math curriculum at grade 6. 6. The implementation of a variety of intervention classes provides extra time and a clear focus for students to mas-

ter the standards. 7. The implementation of accelerated math classes ensures high expectations for students with greater mathematical

ability.

7-12 1. The current math curriculum is aligned to the 2007 State Standard that provides clear and consistent expecta-

tions for all students. 2. The members of the math department all approach teaching with a proactive mentality. 3. The members of the math department seeks opportunities to stay current with trends and issues in mathematics

and in education. 4. The members of the math department are a connected staff that work cooperatively. 5. The members of the math department pride themselves in making time available for students prior to the start of

school, throughout the school day, and after the school day to answer individual questions and to provide indi-vidual instruction and mentoring.

6. The math department is able to provide a variety of course offerings that fit the needs of all students regardless of ability.

7. Students perform well on the American College Testing (ACT). 8. The Minnesota Comprehensive Assessments (MCA) raw score of students tends to exceed the state averages. 9. A proactive remediation plan for the Graduation-Required Assessments for Diploma (GRAD) requirements has

provided for a high success rate to meet the graduation requirement. 10. The math department is able to provide a wide array of technological resources to students including: graphing

calculators; computer applications like spreadsheet programs; Smartboard/Notebook activities, internet applica-tions for presentation; digital copies of some textbooks, etc.

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Limitations of the department:

K-3 1. It is always a struggle to find the time necessary to meet the individual needs of students within the constraints

of the school day, necessary manipulatives, and sufficient supporting materials. 2. There is a lack of time in the school day and the resources necessary to adequately provide time for math inter-

vention for struggling students and enrichment for students who excel. 3. Additional needs exist to assist with differentiated instruction. 4. The current curriculum often times does not provide adequate review and practice of key concepts resulting in

teachers having to create those reviews and practices. 5. The current computer lab is inadequate for the increasing needs of instruction, remediation, enrichment and

testing. 6. There is a lack of specialized training and professional development opportunities both at the local, regional and

state level. 7. With the state’s and nation’s emphasis upon testing and accountability, this takes valuable time away from

instructional time.

4-6 1. It is always a struggle to find the time necessary to meet the individual needs of students within the constraints

of the school day, necessary manipulatives, and sufficient supporting materials. 2. Due to the structure and confines of the school day, instruction and the necessary practice time is severely lim-

ited. 3. Far more instructional time is needed to fully prepare students for the MCA testing. 4. Computer lab space and the resulting limits, prohibit many opportunities that exist for individual and expanded

learning. 5. There is a lack of specialized training and professional development opportunities both at the local, regional and

state level. 6. Greater effort must be placed to insure that a greater percentage of students meet or exceed standards. 7. There is no spiral review or extra math practice readily available within the current math series that results in

the need for individual teachers to create the necessary review and extra practice. 8. The need to have a solid grasp of the basic math facts is vital for the success of 4th grade students as they move

through the curriculum.

7-12 1. The overall math scores by juniors on the ACT do not necessarily reflect on the curriculum, but rather reflect

more on the fact that students have not taken the appropriate college readiness courses in mathematics. 2. The percentage of students achieving proficiency on the MCAs needs to be increased. 3. It is always a struggle to find the time necessary to meet the individual needs of students within the constraints

of time, working within the master schedule, and supporting materials. 4. Time becomes a greater challenge as mathematics requirements increase. 5. There is a lack of specialized training and professional development opportunities both at the local, regional and

state level. 6. Time and scheduling restraints severely limit the options for remediation within the school day. 7. A specific resource area is needed that should be staffed by knowledgeable and capable personnel. 8. The loss of specialized math paraprofessionals has had a significant impact on the amount of one-on-one time

students are receiving in the classroom. 9. There needs to be expanded online or hybrid upper level math courses offered to students by New Ulm High

School math teachers.

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iMproveMent plan

The following improvement plans are recommended for implementation:K-3 1. Create a common block schedule by grade level to allow for differentiated instruction. (Limitation #1) 2. Schedule support staff to be able to assist with differentiated instruction. (Limitation #2) 3. Focus needs to be placed on meeting the needs of all students in the class. (Limitation #3) 4. Working with the technology department, update Jefferson’s technological systems. (Limitation #4) 5. Provide staff development training for current best practices in math. (Limitation #5) 6. Prioritize testing at each grade level. (Limitation #6) 7. Provide consistent curriculum review through PLCs. (Limitation #7)

4-6 1. Focus needs to be placed on meeting the needs of all students in the class. (Limitation #1) 2. Research alternate scheduling formats to find more minutes of instruction and practice within the day.

(Limitation #2) 3. Focus on data collected through formative assessments, to inform instructional decisions for individual

students. (Limitation #3) 4. Research staff development opportunities that focus on math instruction and express the need to share

this information during staff development days. (Limitation #4) 5. Investigate opportunities to increase instructional time prior to the MCA test. (Limitation #5) 6. Foster a mindset among students that they are also responsible for their learning. (Limitation #6) 7. Explore and find quality materials to supplement the lack of extra practice and spiral review in the cur-

riculum. (Limitation #7) 8. Collaborate vertically within the math department to ensure students have the foundational skills neces-

sary for the next level. (Limitation #8)

7-12 1. Math staff and guidance office staff will work together to encourage students to take appropriate math classes

prior to taking the ACT. Use the PLAN test data to direct the discussion. (Limitation #1) 2. Offer and monitor honors class implementation to see impact on the regular classes and honors classes. Revisit

essential learner outcomes on a regular basis and compare to MCA data. (Limitation #2) 3. Seek innovative opportunities and options to meet the individual needs of students and improve overall

student achievement. (Limitation #3 & 4) 4. Research opportunities through professional organizations and technology coordinator. Continue to

share and collaborate. (Limitation #5) 5. Monitor and adjust the resource time framework to include more time for individualized remediation.

(Limitation #6) 6. Explore expanding the resource room to be available for all students. (Limitation #7) 7. Expand co-teaching with special education staff. Look for professional development opportunities for co-

teaching staff to implement teaching strategies. (Limitation #8) 8. Explore training opportunities and research on math online/hybrid courses. Implement and monitor pilot

programs. (Limitation #9)

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GRADE STRAND STANDARD BENCHMARK

Read, write, and represent whole numbers from 0 to at least 31. Representations may include numerals, pictures, real objects and picture graphs, spoken words, and manipulatives such as connecting cubes.

Implementation: Represent the number of students taking hot lunch with tally marks. Create classroom graphs such as favorite color and favorite food.

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Understand the relationship between quantities and whole numbers up to 31.

Recognize that a number can be used to represent how many objects are in a set or to represent the position of an object in a sequence. Implementation: Count students standing in a circle and count the same students after they take their seats. Recognize that this rearrangement does not change the total number, but may change the order in which students are counted. Use the same number of manipulatives to make different arrangements.

Count, with and without objects, forward and back-ward to at least 20.

Implementation: Use websites such as youtube.com to enhance counting. Use manipulatives to count forward and backward.

MatheMatics DepartMent

learner outcoMesKinDergarten

Find a number that is 1 more or 1 less than a given number.

Implementation: Use manipulatives such as number lines, dice, and hundred charts. Also use daily calendar (today, tomorrow, and yesterday).

Compare and order whole numbers, with and without objects, from 0 to 20.

Implementation: Put the number cards 7, 3, 19 and 12 in numerical order. Roll 3 dice and put the numbers in order.

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Kindergarten Mathematics Learner Outcomes continued...


Compose and decompose numbers up to 10 with ob-jects and pictures.

Implementation: A group of 7 objects can be decomposed as 5 and 2 objects, or 2 and 3 and 2, or 6 and 1. Use fact family triangles.

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Use objects nd pictures to represent situations involving combining and separating.

Use objects and draw pictures to find the sums and dif-ferences of numbers between 0 and 10. Implementation: Use dots, dice, tens frames, dominoes and white boards to show sums and differences. Use math games such as domino parking lot.

Identify, create, complete, and extend simple patterns using shape, color, size, number, sounds and move-ments. Patterns may be repeating, growing or shrink-ing such as ABB, ABB, ABB or •, ••, •••.

Implementation: Use numerous manipulatives to create and label patterns. Make patterns using body movement.

Recognize basic two- and three- dimensional shapes such as squares, circles, triangles, rectangles, trapezoids, hexagons, cubes, cones, cylinders and spheres.

Implementation: Use shape manipulatives (two- and three- dimensional) to describe shape attributes. Practice drawing shapes.

Sort objects u sing characteristics such as shape, size, color and thickness.

Implementation: Use manipulatives such as yektti cards, crayons, different sized shapes, etc.

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and extend patterns.

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Recognize and sort basic two- and three- dimensional shapes; use them to model real- world objects.

Use basic shapes and spatial reasoning to model objects in the real-world.

Implementation: A cylinder can be used to model a can of soup. Find as many rectangles as you can in your classroom. Record the rectangles you found by making drawings.

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Kindergarten Mathematics Learner Outcomes continued...


Order 2 or 3 objects using measurable attributes, such as length and weight.

Implementation: Use school supplies to order by length and weight.

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Compare and order objects according to location and measurable attributes.

Use words to compare objects according to length, size, weight and positions. Implementation: Use same, lighter, longer, above, between and next to. Identify objects that are near your desk and objects that are in front of it. Explain why there may be some objects in both groups.

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Read, write, and represent whole numbers up to 120. Representations may include numerals, addition and subtraction, pictures, tally marks, number lines and maniuplatives, such as bundles of sticks and base 10 blocks.

Implementation: Use math manipulatives to show random numbers to 120.

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Count, compare and repre-sent whole numbers up to 120, with an emphasis on groups of tens and ones.

Use place value to describe whole numbers between 10 and 100 in terms of tens and ones. Implementation: Recognize the numbers 21 to 29 as 2 tens and a particular number of ones.

Count, with and without objects, forward and back-ward from any given number up to 120.

Implementation: Count forward and backwaard from 61.

Find a number that is 10 more or 10 less than a given number.

Implementation: Using a hundred grid, find the number that is 10 more than 27.

Compare and order whole numbers up to 120.

Implementation: Put 110, 76, 12, 48 in order from least to greatest.

Use words to describe the relative size of numbers.

Implementation: Use the words equal to, not equal to, more than, less than, fewer than, is about, and is nearly to describe numbers.

Use counting and companion skills to create and ana-lyze bar graphs and tally charts.

Implementation: Make a bar graph of students’ birthday months and count to compare the number in each month.

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Grade 1 Mathematics Learner Outcomes continued...


1

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strategies to solve addition and subtraction problems in read-world and mathematical contexts.

Use words, pictures, objects, length-based models (connecting cubes), numerals and number lines to model and solve addition and subtraction problems in part-part-total, adding to, taking away from and com-paring situations. Implementation: Solve: There were 6 dogs and 7 cats. How many animals in all?

Solve: Lucy has 3 cookies and Dan has 8 cookies. How many more does Dan have?

Compare and decompose numbers up to 12 with an emphasis on making ten.

Implementation: Given 3 blocks, 7 more blocks are needed to make 10.

Create simple patterns using objects, pictures, num-bers and rules. Identify possible rules to complete or extend patterns. Patterns may be repeating, growing or shrinking. Calculators can be used to create and explore patterns.

Implementation: Describe rules that can be used to extend the pattern 2, 4, 6, 8, ¨, ¨, ¨ and complete the pattern 33, 43, ¨, 63, ¨, 83 or 20, ¨, ¨, 17.

Represent real-world situations involving addition and subtraction basic facts, using objects and number sentences.

Implementation: One way to represent the number of toys that a child has left after giving away 4 of 6 toys is to begin with a stack of 6 connecting cubes and then break off 4 cubes.

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Recognize and create pat-terns; use rules to describe patterns.

Determine if equations involving addition and subtrac-tion are true.

Implementation: Determine if the following number sen-tences are true or false.7=7 5+2=2+57=8–1 4+1=5+2

Recognize the relationship between counting and addi-tion and subtraction. Skip count by 2s, 5s, and 10s.

Implementation: Use number line as tool to solve 6 + 3 and 10 – 2. Use hundreds grid to skip count/color.

Use number sentences in-volving addition and subtrac-tion basic facts to represent and solve real-world and mathematical problems, create real-world situations corresponding to number sentences.

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Use addition or subtraction basic facts to represent a given problem situation using a number sentence.

Implementation: 5+5=8 could be used to represent a situation in which 5 red balloons are combined with 3 blue balloons to make 8 total balloons.

1Use number sense and models of addition and subtrac-tion, such as objects and number lines, to identify the missing number in an equation such as:2+4= ¨ 3+ ¨=7 5= ¨–3 Implementation: Use the numbers 5, 4, and 9 to create a fact family.

Describe characteristics of two– and three–dimentional objects, such as triangles, squares, rectangles, circles, rectangular prisms, cylinders, cones and spheres.

Implementation: Triangles have three sides and cubes have eight vertices (corners).

Compose (combine) and decompose (take apart) two– and three–dimentional figures such as triangles, squares, rectangles, circles, rectangular prisms and cylinders.

Implementation: Decompose a regular hexagon into 6 equi-lateral triangles; build prisms by stacking layers of cubes, com-pose an ice cream cone by combining a cone and half of a sphere.Use a drawing program to find shapes that can be made with a rectangle and a triangle.

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Describe characteristics of basic shapes. Use basic shapes too compose and decompose other objects in various contexts.

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Use basic concepts of mea-surement in real-world and mathematical situations involving length, time and money.

Measure the length of an object in terms of multiple copies of another object.

Implementation: Measure a table by placing paper clips end-to-end and counting.

Use number sentences in-volving addition and subtrac-tion basic facts to represent and solve real-world and mathematical problems, create real-world situations corresponding to number sentences.

Tell time to the hour and half hour.

Implementation: Read analog clock set to 9:00 and 11:30.

Identify pennies, nickels and dimes; find the value of a group of these coins, up to one dollar.

Implementation: Use real money to show 7¢, 18¢ and 35¢.

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Read, write, and represent whole numbers up to 1000. Representations may include numerals, addition, sub-traction, multiplication, words, pictures, tally marks, number lines and manipulatives, such as bundles of sticks and base 10 blocks.

Implementation: Use number of the day to read, write and represent whole numbers.

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Compare and represent whole numbers up to 1000, with an emphasis on place value and equality.

Use place value to describe whole numbers between 10 and 1000 in terms of hundreds, tens nd ones. Know that 100 is 10 tens, and 1000 is 10 hundreds.

Implementation: Writing 853 is a shorter way of writing.8 hundreds + 5 tens + 3 ones

Find 10 more or 10 less than a given three-digit num-ber. Find 100 more or 100 less than a given three-digit number.

Implementation: Find the number that is 10 less than 382 and the number that is 100 more than 382.

Round numbers up to the nearest 10 and 100 and round numbers down to the nearest 10 and 100.

Implementation: If there are 17 students in the class and granola bars come 10 to a box, you need to buy 20 bars (2 boxes) in order to have enough bars for everyone.

Compare and order whole numbers up to 1000.

Implementation: Order three, three-digit numbers from least to greatest or greatest to least.

Use strategies to generate addition and subtraction facts including making tens, fact families, doubles plus or minus one, counting on, counting back, and the commutative and associative properties. Use the rela-tionship between addition and subtraction to generate basic facts.

Implementation: Use the associative property to make tens when adding.5+8=(3+2)+8=3+(2+8)=3+10=13

Demonstrate mastery of ad-dition and subtraction basic facts, add and subtract one- and two-digit numbers in real-world and mathematical problems.

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Demonstrate mastery of ad-dition and subtraction basic facts, add and subtract one- and two-digit numbers in real-world and mathematical problems.

Demonstrate fluency with basic addition facts and related subtraction facts.

Implementation: Complete 100 fact timing in addition and subtraction in 5 minutes or less.

Estimate sums and differences up to 100.

Implementation: Know that 23+48 is about 70.

Solve real-world and mathematical addition and sub-traction problems involving whole numbers wit up to 2 digits.

Implementation: Solve--Mrs. Jenkins brings 36 apples to school. The children eat 23. How many apples are left?Solve--Mark has 37 blue stickers and 29 green stickers. How many does he have in all?

Use addition and subtraction to create and obtain in-formation from tables, bar graphs and tally charts.

Implementation: Create a graph showing how the class goes home from school. Analyze the information shown.Create a monthly weather graph and analyze the information shown.

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Identify, create and describe simple number patterns involving repeated addition or subtraction, skip count-ing and arrays of objects such as counters or tiles. Use patterns to solve problems in various contexts.

Implementation: Ski count by 5s beginning at 3 to create the pattern. 3, 8, 13, 18,...Collecting 7 empty milk cartons each day for 5 days will generate the pattern 7, 14, 21, 28, 35, resulting in a total of 35 milk cartons.

Recognize, create, describe, and use patterns and rules to solve real-world and math-ematical problems.

Use mental strategies and algorithms based on knowl-edge of place value and equality to add and subtract two-digit numbers. Strategies may include decomposi-tion, expanded notation, and partial sums and differ-ences.

Implementation: Using decomposition, 78+42, can be thought of as:78+2+20+20=80+20+20=100+20+120 and using expanded notation, 34-21 can be thought of as: 30+4-20-1=30-20+4-1=10+3=13

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Use addition or subtraction basic facts to represent a given problem situation using a number sentence.GRADE STRAND STANDARD BENCHMARK

2Understand how to interpret number sentences involv-ing addition, subtraction and unknowns represented by letters. Use objects and number lines and create real-world situations to represent number sentences. Implementation: One way to represent n+16=19 is by comparing a stack of 16 connecting cubes to a stack of 19 connecting cubes; 24=a+b can be represented by a situation involving a birthday party attended by a total of 24 boys and girls.

Use number sentences involving addition, subtraction, and unknowns to represent given problem situations. Use number sense and properties of addition and sub-traction to find values for the unknowns that make the number sentences true.

Implementation: How many more players are needed if a soccer team requires 11 players and so far only 6 players have arrived? This situation can be represented by the number sen-tence 11–6=p or by the number sentence 6+p=11.

Describe, compare, and classify two- and three-dimen-sional figures according to number and shape of faces, and the number of sides, edges and vertices (corners).

Implementation: Create a chart comparing sides, edges and vertices of two- and three-dimensional shapes.

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Identify, describe and com-pare basic shapes according to their geometric attributes.

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Understand length as a mea-surable attribute; use tools to measure length.

Identify and name basic two- and three-dimensional shapes, such as squares, circles, triangles, rectangles, trapezoids, hexagons, cubes, rectangular prisms, cones, cylinders and spheres.

Implementation: Use Smartboard lesson 2 and 30 shape identification.

Use number sentences in-volving addition, subtraction and unknowns to represent and solve real-world and mathematical problems; create real-world situations corresponding to number sentences.

Understand the relationship between the size of the unit of measurement and the number of units needed to measure the length of an object.

Implementation: It will take more paper clips than white-board markers to measure the length of a table.

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2Demonstrate an understanding of the relationship between length and the numbers on a ruler by using a ruler to measure lengths to the nearest centimeter or inch. Implementation: Draw a line segment that is 3 inches long.

Tell time to the quarter-hour and distinguish between a.m. and p.m.

Implementation: With manipulative clock, show given times to the quarter hour. Identify example as a.m. or p.m. activity.

Identify pennies, nickels, dimes and quarters. Find the value of a group of coins and determine combinations of coins that equal a given amount.

Implementation: 50 cents can be made up of 2 quarters, or 4 dimes and 2 nickels, or many other combinations.

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Understand length as a mea-surable attribute; use tools to measure length.

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Read, write and represent whole numbers up to 100,000. Representations may include numerals, expressions with operations, words, pictures, number lines, and manipula-tives such as bundles of sticks and base 10 blocks.

Implementation: Count the days in school using sticks to represent numbers in the hundreds, tens and ones place.Using number lines and hundreds charts, find the next num-ber in the pattern 3, 6, 9, 12, ¨.

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Compare and represent whole numbers up to 100,000 with an emphasis on place value and equality.

Use place value to describe whole numbers between 1000 and 100,000 in terms of ten thousands, thou-sands, hundreds, tens and ones.

Implementation: Writing 54,873 is a shorter way of writing the following sums: 5 ten thousands+4 thousands+8 hundreds+7 tens+3 ones or 54 thousands+8 hundreds+7 tens+3 ones. The number 4,756 can be represented by writing the following: 4,000+700+50+6. Identify value and place in large numbers: in the number 380,429, the 8 is in the ten thousands place.

Find 10,000 more or 10,000 less than a given five-digit number, Find 1000 more or 1000 less than a given four– or five–digit. Find 100 more or 100 less than a given four– or five–digit number.

Implementation: Using whiteboards to find 10,000 more than the number 58,760 by using place value knowledge. The number 5 is in the ten thousands place so increase the number by 1 making it a 6. Use Study [email protected]; num-bers; Place value games/lessons on Smartboard.

Round numbers to the nearest 10,000, 1000, 100 and 10. Round up and round down to estimate sums and differences.

Implementation: 8726 rounded to the nearest 1000 is 9000, rounded t the nearest 100 is 8700, and rounded to the nearest 10 is 8730.473–291 is between 400–300 and 500–200, or between 100 and 300.Use StudyJams!@ Scholastic.com; Numbers; estimate whole numbers and Funbrain.com. Place Value Puzzler game.

Compare and order whole numbers up to 100,000.

Implementation: Use Smart Notebook Lesson “Comparing Numbers”@Smart Exchange. Using place value models such as base ten blocks, decide which number is greater or less.

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Add and subtract multi-digit whole numbers; represent multiplication and division in various ways; solve real-world and mathematical problems using arithmetic.

Add and subtract multi-digit numbers, using efficient and generalizable procedures based on knowledge of place value, including standard algorithms. Implementation: Compose and decompose numbers for ex-ample, adding 184 and 37 could include 180 and 30, then adding 4 and 7. Using multiples of 10 to add or subtract more efficiently such as, 84+37=(80+4) + (20+10+7).

Represent multiplication facts by using a variety of ap-proaches, such as repeated addition, equal-sized groups, arrays, area models, equal jumps on a number line and skip counting. Represent division facts by using a variety of approaches, such as repeated subtraction, equal sharing and forming equal groups. Recognize the relationship between multiplication and division.

Implementation: Use fact triangles to tie to-gether multiplication and division. For example: Use Math Magician from oswego.org to practice addition, subtraction, multiplication and division facts. Use HSP Math on Location DVD “Inventing Toys” and “The Bakery” with movie guide worksheets to explore multiplication and division concepts.

Solve real-world and mathematical problems involving multiplication and division, including both “how many in each group: and “how many groups” division problems.

Implementation: You have 27 people and 9 tables. If each table sets the same number of people, how many people will yo put at each table? If you have 27 people and tables that will hold 9 people, how many tables will you need? Use of [email protected]; multiplication games. Multiplication Smart Notebook lesson @ SmartExchange.USA

Use addition and subtraction to solve real-world and mathematical problems involving whole numbers. Use various strategies, including the relationship between addition and subtraction, the use of technology and the context of the problem to assess the reasonableness of results.

Implementation: The calculation 117–83=34 can be checked by adding 83 and 34. Use HSP Math on Loca-tion DVD “Wildlife Refuge” video segment and movie guide worksheets. Modeling word problems through the use of math journals and Problem of the Day. For example: There are 11 magicians performing. 6 of the magicians need an assistant. How many magicians do not need an assistant?

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Add and subtract multi-digit whole numbers; represent multiplication and division in various ways; solve real-world and mathematical problems using arithmetic.

Use strategies and algorithms based on knowledge of place value,equality and properties of addition and mul-tiplication to multiply a two-or three-digit number by a one-digit number. Strategies may include mental strate-gies, partial products,the standard algorithm, and the commutative, associative, and distributive properties.

Implementation: 9 x 26 = 9 x (20+6) = 9 x 20+9 x 6 =180 + 54 = 234. Use Jefferson School Third Grade Useful Websites: “Math Journey” - remainders. Use Math arrays to show examples of commutative property. For example: ¨¨ ¨¨¨ 3 x 2 ¨¨ or ¨¨¨ 2 x 3 ¨¨ 3 rows of 2 tiles or 2 rows of 3 tiles.

Read and write fractions with words and symbols. Recognize that fractions can be used to represent parts of a whole, part of a set, points on a number line, or distance on a number line.

Implementation: Parts of a shape (3/4 of a pie), parts of a set (3 out of 4 people), and measurements (3/4 of an inch). Use HSP Math on Location DVD - “The First American video segment with movie guide worksheets to review fraction and decimal concepts. Use “The Rational Number Project” from the U of MN (provides 28 lessons on fraction opera-tions). For example: “Fractions on a Number Line”.

Understand that the size of a fractional part is relative to the size of the whole.

Implementation: One-half of a small pizza is smaller than one-half of a large pizza, but both represent one-half. Solve “Problem of the Day” - The students at Elena’s table ate 5 of a whole pizza. The students at Peter’s table ate 3/4 of a pizza. Who ate less?

Understand meanings and uses of fractions in real-world and mathematical situations.

Order and compare unit fractions and fractions with like denominators by using models and an understanding of the concept of numerator and denominator.

Implementation: Solve A”Problem of the Day”. Use fraction bars to compare numbers such as 2/6 < 3/6 or 5/10 > 1/3.

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3Create, describe, and apply single-operation input-out-put rules involving addition, subtraction and multipli-cation to solve problems in various contexts. Implementation: Describe the relationship between number of chairs and number of legs by the rule that the number of legs is four times the number of chairs. Using a table, extend the pattern using a rule. For example (at right):

Understand how t interpret number sentences in-volving multiplication and division basic facts and unknowns. Create real-world situations to represent number sentences.

Implementation: The number sentence 8 x m = 24 could be represented by the question “How much did each ticket to a play cost if 8 tickets totaled $24?” Also, use of [email protected]; algebra; creating equations from word problems and; determine the missing operation in an equation.

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Use geometric attributes to describe and create shapes in various contexts.

Use multiplication and division basic facts to represent a given problem situation using a number sentence. Use number sense and multiplication and division basic facts to find values for the unknowns that make the number sentences true.

Implementation: Find values of the unknowns that make each number sentence true. 6 = p÷9; 24 = a x b;5 x 8 = 4 x t.How many math terms are competing if there is a total of 45 students with 5 students on each team? This situation can be represented by 5 x a = 45 or 45/5 = n or 45/n = 5.

Use single-operation input-output rules to represent patterns and relationships and to solve real-world and mathematical problems.

Identify parallel and perpendicular lines in various contexts, and use them to describe and create geomet-ric shapes, such as right triangles, rectangles, parallelo-grams and trapezoids.

Implementation: Use [email protected]; geome-try; types of lines. Also, trade book = “The Greedy Triangle”.

in out10 6 4 012 8 ? 4

Rule = subtract 4

Use number sentences in-volving multiplication and division basic facts and unknowns to represent and solve real-world and mathe-matical problems, create real-world situations correspond-ing to number sentences.

Sketch polygons with a given number of sides or vertices (corners), such as pentagons, hexagons and octagons.

Implementation: Create geometry book using construction paper. Cut and fold “nets” for 3-D shape examples.

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3Use half units when measuring distances. Implementation: Measure a person’s height to the nearest half inch. Use [email protected]; measurement; “Tools of Measurement” and “Measure Length”. Use HSP Math on Location DVD: “America’s Teaching Zoo” and movie guide worksheets.

Find the perimeter of a polygon by adding the lengths of the sides. Implementation: Find the perimeter of your math book by measuring in inches and adding all the sides. Repeat by mea-suring in centimeters.

Measure distances around objects. Implementation: Measure the distance around a classroom, or measure a person’s wrist size. Use geoboards or grid paper to create shapes and their measure to find the perimeter.

Use time, money and temper-ature to solve real-world and mathematical problems.

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Determine elapsed time to the minute. Implementation: Your trip began at 9:50 a.m. and ended at 3:10 p.m. How long were you travel-ing? Using T-charts to find elapsed time such as: start time = 11:30 and end time is 45 minutes later.

Understand perimeter as a measurable attribute of real-world and mathematical objects. Use various tools to measure distances.

Know relationships among units of time. Implementation: Know the number of minutes in an hour, days in a week and months in a year. Make conversions using mixed units such as 12 days = 1 week and 5 days.

11:30 Minutes11:35 511:40 1011:45 15 etc.

Make change up to one dollar in several different ways, including with as few coins as possible. Implementation: A chocolate bar costs $1.84. You pay for it with $2. Give two possible ways to make change. Use HSP Math on Location DVD: “Ice Cream the Italian Way” and movie guide worksheets. Coin nd paper manipulatives to count out change.

Use an analog thermometer to determine temperature to the nearest degree in Fahrenheit and Celsius. Implementation: Read the temperature in a room with a thermometer that has both Fahrenheit and Celsius scales. Use the thermometer to compare Celsius and Fahrenheit readings.

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3Collect display and interpret data using frequency tables, bar graphs, picture graphs and number line plots having a variety of scales. Use appropriate titles, labels and units. Implementation: Students make a survey, collect data, dis-play information in several ways and share results with class by using a Smartboard.

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Collect, organize, display, and interpret data. Use labels and a variety of scales and units in displays.

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Demonstrate fluency with multiplication and division facts.

Implementation: 6 x 8 = 48 and 81 ÷ 9 = 9

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Demonstrate mastery of mul-tiplication and division basic facts; multiply multi-digit numbers; solve real-world and mathematical problems using arithmetic.

Use an understanding of place value to multiply a number by 10, 100 and 1000.

Implementation: 8 x 10 = 80 (10s + add a 0)

Multiply multi-digit numbers, using efficient and generalizable procedures, based on knowledge of place value, including standard algorithms.

Implementation: 62 x 14 = 868

Estimate products and quotients of multi-digit whole numbers by using rounding, benchmarks and place value to assess the reasonableness of results.

Implementation: 53 x 38 is between 50 x 30 and 60 x 40, or between 1500 and 2400, and 411/73 is between 5 and 6.

Solve multi-step real-world and mathematical problems requiring the use of addition, subtraction and multipli-cation of multi-digit whole numbers. Use various strat-egies, including the relationship between operations, the use of technology, and the context of the problem to assess the reasonableness of results.

Implementation: The sum of two numbers is 17. Their product is 72. What are the numbers? (8+9)

Use strategies and algorithms based on knowledge of lace value, equality and properties of operations to divide multi-digit whole numbers by one- or two-digit numbers. Strategies may include mental strategies, partial quotients, the commutative, associative, and distributive properties and repeated subtraction.

Implementation: A group of 324 students is going to a mu-seum in 6 buses. If each bus has the same number of students, how many students will be on each bus?

Represent and compare fractions and decimals in real-world and mathemati-cal situations; use place value to understand how decimals represent quantities.

Represent equivalent fractions using fraction models such as parts of a set, fraction circles, fraction strips, number lines and other manipulatives. Use the models to determine equivalent fractions.

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Represent and compare fractions and decimals in real-world and mathemati-cal situations; use place value to understand how decimals represent quantities.

Locate fractions on a number line. Use models to order and compare whole numbers and fractions, including mixed numbers and improper fractions. Implementation: Locate 5/3 and 1 3/4 on a number line and give a comparison statement about these two fractions, such as “5/3 is less than 1 3/4”.

Read and write decimals with words and symbols; use place value to describe decimals in terms of thousands, hundreds, tens, ones, tenths, hundredths and thousandths.

Implementation: Writing 362.45 is a shorter way of writ-ing the sum: 3 hundreds + 6 tens + 2 ones + 4 tenths + 5 hundredths, which can also be written as: three hundred sixty-two and forty-five hundredths.

Compare and order decimals and whole numbers using place value, a number line and models such as grids and base 10 blocks.

Use fraction models to add and subtract fractions with like denominators in real-world and mathematical situ-ations. Develop a rule for addition and subtraction of fractions with like denominators.

Read and write tenths and hundredths in decimal nd fraction notations using words and symbols; know the fraction and decimal equivalents for halves and fourths.

Implementation: 1/2 = 0.5 = 0.50 and 7/4 = 1 3/4 = 1.75, which can also be written as one and three-fourths or one and seventy-five hundredths.

Round decimals to the nearest tenth.

Implementation: The number 0.36 rounded to the nearest tenth is 0.4.

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Use input-output rules, tables and charts to represent patterns and relationships and to solve real-world and mathematical problems.

Create and use input-ouput rules involving addition, subtraction, multiplication and division to solve prob-lems in various contexts. Record the inputs and outputs in a chart or table.

Implementation: If the rule is “multiply by 3 and add 4,” record the outputs for given inputs in a table.

A student is given these three arrangements of dots:

Identify a pattern that is consistent with these figures, create an input-output rule that describes the pattern, and use the rule to find the number of dots in the 10th figure.

•••• ••• •••••• ••• ••••

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4

Use multiplication, division and unknowns to represent a given problem situation using a number sentence. Use number sense, properties of multiplication, and the relationship between multiplication and division to find values for the unknowns that make the number sen-tences true.

Implementation: If $84 is to be shared equally among a group of children, the amount of money each child receives can be determined using the number sentence 84÷n=d.

Find values of the unknown that made each number sentence true: 12 x m =36 and s = 256 ÷ t.

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Name, describe, classify and sketch polygons.

Use number sentences in-volving multiplication, divi-sion and unknowns to rep-resent and solve real-world and mathematical problems; create real-world situations corresponding to number sentences.

Describe, classify and sketch triangles, including equi-lateral, right, obtuse and acute triangles. Recognize triangles in various contexts.

Implementation: Equilateral triangle has 3 equal sides.

Describe, classify and draw quadrilaterals, including squares, rectangles, trapezoids, rhombuses, parallelo-grams and kites. Recognize quadrilaterals in various contexts.

Implementation: Trapezoid

Understand how to interpret number sentences involv-ing multiplication, division and unknowns. Use real-world situations involving multiplication or division to represent number sentences.

Implementation: The number sentence a x b = 60 can be represented by the situation in which chairs are being ar-ranged in equal rows and the total number of chairs is 60.

2 cm 2 cm

2 cm

Understand angle and area as measurable attributes of real-world and mathematical objects. Use various tools to measure angles and areas.

Measure angles in geometric figures and real-world objects with a protractor or angle ruler.

Compare angles according to size. Classify angles as acute, right and obtuse.

Implementation: Compare different hockey sticks according to the angle between the blade and the shaft.

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Find the area of geometric figures and real-word objects that can be divided into rectangular shapes. Use square units to label area measurements.

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Apply translations (slides) to figures.

Implementation:

Apply reflections (flips) to figures by reflecting over vertical or horizontal lines and relate reflections to lines of symmetry.

Implementation:

Understand that the area of a two-dimensional figure can be found by counting the total number of same size square units that co er a shape without gaps or over-laps. Justify why length and width are multiplied to find the area of a rectangle by breaking the rectangle into one unit by one unit squares and viewing these as grouped into rows and columns.

Implementation: How many copies of a square sheet of pa-per are needed to cover the classroom door? Measure the length and width of the door to the nearest inch and compute the area of the door.

Use translations, reflections and rotations to establish congruency and understand symmetries.

Recognize that translations, reflections and rotations preserve congruency and use them to show that two figures are congruent

Use tables, bar graphs, timelines and Venn diagrams to display data sets. The data may include fractions or decimals. Understand that spreadsheet tables and graphs can be used to display data.

Implementation:

Understand angle and area as measurable attributes of real-world and mathematical objects. Use various tools to measure angles and areas.

•

Apply rotations (turns) of 90° clockwise or counter-clockwise.

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Collect, organize, display and interpret data, including data collected over a period of time and data represented by fractions and decimals.

evens221812

21 23

25 27

19 17

15 5

<20>20

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Divide multi-digit numbers, using efficient and generalize procedures, based on knowledge of place value, includ-ing standard algorithms. Recognize that quotients can be represented in a variety of ways, including a whole number with a remainder, a fraction or mixed number, or a decimal.

Implementation: Dividing 153 by 7 can be used to convert the improper fraction 153/7 to the mixed number 21 6/7.

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Divide multi-digit numbers; solve real-world and math-ematical problems using arithmetic.

Consider the context in which a problem is situated to select the most useful form of the quotient for the solution and use the context to interpret the quotient appropriately.

Implementation: If 77 amusement ride tickets are to be distributed equally among 4 children, each child will receive 19 tickets, and there will be one left over. If $77 is to be dis-tributed equally among 4 children, each will receive $19.25, with nothing left over.

Estimate solutions to arithmetic problems in order to assess the reasonableness of results.

Solve real-world and mathematical problems requir-ing addition, subtraction, multiplication and division of multi-digit whole numbers. Use various strategies, including the inverse relationships between operations, the use of technology, and the context of the problem to assess the reasonableness of results.

Implementation: The calculation 117÷9=13 can be checked by multiplying 9 and 13.

Read, write, represent and compare fractions and deci-mals; recognize and write equivalent fractions; con-vert between fractions and decimals; use fractions and decimals in real-world and mathematical situations.

Read and write decimals using place value to describe decimals in terms of groups from millionths to millions.

Implementation: Possible names for the number 0.0037 are:

37 ten thousandths3 thousandths + 7 ten thousandths;

a possible name for the number 1.5 is 15 tenths.

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Read, write, represent and compare fractions and deci-mals; recognize and write equivalent fractions; con-vert between fractions and decimals; use fractions and decimals in real-world and mathematical situations.

Find 0.1 more than a number and 0.1 less than a num-ber. Find 0.01 more than a number and 0.01 less than a number. Find 0.001 more than a number and 0.001 less than a number.

Implementation: Use place value. Order 4.137, 4, and 4.19 from least to greatest.

So, the order from least to greatest is 4, 4.137, 4.19.

Recognize and generate equivalent decimals, fractions, mixed numbers and improper fractions in various con-texts.

Implementation: When comparing 1.5 and 19/12, note that 1.5 = 1 1/2 = 1 6/12 = 18/12, so 1.5 < 19/12.

Step 1 Step 2 Step 3Line up the decimal points. Write equivalent decimals.

4.1374.0004.190

Begin at the left. Compare the digits until they are differ-ent.

4.1374.0004.190

Continue comparing.

4.1374.0004.190

Round numbers to the nearest 0.1, 0.01 and 0.001.

Implementation: Fifth grade students used a calculator to find the mean of the monthly allowance in their class. The calculator display shows 25.80645161. Round this number to the nearest cent.

Add and subtract fractions, mixed numbers and decimals to solve real-world and math-ematical problems.

Add and subtract decimals and fractions, using efficient and generalizable procedures, including standard algo-rithms.

Model addition and subtraction of fractions and decimals using a variety of representations.

Implementation: Represent 2/3 + 1/4 and 2/3 - 1/4 by drawing a rectangle divided into 4 columns and 3 rows and shading the appropriate parts or by using fraction circles or bars.

Estimate sums and differences of decimals and fractions to assess the reasonableness of results.

Implementation: Recognize that 12 2/5 - 3 3/4 is between 8 and 9 (since 2/5 < 3/4).

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Add and subtract fractions, mixed numbers and decimals to solve real-world and math-ematical problems.

Solve real-world and mathematical problems requir-ing addition and subtraction of decimals, fractions and mixed numbers, including those involving measure-ment, geometry and data.

Implementation: Calculate the perimeter of the soccer field when the length is 109.7 meters and the width is 73.1 meters.

Create and use rules, tables, spreadsheets and graphs to describe patterns of change and solve problems.

Implementation: An end-of-the-year party for 5th grade costs $100 to rent the room and 54.50 for each student. Know how to use a spreadsheet to create an input-ouput table that records the total cost of the party for any number of stu-dents between 90 and 150.

Use a rule or table to represent ordered pairs of positive integers and graph these ordered pair on a coordinate system.

Implementation:

Recognize and represent pat-terns of change; use patterns, tables, graphs and rules to solve real-world and math-ematical problems.

Apply the commutative, associative and distributive properties and order of operations to generate equivalent numerical expressions and to solve problems involving whole numbers.

Implementation: Purchase 5 pencils at 19 cents and 7 eras-ers at 19 cents. The numerical expression is 5 x 19 + 7 x 19 which is the same as (5 + 7) x 19.

Determine whether an equation or inequality involving a variable is true or false for a given value of the variable.

Implementation: Determine whether the inequality 1.5 + x<10 is true for x=2.8, x=8.1, or x=9.2.

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Use properties of arithme-tic to generate equivalent numerical expressions nd evaluate expressions involv-ing whole numbers.

Understand and interpret equations and inequali-ties involving variables and whole numbers, and use them to represent and solve real-world and mathematical problems.

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5Understand and interpret equations and inequali-ties involving variables and whole numbers, and use them to represent and solve real-world and mathematical problems.

Represent real-world situations using equations and inequalities involving variables. Create real-world situa-tions corresponding to equations and inequalities.

Implementation: 250 - 27 x a = b can be used to rep-resent the number of sheets of paper remaining from a packet of 250 sheets when each student in a class of 27 a given a certain number of sheets.

Evaluate expressions and solve equations involving vari-ables when values for the variables are given.

Implementation: Using the formula, A = lw, determine the area when the length is 5, and the width 6, and find the length when the area is 24 and the width is 4.

Describe and classify three-dimensional figures includ-ing cubes, prisms and pyramids by the number of edges, faces or vertices as well as the types of faces.

Recognize and draw a net for a three-dimensional figure.

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Determine the area of tri-angles and quadrilaterals; de-termine the surface area and volume of rectangular prisms in various contexts.

Develop and use formulas to determine the area of triangles, parallelograms and figures that can be decom-posed into triangles.

Implementation: 3in

8in

A= l x w

A=8 x 3

A= 24 in2

A= l x w

A=8 x 3

A= 24 in2

Use various tools and strategies to measure the volume and surface area of objects that are shaped like rectangu-lar prisms.

Implementation: Use a net or decompose the surface into rectangles. Measure the volume of a cereal box by using a ruler to measure its height, width and length, or by filling it with cereal and then emptying the cereal into containers of known volume.

3in

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A= l x w

A=8 x 3

A= 24 in2

A= l x w

A=8 x 3

A= 24 in2

3in

8in

A= l x w

A=8 x 3

A= 24 in2

A= l x w

A=8 x 3

A= 24 in2

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5Determine the area of tri-angles and quadrilaterals; de-termine the surface area and volume of rectangular prisms in various contexts.

Understand that the volume of a three-dimensional figure can be found by counting the total number of same-sized cubic units that fill a shape without gaps or overlaps. Use cubic units to label volume measure-ments.

Implementation: Use cubes to find the volume of a small box.

Develop and use the formulas V=lwh and V=Bh to determine the volume of rectangular prisms. Justify why base area B and height h are multiplied to find the vol-ume of a rectangular prism by breaking the prism into layers of unit cubes.

Implementation:

Know and use the definitions of the mean, median and range of a set of data. Know how to use a spreadsheet to find the mean, median and range of a data set. Under-stand that the mean is a “leveling out” of data

Implementation: The set of numbers 1, 1, 4, 6 has mean 3. It can be leveled by taking one unit fro the 4 and three units from the 6 and adding them to the 1s, making four 3s.

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Display and interpret data; determine mean, median and range.

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Create and analyze double-bar graphs and line graphs by applying understanding of whole numbers, fractions and decimals. Know how to create spreadsheet tables and graphs to display data.

6in

6in

6in

V = l x w x h or V = lwh

V = 6 x 6 x 6

V = 216 in3

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Locate positive rational numbers on a number line and plot pairs of positive rational numbers on a coordinate grid.

Implementation: Plot (x,y) when x and y are positive rational numbers.

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Compare positive rational numbers represented in vari-ous forms. Use the symbols <, = and >.

Implementation: 1/2 > 0.36.

Understand that percent represents parts out of 100 and ratios to 100.

Implementation: 75% corresponds to the ratio 75 to 100, which is equivalent to the ratio 3 to 4.

Determine equivalences among fractions, decimals and percents; select among these representations to solve problems.

Implementation: If a woman making $25 an hour gets a 10% raise, she will made an additional $2.50 an hour, because $2.50 is 1/10 or 10% of $25.

Read, write, represent and compare positive rational numbers expressed as frac-tions, decimals, percents and ratios; write positive inte-gers as products of factors; use these representations in real-world and mathematical situations.

Factor whole numbers; express a whole number as a product of prime factors with exponents.

Implementation: 24 = 23 x 3.

Determine greatest common factors and least common multiples. Use common factors and common multiples to calculate with fractions and find equivalent fractions.

Implementation: Factor the numerator and denominator of a fraction to determine an equivalent fraction.

Convert between equivalent representations of positive rational numbers.

Implementation: Express 10/7 as 7+3/7 = 7/7+3/7 = 1 3/7.

Understand the concept of ratio and its relationship to fractions and to the multipli-cation and division of whole numbers. Use ratios to solve real-world and mathematical problems.

Identify and use ratios to compare quantities; under-stand that comparing quantities using ratios is not the same as comparing quantities using subtraction.

Implementation: In a classroom with 15 boys and 10 girls, compare the numbers by subtracting (there are 5 more boys than girls) or by dividing (there are 1.5 times as many boys as girls). the comparison using division may be expressed as a ratio of boys to girls (3 to 2 or 3:2 or 1.5 to 1).

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Understand the concept of ratio and its relationship to fractions and to the multipli-cation and division of whole numbers. Use ratios to solve real-world and mathematical problems.

Apply the relationship between ratios, equivalent frac-tions and percents to solve problems in various contexts, including those involving mixtures and concentrations.

Implementation: If 5 cups of trail mix contains 2 cups of raisins, the ratio of raisins to trail mix is 2 to 5. The ratio corresponds to the fact that the raisins are 2/5 of the total, or 40% of the total.And if one trim mix consists of 2 parts peanuts to 3 parts rai-sins, and another consists of 4 parts peanuts to 8 parts raisins, then the first mixture has a higher concentration of peanuts.

Determine the rate for ratios of quantities with different units.

Implementation: 60 miles for every 3 hours is equivalent to 20 miles for every one hour (20 mph).

Use reasoning about multiplication and division to solve ratio and rate problems.

Implementation: If 5 items and $3.75, and all items are the same price, then 1 item costs 75 cents, so 12 items cost $9.00.

Multiply and divide decimals, fractions and mixed num-bers; solve real-world and mathematical problems using arithmetic with positive ratio-nal numbers.

Add and subtract decimals and fractions, using efficient and generalizable procedures, including standard algorithms.

Implementation: 4.56 x .4 = 1.824 and 1.824 ÷ .4 = 4.56 or 1.824 ÷ 4.56 = .4

Use the meanings of fractions, multiplication, division and the inverse relationship between multiplication and division to make sense of procedures for multiplying and dividing fractions.

Implementation: Just as 12/4 = 3 means 12 = 3 x 4, 2/3 ÷ 4/5 = 5/6 means 5/6 x 4/5 = 2/3.

Calculate the percent of a number nd determine what per-cent one number is of another number to solve problems in various contexts.

Implementation: If John has $45 and spends $15, what per-cent of his money did he keep?

Solve real-world and mathematical problems requiring arithmetic with decimals, fractions nd mixed numbers.

Implementation: How many slices of pizza will you get from 2 3/4 pizzas when each pizza is divided into 8 equal slices?

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Estimate solutions to problems with whole numbers, fractions and decimals and use the estimates to assess the reasonableness of results in the context of the problem.

Implementation: The sum 1/3 + 0.25 can be estimated to be between 1/2 and 1, and this estimate can be used to check the result of a more detailed calculation.

Represent the relationship between two varying quantities with function rules, graphs and tables; translate between any two of these representations.

Implementation: Describe the terms in the sequence of perfect squarest=1, 4, 9, 16,...by using the rule : t=n2 for n=1, 2, 3, 4,...

Multiply and divide decimals, fractions and mixed num-bers; solve real-world and mathematical problems using arithmetic with positive ratio-nal numbers.

Apply the associative, commutative and distributive properties and order of operations to generate equiva-lent expressions and to solve problems involving positive rational numbers.

Implementation: 32/15 x 5/6 = 33 x 5 + 15 x 6 =2 x 16 x 5/3 x 5 x 3 x 2 = 16/9 x 2/2 x 5/5 = 16/9.Use the distributive law to write:1/2 + 1/3 (9/2 - 15/8) = 1/2 + 1/3 x 9/2 - 1/3 x 15/8 = 1/2 + 3/2 - 5/8 = 2 - 5/8 =

Represent real-world or mathematical situations using equations and inequalities involving variables and positive rational numbers.

Implementation: The number of miles m in a k kilometer race is represented by the equation m=0.62 k.

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Recognize and represent relationships between varying quantities; translate from one representation to another; use patterns, tables, graphs and rules to solve real-world and mathematical problems.

Use properties of arithme-tic to generate equivalent numerical expressions and evaluate expressions involv-ing positive rational numbers.

Understand and interpret equations and inequali-ties involving variables and positive rational numbers. Use equations and inequali-ties to represent real-world and mathematical problems; use the idea of maintaining equality to solve equations. Interpret solutions in the original context.

Understand that a variable can be used to represent a quantity that can change, often in relationship to another changing quantity. Use variables in various contexts.

Implementation: If a student earns $7 an hour in a job, the amount of money earned can be represented by a variable and a related to the number of hours worked, which also can be represented by a variable.

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Understand and interpret equations and inequali-ties involving variables and positive rational numbers. Use equations and inequali-ties to represent real-world and mathematical problems; use the idea of maintaining equality to solve equations. Interpret solutions in the original context.

Solve equations involving positive rational numbers us-ing number sense, properties of arithmetic and the idea of maintaining equality on both sides of the equation. Interpret a solution in the original context and assess the reasonableness of results.

Implementation: A cellular phone company charges $0.12 per minute. If the bill was $11.40 in April, how many min-utes were used?

Calculate the surface area and volume of prisms and us appropriate units, such as cm2 and cm3. Justify the formulas used. Justification may involve decomposition, nets or other models.

Implementation: The surface area of a triangular prism can be found by decomposing the surface into two triangles and three rectangles.

Calculate perimeter, area, surface area and volume of two-and three-dimensional figures to solve real-world and mathematical problems.

Calculate the area of quadrilaterals. Quadrilaterals include squares, rectangles, rhombuses, parallelograms, trapezoids and kites. When formulas are used, be able to explain why they are valid.

Implementation: The area of a kite is one-half the product of the lengths of the diagonals, and this can be justified by decom-posing the kite into two triangles.

Estimate the perimeter and area of irregular figures on a grid when they cannot be decomposed into common figures and use correct units, such as cm and cm2.

Implementation: Estimate the area and perimeter of a lake drawn over grid paper.

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Understand and use rela-tionships between angles in geometric figures.

Solve problems using the relationships between the angles formed by intersecting lines.

Implementation: If two streets cross, forming four corners such that one of the corners forms an angle of 120°, determine the measures of the remaining three angles.Recognize that pairs of interior and exterior angles in polygons have measures that sum to 180°.

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6

Understand and use rela-tionships between angles in geometric figures.

Determine missing angle measures in a triangle using the fact that the sum of the interior angles of a triangle is 180°. Use models of triangles to illustrate this fact.

Implementation: Cut a triangle out of paper, tear off the corners and rearrange these corners to form a straight line.

Recognize that the measures of the two acute angles in a right triangle sum is 90°.

Develop and use formulas for the sums of the interior angles of polygons by decomposing them into triangles.

Implementation: Create a formula for finding the sum of the interior angles of any polygon.

Solve problems in various contexts involving conversion of weights, capacities, geometric measurements and times within measurement systems using appropriate units.

Implementation: How many 8 oz glasses of milk can be poured from a gallon?

Estimate weights, capacities and geometric measure-ments using benchmarks in measurement systems with appropriate units.

Implementation: Estimate the height of a house by compar-ing to a 6-foot man standing nearby.

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Choose appropriate units of measurement and use ra-tios to convert within mea-surement systems to solve real-world and mathematical problems.

Determine the sample space (set of possible outcomes) for a given experiment and determine which members of the sample space related to certain events. Sample space may be determined by the use of tree diagrams, tables or pictorial representations.

Implementation: A 6 x 6 table with entries such as (1,1), (1,2), (1,3),..., (6,6) can be used to represent the sample space for the experiment of simultaneously rolling two number cubes.

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Use probabilities to solve real-world and mathematical problems; represent prob-abilities using fractions, deci-mals and percents. Determine the probability of an event using the ratio

between the size of the event and the size of the sample space; represent probabilities as percents, fractions and decimals between 0 and 1 inclusive. Understand that probabilities measure likelihood.

Implementation: Heads and tails are equally likely when flipping a fair coin, but if several different students flipped fair coins 10 times, it is likely that they will find a variety of rela-tive frequencies of heads and tails.

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6Use probabilities to solve real-world and mathematical problems; represent prob-abilities using fractions, deci-mals and percents.

Calculate experimental probabilities from experiments; represent them as percents, fractions and decimals between 0 and 1 inclusive. Use experimental probabili-ties to make predictions when actual probabilities are unknown.

Implementation: Repeatedly draw colored chips with replacement from a bag with an unknown mixture of chips, record relative frequencies, and use the results to make predic-tions about the contents of the bag.D

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Math 7a/b & honors Math 7a/b


Know that every rational number can be written as the ratio of two integers or as a terminating or repeating deci-mal. Recognize that π is not rational, but that it can be approximated by rational numbers such as 22/7 and 3/14.

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Locate positive and negative rational numbers on a number line, understand the concept of opposites, and plot pairs of positive and negative rational numbers on a coordinate grid.

Read, write, represent and compare positive and nega-tive rational numbers, ex-pressed as integers, fractions and decimals.

Compare positive and negative rational numbers ex-pressed in various forms using the symbols <, >, =, ≤, ≥.

Implementation: -1/2 , .36.Honors: List the following rational numbers in order from least to greatest. -1/2, 1.44, 3/4, -1, 5/6.

Calculate with positive and negative rational numbers, and rational numbers with whole number exponents, to solve real-world and math-ematical problems.

Add, subtract, multiply and divide positive and nega-tive rational numbers that are integers, fractions and terminating decimals; use efficient and generalizable procedures, including standard algorithms; raise posi-tive rational numbers to whole-number exponents.

Implementation: 34 x (1/2)2 = 81/4.Honors: (3.4)3 - 7/5 x 22

Understand that division of two integers will always result in a rational number. Use this information to interpret the decimal result of a division problem when using a calculator.

Implementation: 125/30 gives 4.6666667 on a calcu-lator. This answer is not exact. The exact answer can be expressed as 4 1/6, which is the same as 4.16. The calculator expression does not guarantee that the 6 is repeated, but that possibility should be anticipated.

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Recognize and generate equivalent representations of positive and negative rational numbers, including equivalent fractions.

Implementation: -40/12=-120/36=-10/3=-33.–

Use real-world contexts and the inverse relationship between addition and subtraction to explain why the procedures of arithmetic with negative rational num-bers make sense.

Implementation: Multiplying a distance by -1 can be thought of as representing that same distance in the opposite direction. Multiplying by -1 a second time reverses directions again, giving the distance in the original direction.

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Math 7A/B & Honors Math 7A/B Learner Outcomes continued...


7

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Calculate with positive and negative rational numbers, and rational numbers with whole number exponents, to solve real-world and math-ematical problems.

Understand that calculators and other computing tech-nologies often truncate or round numbers.

Implementation: A decimal that repeats or terminates after a large number of digits is truncated or rounded.

Solve problems in various contexts involving calculations with positive and negative rational numbers and posi-tive integer exponents, including computing simple and compound interest.

Use proportional reasoning to solve problems involving ratios in various contexts.

Implementation: A recipe calls for milk, flour and sugar in a ratio of 4:6:3 (this is how recipes are often given in large institu-tions, such as hospitals). How much flour and milk would be needed with 1 cup of sugar?

Honors: A recipe calls for powdered sugar, flour and sugar in a ratio of 4:6:3. If there is a total of 15 cups of these dry ingredi-ents, how much of each is needed?

Understand the concept of proportionality in real-world and mathematical situations, and distinguish between pro-portional and other relation-ships.

Demonstrate an understanding of the relationship between the absolute value of a rational number and distance on a number line. Use the symbol for absolute value.

Implementation: |-3| represents the distance from -3 to 0 on a number line or 3 units; the distance between 3 and 9/2 on the number line is |3-9/2| or 3/2.

Understand that a relationship between two variables, x and y, is proportional if it can be expressed in the form y/x=k or y=kx. Distinguish proportional relationships from other relationships, including inversely proportional rela-tionships (xy=k or y=k/x).

Implementation: the radius and circumference of a circle are proportional, whereas the length x and the width y of a rect-angle with area 12 are inversely proportional, since xy = 12 or equivalently, y=12/x.

Understand that the graph of a proportional relationship is a line through the origin whose slope is the unit rate (constant of proportionality). Know how to use graphing technology to examine what happens to a line when the unit rate is changed.

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Recognize proportional relationships in real-world and mathematical situations; represent these and other relationships with tables, verbal descriptions, symbols and graphs; solve problems involving proportional rela-tionships and explain results in the original context.

Represent proportional relationships with tables, verbal descriptions, symbols, equations and graphs; translate from one representation to another. Determine the unit rate (constant of proportionality or slope) given any of these representations.

Implementation: Larry drives 114 miles and uses 5 gallons of gasoline. Sue drives 300 miles and uses 11.5 gallons of gaso-line. Use equations and graphs to compare fuel efficiency and to determine the costs of various trips.

Honors: From above example, If given a graph of the situation, determine the rates and write a paragraph describing the fuel efficiency of each vehicle. Justify which vehicle is the most cost effective for various trips.

Use knowledge of proportions to assess the reasonableness of solutions.

Implementation: Recognize that it would be unreasonable for a cashier to request $200 if you purchase a $225 item at 25% off.

Represent real-world or mathematical situations using equations and inequalities involving variables and positive and negative rational numbers.

Implementation: “Four-fifths is three greater than the opposite of a number” can be represented as 4/5= –n+3, and “height no bigger than half the radius” can represented as h≤r/2.

Implementation 2: “x is at least –3 and less than 5” can be represented as –3≤<5, and also on a number line.

Solve multi-step problems involving proportional rela-tionships in numerous contexts.

Implementation: Distance-time, percent increase or decrease, discounts, tips, unit pricing, lengths in similar geometric figures, and unit conversion when a conversion factor is given, including conversion between different measurement systems.

Honors: Use irregular geometric figures

Another example: How many kilometers are there in 26.2 miles?

Honors: How many ounces are there in 4 kg?

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7

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Recognize proportional relationships in real-world and mathematical situations; represent these and other relationships with tables, verbal descriptions, symbols and graphs; solve problems involving proportional rela-tionships and explain results in the original context.

Represent proportional relationships with tables, verbal descriptions, symbols, equations and graphs; translate from one representation to another. Determine the unit rate (constant of proportionality or slope) given any of these representations.

Implementation: Larry drives 114 miles and uses 5 gallons of gasoline. Sue drives 300 miles and uses 11.5 gallons of gaso-line. Use equations and graphs to compare fuel efficiency and to determine the costs of various trips.

Honors: From above example, If given a graph of the situation, determine the rates and write a paragraph describing the fuel efficiency of each vehicle. Justify which vehicle is the most cost effective for various trips.

Use knowledge of proportions to assess the reasonableness of solutions.

Implementation: Recognize that it would be unreasonable for a cashier to request $200 if you purchase a $225 item at 25% off.

Represent real-world or mathematical situations using equations and inequalities involving variables and positive and negative rational numbers.

Implementation: “Four-fifths is three greater than the opposite of a number” can be represented as 4/5= –n+3, and “height no bigger than half the radius” can represented as h≤r/2.

Implementation 2: “x is at least –3 and less than 5” can be represented as –3≤<5, and also on a number line.

Solve multi-step problems involving proportional rela-tionships in numerous contexts.

Implementation: Distance-time, percent increase or decrease, discounts, tips, unit pricing, lengths in similar geometric figures, and unit conversion when a conversion factor is given, including conversion between different measurement systems.

Honors: Use irregular geometric figures

Another example: How many kilometers are there in 26.2 miles?

Honors: How many ounces are there in 4 kg?

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Apply understanding of or-der of operations and alge-braic properties to generate equivalent numerical and algebraic expressions con-taining positive and negative rational numbers and group-ing symbols; evaluate such expressions.

Use properties of algebra to generate equivalent numeri-cal and algebraic expressions containing rational num-bers, grouping symbols and whole number exponents. Properties of algebra include associative, commutative and distributive laws.

Implementation: Combine like terms (use the distributive law) to write.

Honors: Combine like terms (using the distributive law) to write 6(2x-1) – 3( x-7) = 12x-6-3x+21= 9x +15

Apply understanding of order of operations and grouping symbols when using calculators and other technologies.

Implementation: Recognize the conventions of using a caret (^ raise to a power) and asterisk (* multiply); pay careful attention to the use of nested parentheses.

Represent relationships in various contexts with equa-tions involving variables and positive and negative rational numbers. Use the properties of equality to solve for the value of a variable. Interpret the solution in the original context.

Implementation: Solve for w in the equation P = 2w + 2l, when P = 3.5 and l = 0.4.

Implementation 2: To post an Internet website, Mary must pay $300 for initial set up and a monthly fee of $12. She has $842 in savings, how long can she sustain her website?

Evaluate algebraic expressions containing rational num-bers and whole number exponents at specified values of their variables.

Implementation: Evaluate the expression 4(x-7)2 when x=5

Honors: Evaluate the expression when x = 5.

Represent real-world and mathematical situations us-ing equations with variables. Solve equations symboli-cally, using the properties of equality. Also solve equations graphically and numerically. Interpret solutions in the original context. Solve equations resulting from proportional relationships

in various contexts.

Implementation: Given the side lengths of one triangle and one side length of a second triangle that is similar to the first, find the remaining side lengths of the second triangle.

Implementation 2: Determine the price of 12 yards of ribbon if 5 yards of ribbon cost $1.85.

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Use reasoning with propor-tions and ratios to determine measurements, justify for-mulas and solve real-world and mathematical problems involving circles and related geometric figures.

Demonstrate an understanding of the proportional re-lationship between the diameter and circumference of a circle and that the unit rate (constant of proportionality) is π. Calculate the circumference and area of circles and sectors of circles to solve problems in various contexts.

Apply scale factors, length ratios and area ratios to deter-mine side lengths and areas of similar geometric figures.

Implementation: If two similar rectangles have heights of 3 and 5, and the first rectangle has a base of length 7, the base of the second rectangle has length 35/3.

Use proportions and ratios to solve problems involving scale drawings and conversions of measurement units.

Implementation: 1 square foot equals 144 square inches.

Honors: 1 cubic foot equals (12)3 cubic feet

Implementation 2: In a map where 1 inch represents 50 miles, ½ inch represents 25 miles.

Honors 2: In a map where 1 inch equals 6.5 miles ½ inch represents 3.25 miles.

Calculate the volume and surface area of cylinders and justify the formulas used.

Implementation: Justify the formula for the surface area of a cylinder by decomposing the surface into two circles and a rectangle.

Analyze the effect of change of scale, translations and re-flections on the attributes of two-dimensional figures.

Describe the properties of similarity, compare geometric figures for similarity, and determine scale factors.

Implementation: Corresponding angles in similar geometric figures have the same measure.

Graph and describe translations and reflections of figures on a coordinate grid and determine the coordinates of the vertices of the figure after the transformation.

Implementation: The point (1, 2) moves to (-1, 2) after reflec-tion about the y-axis.

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Use mean, median and range to draw conclusions about data and make predictions.

Design simple experiments and collect data. Determine mean, median and range for quantitative data and from data represented in a display. Use these quantities to draw conclusions about the data, compare different data sets, and make predictions.

Implementation: By looking at data from the past, Sandy calculated that the mean gas mileage for her car was 28 miles per gallon. She expects to travel 400 miles during the next week. Predict the approximate number of gallons that she will use.

Use reasoning with proportions to display and interpret data in circle graphs (pie charts) and histograms. Choose the appropriate data display and know how to create the display using a spreadsheet or other graphing technology.

Display and interpret data in a variety of ways, including circle graphs and histograms.

Describe the impact that inserting or deleting a data point has on the mean and the median of a data set. Know how to create data displays using a spreadsheet to examine this impact.

Implementation: How does dropping the lowest test score af-fect a student’s mean test score?

Use random numbers generated by a calculator or a spreadsheet or taken from a table to simulate situations involving randomness, make a histogram to display the results, and compare the results to known probabilities.

Implementation: Use a spreadsheet function such as RAND-BETWEEN(1, 10) to generate random whole numbers from 1 to 10, and display the results in a histogram.

Calculate probabilities and reason about probabilities using proportions to solve real-world and mathematical problems.

Calculate probability as a fraction of sample space or as a fraction of area. Express probabilities as percents, decimals and fractions.

Implementation: Determine probabilities for different outcomes in game spinners by finding fractions of the area of the spinner. Use proportional reasoning to draw conclusions about and predict relative frequencies of outcomes based on prob-abilities.

Implementation: When rolling a number cube 600 times, one would predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.

Honors: When rolling a number cube 900 times, predict the number of times a 4 would be rolled. Justify your answer.

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Classify real numbers as rational or irrational. Know that when a square root of a positive integer is not an integer, then it is irrational. Know that the sum of a rational number and an irrational number is irrational, and the product of a non-zero GRADE STRAND STANDARD BENCHMARK

algebra i a/b & honors algebra i a/b

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Read, write, compare, classify and represent real numbers, and use them to solve prob-lems in various contexts.

Compare real numbers; locate real numbers on a num-ber line. Identify the square root of a positive integer as an integer, or if it is not an integer, locate it as a real number between two consecutive positive integers.

Implementation: Put the following numbers in order from smallest to largest: 2, √3 , -4, -6.8, -√37.

Implementation 2: √68 is an irrational number between 8 and 9.

Determine rational approximations for solutions to problems involving real numbers.

Implementation: A calculator can be used to determine that √7 is approximately 2.65.

Implementation 2: To check that 1 5/12 is slight-ly bigger than √2, do the calculation (1 5/12)2 =(17/12)2=289/144=2 1/144.

Implementation 3: Knowing that √10 is between 3 and 4, try squaring numbers like 3.5, 3.3, 3.1 to determine that 3.1 is a reasonable rational approximation of √10.

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Know and apply the properties of positive and nega-tive integer exponents to generate equivalent numerical expressions.

Implementation: 32 x 3(-5) = 3(-3) = (1/3)3 = 1/27.

Classify real numbers as rational or irrational. Know that when a square root of a positive integer is not an integer, then it is irrational. Know that the sum of a rational number and an irrational number is irrational, and the product of a non-zero rational number and an irrational number is irrational.

Implementation: Classify the following numbers as whole numbers, integers, rational numbers, irrational numbers, rec-ognizing that some numbers belong in more than one category: 6/3, 3/6, 3.6, π/2, -√4, √10, -6.7.

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Express approximations of very large and very small numbers using scientific notation; understand how calculators display numbers in scientific notation. Multiply and divide numbers expressed in scientific notation, express the answer in scientific notation, using the correct number of significant digits when physical measurements are involved.

Implementation: (4.2x104)x(8.25x103)=3.465x108, but if these numbers represent physical measurements, the answer should be expressed as 3.5x108 because the first factor, 4.2x104, only has two significant digits.

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Algebra I A/B & Honors Algebra I A/B Learner Outcomes continued...


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Understand the concept of function in real-world and mathematical situations, and distinguish between linear and nonlinear functions.

Understand that a function is a relationship between an independent variable and a dependent variable in which the value of the independent variable determines the value of the dependent variable. Use functional nota-tion, such as f(x), to represent such relationships.

Implementation: The relationship between the area of a square and the side length can be expressed as f (x)=x2. In this case, f (5)=25 , which represents the fact that a square of side length 5 units has area 25 units squared.

Honors: Given the above equation, graph the function. Ex-plain why it’s a function.

Use linear functions to represent relationships in which changing the input variable by some amount leads to a change in the output variable that is a constant times that amount.

Implementation: Uncle Jim gave Emily $50 on the day she was born and $25 on each birthday after that. The function f(x)=50+25x represents the amount of money Jim has given after x years. The rate of change is $25 per year.

Honors: Given the above example graphically, determine the rate of change.

Understand that a function is linear if it can be expressed in the form f (x) = mx+b or if its graph is a straight line.

Implementation: The function f (x)=x2 is not a linear func-tion because its graph contains the points (1.1), (-1.1) and (0.0), which are not on a straight line.

Honors: When given above equation graphically, determine whether it is a function and justify answer.

Understand that an arithmetic sequence is a linear func-tion that can be expressed in the form f(x)=mx+b, where x = 0, 1, 2, 3,….

Implementation: The arithmetic sequence 3, 7, 11, 15, …, can be expressed as f(x) = 4x + 3.

Understand that a geometric sequence is a non-linear function that can be expressed in the form f (x)=abx, where x = 0, 1, 2, 3,….

Implementation: The geometric sequence 6, 12, 24, 48, … , can be expressed in the form f(x) = 6(2x).

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Recognize linear functions in real-world and mathematical situations; represent linear functions and other functions with tables, verbal descrip-tions, symbols and graphs; solve problems involving these functions and explain results in the original con-text.

Represent linear functions with tables, verbal descrip-tions, symbols, equations and graphs; translate from one representation to another.

Identify graphical properties of linear functions includ-ing slopes and intercepts. Know that the slope equals the rate of change, and that the y-intercept is zero when the function represents a proportional relationship.

Represent arithmetic sequences using equations, tables, graphs and verbal descriptions, and use them to solve problems.

Implementation: If a girl starts with $100 in savings and adds $10 at the end of each month, she will have 100 + 10x dollars after x months.

Honors: If a girl starts with $100 in savings and adds $10 at the end of the month, how many months will it take her to have $10,000?

Represent geometric sequences using equations, tables, graphs and verbal descriptions, and use them to solve problems.

Implementation: If a girl invests $100 at 10% annual interest, she will have 100(1.1x) dollars after x years.

Honors: If a girl invests $100 at 10% annual interest, how long will it take for her to have $10,000?

Evaluate algebraic expressions, including expressions containing radicals and absolute values, at specified val-ues of their variables.

Implementation: Evaluate ½ (bh) when b=7 and h=3.

Honors: Evaluate πr2h when r = 3 and h = 0.5, and then use an approximation of π to obtain an approximate answer.

Identify how coefficient changes in the equation f (x) = mx + b affect the graphs of linear functions. Know how to use graphing technology to examine these effects.

Generate equivalent numeri-cal and algebraic expressions and use algebraic properties to evaluate expressions. Justify steps in generating equivalent expressions by

identifying the properties used, including the properties of algebra. Properties include the associative, commuta-tive and distributive laws, and the order of operations, including grouping symbols.

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Represent real-world and mathematical situations using equations and inequalities involving linear expressions. Solve equations and inequali-ties symbolically and graphi-cally. Interpret solutions in the original context.

Use linear equations to represent situations involving a constant rate of change, including proportional and non-proportional relationships.

Implementation: For a cylinder with fixed radius of length 5, the surface area A = 2π(5)h + 2π(5)2 = 10πh + 50π, is a linear function of the height h, but the surface area is not proportional to the height.

Solve multi-step equations in one variable. Solve for one variable in a multi-variable equation in terms of the other variables. Justify the steps by identifying the prop-erties of equalities used.

Implementation: The equation 10x + 17 = 3x can be changed to 7x + 17 = 0, and then to 7x = -17 by adding/subtracting the same quantities to both sides. These changes do not change the solution of the equation.

Implementation 2: Using the formula for the perimeter of a rectangle, solve for the base in terms of the height and perimeter. Express linear equations in slope-intercept, point-slope and standard forms, and convert between these forms. Given sufficient information, find an equation of a line.

Implementation: Determine an equation of the line through the points (-1,6) and (2/3, -3/4).

Honors: Given y= 3/4x + 5 determine the standard form of the same line. Graph the line.Use linear inequalities to represent relationships in vari-ous contexts.

Implementation: A gas station charges $0.10 less per gal-lon of gasoline if a customer also gets a car wash. Without the car wash, gas costs $2.79 per gallon. The car wash is $8.95. What are the possible amounts (in gallons) of gasoline that you can buy if you also get a car wash and can spend at most $35?

Solve linear inequalities using properties of inequalities. Graph the solutions on a number line.

Implementation: The inequality -3x < 6 is equivalent to x > -2, which can be represented on the number line by shading in the interval to the right of -2.Represent relationships in various contexts with equa-tions and inequalities involving the absolute value of a linear expression. Solve such equations and inequalities and graph the solutions on a number line.

Implementation: A cylindrical machine part is manufactured with a radius of 2.1 cm, with a tolerance of 1/100 cm. The radius r satisfies the inequality |r – 2.1| ≤ .01.

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Represent real-world and mathematical situations using equations and inequalities involving linear expressions. Solve equations and inequali-ties symbolically and graphi-cally. Interpret solutions in the original context.

Represent relationships in various contexts using systems of linear equations. Solve systems of linear equations in two variables symbolically, graphically and numerically.

Implementation: Marty’s cell phone company charges $15 per month plus $0.04 per minute for each call. Jeannine’s company charges $0.25 per minute. Use a system of equations to determine the advantages of each plan based on the number of minutes used.

Honors: With the above example, write a paragraph explain-ing which company is the best choice for several situations.

Understand that a system of linear equations may have no solution, one solution, or an infinite number of solu-tions. Relate the number of solutions to pairs of lines that are intersecting, parallel or identical. Check whether a pair of numbers satisfies a system of two linear equa-tions in two unknowns by substituting the numbers into both equations.

Use the relationship between square roots and squares of a number to solve problems.

Implementation: If πx2 = 5, then |x|=√5/π, or equiva-lently, x=√5/π or x=-√5/π. If x is understood as the radius of a circle in this example, then the negative solution should be discarded and x=√5/ π.

Use the Pythagorean Theorem to solve problems involv-ing right triangles.

Implementation: Determine the perimeter of a right triangle, given the lengths of two of its sides.

Honors: Determine the area of a right triangle given the hypot-enuse and one side length.

Implementation 2: Show that a triangle with side lengths 4, 5 and 6 is not a right triangle.

Honors: Create a right triangle- Justify why it must be a right triangle.

Solve problems involving right triangles using the Pythagorean Theorem and its converse.

Determine the distance between two points on a hori-zontal or vertical line in a coordinate system. Use the Pythagorean Theorem to find the distance between any two points in a coordinate system.

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Informally justify the Pythagorean Theorem by using measurements, diagrams and computer software.

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Solve problems involving par-allel and perpendicular lines on a coordinate system.

Understand and apply the relationships between the slopes of parallel lines and between the slopes of perpen-dicular lines. Dynamic graphing software may be used to examine these relationships

Analyze polygons on a coordinate system by determining the slopes of their sides.

Implementation: Given the coordinates of four points, determine whether the corresponding quadrilateral is a paral-lelogram.

Honors: Given the coordinates of 3 points, determine the coor-dinates of a fourth point to create a parallelogram.

Given a line on a coordinate system and the coordinates of a point not on the line, find lines through that point that are parallel and perpendicular to the given line, symbolically and graphically.

Collect, display and interpret data using scatterplots. Use the shape of the scatterplot to informally estimate a line of best fit and determine an equation for the line. Use appropriate titles, labels and units. Know how to use graphing technology to display scatterplots and corre-sponding lines of best fit.

Interpret data using scatter-plots and approximate lines of best fit. Use lines of best fit to draw conclusions about data.

Use a line of best fit to make statements about ap-proximate rate of change and to make predictions about values not in the original data set.

Implementation: Given a scatterplot relating student heights to shoe sizes, predict the shoe size of a 5’4” student, even if the data does not contain information for a student of that height.

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Assess the reasonableness of predictions using scatter-plots by interpreting them in the original context.

Implementation: A set of data may show that the number of women in the U.S. Senate is growing at a certain rate each elec-tion cycle. Is it reasonable to use this trend to predict the year in which the Senate will eventually include 1000 female Senators?

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geoMetry a/b & honors geoMetry a/b

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Calculate measurements of plane and solid geometric figures; know that physical measurements depend on the choice of a unit and that they are approximations.

Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures.

Implementation: Find the volume of a regular hexagonal prism by decomposing it into six equal triangular prisms.

Honors Geometry: The slant height of a right cone is twice the radius of the cone. The surface area of the cone is 75π square inches. Find the slant height and radius of the cone.

Understand that quantities associated with physical measurements must be assigned units; apply such units correctly in expressions, equations and problem solu-tions that involve measurements; and convert between measurement systems.

Implementation: 60 miles/hour = 60 miles/hour × 5280 feet/mile × 1 hour/3600 seconds = 88 feet/second.

Honors Geometry: Copy and complete the statement. 54 cm2 = __?__ m2

Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively.

Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate.

Implementation: Measure the height and radius of a cone and then use a formula to find its volume.

Honors Geometry: What is the height of a cone whose slant height is twice the radius and whose volume is cubic inches?

Make reasonable estimates and judgments about the accuracy of values resulting from calculations involving measurements.

Implementation: Suppose the sides of a rectangle are measured to the nearest tenth of a centimeter at 2.6 cm and 9.8 cm. Because of measurement errors, the width could be as small as 2.55 cm or as large as 2.65 cm, with similar errors for the height. These errors affect calculations. For instance, the actual area of the rect-angle could be smaller than 25 cm2 or larger than 26 cm2, even though 2.6 × 9.8 = 25.48.

Honors Geometry: You hire the same lawn service company to mow your home lawn that mows your company lawn. The two lawns are similar rectangles. Your home lawn is 30 feet by 60 feet and your company lawn is 135 feet by 270 feet. You are charged $90.00 for your home lawn and $1093.50 for the com-pany lawn. Are you charged the same rate for both lawns? If not, which rate is higher, the home rate or the company rate?

2

4

3 34

3

π

page 54

Geometry A/B & Honors Geometry A/B Learner Outcomes continued...


9,10,11,12

Geo

met

ry &

Mea

sure

men

t

Construct logical arguments, based on axioms, defini-tions and theorems, to prove theorems and other results in geometry.

Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments.

Accurately interpret and use words and phrases such as “if…then,” “if and only if,” “all,” and “not.” Recognize the logical relationships between an “if…then” state-ment and its inverse, converse and contrapositive.

Implementation: The statement “If you don’t do your home-work, you can’t go to the dance” is not logically equivalent to its inverse “If you do your homework, you can go to the dance.”

Honors Geometry: Determine whether a statement follows from the Venn diagram. No football players are over 6 feet tall. a. No football players are over 6 feet tall.b. Every football player is over 6 feet tall.

Assess the validity of a logical argument and give coun-terexamples to disprove a statement.

Construct logical arguments and write proofs of theo-rems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justi-fies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations.

Implementation: Prove that the sum of the interior angles of a pentagon is 540˚ using the fact that the sum of the interior angles of a triangle is 180˚.

Honors Geometry: write a two-column proof.GIVEN: LM = JO, MN = ON, m LNK = m JNKPROVE: ΔLNK = ΔJNK

Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dy-namic geometry software, design software or Internet applets.

/ /

page 55



Geo

met

ry &

Mea

sure

men

t

Know and apply properties of geometric figures to solve real-world and mathemati-cal problems and to logically justify results in geometry.

Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a trans-versal, to solve problems and logically justify results.

Implementation: Prove that the perpendicular bisector of a line segment is the set of all points equidistant from the two end-points, and use this fact to solve problems and justify other results

Honors Geometry: Find the value of k such that the line containing point (2,k) is perpendicular to the line y = 2x – 3 at point (4,5).

Know and apply properties of angles, including corre-sponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results.

Implementation: Prove that two triangles formed by a pair of intersecting lines and a pair of parallel lines (an “X” trapped between two parallel lines) are similar.

Honors Geometry: In the diagram, m || n. Find the value of x. Explain how you obtained your answer.

Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results.

Implementation: Use the triangle inequality to prove that the perimeter of a quadrilateral is larger than the sum of the lengths of its diagonals.

Honors Geometry: Baseball A pitcher throws a baseball 60 feet from the pitcher’s mound to home plate. A batter pops the ball up and it comes down just 24 feet from home plate. What can you determine about how far the ball lands from pitcher’s mound? Explain why the Triangle Inequality Theorem can be used to describe all but the shortest and longest possible distances.

9,10,11,12

page 56



Geo

met

ry &

Mea

sure

men

t


Apply the Pythagorean Theorem and its converse to solve problems and logically justify results.

Implementation: When building a wooden frame that is sup-posed to have a square corner, ensure that the corner is square by measuring lengths near the corner and applying the Pythagorean Theorem.

Honors Geometry: Use the Distance Formula and the Con-verse of the Pythagorean Theorem to determine whether ΔABC is a right triangle.

Know and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems and logically justify results.

Implementation: Use 30-60-90 triangles to analyze geomet-ric figures involving equilateral triangles and hexagons.

Implementation 2: Determine exact values of the trigonomet-ric ratios in these special triangles using relationships among the side lengths.

Honors Geometry: Distance Each figure to the right is a30°-60°-90° triangle. Find the value of x. Round to the near-est tenth.

9,10,11,12

page 57



Geo

met

ry &

Mea

sure

men

t


Know and apply properties of congruent and similar figures to solve problems and logically justify results.

Implementation: Analyze lengths and areas in a figure formed by drawing a line segment from one side of a triangle to a second side, parallel to the third side.

Implementation 2: Determine the height of a pine tree by comparing the length of its shadow to the length of the shadow of a person of known height.

Implementation 3: When attempting to build two identical 4-sided frames, a person measured the lengths of corresponding sides and found that they matched. Can the person conclude that the shapes of the frames are congruent?

Honors Geometry: You have a three-dimensional scale model of a Ferris wheel. The scale of the model is 1 inch: 50 feet. When riding the actual Ferris wheel, you travel about 471 feet during one complete rotation. What is the approximate radius of the wheel on the scale model?

Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results.

Implementation: Recognize that a rectangle is a special case of a trapezoid.

Implementation 2: Give a concise and clear definition of a kite.

Honors Geometry: EFGH is a quadrilateral in which ∉HEF and ∉FGH are right angles that are each bisected by EG. What type of quadrilateral is EFGH? Write a paragraph proof.

Know and apply properties of a circle to solve problems and logically justify results.

Implementation: Show that opposite angles of a quadrilateral inscribed in a circle are supplementary.

Honors Geometry: Prove that ABC inscribed in the circle below is a right angle.

/

Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and deter-mine the sine, cosine and tangent of an acute angle in a right triangle.

Solve real-world and math-ematical geometric problems using algebraic methods.

9,10,11,12

page 58



Geo

met

ry &

Mea

sure

men

t


Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios.

Implementation: Find the area of a triangle, given the measure of one of its acute angles and the lengths of the two sides that form that angle.

Honors Geometry: Write an expression for (sin a°)2 + (cos a°)2 in terms of x, y, and z. Then use the Pythagorean Theorem to simplify the expression.

Use calculators, tables or other technologies in connec-tion with the trigonometric ratios to find angle measures in right triangles in various contexts.

Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments.

Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and deter-mine the sine, cosine and tangent of an acute angle in a right triangle.

Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations.

Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90˚, to solve problems involving figures on a coordinate grid.

Implementation: If the point (3,-2) is rotated 90˚ counter-clockwise about the origin, it becomes the point (2, 3).

Honors Geometry: Rotate the line the given number of degrees (a) about the x-intercept and (b) about the y-intercept. Write the equation of each image. y = 2x + 2; 90°.

Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure.

9,10,11,12

page 59

algebra ii a/b anD honors algebra ii a/b


Understand the definition of a function. Use functional notation and evaluate a function at a given point in its domain.

Implementation: If , find f(-4).

Honors: If , find f(2).

Alg

ebra

Distinguish between functions and other relations de-fined symbolically, graphically or in tabular form.

Find the domain of a function defined symbolically, graphically or in a real-world context.

Implementation: The formula f (x) = πx2 can represent a function whose domain is all real numbers, but in the context of the area of a circle, the domain would be restricted to posi-tive x.

Honors example: The function f(x) = (9 – x2)1/2 has a domain of x ≤ 3, but in the context of finding the height of a right triangle, the domain is restricted to 0 ≤ x ≤ 3.

Understand the concept of function, and identify impor-tant features of functions and other relations using sym-bolic and graphical methods where appropriate.

Obtain information and draw conclusions from graphs of functions and other relations.

Implementation: If a graph shows the relationship between the elapsed flight time of a golf ball at a given moment and its height at that same moment, identify the time interval during which the ball is at least 100 feet above the ground.

Honors example: If a graph shows the intersection of two or more inequalities, use that information to maximize the objec-tive function.

Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, us-ing symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f (x) = a(¬x – h)2 + k , or in factored form.

Identify intercepts, zeros, maxima, minima and inter-vals of increase and decrease from the graph of a func-tion.

€

f (x) =−x

2− 3x + 6

1+ 5 − x3

Understand the concept of an asymptote and identify asymptotes for exponential functions and reciprocals of linear functions, using symbolic and graphical methods.

9,10,11,12

page 60


Make qualitative statements about the rate of change of a function, based on its graph or table of values.

Implementation: The function f(x) = 3x increases for all x, but it increases faster when x > 2 than it does when x < 2.

Honors: The function f(x) = 3x increases more quickly than g(x) = 2x and not as quickly as h(x) = 5x. What occurs if we consider graphs like 1x or (-2)x instead?

Alg

ebra

Determine how translations affect the symbolic and graphical forms of a function. Know how to use graph-ing technology to examine translations.

Implementation: Determine how the graph of f(x) = |x – h| + k changes as h and k change.

Honors: Show how the changes in the graph of f(x) = ax2 + bx + c are related to the changes in a, h, and k in the graph of the vertex form f(x) = a(x – h)2 + k.


Represent and solve problems in various contexts using linear and quadratic functions.

Implementation: Write a function that represents the area of a rectangular garden that can be surrounded with 32 feet of fencing, and use the function to determine the possible dimensions of such a garden if the area must be at least 50 square feet.

Honors: Write a function that represents the area of a rect-angular garden that is being extended a uniform width in each direction, and use the function to determine the dimensions of the resultant space, given a required area for that space.

Represent and solve problems in various contexts us-ing exponential functions, such as investment growth, depreciation and population growth.

Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions.

Algebra II A/B & Honors Algebra II A/B Learner Outcomes continued...

Recognize linear, quadratic, exponential and other com-mon functions in real-world and mathematical situations; represent these functions with tables, verbal descrip-tions, symbols and graphs; solve problems involving these functions, and explain results in the original con-text.

9,10,11,12

page 61


Express the terms in a geometric sequence recursively and by giving an explicit (closed form) formula, and express the partial sums of a geometric series recursively.

Implementation: A closed form formula for the terms tn in the geometric sequence 3, 6, 12, 24, ... is tn = 3(2)n-1, where n = 1, 2, 3, ... , and this sequence can be expressed recursively by writing t1 = 3 and tn = 2tn-1, for n ≥ 2.

Implementation 2: The partial sums sn of the series 3 + 6 + 12 + 24 + ... can be expressed recursively by writing s1 = 3 and sn = 3 + 2sn-1, for n ≥ 2.

Honors: Recognize and write a piecewise-defined se-quence. For instance, the terms 5, 18, 9, 30, 15, 48, 24, 12, 6, and 3 can be expressed as What conclusions can you make about the behavior of this sequence of integers?

Alg

ebra

Recognize and solve problems that can be modeled us-ing finite geometric sequences and series, such as home mortgage and other compound interest examples. Know how to use spreadsheets and calculators to explore geo-metric sequences and series in various contexts.

Sketch the graphs of common non-linear functions such as , , , f (x) = x3, and trans-lations of these functions, such as .Know how to use graphing technology to graph these functions.

Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains.



€

an

=

an−1

2, a

n−1 = even

3an−1 + 3, a

n−1 = odd

Generate equivalent algebraic expressions involving polyno-mials and radicals; use alge-braic properties to evaluate expressions.

Add, subtract and multiply polynomials; divide a poly-nomial by a polynomial of equal or lower degree.

Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares.

Implementation: 9x6 – x4 = (3x3 – x2)(3x3 + x2).

Honors: Factor polynomials completely (factor common mono-mials, then factor by grouping, and then factor the difference of two squares): 9x3 + 3m2 – 36m – 12 = 3(3m + 1)(m – 2)(m + 2).

9,10,11,12

page 62


Add, subtract, multiply, divide and simplify algebraic fractions.

Implementation: is equivalent to

Honors: is equivalent to

Alg

ebra

Check whether a given complex number is a solution of a quadratic equation by substituting it for the variable and evaluating the expression, using arithmetic with complex numbers.

Implementation: The complex number is a solution of 2x2 – 2x + 1 = 0, since

Honors: Consider the function f(x) = z2 + c and the infinite list of complex numbers: z0 = 0, z1 = f(z0), z2 = f(z1), … If the absolute values of z0, z1, z2, … are all less than a fixed number N, then c belongs to the Mandelbrot set. Does c = -0.5i belong in the Mandelbrot set?

Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots.

Implementation: Rules for computing directly with radicals may also be used:

Honors:



Justify steps in generating equivalent expressions by identifying the properties used. Use substitution to check the equality of expressions for some particular values of the variables; recognize that checking with substitution does not guarantee equality of expressions for all values of the variables.

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3

x + 52

x − 3+

1

x + 5

€

3(x − 3)

3x + 7

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12x2y6z124 = yz

312x

2y24

9,10,11,12

page 63


Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factor-ing, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities.

Implementation: A diver jumps from a 20-meter platform with an upward velocity of 3 meters per second. In finding the time at which the diver hits the surface of the water, the resulting quadratic equation has a positive and a negative solution. The negative solution should be discarded because of the context.

Honors: For a swimming pool with a rectangular base, Torricel-li’s law implies that the height h of the water in the pool t seconds after it begins draining is given by

where l and w are the pool’s length and width, d is the diameter of the drain, and h0 is the water’s initial height. (All measure-ments are in inches.) Solve for the time required in terms of l, w, d, and h0.

Alg

ebra

Represent real-world and mathematical situations using equations and inequalities involving linear, quadratic, exponential and nth root functions. Solve equations and inequalities symbolically and graphically. Interpret so-lutions in the original context

Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations.

Recognize that to solve certain equations, number systems need to be extended from whole numbers to integers, from integers to rational numbers, from ratio-nal numbers to real numbers, and from real numbers to complex numbers. In particular, non-real complex numbers are needed to solve some quadratic equations with real coefficients.

Represent relationships in various contexts using systems of linear inequalities; solve them graphically. Indicate which parts of the boundary are included in and excluded from the solution set using solid and dot-ted lines.


€

h = h0−2πd2 3

lwt

2

Solve linear programming problems in two variables using graphical methods.

9,10,11,12

page 64


Represent relationships in various contexts using ab-solute value inequalities in two variables; solve them graphically.

Implementation: If a pipe is to be cut to a length of 5 meters accurate to within a tenth of its diameter, the relationship between the length x of the pipe and its diameter y satisfies the inequality | x – 5| ≤ 0.1y.

Honors: In holography, light from a laser beam is split into two beams, a reference beam and an object beam. Light from the object beam reflects off an object and is recombined with the reference beam to form images on film that can be used to create three-dimensional images. Write an equation for the path of the reference beam if it is reflected at a point 5 units horizon-tally and 8 units vertically from the laser.

Alg

ebra

Represent real-world and mathematical situations using equations and inequalities involving linear, quadratic, exponential and nth root functions. Solve equations and inequalities symbolically and graphically. Interpret so-lutions in the original context

Solve equations that contain radical expressions. Rec-ognize that extraneous solutions may arise when using symbolic methods.

Implementation: The equation may be solved by squaring both sides to obtain x – 9 = 81x, which has the solution . However, this is not a solution of the original equation, so it is an extraneous solution that should be dis-carded. The original equation has no solution in this case.

Implementation 2: Solve .

Honors: Solve .

Assess the reasonableness of a solution in its given con-text and compare the solution to appropriate graphi-cal or numerical estimates; interpret a solution in the original context.


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5x + 6 + 3 = 3(x +1) + 4

9,10,11,12

page 65


probability & statistics

9,10,11,12

Dat

a A

naly

sis &

Pro

babi

lity

Display and analyze data; use various measures associated with data to draw conclu-sions, identify trends and describe relationships.

Analyze the effects on summary statistics of changes in data sets.

Implementation: Understand how inserting or deleting a data point may affect the mean and standard deviation.

Implementation 2: Understand how the median and inter-quartile range are affected when the entire data set is trans-formed by adding a constant to each data value or multiplying each data value by a constant.

Use scatter-plots to analyze patterns and describe relation-ships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions.

Describe a data set using data displays, including box-and-whisker plots; describe and compare data sets using summary statistics, including measures of center, location and spread. Measures of center and location include mean, median, quartile and percentile. Measures of spread include standard deviation, range and inter-quartile range. Know how to use calculators, spreadsheets or other technology to display data and calculate summary statistics.

Use the mean and standard deviation of a data set to fit it to a normal distribution (bell-shaped curve) and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets and tables to estimate areas under the normal curve.

Implementation: After performing several measurements of some attribute of an irregular physical object, it is appropriate to fit the data to a normal distribution and draw conclusions about measurement error.

Implementation 2: When data involving two very different populations is combined, the resulting histogram may show two distinct peaks, and fitting the data to a normal distribution is not appropriate.

Evaluate reports based on data published in the media by identifying the source of the data, the design of the study, and the way the data are analyzed and displayed. Show how graphs and data can be distorted to support different points of view. Know how to use spreadsheet tables and graphs or graphing technology to recognize and analyze distortions in data displays.

Implementation: Displaying only part of a vertical axis can make differences in data appear deceptively large.

Explain the uses of data and statistical thinking to draw inferences, make predictions and justify conclusions.

page 66


Identify and explain misleading uses of data; recognize when arguments based on data confuse correlation and causation.

9,10,11,12

Dat

a A

naly

sis &

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babi

lity

Explain the uses of data and statistical thinking to draw inferences, make predictions and justify conclusions.

Design simple experiments and explain the impact of sampling methods, bias and the phrasing of questions asked during data collection.

Select and apply counting procedures, such as the mul-tiplication and addition principles and tree diagrams, to determine the size of a sample space (the number of possible outcomes) and to calculate probabilities.

Implementation: If one girl and one boy are picked at random from a class with 20 girls and 15 boys, there are 20 × 15 = 300 different possibilities, so the probability that a particular girl is chosen together with a particular boy is

Probability & Statistics Learner Outcomes continued...

Calculate probabilities and apply probability concepts to solve real-world and math-ematical problems

Calculate experimental probabilities by performing simulations or experiments involving a probability model and using relative frequencies of outcomes.

Understand that the Law of Large Numbers expresses a relationship between the probabilities in a probabil-ity model and the experimental probabilities found by performing simulations or experiments involving the model.

Use random numbers generated by a calculator or a spreadsheet, or taken from a table, to perform probabil-ity simulations and to introduce fairness into decision making.

Implementation: If a group of students needs to fairly select one of its members to lead a discussion, they can use a random number to determine the selection.

Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems.

Implementation: The probability of tossing at least one head when flipping a fair coin three times can be calculated by looking at the complement of this event (flipping three tails in a row).

Describe the concepts of intersections, unions and complements using Venn diagrams. Understand the relationships between these concepts and the words AND, OR, NOT, as used in computerized searches and spreadsheets.

page 67


Understand and use simple probability formulas involving intersections, unions and complements of events.

Implementation: If the probability of an event is p, then the probability of the complement of an event is 1 – p; the probability of the intersection of two independent events is the product of their probabilities.

Implementation 2: The probability of the union of two events equals the sum of the probabilities of the two individual events minus the probability of the intersection of the events.

9,10,11,12

Dat

a A

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Probability & Statistics Learner Outcomes continued...

Calculate probabilities and apply probability concepts to solve real-world and math-ematical problems

Apply probability concepts to real-world situations to make informed decisions.

Implementation: Explain why a hockey coach might decide near the end of the game to pull the goalie to add another forward position player if the team is behind.

Implementation 2: Consider the role that probabilities play in health care decisions, such as deciding between having eye surgery and wearing glasses.

Use the relationship between conditional probabilities and relative frequencies in contingency tables.

Implementation: A table that displays percentages relating gender (male or female) and handedness (right-handed or left-handed) can be used to determine the conditional probability of being left-handed, given that the gender is male.

page 68


pre-calculus a

10,11,12

Alg

ebra


Distinguish between functions and other relations de-fined symbolically, graphically or in tabular form.


Implementation: The formula f (x) = πx2 can represent a function whose domain is all real numbers, but in the context of the area of a circle, the domain would be restricted to positive x.

Understand the definition of a function. Use functional no-tation and evaluate a function at a given point in its domain.

Implementation: If , find f (-4).



Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, us-ing symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f (x) = a(x – h)2 + k , or in factored form.







page 69




10,11,12

Alg

ebra

Pre-Calculus A Learner Outcomes continued...




Sketch the graphs of common non-linear functions such as , , , f (x) = x3, and translations of these functions, such as . Know how to use graphing technol-ogy to graph these functions.

Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains.






Implementation: is equivalent to .


Implementation: The complex number ,is a solution of 2x2 – 2x + 1 = 0, since

page 70



Implementation: Rules for computing directly with radicals may also be used:

10,11,12

Alg

ebra


Represent relationships in various contexts using qua-dratic equations and inequalities. Solve quadratic equa-tions and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities.

Implementation: A diver jumps from a 20 meter platform with an upward velocity of 3 meters per second. In finding the time at which the diver hits the surface of the water, the resulting quadratic equation has a positive and a negative solution. The negative solution should be discarded because of the context.



Recognize that to solve certain equations, number systems need to be extended from whole numbers to integers, from integers to rational numbers, from ratio-nal numbers to real numbers, and from real numbers to complex numbers. In particular, non-real complex numbers are needed to solve some quadratic equations with real coefficients. .


Represent real-world and mathematical situations using equations and inequalities involving linear, quadratic, exponential and nth root functions. Solve equations and inequalities symbolically and graphically. Interpret solutions in the original con-text.


page 71




10,11,12

Alg

ebra


Solve equations that contain radical expressions. Rec-ognize that extraneous solutions may arise when using symbolic methods.

Implementation: The equation may be solved by squaring both sides to obtain x – 9 = 81x, which has the solution . However, this is not a solution of the original equation, so it is an extraneous solution that should be discarded. The original equation has no solution in this case.

Implementation 2: Solve


Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments.

Represent real-world and mathematical situations using equations and inequalities involving linear, quadratic, exponential and nth root functions. Solve equations and inequalities symbolically and graphically. Interpret solutions in the original con-text.

Know the equation for the graph of a circle with radius r and center (h, k), (x - h)2 + (y-k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations.

page 72


pre-calculus b

10,11,12

Graph logarithmic functions. Graph translations and reflections of these functions.

Use the inverse relationship between exponential and loga-rithmic functions to solve equations and problems.

Implementation: Solve the following equation.

Solve exponential and logarithmic equations when pos-sible. For those that cannot be solved analytically, use graphical methods to find approximate solutions.

Implementation:

Explain how the parameters of an exponential or loga-rithmic model relate to the data set or situation being modeled. Find an exponential or logarithmic function to model a given data set or situation. Solve problems involving exponential growth and decay.

Implementation: Write a model that represents a popula-tion growth of 6% yearly given that the 2010 population was 60 million. Let t=0 in 2010. Use the model to predict the population in 2020.

Define (using the unit circle), graph, and use all trigo-nometric functions of any angle. Convert between radian and degree measure. Calculate arc lengths in given circles.

Use unit conversions and the ideas of arc length and angle measurements to solve problems that deal with linear velocity and angular velocity.

Implementation: A child 4 feet from the center of a merry-go-round is being spun at 20 rpm, what is the linear velocity of the child?

Apply the any of the six basic trigonometric functions to solve applications problems that deal with right triangles.

Implementation: From a point on level ground to the base of a tower is 125 feet, the angle of elevation is 57.2°. Approxi-mate the height of the tower to the nearest foot.

Graph transformations of the six trigonometric func-tions and explain the relationship between constants in the formula and transformed graph. For sine and cosine functions involve changes in amplitude, period, equilib-rium, and phase shift.

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log2 x( ) = 8

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5x

= 3x+1

Expo

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ial &

Log

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mic

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rigon

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ric F

unct

ions

page 73


10,11,12 Know the basic trigonometric identities for sine, cosine,

and tangent (e.g., the Pythagorean identities, sum and dif-ference formulas, co-functions relationships, double- angle and half-angle formulas).

Know the basic properties of the inverse trigonometric func-tions, including their domains and ranges. Recognize their graphs.

Solve trigonometric equations using basic identities and inverse trigonometric functions.

Implementation: Find the solutions for

Prove trigonometric identities and derive some of the basic ones (e.g., double-angle formula from sum and difference formulas, half-angle formula from double- angle formula, etc.).

Find a sinusoidal function to model a given data set or situation and explain how the parameters of the model relate to the data set or situation.

Implementation: Use sine or cosine to write a model that represents the simple harmonic motion of a buoy in the ocean if every 10 seconds the buoy raises from its lowest point 30 feet above the ocean floor to 45 feet above the ocean floor.

Use the law of sines/cosines to solve problems that involve non-right angle triangles.

Pre-Calculus B Learner Outcomes continued...

Trig

onom

etric

Fun

ctio

ns

€

0 ≤ x < 2π

€

cos2x + 3cos x + 2 = 0

page 74


calculus a/b

11, 12Demonstrate how to solve absolute equations, inequalities, trigonometric, exponential and logarithmic equations.


Identify, state the domain and range, and sketch the graph of elementary functions.

Implementation: State the domain and range of

Use different techniques of graphing such as shifting, moving, stretching, or shrinking.

Implementation: Given f(x), sketch –f(x + 4)

Demonstrate the understanding of limit properties, e.g., limit of a constant, sum, product and quotient.

Use of L’Hopital rule to find a limit.

Implementation: Evaluate

Show different types of non-existent limits.

Implementation: Evaluate and graph.

Identify different cases of discontinuity and differenti-ate between removable and non-removable discontinu-ity using limit concepts, including one-sided limits.

Implementation: Find the values of discontinuity and determine if the discontinuity is removable.

Elem

enta

ry F

unct

ions

Lim

its

Demonstrate their knowledge of algebraic, trigonometric, exponential, and logarithmic functions.

€

56x

= 8320

€

f x( ) = x − 5

Demonstrate their knowledge of the limits and continuity.

€

limx→ 2

x2

+ x − 6

x2− 4

€

limx→1

1

x −1

Use limits to find horizontal, vertical and slant asymp-totes.

Demonstrate their knowledge of the derivative and prob-lem solving using derivative concepts.

Diff

eren

tial C

alcu

lus

Students demonstrate an understanding and applica-tion of the formal definition of derivative of a function at a point and the notation of differentiability.

For Example: Use the limit process to differentiate f(x) = x - 5

Find the derivatives of elementary, sum, product, quotient, composite function (chain rule) and implicitly defined functions.

Implementation: Find the derivative of

€

f x( ) = x2− 5

page 75


11, 12Evaluate the average and instantaneous rates of change and find the velocity and acceleration of a particle moving along a line.

Implementation: A ball is dropped from 600 feet. Find the instantaneous rate of change and average rate of change when the ball hits the ground.

Demonstrate the understanding and application of the Intermediate Value Theorem, Mean Value Theorem, and Rolle’s Theorem.

Find the high order derivatives, relation between dif-ferentiability and continuity.

Implementation: Find the third derivative of f(x) = sinx

Solve problems involving rates and change.

Implementation: A 20 foot ladder leaning against a wall has its base is being pulled away from the wall at a rate of 2 feet per second. What rate is the top of the ladder falling when the base is 10 feet from the wall

Use derivatives to find the increasing and decreasing intervals, the critical points and relative extrema, the concavity and inflection points of a function. Utilize the above findings to sketch the graph of the function.

Use derivatives to maximize/minimize a function relat-ing to an application.

Implementation: An industrial tank is formed by adjoining two hemi-spheres to the ends of a right circular cylinder. The total volume of the tank is 3000 cubic feet. The hemispherical ends cost twice as much per square foot of surface area as the sides. Find the dimension that will minimize the cost.

Diff

eren

tial C

alcu

lus

Demonstrate their knowledge of the derivative and prob-lem solving using derivative concepts.

Use Newton’s method to appropriate the zeros of a function.

Implementation: Use Newton’s method to find the zero(s) of the function to within .001

Calculus A/B Learner Outcomes continued...

€

f x( ) = x3 + x −1

page 76


11, 12Use techniques of integration including basic integration formulas, integration by substitution, and change of vari-ables.

Implementation: Evaluate

Use integration techniques to find distance and velocity from acceleration with initial conditions and problem solving in growth and decay.

Implementation: A population of bacteria is changing at a rate of

where t is the time in days. Write an equation that gives the population at any time t, and find the population when t = 3 days.

Apply the definite integral concepts such as area, approximations to the definite integral by using rect-angles and the limit as a sum.

Implementation: Find the upper and lower sums for the region bounded by the graph and the x-axis between x=0 and x=2. Use 4 subintervals.

Use the fundamental theorems of Calculus.

Implementation: Evaluate Inte

gral

Cal

culu

s

Students demonstrate their knowledge of anti-derivative techniques and problem solv-ing in integral calculus.

Use Riemann and Trapezoidal sums to approximate definite integrals of functions represented algebraically, geometrically and by tables of values.

Implementation: Use the Trapezoidal sums to approxi-mate .

Compare the results for n=4 and n=8 to the results using the fundamental theorem of calculus.

Calculus A/B Learner Outcomes continued...

€

x

1− x 2( )3∫ dx

€

dP

dt=

3000

1+ 0.25t

€

f x( ) = x2

€

x

x2

+10

3

∫ dx

€

sin x0

π

∫ dx

page 77


algebra iic

10, 11, 12

Understand the definition of a function. Use functional no-tation and evaluate a function at a given point in its domain.

Implementation: If , find f (-4).

Distinguish between functions and other relations defined symbolically, graphically or in tabular form.


Implementation: The formula f (x) = πx2 can represent a function whose domain is all real numbers, but in the context of the area of a circle, the domain would be restricted to posi-tive x.



Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, us-ing symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f (x) = a(¬x – h)2 + k , or in factored form.


Alg

ebra

Understand the concept of function, and identify impor-tant features of functions and other relations using symbolic and graphical methods where appropriate.






page 78


10, 11, 12





Express the terms in a geometric sequence recursively and by giving an explicit (closed form) formula, and ex-press the partial sums of a geometric series recursively.

Implementation: A closed form formula for the terms tn in the geometric sequence 3, 6, 12, 24, ... is tn = 3(2)n-1, where n = 1, 2, 3, ... , and this sequence can be expressed recur-sively by writing t1 = 3 and tn = 2tn-1, for n ≥ 2.

Implementation 2: The partial sums sn of the series 3 + 6 + 12 + 24 + ... can be expressed recursively by writing s1 = 3 and sn = 3 + 2sn-1, for n ≥ 2.

Alg

ebra

Recognize linear, quadratic, exponential and other com-mon functions in real-world and mathematical situations; represent these functions with tables, verbal descriptions, symbols and graphs; solve problems involving these func-tions, and explain results in the original context.

Recognize and solve problems that can be modeled using finite geometric sequences and series, such as home mortgage and other compound interest ex-amples. Know how to use spreadsheets and calculators to explore geometric sequences and series in various contexts.

Sketch the graphs of common non-linear functions such as , , , f (x) = x3, and translations of these functions, such as . Know how to use graphing technology to graph these functions.

Algebra II C Learner Outcomes continued...

page 79


10, 11, 12

Evaluate polynomial and rational expressions and expres-sions containing radicals and absolute values at specified points in their domains.





Implementation: is equivalent to .

Alg

ebra



Implementation: The complex number is a solution

of 2x2 – 2x + 1 = 0, since


Implementation: Rules for computing directly with radicals may also be used: .


Justify steps in generating equivalent expressions by identifying the properties used. Use substitution to check the equality of expressions for some particular values of the variables; recognize that checking with substitution does not guarantee equality of expressions for all values of the variables.

page 80


10, 11, 12

Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities.

Implementation: A diver jumps from a 20 meter platform with an upward velocity of 3 meters per second. In finding the time at which the diver hits the surface of the water, the resulting quadratic equation has a positive and a negative solution. The negative solu-tion should be discarded because of the context.

Alg

ebra

Represent real-world and mathematical situations using equations and inequalities involving linear, quadratic, exponential and nth root func-tions. Solve equations and inequalities symbolically and graphically. Interpret solutions in the original context.


Recognize that to solve certain equations, number systems need to be extended from whole numbers to integers, from integers to rational numbers, from ratio-nal numbers to real numbers, and from real numbers to complex numbers. In particular, non-real complex numbers are needed to solve some quadratic equations with real coefficients.






page 81


10, 11, 12

Solve equations that contain radical expressions. Recognize that extraneous solutions may arise when using symbolic methods.

Implementation: The equation may be solved by squaring both sides to obtain x – 9 = 81x, which has the solution . However, this is not a solution of the original equation, so it is an extraneous solution that should be discarded. The original equation has no solution in this case.

Implementation 2: Solve

Alg

ebra

Represent real-world and mathematical situations using equations and inequalities involving linear, quadratic, exponential and nth root func-tions. Solve equations and inequalities symbolically and graphically. Interpret solutions in the original context.


Apply the Pythagorean Theorem and its converse to solve problems and logically justify results.

Implementation: When building a wooden frame that is supposed to have a square corner, ensure that the corner is square by measuring lengths near the corner and applying the Pythagorean Theorem.


Know and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems and logically justify results.

Implementation: Use 30-60-90 triangles to analyze geo-metric figures involving equilateral triangles and hexagons.

Implementation 2: Determine exact values of the trigo-nometric ratios in these special triangles using relationships among the side lengths.

Understand how the properties of similar right tri-angles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle.

Apply the trigonometric ratios sine, cosine and tan-gent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use cal-culators, tables or other technology to evaluate trigono-metric ratios.

Implementation: Find the area of a triangle, given the measure of one of its acute angles and the lengths of the two sides that form that angle.

Geo

met

ry &

Mea

sure

men

t

Geometry & Measurement Know and apply properties of geometric figures to solve real-world and mathemati-cal problems and to logically justify results in geometry.


page 82


10, 11, 12

Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts.

Geo

met

ry &

Mea

sure

men

t

Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and jus-tify this equation using the Pythagorean Theorem and properties of translations.


Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equa-tion for a length in a geometric figure.


page 83


graD Math

12

Students will apply the correct order of operations to sim-plify and evaluate numeric expressions.

GRAD Content Limit:Items must use positive rational numbers. Items will use addition, subtraction, multiplication, division and grouping symbols only. Fraction denominators are limited to 2, 3, 4, 5, 6, 8, 10 and 12.Subtraction cannot be a mixed number minus a mixed number re-quiring regrouping (e.g., 3 14 − 1 78 is not acceptable) Multipli-cation may be expressed as raised dot, x, or ( ) (e.g., 5 • 6, 5 x 6, 5(6) ). Division may be expressed using division symbol or fraction bar (e.g., 6 ÷2 or 6/2 ). Division items must have a whole number divisor. For multiplication and division items, mixed numbers must be expressed as improper fractions. No nested grouping symbols are allowed (e.g., 3[292 + (100/2)] is not allowed). Items may require the identification of the correct order of operations shown (calculation not required). Items may include exponents. Problems may include context.

Students will use rational numbers in complex ways to solve multi-step, real-world and mathematical problems.

GRAD Content Limit:Rationals are limited to positive rationals. Non-integer ra-tionals will be represented in decimal form. Items may require integer approximations of square roots of positive integers. Squares must be less than or equal to 150.N

umbe

r Sen

se

Appropriately use calculators and other technologies to solve algebraic, geometric, probabi-listic and statistical problems.

Students will use fractions, decimals and percents in multiple representations for estimation and computa-tion to solve real- world and mathematical problems.

GRAD Content Limit: 1–2 Fraction denominators are limit-ed to 2, 3, 4, 5, 6, 8, 10, 12, 16, 20, 25 and 100. Restrictions on denominators apply to problem and answer options. (e.g., 1/3 + 1/5 is not allowed; 1/15 – 1/3 is not allowed). Items may include positive and negative fractions, decimals and percents.

Students will use proportional reasoning to solve real-world and mathematical problems.

GRAD Content Limit: Items may involve:•Rates •Scale drawings and maps • Similar figures •Ratio • Unit pricing •Showing how changing one or more dimensions affects change in area. Shapes are limited to circles, parallelograms and triangles. Items are limited to two-dimen-sional figures. Pictures or diagrams may be used but are not required. Similarity may be shown using similarity symbol (~) or using markings on figures. Items may include context.

page 84


12

Students will know the numeric, graphic and symbolic properties of linear, step, absolute value and quadratic functions.

Content Limit: Items may include rates of change, inter-cepts, maxima and minima. Items may include intersection between two graphs. Step functions must model real-world situations. Step functions will not be represented symbolically.

GRAD Content Limit: Items may not include step or absolute value functions. Items may include rates of change and intercepts. Items that assess quadratics are limited to graphical properties. Increments of x and y-axes must be integers.

Students will model exponential growth and decay.

Content Limit: Models may be numeric, graphic and symbolic. When calculation is required, exponents must be integers. Items may have real-world context (e.g., bacterial growth, half-life, compound interest).

GRAD Content Limit: Models may be numeric or graphic. Items may have real-world context (e.g., bacterial growth, half-life, compound interest).

Patt

erns

, Fun

ctio

ns, a

nd A

lgeb

ra

Represent and analyze real-world and mathematical prob-lems using numeric, graphic and symbolic methods for a variety of functions.

Students will apply basic concepts of linear, quadratic and exponential expressions or equations in real-world problems.

Content Limit: Exponents must be integers.

GRAD Content Limit: Items will be limited to linear and exponential. Expressions and equations must be able to be solved numerically. Table or graph required. Items will not require expressions or equations to be solved symbolically.

Students will generate a table of values from a formula or equation. Students will graph the result of a formula or linear equation in ordered pair format on a grid.

GRAD Content Limit: x and y axes may have different scales. Items do not require students to graph or generate a table of a non-linear relation. Formulas will only have unknowns to the first degree. Items may not require generating a linear equation from a table of values. Items may include real-world context (e.g., converting temperature). Given a continuous (i.e., individual points not indicated) linear graph, students will generate a table of values. Linear equations will be given in y = mx + b form.

GRAD Math Learner Outcomes continued...

page 85


12

Students will know the numeric, graphic and symbolic properties of linear, step, absolute value and quadratic functions.

Content Limit: Items may include rates of change, inter-cepts, maxima and minima. Items may include intersection between two graphs. Step functions must model real-world situations. Step functions will not be represented symbolically.

GRAD Content Limit: Items may not include step or ab-solute value functions. Items may include rates of change and intercepts. Items that assess quadratics are limited to graphical properties. Increments of x and y-axes must be integers.

Students will translate a problem described verbally or by tables, diagrams or graphs, into suitable mathematical language, solve the problem mathematically and interpret the result in the original context.

GRAD Content Limit: Items may include real world context.

Patt

erns

, Fun

ctio

ns, a

nd A

lgeb

ra

Represent and analyze real-world and mathematical prob-lems using numeric, graphic and symbolic methods for a variety of functions.

Students will translate among equivalent forms of expressions.

Content Limit: Items may include simplifying algebraic expres-sions involving nested pairs of parentheses and brackets; simplify-ing rational expressions; factoring a common monomial term from an expression; applying associative, commutative and distributive laws. A simplified expression should contain at most four terms with at most two variables per term.

GRAD Content Limit: Items may include simplifying algebraic expressions; simplifying rational expressions; factoring a common monomial term from an expression; applying associative, commuta-tive and distributive laws. When applying distributive law, expres-sions may not contain 2 binomials. Expressions may not include nested pairs of parentheses or brackets. A simplified expression should contain at most two terms with at most one variable per term

Students will find equations of a line.

Content Limit: Items will provide two points on the line, a point and the slope of the line or the slope and y-intercept of the line. All answer options will be given in the same form within a MC item, either slope-intercept (y = mx + b) or standard form (ax + by = c). CR items must represent real-world contexts.

GRAD Content Limit: Items may require the student to generate the equation from the graph or identify the graph given the equation. Items will provide the slope and y-intercept of the line, when graph is not provided. Equations must be presented in slope-intercept form.


Solve simple equations and in-equalities numerically, graphi-cally and symbolically. Use recursion to model and solve real-world and mathematical problems.

page 86


12

Students will solve linear equations and inequalities in one variable with numeric, graphic and symbolic methods.

Content Limit: Forms of the linear equations or inequalities are not limited (e.g., 4(x + 5) – 3x = 6(x + 10) is accept-able). Items may include context.

GRAD Content Limit: Items may include at most one application of the distributive property. Items will not include inequalities. Forms of the linear equations are limited to at most a binomial equaling a binomial. Items must have a numeric solution.

Patt

erns

, Fun

ctio

ns, a

nd A

lgeb

ra

Students will determine solutions to quadratic equations in one variable with numeric, graphic and symbolic methods.

Content Limit: All solutions are real. Solutions determined from a graph will be integer solutions. Items may include context.

GRAD Content Limit: All solutions are integers. Coefficient on second-degree term will always be 1.

Students will solve systems of two linear equations and inequalities with 2 variables using numeric, graphic and symbolic methods.

Content Limit: Inequalities will only be solved graphically. Items may include context.

GRAD Content Limit: Items may include at most one application of the distributive property. Items will not include inequalities. Forms of linear equations are limited to at most a binomial equaling a binomial. Systems may be represented using graph, slope-intercept and table format. Systems are consistent and independent (i.e., solution is one ordered pair).



Students will understand how slopes can be used to deter-mine whether lines are parallel or perpendicular and deter-mine equations for parallel lines and perpendicular lines.

Content Limit: Items may provide a line and a point not on that line. Items may require students to determine the equation of the line passing through a given point and parallel to a given line. Items may require students to determine the equation of the line passing through a given point and perpendicular to a given line. Items may include context.

GRAD Content Limit: Items may not require students to determine the equation of a line. Equations in problem and answer options must be in slope-intercept form.

page 87


12

Students will use formulas with more than one variable to solve real-world and mathematical problems.

GRAD Content Limit: Formulas must be from a real-world context and may include powers (e.g., area, volume, I=prt or d=rt). Items may contain formulas with at most four variables. Roots are limited to square roots. Formula notation may not include subscripts. Formulas must be included within the item.

Patt

erns

, Fun

ctio

ns,

and

Alg

ebra

Students will use measures of central tendency and variability to describe, compare and draw conclusions about sets of data.

Content Limit: Measures may be mean, median, maximum, minimum, range, standard deviation, quartile, percentile, mode or interquartile range (IQR).

GRAD Content Limit: Measures may be mean, median, maximum, minimum, range, quartile, mode or interquartile range (IQR). The interquartile range may be referred to conceptu-ally, but the terminology “interquartile range” will not be used.

Students will determine approximate line of best-fit and use the line to draw conclusions.

Content Limit: Items will provide a scatter plot (coordinates of points on scatter plot are integers) or data set.

GRAD Content Limit: Equations are limited to linear equations only. Items will provide a scatter plot (coordinates of points on scatter plot are integers). Line of best fit may be pro-vided and asked to draw conclusions.



Students will analyze histograms, bar graphs, circle graphs, stem-and-leaf plots and box-and-whisker plots.S

GRAD Content Limit: Graphics may have at most ten data categories. Circle graphs may have at most eight sectors. Scales are in increments appropriate to the application. Histogram intervals must be consistent. Items may involve: Reading and interpret-ing Identifying trends and patterns and make predictions Solve problems using information presented in the graph

Dat

a A

naly

sis, S

tatis

tics,

and

Prob

abili

ty

Represent data and use vari-ous measures associated with data to draw conclusions and identify trends. Understand the effects of display distortion and measurement error on the interpretation of data.

Students will understand the meaning of and be able to compute minimum, maximum, range, median, mean and mode of a data set.

GRAD Content Limit: At most twenty numbers in the data set. Numbers used are less than 300. Items may ask which values (mean, median, mode, range) “best” describes a data set in context and identify justification. Items may require the calculation of quartiles. The interquartile range may be referred to conceptually, but the terminology “interquartile range” will not be used.

page 88


12

Students will select and apply appropriate counting pro-cedures to solve real-world and mathematical problems.

Content Limit: Items may involve computing probabilities. Items may include combinations and permutations.

GRAD Content Limit: Items may involve determining sam-ple space and/or computing probabilities. Items may not include formulas. Solutions may have at most 24 possibilities.

Students will calculate probabilities and relate the results in real-world and mathematical problems.

Content Limit: Items may use area, trees, unions and intersec-tions to calculate probabilities. Items may involve both the concept of mutually exclusive events or not mutually exclusive events. Items may involve independent or dependent events. Items may involve conditional probability.

GRAD Content Limit: Items may involve independent events. Items will not involve conditional probability.


Students will use probability models in real-world and mathematical problems.

Content Limit: Models may include area and binomial models. Binomial probabilities will involve at most 4 events.

GRAD Content Limit: Binomial probabilities will involve at most 3 events.

Dat

a A

naly

sis, S

tatis

tics,

and

Prob

abili

ty

Use appropriate counting procedures, calculate prob-abilities in various ways and apply theoretical probability concepts to solve real-world and mathematical problems.

Students will use models and visualization to under-stand and represent various three-dimensional objects and their cross sections from different perspectives.

Content Limit: Items are limited to top view, side view, front view or net. Shapes are limited to polyhedra, combina-tions of polyhedra, cylinders and cones. No figures will be oblique. All visible sides of views are clearly labeled. Prisms will have a base with at most six sides. Pyramids will have a base with at most six sides. Cross sections are limited to rectangular prisms, cones, cylinders, rectangular pyramids and triangular pyramids.

GRAD Content Limit: Shapes are limited to prisms, pyramids, cylinders and cones. Prisms will have a base with at most four sides. Pyramids will have a base with at most four sides.

Geo

met

ry

Use models to represent and understand two- and three-dimensional shapes and how various motions affect them. Recognize the relationship be-tween different representations of the same shape.

page 89


12

Students will know and use theorems about triangles and parallel lines in elementary geometry to justify facts about various geometrical figures and solve real-world and mathematical problems.

Content Limit: Theorems may include criteria for two triangles to be congruent or similar. Theorems may include facts about angles formed by parallel lines cut by a transversal. Items may involve the application of these theorems to solve real-world and mathematical problems involving other plane figures.

GRAD Content Limit: Items will require knowledge and use of theorems. Items will not require students to use theorems for justification. Items must include context or diagram.


Students will use properties of two- and three-dimen-sional figures to solve real-world and mathematical problems.

Content Limit: Use 3.14 as an approximation for π. Situ-ations may include finding area, perimeter, volume and surface area. Situations may include applying direct or indirect methods of measurement. Situations may include applying the Pythagorean theorem and its converse. Situations may include properties of 45-45-90 and 30-60-90 triangles.

GRAD Content Limit: Situations may include applying the Pythagorean theorem but not its converse. Situations will not include properties of 45-45-90 and 30-60-90 triangles. Limits on shapes in V.C.G.2 apply.

Apply basic theorems of plane geometry, right triangle trigo-nometry, coordinate geometry and a variety of visualization tools to solve real-world and mathematical problems.

Students will apply the basic concepts of right triangle trigonometry to determine unknown sides or unknown angles when solving real-world and mathematical prob-lems.

Content Limit: Concepts may include sine, cosine and tan-gent. Items will not require the use of the reciprocals or inverses of sine, cosine and tangent. Items will provide a table of deci-mal approximations of three trigonometry values for each angle given in the item or students may use trigonometry values from a calculator.

GRAD Content Limit: Items must include diagram.

Geo

met

ry

page 90


12

Students will use formulas to solve real world and mathematical problems.

GRAD Content Limit: Items may include determining the surface area or volume of shapes. • Shapes are limited to cubes, prisms and cylinders. • Pictures or diagrams may be used but are not required. • The radius or diameter is supplied for cylinders.• Answer options may be left in terms of π (e.g., 7π).• Non-rectangular prisms must provide the area of the base. Items may include calculating perimeter and area of two-dimensional figures obtained by putting together triangles, parallelograms and sectors of circles to solve real-world and mathematical problems.• Items must provide a picture or diagram. Items may include calculating the radius, diameter, circumference and area of a circle.• Given the diameter or radius, items may require students to determine area or circumference. • Given the circumference, items may require students to determine radius, diameter or area. • Radii must be greater than 2.Grade 11 Formula Sheet will be provided. (See copy of grade 11 formula sheet in Grade 11 Item Sampler).


Apply basic theorems of plane geometry, right triangle trigo-nometry, coordinate geometry and a variety of visualization tools to solve real-world and mathematical problems.

Geo

met

ry

page 91


trigonoMetry & special functions

11, 12

Use the inverse relationship between exponential and logarithmic functions to solve equations and problems.


Graph logarithmic functions. Graph translations and reflections of these functions.

Expo

nent

ial &

Log

arith

mic

Fun

ctio

ns

Solve exponential and logarithmic equations when pos-sible. For those that cannot be solved analytically, use graphical methods to find approximate solutions.

Implementation:

Explain how the parameters of an exponential or loga-rithmic model relate to the data set or situation being modeled. Find an exponential or logarithmic function to model a given data set or situation. Solve problems involving exponential growth and decay.

Implementation: Write a model that represents a popula-tion growth of 6% yearly given that the 2010 population was 60 million. Let t=0 in 2010. Use the model to predict the population in 2020.

€

log2 x( ) = 8

€

5x

= 3x+1

Trig

onom

etric

Fun

ctio

ns

Define (using the unit circle), graph, and use all trigo-nometric functions of any angle. Convert between radian and degree measure. Calculate arc lengths in given circles.

Use unit conversions and the ideas of arc length and angle measurements to solve problems that deal with linear velocity and angular velocity.

Implementation:: A child 4 feet from the center of a merry-go-round is being spun at 20 rpm, what is the linear velocity of the child?

Apply the any of the six basic trigonometric functions to solve applications problems that deal with right triangles.

Implementation: From a point on level ground to the base of a tower is 125 feet, the angle of elevation is 57.2°. Approxi-mate the height of the tower to the nearest foot.

page 92


11, 12

Graph transformations of the six trigonometric functions and explain the relationship between constants in the for-mula and transformed graph. For sine and cosine functions involve changes in amplitude, period, equilibrium, and phase shift.

Know the basic properties of the inverse trigonometric functions, including their domains and ranges. Recog-nize their graphs.

Know the basic trigonometric identities for sine, cosine, and tangent (e.g., the Pythagorean identities, sum and difference formulas, co-functions relationships, double- angle and half-angle formulas).

Trig

onom

etric

Fun

ctio

ns

Prove trigonometric identities and derive some of the basic ones (e.g., double-angle formula from sum and difference formulas, half-angle formula from double- angle formula, etc.).

Find a sinusoidal function to model a given data set or situation and explain how the parameters of the model relate to the data set or situation.

Implementation: Use sine or cosine to write a model that represents the simple harmonic motion of a buoy in the ocean if every 10 seconds the buoy raises from its lowest point 30 feet above the ocean floor to 45 feet above the ocean floor.

Trigonometry & Special Functions Learner Outcomes continued...

Solve trigonometric equations using basic identities and inverse trigonometric functions.

Implementation: Find the solutions for

€

0 ≤ x < 2π

€

cos2x + 3cos x + 2 = 0

Use the law of sines/cosines to solve problems that involve non-right angle triangles.

Use concepts of trigonometry to convert between rect-angular and polar coordinates, equations, and graphs.

Apply trigonometry to represent and manipulate vector forces to solve work and force problems.

Implementation: If a box weighs 100 pounds, find the magnitude of the force needed to pull it up a ramp that has a 18° incline.

Documents

K-12 MatheMatics curriculuM · K-12 MatheMatics DepartMent 2012-2013 Mission stateMent The aim of the Mathematics Department is to expose students to the benefits and enjoyment of