Learning Outcomes/Standards and Content/Activities s1

MATHEMATICS -- GRADE 5

PLANNED COURSE CURRICULUM GUIDE

I. COURSE DESCRIPTION AND INTENT:

The goal of the mathematics program is to provide an appropriate curriculum for every student to value mathematics; to become confident in one’s ability; to become a mathematical problem solver; and to learn to communicate and reason mathematically.

The Fifth Grade Mathematics Curriculum provides an extension of previously learned skills. The objectives expand upon skills in numeration, computation, fractions, graphing, measurement, money, geometry, time, and decimals.

The curriculum introduces concepts of statistics, probability, algebra, trigonometry, and calculus.

The curriculum follows the PDE Academic Standards for mathematics.

Varied teaching-learning activities are suggested to afford opportunities for students to apply basic mathematics knowledge through problem-solving activities. Students are expected to meet minimal achievement levels as measured in the expected student learning outcomes.

II. INSTRUCTIONAL TIME:

Class Periods: Daily Length of Class Periods (minutes): 60 Length of Course: 180 days; 180 clock hours Unit of Credit: Updated: June 2006 PLEASANT VALLEY SCHOOL DISTRICT Brodheadsville, Pennsylvania 18322 PLANNED COURSE ADAPTATIONS/MODIFICATIONS

Introduction

The instructional adaptations that follow are provided as suggestions to be implemented with all students, particularly with those in need of special education services including the gifted. This listing is in no way intended to be exhaustive. Rather, it is reflective of some major considerations in the area of curriculum adaptations/modifications.

These instructional adaptations will work with any student, but are especially beneficial to those in need of learning support. Some may argue that these modifications are simply good teaching. Indeed, modifications of this type do represent good teaching. These principles of good teaching become instructional modifications whenever: (1) certain students in a particular class require such modifications above and beyond what is typically required by most students in that class and (2) without these modifications, these same students would not succeed.

Assessment is an integral part of instruction reflecting student progress as well as achievement. Therefore, also included are suggestions for assessment modifications.

. Peer Support . Extended test time . Cooperative learning among peers . Test read to student by teacher or peer . Modeling . Oral testing (i.e., student retelling of information) . Development of Information Organizer . Open book/note test . Development of Graphic Organizer . Alternate testing (any demonstration of a student's understanding of concepts) . Development of structured study guides . Retesting . Student selection of instructional material (i.e., reading, writing, math) . Reduce the number of responses required on tests . Taped lessons . Use of curriculum based assessment . Copy notes (peer or teacher) . Vary test format . Student conferencing . Objectively define mastery as related to each task. Tasks should be learned to . Combine and vary modes of lesson presentation mastery . Adjust language level to match the developmental and intellectual levels of students . Reduce or remove distracting stimuli . Let student practice given examples first. Then assign tasks to be completed. . Use of concrete objects and manipulatives in all stages of instruction and . Provide opportunity for guided and independent practice in a variety of situations assessment . Limit number and length of directions . Emphasize important information . Have students repeat/review directions (i.e., peer to peer, student to teacher) . Allow extra time to complete assignments/projects . Give feedback that is as immediate, specific, and objective as possible . Limit the number of assigned tasks in the initial stages of learning. As the student's . Clarify error responses so that students do not make the same errors over and over competency increases, expect the student to complete the same number of tasks as again the rest of the class . Reinforce progress towards desired outcomes . Use supplemental materials . Breakdown complex tasks into smaller, more manageable units . Alternate assignments accepted (i.e., modification to homework assignments) . Use verbal prompts to elicit desired results . Flexible grouping/individual assistance . Use manual guidance (i.e., hand over hand) to facilitate correct responses . Seating to accommodate needs . Computer assisted instruction . Teacher proximity . Assessment based upon teacher observation of student performance (i.e., daily . Use behavioral management techniques (i.e., contracts, time-out, token system, work, portfolio, artifacts, projects) charts)

2 PREFACE

Users and information seekers should familiarize themselves with the purpose and terminology of this Planned Course Curriculum Guide (PCCG). We suggest that you first read the following:

• PCCG PURPOSE AND INTENT • PCCG DEFINITIONS

The PCCG specifies the unit lesson outcome, essential content, standards, activities, resources, and evaluation of student performance. This sector provides the means to initiate the learning activities to attain the program goal as identified in the course description and intent.

The standards and outcomes are minimal expectations; further embellishment of the course is discretionary with the instructor depending upon the capability of the students.

This PCCG is designed as an ACTIVE document capable of technological modification as required.

The instructional delivery of this curriculum is quality controlled through the lesson plan development of the teacher.

3 PLANNED COURSE CURRICULUM GUIDE (PCCG) PURPOSE AND INTENT

The Planned Course Curriculum Guide (PCCG) is a multi-purpose document:

• All staff, particularly new teachers, can understand instructional expectations through the WRITTEN curriculum

• A continuing district-wide instructional process and scope and sequence of subject matter are enhanced. The WRITTEN curriculum is delivered through the TAUGHT curriculum [instructional content and learning activities] and is evaluated through the TESTED curriculum [expected levels of student achievement - learning outcomes]

• Priority student-centered outcomes are identified and attained through suggested learning activities and content designed to help insure a balanced and comprehensive basic curriculum

• Essential content and course standards provide an efficient basis for selecting appropriate instructional materials and resources

• Staff development areas for curriculum improvement are provided

• The PCCG conforms with current Pennsylvania Department of Education curriculum regulations and serves the dual feature of providing both an administrative document and an instructional guide

• Content and subject format remain flexible and adaptable to modification - an “active” document

• Special Pennsylvania Department of Education [PDE] legislation is identified

• Parents and students are provided with an overview of the instructional program and each course in particular

4 PLANNED COURSE CURRICULUM GUIDE (PCCG) DEFINITIONS

• Course Description and Intent: a brief overview of the course and program goals

• Instructional Time: frequency of class meetings and time/appropriate credit at the secondary level

• Special Notes: emphatic features or highlights and identification of Department of Education mandates found in the course

• Unit Lesson Outcome: describes the knowledge, skills, attitudes, student performance behaviors and areas of study that have been identified as appropriate to help the student attain the rigorous standards of a quality education

• Teaching-Learning Activities: suggested activities designed to help all students achieve the learning outcomes and standards

• Outcomes/Standards: statements establishing the minimal knowledge, skills, performance behaviors, and essential learning (content) a student must attain

• Expected Levels of Achievement (Learning Outcomes): what students will be expected to do as a result of the application of teaching-learning activities and content

• Evaluation Criteria (Actual Level of Attainment): student performance level achieved and measured through specified evaluation criteria.

5 LEARNING OUTCOMES/STANDARDS AND CONTENT/ACTIVITIES Statements of student learning expectations achieved through suggested teaching-learning activities and selected content to help reach standards and graduation requirements.

Subject Title: MATHEMATICS

Discipline/Grade Level: GRADE 5

UNIT LESSON OUTCOME: 1

The learner will read, write, order, and compare numbers and equivalent forms of integers and rational numbers.

RELATIONSHIP TO PA OUTCOMES/STANDARDS (Check Appropriate Graduation Outcomes) Communications 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

Mathematics 2.1 X 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11

Science & Technology 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9

Environment & Ecology 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9

Civics & Government 5.1 5.2 5.3 5.4

Economics 6.1 6.2 6.3 6.4 6.5

Geography 7.1 7.2 7.3 7.4

History 8.1 8.2 8.3 8.4

Arts & Humanities 9.1 9.2 9.3 9.4

Health, Safety & PE 10.1 10.2 10.3 10.4 10.5

Family & Consumer Science 11.1 11.2 11.3 11.4

World Language 12.1 12.2 12.3 12.4 12.5 12.6

Career Education & Work 13.1 13.2 13.3 13.4

ESSENTIAL CONTENT CONTENT & INSTRUCTIONAL ACTUAL LEVEL OF RESOURCES AND MATERIALS OUTCOMES/STANDARD ACTIVITIES/STRATEGIES WITH ATTAINMENT (EVALUATION CORRECTIVES AND CRITERIA) ASSESSMENT EXTENSIONS (Individually created teaching activities may be used to achieve the standards; however, listed below are activities which may be helpful: STANDARD 1

A. Write the standard, word and A. Have one student say a number,  Teacher-made or commercial tests of A. Chalkboard expanded form of whole numbers to or write it on the board, and skills identified by the standards. B. Mathematics textbook, hundred billions or decimals through challenge another student to  Mathematics homework. multiplication and division thousandths and identify the place write it in expanded notation. If  Mathematics quizzes. flashcards. value of a digit. correct, that student challenges  Non-routine problems, including C. Cuisennaire rods. another student, and so on. If explanation of solution. D. Cuisennaire rods, counting chips, B. Compare and order whole numbers incorrect, the teacher can  Mathematics writing journals or parent-baked food (brownies, (through hundred billions) and remedial. notebooks. cupcakes), counting animals, ¼”- decimals (through thousandths).  Oral presentations. ruled graph paper. Compare proper fractions with like B. Use Cuisennaire rods or  Use of manipulatives or pictorial E. 1-100 number chart. and unlike denominators. counting chips to demonstrate representations to explain skills and F. Student number lines from –20 to the meaning of fractions by concepts defined by standard. +20, classroom demonstration  C. Use and create models to represent placing a given number of Portfolios. thermometers. fractions, mixed numbers and  PSSA manipulatives over other G. Chalkboard, index cards. decimals through hundredths. amounts (illustrating numerator D. Explain the concepts of prime and and denominator). composite numbers. Use brownies, pizzas, cupcakes E. Use simple concepts of negative or small identical toys (counting numbers such as on a number line, in bears) to illustrate parts of a counting, and temperature. whole and parts of a group.

F. Develop and apply number theory Use graph paper to illustrate concepts (e.g. primes, factors, decimals as parts of ten, parts of multiples, and composites) to 100, and parts of 1,000. represent numbers in various ways. C. Use a Sieve of Erastothenes to find number patterns, crossing out all numbers that are evenly divisible by other numbers. Using a 1-100 7 LEARNING OUTCOMES/STANDARDS AND CONTENT/ACTIVITIES Statements of student learning expectations achieved through suggested teaching-learning activities and selected content to help reach standards and graduation requirements.

number chart, cross out all multiples of 2, then 3, then 4, etc. The crossed out numbers are composite numbers, and those not crossed out are prime numbers. Encourage students to discover that the prime numbers are plentiful in the low range (2, 3, 5, 7, 11, 13, 17, 19), and become more sparse as the grid reaches 100.

D. Distribute to all students a number line from –20 to +20, and have them calculate simple addition and subtraction problems by moving their fingers along the number line. For example, -10 + 15, the student starts at –10 and physically moves/counts 15 toward the positive numbers to arrive at +5.

Use student quizzes and test papers with extra credit items to have students calculate their final grade, counting the points for wrong items as negative numbers and extra credit as positive numbers.

Use both Celsius and Fahrenheit classroom demonstration thermometers to show temperatures rising and falling through 0°.

E. Use prime factorization to make number trees for composite numbers: 24 ^ 8 LEARNING OUTCOMES/STANDARDS AND CONTENT/ACTIVITIES Statements of student learning expectations achieved through suggested teaching-learning activities and selected content to help reach standards and graduation requirements.

6 x 4 ^ ^ 2 x 3 x 2 x 2

Students rewrite small whole numbers (<32) as the sums of binary multiples (1, 2, 4, 8, 16). For example, 13 = 1 + 4 + 8).

Students can make a set of “magic mind reader cards” to “guess” another student’s selected secret number from the cards. The five cards have the binary multiples in the upper left corner and all of the numbers that use that multiple on the card. The “magician” asks if the chosen secret number is on each card, mentally adding the upper left amount for every “yes” response. For example, if 13 were the selected secret number, it would appear on the 1, 4 and 8 cards, but not on the 2 or 16 card.

Correctives:  Students are given index cards with numbers to compare and order from least to greatest.

Extensions:  Students research statistical facts using an Almanac to compare and order numbers (ex: populations of U. S. states from largest to smallest).

The learner will use computation and estimation to solve theoretical and practical problems.

Mathematics 2.1 2.2 X 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11

Economics 6.1 6.2 6.3 6.4 6.5

Geography 7.1 7.2 7.3 7.4

History 8.1 8.2 8.3 8.4

Health, Safety & PE 10.1 10.2 10.3 10.4 10.5

World Language 12.1 12.2 12.3 12.4 12.5 12.6

Career Education & Work 13.1 X 13.2 X 13.3 13.4

ESSENTIAL CONTENT CONTENT & INSTRUCTIONAL ACTUAL LEVEL OF RESOURCES AND MATERIALS OUTCOMES/STANDARD ACTIVITIES/STRATEGIES WITH ATTAINMENT (EVALUATION Upon completion of teaching-learning activities, CORRECTIVES AND CRITERIA) ASSESSMENT students will be expected at minimum to: EXTENSIONS (Individually created teaching activities may be used to achieve the Upon completion of teaching-learning activities, standards; however, listed below are activities students will be expected at minimum to which may be helpful: complete: STANDARD 2

A. Create and solve word problems A. Instruct on the five-step  Teacher-made or commercial tests of A. Flashcards in all operations; involving addition, subtraction, problem-solving process: skills identified by the standards. chalkboard. multiplication and division of whole  Understand  Mathematics homework. B. Baseball cards; models or numbers and decimals.  Plan  Mathematics quizzes. drawings of pizza; chalkboard.  Try  Non-routine problems, including C. Student number lines from 5 B. Develop and apply algorithms to explanation of solution.  Check through 24, and from 0.05 solve word problems that involve  Mathematics writing journals or  Extend through 0.24; rulers in increments addition, subtraction, and/or notebooks. of 1/16”; overhead number lines; multiplication with decimals with and Use number sentences from each  Oral presentations. operation to demonstrate overhead rulers; overhead without regrouping fractions and  Use of manipulatives or pictorial projector; chalkboard. mixed numbers that include like and conversion to a word problem, in representations to explain skills and which each number is a fact and D. Classroom calculators; overhead unlike denominators. concepts defined by standard. calculator; overhead projector. the question dictates the  Portfolios. E. Fraction calculators (TI Math C. Round numbers and decimals. operation: 3 x 4 = 12 “You  PSSA have three shelves. Each shelf Explorer); overhead fraction calculator; overhead projector. D. Determine through estimations the has 4 toys. How many toys do reasonableness of answers to you have in all?” Instruct that F. Newspapers; calendars; problems involving addition, answers should take the question manipulative money; gift catalogs; subtraction, multiplication and as a starting point and be chalkboard. division of whole numbers and complete sentences: “I have 12 G. Multiplication and division decimals. toys in all.” workmats (which can be made from 11x17 sheets of paper); small E. Demonstrate skills for using fraction B. After instruction in basic skills of manipulatives (buttons, counters, calculators to verify conjectures, adding, subtraction, and tiles); chalkboard. H. Chalkboard. confirm computations, and explore multiplication, with fractions and complex problem solving situations. mixed numbers, provide real-life examples as sources for word F. Apply estimation strategies to a problems. variety of problems including time and money.

G. Explain multiplication and division For fractions, students can 12 LEARNING OUTCOMES/STANDARDS AND CONTENT/ACTIVITIES Statements of student learning expectations achieved through suggested teaching-learning activities and selected content to help reach standards and graduation requirements.

algorithms. calculate fractional slices of pizza multiplied by the number of H. Select a method for computation and tables in a restaurant with slices explain why it is appropriate. on them (as at a party): ¾ x 8 = 6 becomes “Roma Pizza has 8 tables. Customers have left 3/4 of a pizza on each table. How much pizza is on all 8 tables?”

Have students record vocabulary words useful to each operation. Students list words such as all, and, both, in all, sum, and total for addition. Difference, fewer, take away, gave away, how many fewer, how much shorter for subtraction. Total, times as many, times more for multiplication. Every, each, half as many, a third as many, how many times, per for division.

C. To round whole numbers, have students make a number line from 5 through 24. Round each whole number to the nearest 10, demonstrating that 5-14 will round to 10 while 14-24 will round to 20. Repeat as needed for greater place values.

To round decimals, review rounding whole numbers. Review the steps: Identify the digit to be rounded, underline it, circle the digit to its right, and compare that digit to 5 (0-4 add nothing to digit being rounded; 5-9 add 1 to digit being rounded). Reapply these same steps to a decimal number,

identifying the place value to which you are rounding. Ensure that students drop trailing 0s to the right of the digit being rounded.

Have students create a number line from 0.05 to 0.24 and round to the nearest tenth (0.05 through 0.14 will round to 0.1, while 0.15 through 0.24 will round to 0.2.

To round fractions, students can use a standard measure ruler with 1/16ths and explore how each fraction rounds to the nearest whole inch.

D. Students may use calculators to explore the accuracy of random guesses for all operations. Then instruct on rounding to estimate, with students verifying accuracy with calculators. Have students use statistics such as number of lunches sold in the cafeteria, number of books checked out from the library each day of the week, number of absent students each Monday in a month. For addition and subtraction, both numbers round to the greatest place value of the smaller number. For multiplication, students round each number to its greatest place value. For division, demonstrate compatible numbers in which the dividend and divisor are modified so that the dividend is

a multiple of the divisor.

E. After instruction in fractions and mixed numbers, demonstrate features of a fraction calculator using an overhead calculator and at least one fraction calculator for each pair of students. Have students measure five items in the classroom, listing them vertically. After rounding to estimate and using pencil and paper to compute, students use fraction calculator to verify their answers.

F. Use nightly television schedules and weekend movie schedules from newspapers to estimate elapsed time of programs and movies. Have students estimate the number of full weeks in a year and explore why the actual number of days is slightly longer (by 1 day).

Have students use manipulative money to “buy” items in a classroom store based on predetermined shopping lists. Before shopping students use estimation to determine if they have enough money.

Use gift catalogs and a fictitious gift budget to have students first estimate the cost of gifts for their family and then compute the actual amount.

G. Use paper workmats to display

arrays of small manipulatives (buttons, counters, tiles). Students record the number of columns, or number of items in each row, at the top of the paper. They record the number of rows, or number of items in each column, on the left side of the paper. They count the items in the columns and rows to demonstrate multiplication. To demonstrate division, the workmat locates the dividend, divisor and quotient in their correct positions in the abstract problem. 4 2  

Students can see that multiplication is repeated addition of sets of numbers, while division is repeated subtraction of a given amount from a large number.

Have students write solutions to word problems in which they explain arrays and sets being composed and broken down, as in “Sam has nine statues of mathematicians on each shelf in his room. He counts six shelves. Explain how Sam can find the total number of statues in his room.”

“Sally has 56 pictures of Albert Einstein that she wants to put on the four walls of her room.

Explain how Sally can best decide how many pictures should go on each wall.”

H. For a given word problem, demonstrate the four operations that could be applied to the numbers in the one-step problem. Have students determine the reasonableness of each answer, as in “Karen Uhlenbeck has 35 books on algebra. She puts them on 7 shelves as equally as possible. How many will be on each shelf?”

Answers using the four operations would be 42 (adding), 28 (subtracting), 245 (multiplying), and the correct answer, 5. Have students write explanations of how the first three choices could not be possible, referring back to the problem as needed.

Correctives:  Have students work in pairs. Each student writes 2,157 + 643 + 4,922 in vertical format on a place-value chart. Have one student underline the greatest number and the other student underline the least number. Have them both circle the greatest place of the underlined numbers and round each number to that place. Then have them rewrite the problem using the rounded numbers and find the estimated sum.

Extensions:  Students will estimate the number of blades of grass on a soccer or football field by counting the blades of grass on a small sample area (using a cardboard "window" tossed randomly on the field) and then multiplying the area by the appropriate numbers.

The learner will measure, calculate, and estimate using units of length, area, volume, mass, time, and temperature.

Mathematics 2.1 2.2 2.3 X 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11

Economics 6.1 6.2 6.3 6.4 6.5

Geography 7.1 7.2 7.3 7.4

History 8.1 8.2 8.3 8.4

Health, Safety & PE 10.1 10.2 10.3 10.4 10.5

World Language 12.1 12.2 12.3 12.4 12.5 12.6

ESSENTIAL CONTENT CONTENT & INSTRUCTIONAL ACTUAL LEVEL OF RESOURCES AND MATERIALS OUTCOMES/STANDARD ACTIVITIES/STRATEGIES WITH ATTAINMENT (EVALUATION CORRECTIVES AND CRITERIA) ASSESSMENT Upon completion of teaching-learning activities, EXTENSIONS (Individually created students will be expected at minimum to: teaching activities may be used to achieve the Upon completion of teaching-learning activities, standards; however, listed below are activities students will be expected at minimum to which may be helpful: complete: STANDARD 3

A. Select and use appropriate A. Use a measuring tape (in  Teacher-made or commercial tests of A. Measuring tapes in metric and instruments and units for measuring customary and metric units) to skills identified by the standards. customary units; yardsticks and quantities such as perimeter, volume, measure classroom items, such  Mathematics homework. meter sticks; balance scales; area, weight, time, and temperature. as the distance around desk tops,  Mathematics quizzes. Judy clocks; biography books, and other flat, rectangular  Non-routine problems, including encyclopedias (Webster’s New B. Select and use standard tools to objects, for perimeter. explanation of solution. Biographical Dictionary, measure the size of figures with  Mathematics writing journals or R920.02 in the library), specified accuracy, including length, Use measuring sticks (in notebooks. chalkboard. width, perimeter, and area. standard and metric units) to  Oral presentations. B. Yardsticks, meter sticks, measure the volume of  Use of manipulatives or pictorial chalkboard, chart paper. C. Estimate, refine, and verify specified classroom items such as file representations to explain skills and C. Yardsticks, meter sticks, measurements of objects. cabinets, boxes, and the desk concepts defined by standard. chalkboard, chart paper.  space within a student’s empty Portfolios. D. Rulers in metric and customary D. Convert linear measurements,  PSSA desk. units, yardsticks, meter sticks, capacity and weight (mass) within the chalkboard, chart paper. same system within the same system. Use measuring sticks and tapes E. Measuring tapes in metric and E. Add and subtract linear measurements (in customary and metric units) customary units; Guinness Book or units of time. to measure and calculate the of World Records (R031 in the area of desktops, chalkboard, school library), chalkboard. bulletin board, countertops.

Estimate the weight of small classroom items and use balance scales to measure the weight. Compare estimates and actual weights.

Create word problems based on daily situations such as movie schedules, waiting for the bus, etc. Use Judy clocks

(manipulative clocks) to demonstrate passage of time and calculate the elapsed time in the word problems.

Use the birth and death dates of famous people to calculate the years and months of their lives.

B. Use yard and meter sticks to measure objects around the room to the nearest inch, ½”, ¼” and 1/8” and to the nearest cm. Calculate lengths, widths, perimeters, and areas of objects.

C. The class estimates the height of students. First they estimate and record their estimates however they express them (often students are unreasonably exact in their estimates). Then use students who have already been measured as known quantities to use in arriving at second, more reasonable estimates. Finally, use yard and meter sticks to measure the height of students and compare estimates and actual measurements.

D. Use rulers, yard and meter sticks to measure items in the classroom. Change inches to feet and inches. Change short feet and inch measurements (under 5 feet) to inches. Repeat for metric units from centimeters to meter and centimeters; short meter and centimeter measurements to centimeters.

E. Add students’ individual heights in feet and inches to find the “total height” of groups of 3 to 4 students. Compare groups’ total heights. Subtract students’ heights to find differences between two students.

Use the Guinness Book of World Records (R031 in PVI’s library) to compare, through addition and subtraction, students’ heights to known world records for women and men. Students can work to find a combination of students’ heights that comes closest to the records.

Correctives:  Distribute square centimeter paper to students. Staying on the grid lines, students draw four or five shapes. Students exchange papers and count the squares to find the area of each shape.

Extensions:  Students decide on a common time to check the temperature daily at home. Record the results for one week. Make a line graph to show the results. Students share graphs and results with class. Compare and contrast temperature in different microclimates.

The learner will apply mathematical reasoning and logic in practical and theoretical situations.

Mathematics 2.1 2.2 2.3 2.4 X 2.5 2.6 2.7 2.8 2.9 2.10 2.11

Economics 6.1 6.2 6.3 6.4 6.5

Geography 7.1 7.2 7.3 7.4

History 8.1 8.2 8.3 8.4

Health, Safety & PE 10.1 10.2 10.3 10.4 10.5

World Language 12.1 12.2 12.3 12.4 12.5 12.6

 Teacher-made or commercial tests of A. Index cards or 2” x 3” flashcard A. Distinguish between relevant and D. Use word problems to identify skills identified by the standards. blanks. irrelevant information in a and cross out information that  Mathematics homework. B. and C. String or yarn (30’ per mathematical problem. does not help in solving the  Mathematics quizzes. class); chalkboard. problem. Use newspaper articles  Non-routine problems, including D. Newspapers or Internet news B. Interpret statements made with to construct simple one-step explanation of solution. articles. precise language of logic (i.e. all, word problems, including  Mathematics writing journals or E. Dr. Seuss books or Edward Lear every, none, some, or many). extraneous information. Have notebooks. poetry books from the library; students write word problems  Oral presentations. overhead projector and film; based on their own activities,  Use of manipulatives or pictorial chalkboard. including extraneous representations to explain skills and F. Logbook; working outdoor information. Students swap with concepts defined by standard. thermometer; graph paper.  classmates to identify the Portfolios.  PSSA irrelevant information and solve the problem.

E. Define vocabulary of logic: all, every, none, some, many, always, never. Read nonsense verse such as Edward Lear or Dr. Seuss to illustrate a ridiculous situation or conclusion. Make exaggerated statements concerning students, school life, etc. and have student rewrite the sentences to use logical vocabulary to make the sentences true. Continue, moving from practical situations to mathematical theoretical situations, such as “All prime

numbers are odd,” which is not true (2 is prime), or “All right triangles are scalene,” which is also not true, since a right isosceles triangle has two equal sides. Students construct their own logic statements in small groups and test them on the board or overhead.

Correctives:  Given index cards with various multiplication facts, students will group them according to their products (ex: 5 x 3 = 15, 3 x 5 = 15).

Extensions:  Have students create word problems with additional information, then swap with neighbor to identify extraneous information.

The learner will use and explain mathematical problem solving in theoretical and practical situations.

Mathematics 2.1 2.2 2.3 2.4 2.5 X 2.6 2.7 2.8 2.9 2.10 2.11

Economics 6.1 6.2 6.3 6.4 6.5

Geography 7.1 7.2 7.3 7.4

History 8.1 8.2 8.3 8.4

Health, Safety & PE 10.1 10.2 10.3 10.4 10.5

World Language 12.1 12.2 12.3 12.4 12.5 12.6

A. Develop a plan to analyze a problem, A. Use sample open-ended  Teacher-made or commercial tests of A. PSSA open-ended questions as identify the information needed to questions released from the skills identified by the standards. released from the PDE. solve the problem, carry out the plan, PSSA assessment each year. Use  Mathematics homework. B. Graph paper; overhead projector; check whether an answer makes sense the five-step problem-solving  Mathematics quizzes. overhead graph film. and explain how the problem was plan (understand, plan, try,  Non-routine problems, including C. Graph paper; large circle solved. check, extend) to solve these explanation of solution. templates for pie charts; overhead questions.  Mathematics writing journals or projector; chalkboard; small B. Use appropriate mathematical terms, notebooks. manipulatives with two attributes vocabulary, language symbols and B. Create a multiple-line graph  Oral presentations. (to express numerator and graphs to clearly and logically explain from a table that shows average  Use of manipulatives or pictorial denominator), such as isomorphic solutions to problems. attendance at four Major League representations to explain skills and animal counters in different Baseball stadiums for the years concepts defined by standard. colors.  C. Show ideas in a variety of ways, 1994, 1995, 1996. Include such Portfolios. D. Chart paper; graph paper; meter including words, numbers, symbols,  PSSA items as vertical and horizontal sticks and yardsticks; chalkboard. pictures, charts, graphs, tables, axes, title, and legend. Select E. PSSA open-ended questions as diagrams and models. colors or symbols to show released by the PDE. D. Connect, extend, and generalize respective stadiums. Have the F. Chalkboard or overhead projector. problem solutions to other concepts, students compare the data from problems and circumstances in the multiple-line graph and mathematics. make up three questions that they could ask other students. E. Select, use, and justify the methods, “What is the difference between materials and strategies used to solve the stadium with the highest problems. attendance versus the lowest?”

F. Use appropriate problem solving C. Use the same simple data to have strategies such as solving a simpler students construct pictographs, problem or drawing a picture or pie, line, and bar graphs. Rewrite Age in Years Boys Girls diagram. the data in table form. 7 1.45 1.35 8 1.3 1.29 9 1.33 1.24 28 10 1.28 1.18 11 1.23 1.13 12 1.19 1.08 LEARNING OUTCOMES/STANDARDS AND CONTENT/ACTIVITIES Statements of student learning expectations achieved through suggested teaching-learning activities and selected content to help reach standards and graduation requirements.

Have students construct a model to explain addition of fractions with like denominators. The model should show each step without words.

D. Allow students to use a statistical chart to predict future outcomes. Students can use a statistical table to predict the adult height a child will attain. Students can work in pairs and measure each others’ heights in cm. Record results and multiply their height by the figure from the chart:

For example, if Jane is 122 cm tall and 8 years old, she multiples 122 x 1.29 and gets 157.38 cm.

E. Use sample open-ended questions released from PSSA assessment.

F. Solve this problem by drawing a diagram: Using red, green, blue and yellow blocks only once in each row, how many four-block patterns can you make that have a red block in the first position? [Correct answer is 6].

Correctives:  Each student is given an ad from a local grocery store and a $50 amount of play money to spend. Students need to make a list of items that could be purchased with the $50.

Extensions:  Have students determine the amount and cost of carpeting needed for their classroom. They will need to determine area and total cost, using local carpet company prices from advertisements.

The learner will collect, organize, analyze, interpret, and make mathematical inferences from data.

Mathematics 2.1 2.2 2.3 2.4 2.5 2.6 X 2.7 2.8 2.9 2.10 2.11

Economics 6.1 6.2 6.3 6.4 6.5

Geography 7.1 7.2 7.3 7.4

History 8.1 8.2 8.3 8.4

Health, Safety & PE 10.1 10.2 10.3 10.4 10.5

World Language 12.1 12.2 12.3 12.4 12.5 12.6

ESSENTIAL CONTENT CONTENT & INSTRUCTIONAL ACTUAL LEVEL OF RESOURCES AND MATERIALS OUTCOMES/STANDARD ACTIVITIES/STRATEGIES WITH ATTAINMENT (EVALUATION CORRECTIVES AND CRITERIA) ASSESSMENT Upon completion of teaching-learning activities, EXTENSIONS (Individually created students will be expected at minimum to: teaching activities may be used to achieve the Upon completion of teaching-learning standards; however, listed below are activities activities, students will be expected at minimum which may be helpful: to complete: STANDARD 6

A. Organize and display data using A. On grid paper label 12 columns  Teacher-made or commercial tests A. Graph paper; chart paper. pictures, tallies, tables, charts, bar with the twelve months of the of skills identified by the standards. B. Unifix cubes or similar counting graphs, and circle graphs. year. Name each month and have  Mathematics homework. cubes. students draw a candle on the grit  Mathematics quizzes. C. Plain paper for Venn diagram. B. Describe data sets using mean, when their birth month is named.  Non-routine problems, including D. Dice (3 per pair of students); median, mode, and range. Tally birthdays by months. Make explanation of solution. graph paper, chalkboard or tables, charts, bar graphs and  Mathematics writing journals or overhead projector. C. Sort data using Venn diagrams. circle graphs using the data. notebooks. E. Dice (2 per pair of students);  Oral presentations. counting chips (9 per pair of B. Use cubes to build stacks to  Use of manipulatives or pictorial students); plain paper; relevant represent the birthday months representations to explain skills and historical data such as a data. Ensure that students see concepts defined by standard. population, precipitation, or  that 0 can be a possible answer. Portfolios. resources map; line graphs.  PSSA Compare the tallest and shortest stacks. What is the range? (The difference between the tallest and shortest).

Remove and preserve the three “summer month” stacks, leaving 9 stacks, an odd amount. Arrange the stacks in order from shortest to tallest. What is the median? (The number of cubes in the middle stack).

Do any of the stacks have the same number of cubes? (The number that occurs most often is the mode. There may be 0, 1 or

more modes).

Take all 12 stacks and rearrange the cubes to make 12 even stacks. The number of cubes in each stack is the average, or mean.

C. Take a survey of siblings in the class. Make a Venn diagram to show the information. Ensure students understand that the intersection of the two sets shows students having at least one brother and sister.

have at least one brother have at least one sister

Correctives:  Collect student data on pets (how many, what type, number of legs, etc.). Students draw Venn diagrams relating two data sets.

Extensions:  Collect data from classmates (favorite singer, favorite pizza topping, student height, number of siblings, number of states visited, etc.). Draw pictograph, bar graph and line graph to represent data.

The learner will use probability to make predictions.

Mathematics 2.1 2.2 2.3 2.4 2.5 2.6 2.7 X 2.8 2.9 2.10 2.11

Economics 6.1 6.2 6.3 6.4 6.5

Geography 7.1 7.2 7.3 7.4

History 8.1 8.2 8.3 8.4

Health, Safety & PE 10.1 10.2 10.3 10.4 10.5

World Language 12.1 12.2 12.3 12.4 12.5 12.6

A. Perform simulations with concrete A. Students break into small groups  Teacher-made or commercial tests of A. Dice (2 per student group) or a devices (dice, spinner, etc.) to predict to roll dice or spin a spinner 50 skills identified by the standards. spinner (1 per student group), the chance of an event occurring. times. Students record data  Mathematics homework. plain paper. using tally marks. Students  Mathematics quizzes. B. Spinners (1 per pair of students). B. Determine the fairness of the design predict beforehand which color  Non-routine problems, including C. Spinners (1 per pair of students); of a spinner. or number will appear the most. explanation of solution. calculators; overhead calculator;  Mathematics writing journals or overhead projector. C. Express probabilities as fractions and B. Students observe and describe notebooks. D. Four-colored spinners (1 per decimals. the physical appearance of a  Oral presentations. student group); graph paper. spinner. Students notice now  Use of manipulatives or pictorial E. Dice (1 per student); plain paper; D. Compare predictions based on each area is equal in size to representations to explain skills and chalkboard. theoretical probability and ensure fairness. concepts defined by standard. F. Yellow blocks (8 per student  experimental results. Portfolios. group); red blocks (4 per group);  PSSA C. Students use the number of blue blocks (6 per group); green E. Calculate the probability of a simple possible outcomes in a situation blocks (2 per group); paper bags event. (spinner) as the denominator, (1 per group); plain paper; graph F. Determine patterns generated as a and the number of favorable paper; chart paper; chalkboard. result of an experiment. outcomes as the numerator. G. Number dice (2 per pair of Students convert percentages students); plain paper; G. Determine the probability of an event with possible outcomes of 100% chalkboard. involving "and", "or" or "not". to decimals. H. Four-colored spinners (2 per student group); plain paper; H. Predict and determine why some D. Students predict based on graph paper; chalkboard. outcomes are certain, more likely, less possible and favorable outcomes I. Lined paper; chalkboard. likely, equally likely or impossible. for a four-colored spinner. They J. Four-colored spinners (2 per experiment and record results class); chalkboard or overhead I. Find all possible combinations and after a determined number of projector; chart paper. arrangements involving a limited spins. Students compare results number of variables. and predictions.

J. Make a tree diagram and list the 35 LEARNING OUTCOMES/STANDARDS AND CONTENT/ACTIVITIES Statements of student learning expectations achieved through suggested teaching-learning activities and selected content to help reach standards and graduation requirements. elements in the sample space E. Use a six-sided die to calculate the amount of possible outcomes (6). Explain each favorable outcome (1 out of 6) as the numerator of a fraction, as in 1 will appear 1/6 times.

F. Students place 8 yellow, 4 red, 6 blue and 2 green blocks in a bag. Students take turns picking a block from the bag, recording its color, and replacing it. Students determine, using recorded results, which color is most/least likely to be picked.

G. Use 2 number dice. Students work in small groups to find the probability of rolling  2;  an odd number;  3 and 4;  1 or 6;  anything but a 3; or  other variations as needed.

H. Use two four-color spinners. Students label them 1 and 2, then independently spin them and record results. They find the probability of spinning  blue-blue;  1 spins a green;  1 and 2 same color;  no yellow; or  other variations as needed.

I. Arrange two students at the front of the room. Have the class determine all the ways they can stand in a line. Then arrange three students at the

front of the room. Have students rearrange the students in as many combinations as possible, recording answers. Have them arrange them with one student stationary. Repeat for four students. Demonstrate making an organized list of permutations with limited variables.

J. This is a teacher-directed activity. Take two four-color spinners to create a tree diagram. Begin by listing all the colors possible on the first spinner. Next to each color from the first spinner, list all the colors possible on the second spinner. Combine the first spin with the second spin to form all outcomes (16 total outcomes).

Correctives:  Students work with partners using a four-color spinner to make predictions and find probability. List all the possible outcomes.

Extensions:  Students track the weather for one week, or access the National Weather Service database for a given time period. Then students make predictions for the upcoming week or month.

The learner will use algebra and functions to solve theoretical problems.

Mathematics 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 X 2.9 2.10 2.11

Economics 6.1 6.2 6.3 6.4 6.5

Geography 7.1 7.2 7.3 7.4

History 8.1 8.2 8.3 8.4

Health, Safety & PE 10.1 10.2 10.3 10.4 10.5

World Language 12.1 12.2 12.3 12.4 12.5 12.6

ESSENTIAL CONTENT CONTENT & INSTRUCTIONAL ACTUAL LEVEL OF RESOURCES AND MATERIALS OUTCOMES/STANDARD ACTIVITIES/STRATEGIES WITH ATTAINMENT (EVALUATION CORRECTIVES AND CRITERIA) ASSESSMENT Upon completion of teaching-learning activities, EXTENSIONS (Individually created students will be expected at minimum to: teaching activities may be used to achieve the Upon completion of teaching-learning activities, standards; however, listed below are activities students will be expected at minimum to which may be helpful: complete: STANDARD 8 A. Recognize, reproduce, extend, create A. Students copy a pattern block  Teacher-made or commercial tests of A. Critical Thinking Activities (Dale and describe patterns, sequences and array from the overhead, then skills identified by the standards. Seymour Publications, Palo Alto, relationships verbally, numerically, extend it to predict the pattern  Mathematics homework. CA, 1988); pattern blocks; overhead symbolically and graphically, using a created by the blocks. Challenge  Mathematics quizzes. pattern blocks; chalkboard or variety of materials. students to complete patterns  Non-routine problems, including overhead projector. that rely on words, numbers, and explanation of solution. B. Fraction circles; counting tiles; B. Connect patterns to geometric symbols:  Mathematics writing journals or Measuring Up to the relations and basic number skills.  J A S O N (July, August, notebooks. Pennsylvania Academic September, October,  Oral presentations. Standards, Level E (The Peoples C. Form rules based on patterns (e.g., an November);  Use of manipulatives or pictorial Publishing Group, Inc., Saddle equation that relates pairs in a representations to explain skills and  O, T, T, F, F, S (One, Two, Three, Brook, NJ, 2001); drawing sequence). concepts defined by standard. etc.); paper; chalkboard or overhead   1, 1, 2, 3, 5, 8 (the Fibonacci Portfolios. D. Use concrete objects and  PSSA projector. number series in which the next combinations of symbols and C. Measuring Up to the number is the sum of the numbers to create expressions that Pennsylvania Academic model mathematical situations. previous two); Standards, Level E; pattern  60, 60, 24; 7 (seconds in a blocks; computers with word E. Explain the use of combinations of minute, minutes in an hour, processing programs that symbols and numbers in expressions, hours in a day, etc.); contain symbols; markers and equations, and inequalities.  3, 3, 5, 4, 4, 3, 5 (the number of crayons; lined paper. letters in the words one, two, D. Counting tiles; index cards; lined F. Describe a realistic situation using three, etc.); paper. information given in equations,                 . E. Chalkboard; drawing paper; inequalities, tables or graphs. markers or crayons; lined paper; Students can solve abstract word cover stock for booklets. G. Select and use appropriate strategies, problems based on described F. Chalkboard or overhead including concrete materials, to solve problems: “If you had a color projector. number sentences and explain the pattern of beads arranged white, G. Counting chips, counting tiles; method of solution. white, blue, blue, red, white, lined paper; chalkboard; H. Locate and identify points on a white, blue, blue, what would be overhead projector. coordinate system. the next color?” H. Graph paper; counting chips or

I. Generate functions from tables of data other small manipulatives; to corresponding graphs and Students can complete roadmaps. functions. worksheets from Critical I. Measuring Up to the Thinking Activities that Pennsylvania Academic gradually increase in difficulty Standards, Level E; Sentence and include graphic, symbolic strips with labels for graph parts; numeric and verbal patterns. chalkboard.

B. In fraction instruction, demonstrate addition of fractions with like denominators and encourage students to continue a visual pattern of the addition using color tiles or fraction circles.

Provide shapes that progress from a single line segment to a complex figure, as in the following: Students create the next shape in the progression.

To discover patterns in basic number operations, students solve Cloze passages in which the basic operation is easily determined through guess and check:  1, 7, 13, 19, 25, 31… (start at 1 and add 6 each time);  0, 9, 18, 27, 36, 45 … (multiplication of 9);  100, 89, 78, 67, 56 … (subtract 11 each time);  100,000; 10,000; 1,000; 100 … (divide by 10 each time).

C. Provide patterns of numbers or symbols and have students write a description of the pattern. Use

Measuring Up to the Pennsylvania Academic Standards to practice this skill in a realistic PSSA-type setting.

Have students develop patterns of numbers, symbols or pattern blocks and have other students determine the pattern. This can be done competitively or cooperatively. These may be done freehand or on a computer using a word processing program that allows insertion of symbols (as with Microsoft Word).

In art class, have students rotate block print designs to create a pattern based on a logical progression (90° turns at each step, for example).

C. Students rewrite flash cards to contain an algebraic unknown within the sentence: 2 x 8 =  becomes 2 x  = 16.

Have students develop fact family cards in which a number is replaced with a picture or symbol. Other students must guess the meaning of the symbol. 2 x . = 16; . x 2 = 16; 16  . = 2; 16  2 = ..

Use counting tiles to illustrate simple algebraic equations, such as 17 + x = 21.

Write a realistic word problem 42 LEARNING OUTCOMES/STANDARDS AND CONTENT/ACTIVITIES Statements of student learning expectations achieved through suggested teaching-learning activities and selected content to help reach standards and graduation requirements.

based upon an existing algebraic equation. For 48  N = 6, students could write, “I have 48 cupcakes for a bunch of friends. If each friend gets 6 cupcakes, how many friends did I feed?” Students work in groups to write word problems for other equations, then compare and discuss the various possible situations for each equation.

D. Students invent symbols to stand for a particular function, such as “add 1 and multiply the sum by 3”. Students then invent symbols to stand for comparatives, such as “greater than or equal to,” or “not equal.” After they construct their own symbols, instruct on the correct symbols for these inequalities.

Use Cloze passage number sentences in a chalkboard game in which students must insert the correct symbol of inequality. A student from one half of the class writes a number sentence on the board, such as “8 x 4  3 x 2 x 4” and the other team’s player must insert < to score a point for their team.

Students work in groups to write a description of a particular symbol, such as > or <, so that a student in a lower grade could comprehend its use. The various groups assemble the descriptions into a Dictionary of 43 LEARNING OUTCOMES/STANDARDS AND CONTENT/ACTIVITIES Statements of student learning expectations achieved through suggested teaching-learning activities and selected content to help reach standards and graduation requirements.

Mathematical Symbols, which is reproduced and distributed to the students in the class to use as a resource.

Have students write an advertisement explaining why their product, on sale at 2 packages of 12, is a better value than an equally priced competitor’s product, on sale at 4 packages of 5. Replace the units with similar comparisons (5 for $1.00 versus 8 for $2.00; Buy 3 get 1 free versus Buy 5 get 2 free) and repeat the activity.

E. Students compete in teams to match equations on board to descriptions of situations, such as “Sally had 5 tomato plants, and each one grew the same number of tomatoes. She harvested 35 tomatoes; how many did each tomato plant grow?” Choices on the board could include 5 + 35 = X; 5 x X = 35; 35 – X = 5.

Students work in teams to develop situations matching an inequality provided to the whole class, such as 2 x 12 - 1  5 x 4 + 3. An accurate response might be, “Two boys each had 12 highlighters, but 1 didn’t work. In the same class, five girls each had 4 highlighters, but one girl had 3 more. Which group had the most highlighters?” An incorrect response might be,

“Two boys each had 12 frogs but each lost a frog. Five girls each had 4 frogs and each girl bought 3 other frogs. How many frogs did the girls have all together?” Read and discuss all solutions and clarify misunderstandings, based on identifying the comparative, not a final answer.

F. Demonstrate pattern division and pattern multiplication as a way to make simpler problems. Students can work in teams to identify the basic fact needed to solve algebraic word problems, such as “Each player on the soccer team runs backwards 50 yards to warm up before practice; if the team ran 400 yards backwards in practice today, how many players were at practice?” Students must find the algebraic sentence, 50 x N = 400, then find the basic fact, 5 x N = 40, to solve.

Have students use counting chips or tiles as models for algebraic problems, such as “Fred wants to give his 36 trading cards to his 9 friends. How many cards does each friend receive?” Students should array the counting chips to show 36 chips in sets of 9; the number of sets identifies the number of cards each child receives.

Students play Judge Al (as in Algebra), and must write a

decision in the case of Kang versus Aroo. Kang and Aroo make didjeridus in equal amounts. For every didjeridus Kang produces, he gets $2.00. For every one Aroo produces, he gets $3.00. Their combined pay this week was $60. Kang says he should get $26 this week. Aroo says he’s wrong, that Kang should only get $18. The judge (each student) must write a fair and correct decision and determine how much each worker gets, using math to support the decision. The correct answer is, ($2.00 + $3.00) x N = $60.00; N = 12; Kang made 12 didjeridus and should get $24; Aroo made 12 and should get $36.

G. Have students make and play Battleship, using coordinate grid points.

Distribute and use roadmaps to identify historic sites, cities and parks by coordinate grid points.

Students make a schematic map of their bedrooms and make an index using coordinate points to find furniture.

In Social Studies, students can use coordinate points on battlefield maps or National Park maps.

H. Use Measuring Up to the

Pennsylvania Academic Standards, Level E to practice PSSA-type responses to multiple-choice questions.

Play a cooperative game in which a chalkboard bar or line graph lacks a title and labels for its axes. Have students place sentence strips on the appropriate parts. Then have students answer questions about the content of the graph, including extrapolating trends in the data.

Correctives:  Use pattern puzzles with limited (2- 5) variables. For example:

______, ______, ______. Or 1, 3, 5, 7, ____, 11, 13, ____, ____

 Make a set of cards with descriptions matching equations, then play concentration in small group. Example: One card says 2 + 5 = 7, another card says "Two puppies plus five puppies makes ___ puppies in all."

Extensions:  Students generate 3 repetitive patterns (1 visual, 1 word, and 1 number) and challenge classmates to add to the pattern. Students swap and keep swapping, correcting others' work if they think it is wrong. 47 LEARNING OUTCOMES/STANDARDS AND CONTENT/ACTIVITIES Statements of student learning expectations achieved through suggested teaching-learning activities and selected content to help reach standards and graduation requirements.

 Students draw cards from a deck so that Jack = add, Queen = multiply, King = subtract. They use the number cards to create equations such as A J 5 =? [6]

The learner will identify and classify geometric figures.

Mathematics 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 X 2.10 2.11

Economics 6.1 6.2 6.3 6.4 6.5

Geography 7.1 7.2 7.3 7.4

History 8.1 8.2 8.3 8.4

Health, Safety & PE 10.1 10.2 10.3 10.4 10.5

World Language 12.1 12.2 12.3 12.4 12.5 12.6

A. Give formal definitions of geometric A. Each student makes a Geometry  Teacher-made or commercial tests of A. Lined paper, cover stock, stapler. figures. Dictionary. As terms are skills identified by the standards. B. Geoboards (1 per student pair); introduced, students write the  Mathematics homework. rubber bands. B. Classify and compare triangles and word, the definition, and the  Mathematics quizzes. C. Chalkboard compass; string; quadrilaterals according to sides or geometric figure on a one sheet  Non-routine problems, including rulers. angles. of paper. At the end of the unit, explanation of solution. D. Pattern blocks; drawing paper; file place the words in alphabetical  Mathematics writing journals or folders used as screens. C. Identify and measure circles, their order and bind the sheets to notebooks. E. Pattern blocks; drawing paper; diameters and radii. form a booklet.  Oral presentations. solids patterns; tape; markers or  Use of manipulatives or pictorial crayons. D. Describe in words how geometric B. Working in pairs, students form representations to explain skills and F. Magazines or newspapers; digital shapes are constructed. the different triangles and concepts defined by standard. camera; computer printer.  quadrilaterals on geoboards Portfolios. G. Pattern blocks; oaktag; drawing E. Construct two and three dimensional  PSSA when given clues describing paper; scissors. shapes and figures using them. Example: “Using your H. Graph paper; chalkboard or manipulatives, geoboards and computer software. geoboard, make a triangle that overhead projector. has one obtuse angle.” Students I. Rulers; chalkboard. F. Find familiar solids in the make the figure, then draw it in J. Index cards with geometric figures environment and describe them. their notebook and label it on them. “obtuse triangle,” with a written K. Overhead geoboard; rubber G. Describe the relationship between the description. bands; pattern blocks; drawing perimeter and area of triangles, For teacher information: paper. quadrilaterals and circles.  Triangles identified by L. Index cards prepared with angles: acute (3 acute geometric figures, definitions, and H. Identify, draw and label points, lines, angles); right (1 right angle); names. line segments, rays, and planes. obtuse (1 obtuse angle).  Triangles identified by sides: I. Define the basic properties of squares, scalene (no sides equal); pyramids, parallelograms, isosceles (2 sides equal); quadrilaterals, trapezoids, polygons, equilateral (3 sides equal). rectangles, rhombi, circles, triangles, 50 LEARNING OUTCOMES/STANDARDS AND CONTENT/ACTIVITIES Statements of student learning expectations achieved through suggested teaching-learning activities and selected content to help reach standards and graduation requirements.

cubes, prism, spheres, and cylinders.  Quadrilaterals: trapezoid, parallelogram, rhombus, J. Draw or identify a translation (slide), rectangle and square. reflection (flip), or rotation (turn).. C. Place a large circle on the board and identify the center, K. Identify properties of geometric diameter, radius, chord and figures (i.e., parallel, perpendicular, circumference. Students bring in similar, congruent, symmetrical). cans from home. They locate the center by drawing a dot. They use string to locate the radius, then measure the string with a ruler. They repeat to measure the diameter and discover the diameter is twice the radius. They can also use the string to measure the circumference and find that it is approximately three times the diameter or six times the radius.

D. Each student in a group draws a simple picture made with geometric shapes (or designs one with pattern blocks) and keeps it a secret from the other students. Students take turns describing their pictures using only geometry terms and the other group members try to draw it from the description. When finished, compare the drawings/block designs with the original. Who described the shapes most accurately?

E. For two-dimensional figures, students make Geobots with pattern blocks. Geobots are imaginary robots designed by students using many geometric figures. Students draw their

figures on paper and label the various polygons. As an extension, students can write a paragraph explaining what the Geobot can do.

For three-dimensional shapes, students can use patterns to cut, fold and tape geometric solids (cube, rectangular prism, square pyramid, cylinder, triangular pyramid, etc.) These can be used as classroom or holiday decorations.

F. Place the names of different solids on the bulletin board. Students find pictures in newspapers or magazines of objects in our world that are representative of the solids. Students can also use digital cameras to take pictures around the school and print them using computer technology. Students write riddles for different solids, being sure to give enough information so that classmates can solve the riddle. Example: “I have only two edges and three faces.” [Cylinder]

G. Students use graph paper to make as many different rectangles as they can when given a specific area. Students will note that although the areas are the same, each rectangle has a different perimeter.

Students draw and label the sides of as many rectangles as they can think of when given a specific perimeter. Students will note that although the perimeters are the same, the areas are different when applying the formula A=L x W.

H. Students use two rulers or sticks to represent the concept of intersecting, perpendicular and parallel lines. Draw these representations on the board, exaggerating the arrows on the lines indicating that the lines go on and on, infinitely.

Students name objects that suggest each shape, such as a road (line), pencil point (point), lake surface (plane), laser beam (ray), railroad tracks (parallel lines).

I. Distribute index cards with pictures of each geometric figure to groups of students. Students group pictures according to common properties. Each group shares with the class why they categorized their geometric figures as they did. Groupings of figures may vary amongst the student groups.

J. Teacher or student creates a geometric figure on the overhead geoboard. Using slides, flips or turns, students represent the figure on their individual

geoboards. Students compare, discuss and analyze one another’s representations.

Students create geometric designs by making slides, flips and turns with a figure, such as a polygon created by tracing around several different pattern blocks (creating an irregular shape).

K. The teacher prepares three sets of index cards. On one set, draw geometric figures (one per card). On the second set, write the names of the figures. On the third set, write the definitions and/or properties of each figure. Distribute the cards to the students, one per student. Students locate their partners and form triads—drawing, name of figure, and definition/properties.

Correctives:  Trace pattern blocks into notebook, label each shape.  Play geometry dominoes to match names with shapes.

Extensions:  Make string art designs based on geometric shapes.  Create abstract artwork using geometric shapes. Compare and contrast student work to cubists (Picasso, Braque).

The learner will identify and create triangles by their sides or angles to explore the basic concepts of trigonometry.

Mathematics 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 X 2.11

Economics 6.1 6.2 6.3 6.4 6.5

Geography 7.1 7.2 7.3 7.4

History 8.1 8.2 8.3 8.4

Health, Safety & PE 10.1 10.2 10.3 10.4 10.5

World Language 12.1 12.2 12.3 12.4 12.5 12.6

A. Identify and compare parts of right A. Use Cuisennaire rods or similar  Teacher-made or commercial tests of A. Cuisennaire rods or other sticks; triangles including right angles, acute sticks to model right triangles. skills identified by the standards. counting tiles. angles, hypotenuse, and legs. Place a counting tile in the  Mathematics homework. B. Rubber bands; geoboards; corner that forms the right angle.  Mathematics quizzes. chalkboard or overhead projector. B. Create various triangles on a Draw and label all the parts  Non-routine problems, including geoboard. (legs, hypotenuse, right angle, explanation of solution.  Right acute angles) of a right triangle.  Mathematics writing journals or  Acute Students draw examples and notebooks.  Obtuse label parts in notebooks.  Oral presentations.  Isosceles Students are asked to locate  Use of manipulatives or pictorial  Equilateral parts on an unlabeled drawing representations to explain skills and  Scalene on the board. concepts defined by standard.  Portfolios.  PSSA B. Students use rubber bands to create triangles on geoboards. Students exchange geoboards with a partner to check if the polygon is the specified triangle.

Correctives:  Students work with partners to create right triangles using string/wooden sticks of lengths 3-4- 5 (automatically makes a right triangle).

Extensions:  Students identify as many right triangles as they can at school or at home (describe the part in relation

to the object).

 Students identify as many right triangles as they can in an op art or line drawing such as the one below. They must include the larger triangles created by four, nine and sixteen little triangles.

The learner will use concepts of calculus to compare numbers, explore relationships of unit of measurement and accuracy, and identify maximum and minimum.

Mathematics 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 X

Economics 6.1 6.2 6.3 6.4 6.5

Geography 7.1 7.2 7.3 7.4

History 8.1 8.2 8.3 8.4

Health, Safety & PE 10.1 10.2 10.3 10.4 10.5

World Language 12.1 12.2 12.3 12.4 12.5 12.6

A. Make comparisons of numbers such A. Give an imaginary situation: The  Teacher-made or commercial tests of A. Poster board to create a fictitious as more, less, same, least, most, local volunteer firefighters are skills identified by the standards. sign; chalkboard or overhead greater than, and less than. having a pancake breakfast. The  Mathematics homework. projector; lined paper. B. Identify least and greatest values diners spin a spinner numbered  Mathematics quizzes. B. Circle templates to make circle represented in bar and circle graphs. 1-9 to determine how many  Non-routine problems, including graphs; rulers or yardsticks; C. Identify maximum and minimum. pancakes they get for their $6 explanation of solution. calculators; graph paper; a D. Describe the relationship between ticket. The firefighters sold 537  Mathematics writing journals or computer spreadsheet program rates of change and time. tickets. Students determine the notebooks. capable of creating charts from  Oral presentations. E. Estimate areas and volumes as the maximum and minimum (most data.  Use of manipulatives or pictorial sums of areas of tiles and volumes pancakes required, fewest C. Sets of data, such as the World representations to explain skills and of cubes. required, with explanation) Almanac and Book of Facts (317.3 [4,833 is the most, 537 is the concepts defined by standard. in our library) or Guinness Book F. Describe the relationship between  least, because each diner could Portfolios. of World Records (R031 in our the size of the unit of measurement  PSSA and the estimate of the areas and spin a 9—9 x 537=4,833—or a 1]. library). volumes. D. Measuring Up to the Pennsylvania Imagine a pet shop with X cats Academic Standards, Level E; tree and Y birds. Have students growth chart; student growth describe the number of legs for chart; newspapers; Internet access each type of animal. Students via school computer. note how a change in the E. Counting tiles; centimeter cubes; number of animals changes the drawings; empty containers. relationship of legs. For example, with 5 cats and 10 birds, the numbers of feline and bird legs are equal. With one less cat, the bird legs are greater. With three fewer birds, the feline legs are greater.

In pairs, students write a whole number, a decimal, and a mixed number. Taking turns with the partner, they put down their numbers in order, making an organized list from least to greatest or greatest to least.

B. Organize data into a circle graph. Identify the largest wedge has having the greatest value. Identify the smallest wedge as having the

least value. Organize data into a bar graph. Identify the tallest bar as having the greatest value. Identify the shortest bar as having the least value. Repeat as needed for other formats, such as pictographs, double-bar graphs, 3-d circle graphs, etc.

C. To identify the maximum, find the greatest number in a set of data. Repeat with various types of numbers, such as fractions, decimals, whole numbers, etc. To identify the minimum, find the smallest number in a set of data.

D. Use Measuring Up to the Pennsylvania Academic Standards, Level E to practice PSSA-type questions regarding this skill.

Use a line graph to demonstrate a steady growth over time (such as a tree’s height). Have students plot the two axes, so that for a given time (year, month, etc.) they can identify the quantity associated with it. Then have them compare any two points on the graph to describe the difference between them. Students work in groups to define the reason for the difference (things such as population change over time). Identify rate and ratio in connection with this change.

Use a student growth chart to identify various rates of change, in which a small child grows rapidly while a teen’s growth slows. Students write descriptions of each time period in terms of the rate of growth.

Students look at a growth chart and must identify a particular year or span of years from teacher-provided clues. The clues, such as “the rate of growth from this point was less than an earlier time, but much faster than the last two years’ rate,” encourage the students to look for trends, not specific numeric answers.

Have students track two stocks using the newspaper or Internet. Using the Internet, print an earnings chart for a company as compared to the S&P 500. Students must write a promotional paragraph for the company, identifying and comparing its growth rate against the S&P’s.

E. Use counting tiles to fill, neatly, an irregular plane figure, such as a cartoon drawing of a face, animal, or similar curvilinear shape. Students count the tiles to estimate the area of the figure in square inches. Use centimeter cubes to fill, neatly, an irregular 3-D shape such as a small vase or jar. Students count the cubes

arrayed in rows and columns to estimate the volume in cubic cms.

Correctives:  Students arrange a given set of numbers from least to greatest. They circle the least and the greatest number, and identify these as minimum and maximum, respectively.

 Use large (1") square tiles to fill a large 2-D geometric figure (6" or larger). Repeat the filling with smaller sizes of tiles (cm cubes, for example) to illustrate that the smaller measurement unit increases the accuracy and approaches the limit of the geometric space.

Extensions:  Explore area of irregular curves such as cardoid curves, sine waves and ellipses.

 Students search for and collect three different line graphs from print media. They create questions for classmates to solve about the collected graphs.