a framework for achieving the...

. . ,.,.

Dr. Terry Bergeson State Superintendent of

Public Instruction

A Framework for Achieving the Essential Academic Learning Requirements in

Mathematics Grades 5-7

This framework is designed to assist teachers in planning and implementing the mathematics curriculum. It provides a focus for assessment that emphasizes achievements in mathematics understanding and skills appropriate for students at the middle years. It also serves as a basis for documenting and reporting students' progress in mathematics understanding and skills to colleagues and parents.

Prepared by Martha Hopkins Ph.D. for the Commission on Student Learning

Introduction

Framework

Fifth Grade

Sixth Grade

Seventh Grade

Glossary

Table of Contents

1

3

11

20

29

May 14. 1999 Office of Superintendent of Public hsmction

Introduction

The intent of this framework is to support the Essential Academic Learning Requirements in Mathematics. This framework is designed to assist teachers in planning and implementing the mathematics curriculum. It provides a focus for assessment that emphasizes achievements in mathematics understanding and skills appropriate for students at the middle grade levels. It also serves as a basis for documenting and reporting students' progress in mathematics understanding and skills to colleagues and parents.

Although students begin the school year with varied background expe'riences and knowledge and progress at different rates on different skills, this framework is organized by grade level to indicate goals for learning. The chart for each Suggested Assessment Evidence for'the End of (Fifth, Sixth, Seventh) Grade contains a listing of grade-appropriate evidence of learning, a correlation of that evidence with corresponding links to the'Essential Academic Learning Requirements, and suggestions for methods of gathering evidence in order to assess the level of attainment.

Teachers are reminded that the framework offers suggested goals for each grade level. To assure mathematical achievement, it is important that content and processes introduced at any level continue to be practiced, refined, and extended at subsequent levels.

Problem P osing/frob/em Solving, Mathematical Reasoning, and Communication

Primary goals of the Essential Academic Learning Requirements in Mathematics include posing and solving problems, reasoning mathematically, and communicating knowledge and understanding in everyday and mathematical language. Problem solving, reasoning, and communication shouid be infused as critical components of all mathematics instruction.

While engaged in meaningful mathematical experiences, students define and solve problems, reason logically, and communicate their understanding and knowledge to parents,

with a real world context in which to learn the concepts and mechanics of mathematics. peers, teachers, and themselves. Appropriate problem-solving experiences provide students

Students are encouraged not only bthink, but also to reflect on their thinking. They communicate their understanding through words, pictures, or numbers; develop an appreciation for the fact that there are many ways to solve problems; and develop an understanding of the role mathematics plays in their lives.

The charts are based on the assumption that concepts and topics in mathematics will be presented within a problem-solving setting. Students will be encouraged to solve problems arising from accessible situations rather than from artificial models. Similarly appropriate assessment techniques will include activities in which the students demonstrate their understanding of the concepts within problem-solving contexts. These techniques are indicated in the charts and described at the end of each grade level in the "Key for Assessment Processes." Not only will the teacher assess the students' abilities to apply mathematical knowledge, compute and estimate in this setting, but the teacher will also pay

particular attention to their abilities to conjecture, investigate, reflect, explore, justify, clarify and model mathematical ideas.

"Quick Checks" provided at each grade level are designed to serve two purposes: (1) to identify mathematical content to be included in each grade level, and (2) to identify the critical thinking,' problem-solving, communication, and reasoning processes. Successful implementation of the Essential Academic Learning Requirements depends upon the degree to which all students are given the opportunity to develop the above described thought processes while participating in activities that develop the concepts and procedures of mathematics.

In this document we have attempted to use the same vocabulary found in the Essential Academic Learning Requirements. For clarity, we use the words "geometric figure" while "shape" is used in the Essential Academic Learning Requirements.

M a y 14.1599

FIFTH GRADE

Content Overview: By the end of the fifth grade, students will demonstrate an understanding of the meaning of non-negative whole numbers, fractions, decimals, and percents. They can identify, compare, and order non-negative whole numbers and can describe the relationship between fractions and decimals. Students can add, subtract, multiply, and divide non-negative whole numbers; add and subtract decimals, fractions, and mixed numbers; and determine and justify the reasonableness of their answers by using the estimation strategy most appropriate to the situation. They can choose and use the most appropriate standard units of measurement, and they are comfortable using both the U.S. Customary System and the Metric System.

Students can identify and describe geometric properties such as parallel, perpendicular, congruent, and similar. They can describe the location of points located on a coordinate grid and can describe simple transformations of two-dimensional geometric figures. Students can make predictions based on experiments and determine the probability of possible outcomes. They can create, organize, read, and interpret tables, charts, and graphs when analyzing and comparing data. Students can describe and explain how to use specific measures of central tendency (mean, median, mode, and range) for simple data. They recognize number patterns and sequences and translate problem-solving situations into simple mathematical equations containing a single variable.

Process Overview: By the end of fifth grade, students can solve problems by identifying the unknowns; using a variety of strategies (acting it out, making a model, looking for a pattern); and applying appropriate methods, operations, and processes. They can make and test conjectures baied on data they have collected; validate their thinking using models, facts, patterns, and relationships; and evaluate their procedures. Students use a variety of sources (print and nonprint) to locate and retrieve mathematical information; organize, clarify, and analyze the information they gather; and express mathematical ideas using everyday language, models, charts, tables, and symbols. They recognize relationships within mathematics, use mathematics in familiar settings in other disciplines, A d identify how mathematics is used in career settings.

h y 14.1999

FIFIM GRADE Content Quick Check

Does the student:

Use models to describe the meaning of fractions, decimals, and percents?

Identify, compare, and order non-negative whole numbers?

Orders fractions with fractions and decimals with decimals?

Identify equivalent fractions and simplify fractions to lowest terms?

Use models to describe prime and composite numbers, factors and multiples, and determine divisibility by 2,5, and lo?

Uses objects, pictures, and symbols to illustrate equivalent ratios?

Makes comparisons between two part.part relationships?

Add, subtract, multiply, and divide non-negative whole numbers?

Add and subtract deiimals, fractions, and mixed numbers?

Use models to demonstrate and explain the meaning of multiplication of a fraction by a whole number and a fraction by a fraction?

Use mental arithmetic, paper and pencil, calculator, or computer as appropriate for a given situation?

Determine and justify the reasonableness of answers by estimating prior to actual computation with whole numbers?

* * *

Choose the appropriate standard unit and tool and measure objects directly?

Choose standard units of measure yielding the most appropriate measurement?

Explain the advantages of standard units of measure?

Make conversions wilhin the US. Customary System and within the Metric System?

Use estimation to obtain reasonable approximations? * * *

May 14, 1999

a

0

0

0

P

P

P

P

Q

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P

0

0

a

Identify and describe properties of geometric figures and find examples in the physical world?

Describe the location of points on coordinate grids in the first quadrant?

Identify simple transformations using combinations of translations, reflections, or rotations?

Construct angles using tools such as a compass, straightedge, and/or transparent mirror?

* * *

Prepare and organize displays of all possible results for a given probability experiment?

Carry out experiments to determine probabilities and compare predictions to experimental results?

Use and describe strategies for determining the probability of an event?

Identify a random sample taken from a described population?

Organize and display data using frequency tables?

Identify outliers in a set of data?

Describe mean,'median, mode, and range for simple data?

* * *

Reiognize number patterns and sequences?

Use physical and visual materials to model operations performed on both sides of an equation? 8

Use variables to describe patterns and sequences?

Translate problem-solving situations into simple mathematical equations containing a single variable?

Evaluate simple expressions using manipulatives?

h y 14,1999

Process Ouick Check

infused throughout the curriculum. Although the processes are the same for each grade level, Note: Problem solving, mathematical reasoning, communication, and connections are necessarily

implementation of them will vary based on the developmental level of the students. Specific examples appropriate for Grade 5 can be found on the Suggested Assessment Evidence for the End of Fifth Grade chart.

Does the student:

0 Develop and use a variety of strategies,’such as act it out, make a physical model, and look for a pattern?

0 Identify missing or extraneous information?

0 Define questions to be answered in new situations?

CI Organize relevant information from multiple sources?

* * *

0 Validate thinking and mathematical ideas using models, known facts, and patterns?

a 0

a

0

0

0

Make conjectures and inferences based on analysis of new problem situations?

Test conjectures and inferences and explain why they are true or false?

Check for reasonableness of results?

* * *

Develop a plan for collecting mathematical information?

Organize and clarify mathematical information by reflecting and verbalizing?

Express ideas clearly and effectively using everyday and mathematical language appropriate to the audience?

* * *

Relate and use different models and representations for the same situation?

Identify mathematical patterns and relationships in other disciplines?

Use mathematical thinking and modeling in other disciplines?

.-

May 14. 1999

D Describe examples of contributions to the development of mathematics?

D Recognize the extensive use of mathematics outside the classroom?

D Investigate the use of mathematics within several occupationaVcareer areas?

May 14.1999

Suggested Assessment Evidence for the End of Fifth Grade

LINKS TO PROCESSES*' ASSESSMENT

EALRs 1 2 3 4 5

CONCEPTS AND PROCEDURES Number Sense

Uses visual and physical models to describe the meaning of fractlons,

. . . ~ ~ ~ ~ ~ ~~ ~

decimals, and percents Identifies, compares, and orders non-negative whole numbers Orden fractions with fractions and decimals with decimals

. .

(ex: Which is bigger 3/8 or 2/32 Identifies equivalent fractions and simplifies fractions to lowest term!

Uses visual and physical models to describe prime and composite numbers. facton and multiples. and determines divisibility by 2 .5 ,

Uses objects, pictures, and symbols to illustrate equivalent ratios and IO

(ex: 1:2 is equivalent to 4 8 ) Makes comoarisons between two part:part relationships (ex: Which

multiplication of a fraction by a whole number and a fraction by a

Determines and iustifies the reasonableness of answers by estimating "or to actual computation with whole numbers

Measurement

measurement Explains the advantages of standard units of measure ~ a k e s conversions within the US. Customary System and wifhin

I

1.2, 2.3 I X I 1.2.4.3 X

the 1.2.5.1

uses estimation to obtain reasonable approximations I .2

**Key for Assessment Processes 1. Illustrated journals 2. Focused observatiodanecdotal records

4. Performance assessment 5 . Traditional paper and pencil tests

3. Individual interviews

May 14.1559

Suggested Assessment Evidence for the End of Fifth Grade (continued)

I I I ASSESSMENT I LlNKS TO PROCESSES**

E A L k 1 1 2 1 3 1 4 1 5 Geotnem'c Sense

Identifies and describes propenies of geometric figures (ray. angle. line segment. parallel, symmetric. perpendicular. similar. and congruent) and finds examples in the physical world

x x x 1.3.4.3

x x x 1.3.4.3 Identifies simple transformations using combinations Of translalions.

X X 1.3 Describes the location of points on coordinate grids in first quadrant

Uscs variables to describe patterns and sequences 1.5. 2.1.4.3 x X Translates problem-solving situations into simple mathematical I .5 X x x Evaluates simple expressions using mnipulatives equations containing a single variable

1.5 x x

**Key for Assessment Processes 1 . Illustrated journals 2. Focused observatiodanecdotal records

4. Performance assessment 5 . Traditional paper and pencil tests


May 14.1999

Suggested Assessment Evidence for the End of Fifth Grade (continued)

I I I

I PROBLEM SOLVING Develops and uses a variety of strategies. such as act i t out. make a

physical model. look for a pattern Identifies missing or extraneous information

2. I

2.2 Defines questions to be answered in new situations (i.e.. after being 2. I

presented with new information or witnessing an unfamiliar event) Organizes relevant information from multiple sources (firsthand experimental data data reponed by others. books. Internet. etc.) MATHEMATICAL REASONING Validates thinking and mathematical ideas using models, known facls and patterns (ex: uses manipulatives to demonstrate addition of fractions with unlike denominators) Makes conjectures and inferences based on analysis of new problem situations (ex: makes a hypothesis when asked if there is a

2.3

3.1

3.2

relationship between the &ea and perimeter of quadrilaterals) I ~ e s t s conjectures and inferences and explains why they are uue or I 3.3 All of the Assessment false (ex:bevises, carries out and evaluates a plan to test the hypothesis that an increasc in area results in an increase in perimeter)

3.3 Checks for reasonableness of results

m a y be used for Processes

gathering evidence

I

COMMUNICATION

Develops a plan for collecting mathematical information (from both

4.2 Organizes and clarifies mathematical information by reflecting and

4.1 print and nonprint sources)

verbalizing (ex: after a class discussion on measurement explains

Clearly and effectively cxpresscs ideas using both everyday and precision in own words)

4.3 mathematical language (models. tables, charts) appropriate to the

2. 1 .

3.

audience I CONNECTIONS Relam and uses different models and representations for the m e

physical and visual models) situation (ex: explains the meaning of multiplication of fractions using

5.1

(ex: understands patterns, shapes. time, distances. and relative 5.2 Identifies mathematical patterns and relationships in other disciplines

occupationallcareer areas (ex: aerospace. medicine. carpcnuy. Investigates the use of mathematics within several

5.3 Recognizes the extensive use of mathematics outside the classroom ~ mathematics (such as the'contributions of women)

5.2 D e r r i b e s examples of contributions to the development of 5.2 Uses mathematical thinking and modeling in other disciplines

distances to other objects within our solar system)

5.3

banking. sales) **Key for Assessment Processes Illustrawd journals Focused observatiordanccdolal records

4. Performance assessment

Individual interviews 5. Traditional paper and pencil tests

SIXTH GRADE

Content Overview: By the end of sixth grade, students will demonstrate an understanding of the meaning of non-negative numbers, decimal numbers, and fractions or mixed numbers. They can add, subtract, multiply, and divide whole numbers; add, subtract, and multiply decimals, fractions, and mixed numbers; demonstrate the meaning of division of simple fractions and decimals using visual models; and choose the most appropriate strategy for solving computation problems (mental math, calculator, or paper and pencil). Students can describe primes, composites, factors, and multiples and use ratios to compare quantities. They can find area and volume of regular geometric figures and area of irregular shapes. They understand that the unit of measure chosen determines the precision of a measurement, and they apply the concept of ratio to construct scale models.

Students can graph points on a coordinate grid, identify geometric transformations of 2-D geometric figures, and describe, compare and contrast geometric figures using concepts of similarity, congruence, symmetry. and number of degrees. They can determine the probability of events by designing and implementing probability experiments and they can organize displays of their results. Students can choose mean, median, mode and/or range to describe data and can perform the computation most appropriate for their choice. They can find patterns in data located in t-tables, extend those patterns, and write rules to describe them.. Students can express relationships between numbers symbolically and can use pictures and/or words to describe solutions to single variable equations.

Process Overview: By the end of sixth grade, students can solve problems (drawing a diagram, making a chart or table, looking for a pattern) and choose the most appropriate mathematical tools. They can gather, interpret, compare and contrast pictures, diagrams or physical models found in a variety of sources when solving problems and use inductive reasoning to support arguments or validate solutions. Students are comfortable using both everyday and mathematical language to describe and justify their thinking and can express their ideas using charts, graphs and journals. They connect concepts and procedures among different mathematical content areas, use mathematics in familiar settings in other disciplines, identify how mathematics is used in career settings, and describe the contributions of men and women to the development of mathematics.

M a y 14.1999

SIXTH GRADE

Content Quick Check

Does the student:

0

0

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P

0

0

0

0

P

c1

0

0

0

0

0

0

Use models to explain equivalencies of fractions, decimals, and percents?

Identify, compare, and order non-negative whole numbers, fractions, and decimals?

Use models to describe primes, composites, factors, and multiples, and determine divisibility by 2,4,5,8, and lo?

Use objects, pictures, and symbols to create equivalent ratios in parkwhole context?

Find missing values within proportional conditions using ratios and rates?

Add, subtract, multiply, and divide whole numbers?

Add, subtract, and multiply decimals, fractions, and mixed numbers?

Use models to demonstrate the meaning of division of simple fractions and decimals?

Use order of operations to simplify arithmetic expressions with whole numbers?

Justify the use of mental arithmetic, paper and pencil, calculator or computer as appropriate for a given situation?

Determine and justify the reasonableness of answers by estimating results prior to actual computation with whole numbers and fractions?

* * * Determine area and volume when given dimensions of the object o r space measured in US. or metric units of measurement?

Determine the area of irregular shapes using customary and metric units of measurement?

Apply the concept of ratio when constructing scale models using customary or metric . units of meaiurement?

Determine which US. or metric unit of measurement will result in the most appropriate measurement for a given situation?

Explain how precision depends on the calibration of the measurement tool? * * *

May 14.1999

0

a

a

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a 0

a a 0

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Identify and describe figures that are similar, congruent, or symmetric?

Describe the location of points on coordinate grids using letters and numbers on axes?

Identify the number of degrees in a circle, triangle, and quadrilateral?

Compare, contrast, and construct isosceles, equilateral, and scalene triangles?

Describe simple transformations using combinations of translations, reflections, and rotations?

* * * Display the sample space of a probability experiment by making a table or using a diagram?

Conduct simulations to determine probabilities?

Predict outcomes of simple experiments and simulations and compare the predictions to experimental results?

Form a random sample from a described population?

Collect, organize, and display data' using the appropriate forms?

Identify the effects of outliers on the mean and median?

Compute mean,'median, mode, andlor range as appropriate in describing simple data?

Make inferences based on experimental results?

* * *

Recognize and extend number patterns and sequences?

Use relationships found among sets of numbers to extend patterns on t-tables and function machines?

Write rules for data found on t-tables and function machines?

Express relationships between numbers using =, #, >, or <?

Describe variables found in simple inequalities and formulas?

Evaluate simple expressions using pictorial representations?

Use pictures and/or words to describe solutions to single-variable equations?

Process Ouick Check

Note: Problem solving, mathematical reasoning, communications, and connections are necessarily infused throughout the curriculum. Although the processes are the same for each grade level, implementation of them will vary based on the developmental level of the students. Specific examples appropriate for Grade 6 can be found on the Suggested Assessment Evidence for the End of Sixth Grade chart.

Does the student:

a a

a a

a

a a a a

a

a

a a

a

0

Search systematically for patterns in simple situations?

Develop and use a variety of strategies, such as draw a diagram, make a chart or table, and look for a pattern?

Identify unknowns in new situations?

Select and use appropriate mathematical tools to construct solutions to problems?

* * *

Interpret, compare, and contrast information from a variety of sources, such as books, personal investigations, and/or a computer?

Validate thinking and mathematical ideas using models, patterns, and relationships?


Support arguments and justify results using inductive reasoning?

Check for reasonableness of results? ' * * *

Use reading, listening, and observation skills to access and extract mathematical information?

Use available technoloa to browse, select, and retrieve relevant mathematical information?

Organize and clarify mathematical information by reflecting and writing?

Express ideas clearly and effectively using both everyday and mathematical language?

* * * Connect conceptual and procedural understandings among different mathematical content areas?


M y 14.1999

0 Use mathematical thinking and modeling in other disciplines?

0 Describe examples of contributions to the development of mathematics?

0 Recognize the extensive use of mathematics outside the classroom?

0 Investigate the use of mathematics within several occupationaVcareer areas?

h b y 14. 1999

Suggested Assessment Evidence for the End of Sixth Grade

LINKS TO ASSESSMENT

EALRs PROCESSES"

1 2 3 4 5

ZONCEPTS AND PROCEDURES

Number Sense

Jses visual and physical models to explain equivalencies of iactions. decimals, and percents dentifies. compares, and orders non-negative whole numbers.

Jses visual and physical models to describe primes. composites. iactions. and decimals

IO actors. and multiples. and determines divisibility by 2.4, 5. 8. and

Jses objects. pictures. and symbols to create equivalent ratios in ,art:whole context (ex: in probability, 5 wins in 50 draws is quivalent to 10 wins in 100draws) ?in& missing values within proportional conditions (ex: if 2 cm = I N , then 8 cm = ? mi) using ratios and rates 4dds~ whlractc. multiolies. and divides whole numbers Adds. subtracts. and multiplies decimals. fractions, and mixed numbers Uses visual and physical models to demonstrate the meaning of division of simple fractions and decimals Uses order of operations to simplify arithmetic expressions with whole numbers (multiplication and division, addition and subtraction) ~ustifies the use of mental arithmetic. paper and pcncil. calculator or computer as appropriate for a given Situation Determines and justifies the reasonableness of answers by estimating results prior to actual computation with whole numbers and fractions

1.1.3.1.4.3

1 . 1

1.1. 3.1.4.3

1.1

1.1

1.1 1.1

l . l . 3 .1 .4 .?

1 . 1

1.1.4.1

1.1.3.3.4.:

Measurement

Determines area and volume when given dimensions of the object or 1.2 space measured in customary or metric units of measurement Determines the area of irreaular shapes using customary and metric I 1.2 - units of measurement I Applies the concept of ratio when constructing Scale models Using I 1.2.5.1 .. customary or metric units of m u r c m e n t I Determines which US. or metric unit of measurement will result in I I .2.2.3.4.3 the most appropriate w u r e m e n t for a given situation I Explains how precision depends on the calibration of the,measuring I 1.2

qff x x

**Key for Asscssmeot Processes

2. Focused observationlanecdotal records I . Illusuatcd/handwritten journals 4. Performance assessment

3. Individual interviews 5. Traditional paper and pencil tests

office of svpcnnlcndent of Public hrmction 16

Suggested Assessment Evidence for the End of Sixth Grade (continued)

**Key for Assessment Processes

2. Focuscd observatiodanecdotaI records I . Illustratedhandwitten journals 4. Performance assessment


May 14.1999


**Key for Assessment Processes 1. Illustratedlhandwitten journals 4. Performance assessment 2. Focused observationlanecdolai records 5. Traditional paper and pencil tests 3. Individual interviews


I I LINKSTO I ASSESSMENT

**Key for Asssssment Procesg 1. Illustratcd/handwrinn journals 2. Focused observationlanecdotal records

4. Performance assessment


.

May 14.1999

SEVENTHGRADE

Content Overview: By the end of seventh grade, students can explain the relationships among rational numbers; identify equivalencies among fractions, decimals, ratios, and percents; and express numbers in standard and exponential form. They can identify properties of rational numbers and can apply them when predicting the sum, difference, product, and quotient of whole numbers, decimals, fractions, and mixed numbers. Students can justify the use of mental math, paper and pencil, calculator or computer when given situations involving rational numbers and can determine when estimation (not computation) is sufficient. Students can use ratios and proportions to measure indirectly and can justify their choice of methods when obtaining reasonable approximations rather than exact measures, They can solve problem using rates and make conversions wirhin the U.S. Customary or Metric systems.

Students can describe and construct congruent, symmetric, and similar geometric figures as well as simple geometric transformations using a compass, straightedge, and/or computer software. They can calculate the probability that an event will occur and can compare experimental and theoretical results. Students justify conclusions based on data they have organized using stem-and-leaf plots or other appropriate forms. They can represent number patterns using graphs, evaluate simple expressions, solve simple equations and inequalities containing one variable, and solve problems after translating the information into algebraic expressions.

Process Overview: By the end of seventh grade, students can use and explain a variety of strategies (guesdcheckhevise, work backwards, solve a simpler problem and generalize, write an equation). They recognize the need to modify or abandon an unproductive approach, and they can design and conduct simple open-ended investigations. Students can validate their thinking using patterns, relationships and' counterexamples, and they can reflect on and evaluate procedures and results in new problem situations. Students are comfortable with available technology and can use it to retrieve information helpful to their understanding of mathematics. They can express their mathematical understandings in both everyday and mathematical language, describe the contributions of other cultures to the development of mathematics, and recognize the extensive use of mathematics outside the classroom.

May 14.1999

SEVENTH GRADE

Content Quick Check

Does the student:

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Use models to show understanding of non-negative fractions, decimals, percents, and place value?

Use pictures and symbols to demonstrate properties of the rational number system?

Identify fraction, decimal, ratio, and percent equivalencies?

Compare and order whole numbers, fractions, and decimals?

Use models to describe prime and composite numbers, factors and multiples, and determine divisibility?

Express numbers in factored form including all factor pairs?

Add, subtract, multiply, and divide non-negative whole numbers, decimals, fractions, and mixed numbers using order of operations?

Justify the use of mental arithmetic, paper and pencil, calculator or computer as appropriate for a given situation involving non-negative rational numbers?

Identify situations involving non-negative rational numbers in which estimation is suffkient and computation is not required?

Determine and justify the reasonableness of answers by estimating results prior to actual computation with non-negative rational numbers?

* * *

Measure indirectly using formulas for perimeter, area, and volume?

Solve problems using rates and determine the appropriate units?

Describe and justify methods used to obtain reasonable approximations when given no exact measures?

Make conversions within US. Customary and within Metric systems?

* * *

May 14.1599 Office of Supcrinlcndenl of Public lnsrmction 21

0

0

0

0

0

0

0.

0

0

0

0

0

0

0

Identify and describe geometric shapes found in the environment?

Describe the location of points on coordinate grids (first quadrant)?

Construct and describe symmetric, congruent, and similar geometric figures using appropriate tools and computer software?

Describe and construct simple transformations using combinations of translations, reflections, and rotations?

* * *

Calculate the probability that an event will occur in experimental and theoretical ,

situations?

Compare experimental and theoretical results?

Implement an investigation in which a random sample of data representing a described population is collected?

Collect, organize, and display data using appropriate form?

Calculate and demonstrate the appropriate use of mean, median, mode, and range as appropriate in describing a set of data?

Predict outcomes of experiments and simulations and compare the predictions to experimental results?

Make and justify inferences based on experimental results?

* * *

Recognize, extend, create, and represent number patterns using tables, graphs, and rules?

Translate a given problem situation into a simple mathematical equation and find the solution?

Evaluate simple expressions and formulas?

Solve simple equations and inequalities containing one variable?

May 14. 1 9 9 9 Offlee of Superintenden1 of Public hrmction 22

Process Ouick Check

Note: Problem solving, mathematical reasoning, communications, and connections are necessarily infused throughout the curriculum. Although the processes are the same lor each grade level, implementation of them will vary based on the developmental level of the students. Specific examples appropriate for Grade 7 can be found on the Suggested Assessment Evidence for the End of Seventh Grade chad.

Does the student:

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Develop and use a variety of stra!egies, such as guess/check/revise, work backwards, solve a simpler problem and generalize, or write an equation?

Recognize the need to modify or abandon an unproductive approach?

Define problems in new situations?

Apply appropriate methods, operations, and processes to construct a solution?

* * * Validate thinking and mathematical ideas using patterns, relationships, and counterexamples?


Check for reasonableness of results?

Reflect on and evaluate procedures and results in new problem situations? .=

* * *

Use r iding, listening, and observation skills to access and extract mathematical information?

Use available technology to browse, select, and retrieve mathematicai information?

Organize and clarify mathematical information by reflecting and discussing?

Express ideas clearly and effectively using both everyday and mathematical language appropriate to the audience?

* * * Connect conceptual and procedural understandings among different mathematical content areas?


Use mathematical thinking and modeling in other disciplines?

0 Describe examples of contributions to the development of mathematics?

Ll Recognize the extensive use of mathematics outside the classroom?

0 Investigate the use of mathematics within several occupationaVcareer areas?

Suggested Assessment Evidence for the End of Seventh Grade

LINKS TO ASSESShlENT

EALRs PROCESSES"

1 2 3 4 5

CONCEPTS AND PROCEDURES

Number Sense

Uses visual and physical models 10 show understanding of non- negative fractions. decimals, percents. and place value Uses pictures and symbols to demonsuate propenies of the rational number system (propenies of addition: commutative. associative. zero; propenies of multiplication: commutative. associative. identity. zero: distributive propeny of multiplicalion and addition) Identifies fraction. decimal. ratio. and wrcent equivalencies

numbers. factors-and multiples. and determines divisibility Expresses numbers in factored form including all factor pairs

~~

Adds, subtracts. multiplies. anddivides non-negative whole numbers. decimals. fractions. and mixed numbers using order of operations Justifies the use of mental arithmetic. paper and pencil. calculator or computer as appropriate for a.given situation involving non-negative rational numbers Identifies situations involving non-negative rational.aumbers in which estimation is sufficient and computation is not required Determines and justifies the reasonableness of answers by estimating results prior to actual computation with non-negative rational numbers

Ueasuremenl

I . l . 3 . 1 . 4 . 3

1 . 1 . 5.1

1.1

1.1

1.1.3.1.4.3

1 . 1 1 . 1

1.1.4.1

1.1

1.1. 3.3.4.3

Measures using formulas for perimeter. area. and volume 1.2.2.3 X Selects and uses tools with calibrations which provide appropriate I .2 X degree of precision Understands relationship among perimeter, area. and volume (area is 1.2 x X X measured in square units while perimeter is linear) Solves problems using rates (speed. money exchange) and 1.2, 2.3. 5.1 X x x x determines the appropriate units I I I I I I Describes and justifies methods used to obtain reasonable I 1.2. 2.3.4.3 I X 1 Ix I approximations when given no exact measures I I I I I I Makes conversions within V.S. Customary and within Metric 1.2. 5.1 x x systems (lengthninear. weighdmass. volumelcapacity)

**Key for Assessment Processes

2. Focused observatiodanecdotal records I . Illusuatedhandwritten journals 4. Performance assessment

3. Individual interviews 5 . Traditional paper and pencil tests

M a y 14. 1 9 9 9

Suggested Assessment Evidence for the End of Seventh Grade (continued)

I

Recognizes, extends, creates, and represents number patterns using

X X 1.5 Evaluares simple expressions and formulas equation and finds the solution

x x X 1.5.2.3 Translates a given problem situation into a simple mathematical tables, graphs, and rules

x x 1.5. 2.1. 4.3

Solves simple equations and inequalities containing one variable I .5 X

**Key for Assessment Processes 1. Illusuatedihandwritten journals 4. Performance assessment 2. Focused obse.rvation/anecdotal records 5 . Traditional paper and pencil tests 3. Individual interviews

May 14.1599


LINKS TO EALRs

ASSESSMENT PROCESSES**

PROBLEM SOLVING I Develops and uses a variety of strategies. such as guessfcheckkevise. 2.1 work backwards, solve a simpler problem and generalize, write an equation. organized list. etc. Recognizes the need to modify or abandon an unproductive approach I 2. I

~~

I I (in computation as well as in problem solving) I I Defines problems in new situations I 2.2 Applies appropriate methods, operations. and processes to construct I 2.3 I a Solution I MATHEMATICAL REASONING Validates thinking and mathematical ideas using pauerns. 3.1 relationships, and counterexamples (ex: identifies and creates examples of proportions) Makes conjectures and inferences based on analysis of new problem 3.2.3.3 situations Checks for reasonableness of results Reflects on and evaluates procedures and results in new problem

3.3

Processes situations (ex: after completina an investigation. students evaluates All of the Assessment 3.3

-

- - -

- - - . -

the appropriateness of t ie method used)

COMMUNICATION Uses reading, listening, and observation skills to access and extract

- I may be used for

atherinp. evidence

_ _ -

-- -

-. -

I

mathematical information (diagrams, oral narratives. symbolic' representations) Uses available technology to browse, selecr and retrieve mathematical information (ex: uses the internet and/or CD-ROM! to find information regarding procedures used to compute distances - .. between planets) Organizes and clarities mathematical information by reflecting and dixussing (ex: during class discussion about probability. student presents oral justification for inferences made from experimental

Clearly and effectively expresses ideas using both everyday and mathematical language appropriate to the audience (ex: written logs,

4.1

4.1

4.2

4.3

algebraic notation)

**Key for Assessment Processec I , Ulustratedhandwritten journals 4. Performance assessment 2. Focused observatiodanecdotal records 5. Traditional paper and pencil tests 3. Individual interviews

May 14, 1999


I I I ASSESSMENT PROCESSES.' I

CONNECTIONS I Connects conceptual and procedural understandings among different I 5.1 mathematical content areas (ex: applies ratios and proponions to indirect measurement tasks) Identifies mathematical patterns and relationships i n other disciplines

collecting car speed and acceleration data and using that data to (ex: explores the relationship bctween force and motion of objects by

calculate balanced and unbalanced forces on motion along a straight line) Uses mathematical thinking and modeling in other disciplines (ex: uses probability to describe results when repeating Mendel's experiments with pea plants) Describes examples of contributions to the development of mathematics (such as the conlributions of different cultures) Recognizes the extensive use of mathematics outside the classroom Investigates the use of mathematics within several occupationaVcarccr areas (ex: banking, accounting. stock market,

5.2 All of the Asswment Processes

gathering evidence may be used for

**Key for Assessmen1 Processes

2. Focused observatiodanecdotal records . 5 . Traditional paper and pencil tests I . Illustrated/handwitten journals 4. Performance assessment


May 14. 1 9 9 9 Office of Supainandsnt of Pvblic Insrmctian 28

Terms Introduced in the Grades K-4 Glossary:

acute addend angle associative properly of addition associative properly of multiplication average axes bar graph capacity cardinal number central tendency circle circumference common denominator common multiple commutative properly of addition commutative property of multiplication COO0

congruent figures cylinder data denominator difference discrete dividend divisible divisor edge equation equilateral estimate

expanded form even number

face fact factor figure flip fraction fraction families grid hexagon horizontal

Copyright 8 1 9 9 9 5tsfc

identity property of addition identity property of multiplication inequality integer intersecting lines isosceles triangle line line graph line of symmetry mean measure of central tendency median mixed number mode multiple nonstandard units of measure number line number sentence

obtuse numerator

octagon odd number operation ordered pair ordinal number parallel lines parallelogram pattern pentagon perimeter perpendicular.lines point

polyhedron probability product pyramid quadrilateral quotient

rectangle reflection regular polygon

polygon

ray

d Wa4hingfon. 5vpcdnfcndcm d Public Instrumon. All nghts need. 29

remainder rhombus right angle right triangle rotation sample scalene side similar figures slide solid sphere square standard form standard units of measure

sum survey symmetrical transformation translation trapezoid trend triangle turn variable vertex vertical volume whole number word form

Terms introduced in the Grades 5-7 Glossary:

analyze approximate argument chart combination compare conclude conjecture contrast cube diagram distributive property of multiplication

over addition equivalent fractions evaluate evidence exponent expression function machine gram graph greatest common factor improper fraction infer interpret investigate irrational number justiw least common multiple liter meter method metric prefixes order of operations outlier percent place value population precision predict prime number prism proportion

m h r 0 1 5 9 9 S u e d wa&hi-. 5upcnn&nr d tl,Gbllc hstructon. All yhc, reecrucd

random sample ratio rational number real number reasonable represent revise rule sample space sequence simplest form solution solve stem-and-leaf plot strategy

table symbol

t-chart three-dimensional figure two-dimensional figure unknown validate verity zero property of addition zero property of multiplication

31

acute

addend

analyze

angle

A Framework for Achieving the Essential Academic Learning Requirements in Mathematics Grades 5-7 Glossary

see angle and triangle

any number that is added

addend + addend = sum

to break up a whole into its parts; to'examine in detail so as to determine the nature of

two rays that share an endpoint; named according to the number of degrees of its measure

acute angle right angle obtuse angle straight angle (greater than Oo (equal to 90") (greater than 90° (equal to 180') but less than 90a) but less than 180")

approximate to obtain a number close to an exact amount

argument a reason or reasons offered for or against something; suggests the use of logic and facts to support or refute a point

assoclatlve property the sum stays the same when the grouping of addends is changed of addition

Example: (a + b) + c = a + (b + c) (30 + 4) + 20 = 30 + (4 + 20j

associative property the product stays the same when the grouping of factors is changed of multiplication

Example: (a x b) x c = a x (b x c) ( 2 X 3 ) X 4 = 2 x ( 3 X 4 )

average a measure of central tendency; a number somewhere in the middle of data ordered lrorh least to greatest, or a number with a lot of data clustered around it. There are three types of averages: mean, median, and mode.

32

axes

bar graph

perpendicular lines used as reference lines in a coordinate system or graph; the horizontal line is the x-axis; the vertical line is the y-axis

Y

3 i 2 1 4 : : : : : I : : : : : v

- 3 - 2 - 1 0 1 2 3 x

-1

a graph that uses the length of solid bars to represent numbers and compare data

Favorite Dog Breed Survey

t

Breed of Dog

capacity the volume of material or liquid that can be poured into a container

cardinal number a number that designates the "manyness" of a set of objects, or the number of units in the set; answers the question "How many...?"

Example: 34 and 50,098

33

central tendency a single number that describes all the numbers in a set

Example: For the set of numbers 95, 86, 82, and 83, the mean is 89.

chart

circle

circumference

a method of displaying information in the form of graphs or tables

a set of points in a plane that are all the same distance from the center point

the boundary line, or perimeter, of a circle; also, the length of the perimeter of a circle

combination a group of objects, numbers, or events: changing the order does not create a new combination (1,2, 3 is the same combination as 3, 1, 2)

common denominator a number divisible by all of the denominators being considered; also known as a common multiple of the denominators

Example: 4 + 3 =

Multiples of 2 = (0, 2, 4, 6, 8. 10, 12,. ..I Multiples of 3 = (0, 3, 6, 9, 12. 15 .___ I

Multiples common to both sets, other than zero, include 6 and 12. (There are infinite numbers of common multiples as the sets continue, i.e., 18. 24, 36, etc.) Any of these numbers can be used as common denominators for the two fractions.

1 - 3 2 6

OR 2 12

1 - 2 _ - - 1 - 4 3 6 3 12

10 12

_ - _ _ _ -

- 5 6

-

C q y i g h t 0 I 9 9 9 Statz of WlshfqfOn. Supcrintecndcnt d FWi& Insu~cT#on. Ni rlghM r e s c m d . 54

common multiple a number that is a multiple of each of two or more numbers; used to find a common denominator when operating with fractions having unlike denominators

Example: 24 is a common multiple of 2, 3, and 8

commutative property it makes no difference in which order two numbers are added (the commutative of addition property does not apply to subtraction)

Example: a + b = b + a 4 + 50= 50+ 4

commutative property it makes no difference in which order two numbers are multiplied (the commutative of multiplication properly does not apply to division)

Example: a x b = b x a 3 x 5 = 5 x 3

compare to look for similarities and/or differences

conclude to make a judgment or decision after investigating or reasoning; to infer

cone a three-dimensional figure with one circular or elliptical base and a curved surface that joins the base to the vertex

congruent figures figures that have the same shape and size

conjecture

contrast

cube

inference or judgment based on inconclusive or incomplete evidence; guesswork

to emphasize differences

a rectangular prism having six congruent square faces

35

cylinder a solid figure with two circular or elliptical bases that are congruent and parallel to each other

data

denominator

diagram

difference

discrete

distributive

multiplication over addition

dividend

Property of

divisible

divisor

collected pieces of information

the number below the fraction bar; indicates the number of equivalent pieces into which something is divided

a drawing that represents a mathematical situation

the number found when subtracting one number from another; the result of

than another number a subtraction operation; the amount by which a quantity is more or less

that has no limit points composed of distinct parts or discontinuous elements; a set of numbers, or points,

Example: discrete - taking coins out of yourpocket one at a time; non-discrete (or continuous) - pouring water from one container to another container

a properly of real numbers stating that a x (b + c) = (a x b) + (a X c) where a, b. and c stand for any real numbers

Example: 3 x (40 + 5) = (3 x 40) + (3 X 5)

a number which is to be divided by another number

dividend + divisor = Quotient 15+ 3 = 5

divisor)- quotient

3)15 5

a whole number is divisible by another whole number if the remainder equals zero when you divide

the number by which the dividend is to be divided; also a factor

divisor)- dividend - divisor = quotient quotient

36

edge the line segment formed by the intersection of two faces of a solid figure; acubehas12edges

equation

equilateral

equivalent fractions

estimate

evaluate

even number

evldence

expanded form

exponent

expression

a number sentence which shows equality between two sets of values

Example: 4 + 8 = 6 + 6

see triangle

fractions that name the same number

Example:: and: and$ are equivalent fractions

to find an approximate value or measurement of something

to examine and judge carefully; appraise

a whole number divisible by two

Example: 0. 4, 678

models, known facts, patterns, relationships, and counterexamples

a number written in component parts showing the cumulative place values of each digit in the number

Example: 546 = 500 + 40 + 6

a numeral written above and lo the right of another numeral to indicate how many times the original number is used as a factor

Example: The exponent “3” in 4’ means 4 is a factor 3 times. 4 X 4 x 4

a variable or combination 01 variables, numbers, and symbols that represent a mathematical relationship

face a flat surface, or side,-of a solid figure

fact

factor

figure

flip

fraction

a basic mathematical statement involving numbers and operations; ex. 3 i 5 = 8, 1 0 + 2 = 5

one of two or more numbers that are multiplied together to obtain a product; factor x factor = product

Example: 4 x 3 = 12, 4 and 3 are factors

a closed geometric shape in 2 or 3 dimensions

the effect of a flip is a reflection; see reflection

a way of representing part of a whole or part of a group by telling the number of equal parts in the whole and the number of those parts you are describing; it is written in the form zzzL where the numerator can be any integer and the denominator can be any integer except zero

Example:- - - - - 2 5.5 1.5 o e 3, 23, 765, 34. 1

fraction famllles fractions having denominators that are multiples of a single number; ex. halves, fourths, eighths, and sixteenths; thirds, sixths, and ninths

Example: $, i, 4 have denominators that are multiples of 4

function machine

applies a function rule to a set of numbers which determines a corresponding set of numbers

Example:

l f you app/y the function w/e "multiply by 7" to the values 5, 7, and 9, the corresponding values would be:

5+ 35 7+ 49 9+ 63

30

gram

greatest common factor

grid

hexagon

horizontal

Identity property of addition

Identity property of multiplication

improper fraction

Inequality

infer

integer

a basic unit in the metric system measuring mass/weight; the mass of 1 cubic centimeter of water at 4 degrees Celsius

a "picture" showing how certain facts are related to each other or how they compare to one another

the largest factor of two or more numbers; onen abbreviated as GCF

Example: to find the greatest common factor of24 and 36 factors of24 = (1 , 2. 3, 4, 6, 8, 12, 24}

common factors of 24 and 36 are (1 , 2, 3, 4, 6. 121, the largest being 12 factors of 36 = ( 1 , 2. 3, 4, 6. 9. 12, 18. 361

12 is the greatest common factor of 24 and 36

a pattern of regularly spaced horizontal and vertical parallel lines drawn on a map or chart with ordered pairs of numbers that can be used to locate points

a six-sided polygon

regular hexagon nonregular hexagons

extending side to side, parallel to the horizon

adding zero to a number gives a sum identical to the given number

multiplying a number by 1 gives a product identical to the given number

a fraction in which the numerator is equal to or greater than the denominator

Example: +, 5 two sets of values that are not equal

to draw a conclusion from facts or evidence

the counting numbers (1, 2. 3 ,.., ), their opposites (-1. -2. -3 ,.__ ), and zero

tOWnghf B 1 9 9 9 5Utc of Washinqfon Supcnnfcndcnt of h!Ac InaUucUoo. NI nghts rcrmd. 39

interpret to explain the meaning of information, facts. and/or observation

intersecting lines lines that meet at a point

Investigate to research using careful observation and examination of the facts; to inquire

Irrational number a number that cannot be expressed as a ratio of two integers

Example: fi Isosceles triangle a triangle with 2 congruent sides; an alternate definition is a triangle with at least

2 congruent sides (there is no common agreement on a definition of an isosceles triangle)

justify

least common multiple

line

to prove or show to be true or valid using logic and/or evidence

the smallest number, besides zero. that is a multiple of a set of two or more numbers; often abbreviated LCM

Example: to find the least common multiple of 4 and 12 multiples of 4 = {O, 4, 8, 12, 16,. ..) multiples of 12 = {O. 12. 24, 36 ,... ) The lowest common number besides 0 in both sets is 12, so the LCM of 4 and 12 is 12.

a set of points extending infinitely in opposite directions - *

40

line graph a graph that uses a line to show that something is increasing, decreasing, or staying the same over time

Amount of Snow Fall

12

Nov. Dec. Jan. Feb. Mar. Apr.

line of symmetry a line on which a figure can be folded into two parts that are congruent mirrol images of each other /Jml /7/.. __-. ...-

'. __.. ...

_..- __.- _ _- - not a line line of

i not a line of

of symmetry symmetry symmetry

.- liter

mean

a basic unit of measure in the metric system measuring capacitylvolume; one cubic decimeter; a little more than a quart; 1000 cubic centimeters

an average obtained by dividing the sum of the data items by the number of data items

Example: If there are three classes: A = 24 children, B = 25 children, and C = 23 children, the classes would be balanced by moving one student from class B to class C. thus making each class the same size (24). This number would indicate the average class size. Arithmetlcall~ it is obtained by adding all data points together and dividing the sum by the number of points (24 + 25 + 23 = 72; 72 divided by 3 =24).

measure of central see average tendency

41

median

meter

method

metric prefixes

mixed number

mode

multiple

nonstandard units of measure

the number in the middle of a set of data arranged in order from least to greatest or from greatest to least; or the average of the two middle terms if there is an even number of terms

Example:

For the data: 6. 14. 23, 46, 69, 72, 94 + The median is 46 (the middle number)

for the data: 6. 14, 23, 69, 72, 94 -b The median is also 46 (the average of the two middle numbers in the list)

a unit of linear measure in the metric system, a little more than a yard

a systematic way of accomplishing a task I

prefixes used to describe the relationship between measures in the metric system; the most common prefixes are: kilo- hecto- deka- (unit) deci- centi- milli- (k) (h) 1000 100

@a) (g.m.1) ( 4 10 1 1/10 11100 111000

(C) (m)

Example: 2 kg = 2 kilograms = 2000 grams 3 cm = 3 centimeters = .03 m

a number expressed as the sum of an integer and a fraction

Example: 6 $

the number that’occurs most frequently in a set of data

Example: For the set of data (42.36, 75, 75, 80). 75 is the mode because it occurs most oflen. For the set of data (25, 45.25, 55, 45, 651.25 and 45 are the modes. I f no number occurs most often, then the set of data has no mode.

a number that is divisible by a given factor; a multiple of a whole number can be found by multiplying it by any whole number other than one

Example: 56 is a multiple of 7; 0 is a multiple of 34

measurement units that are not commonly accepted as standard but are applied uniformly when measuring

Example: paperclips, pencils. cubes

42

number line a line that shows numbers ordered by magnitude from left to right or bonom to top; an arrowhead at each end indicates that the line continues endlessly in both directions; points are marked to subdivide the line into intervals that correspond to indicated numbers

-2 -1 0 1 2 4 ' ' I ' 1 : ' ' '

-1 i i 2.7 +

number sentence an expression of a relationship between quantities as an equation or an inequality

Example: 7 + 7 = 8 + 6 14 < 92 5 6 + 4 > 5 9

numerator the number above the line in a fraction; indicates the number of equivalent parts being considered

obtuse see angle and triangle

octagon an eight-sided polygon

odd number

operation

regular nonregular octagon octagons

a whole number that is not divisible by two

Example: 53, 70 1

a mathematical process that combines numbers; basic operations of arithmetic include addition, subtraction, multiplication. and division

43

ordered pair

order of operations

two numbers for which their order is important when used to locate points on a coordinate graph; the first element indicates distance along the x-axis (horizontal) and the second indicates distance along the y-axis (vertical); see illustration for grid

rules describing the order to use in evaluating numerical expressions: the order is parentheses, exponents, multiplyldivide, addkubtract

Example: 7 + 3 x 8 = 31 [multiply 3 x 8 before adding 71

ordinal number a number that designates the position of an object in order; first. second, and third are examples of ordinal numbers

' Example: Eraser is the SECOND element in the set (pencil, eraser, desk, chalkboard, book, file, paper); Z is the WENWSIXTH element in the set (a, b, c, d ,... , z).

outlier

parallel lines

parallelogram

pattern

a number in a set of data that is much larger or smaller than most of the other numbers in the set

lines that lie in the same plane and never intersect

a quadrilateral with opposite sides parallel

parallelograms

the arrangement of numbers, pictures, etc. in an organized and predictable way

Example: 3, 6, 9, 12, .._ O A O A O A

44

pentagon a five-sided polygon

percent

regular pentagon

nonregular pentagon

a special ratio that compares a number to 100 using the symbol '10

Example: 40% =

perimeter the distance around the outside of a shape or figure

perpendicular lines lines that lie on the same plane that intersect to form right angles (90 degrees)

place value

point

Polygon

polyhedron

population

precision

the value of a digit as determined by its place in a number

Example: In the number 135, the 3 means 3 10 or 30; in the number 356, the 3 means 3 100 or 300.

an exact position in space

a closed plane figure having three or more straight sides

a solid figure, the sides of which are polygons

Example:

a group of people, objects, or events that fit a particular description

an indication of how finely a measurement is made; related to the unit of measurement and the calibration of the tool

Example: Was the measurement made using a ruler marked in increments off or increments of & '?

predict

prime number

prism

to tell about or make known in advance, especially on the basis of special knowledge or inference; to make an educated guess

a number having exactly 2 factors (1 and itself); the first five prime numbers are 2, 3, 5, 7 , 11

a 3-dimensional figure that has 2 congruent and parallel faces that are polygons and the remaining faces are parallelograms

Example:

probability

product

proportion

pyramid

the numerical measure of the chance that a particular event will occur, depending on the possible events: the probability of an event is always between 0 and 1, with 0 meaning that there is no chance of occurrence and 1 meaning a certainty of Occurrence

the result of a multiplication expression; factor X factor = product

Example: 3 x 4 = 12, 12 is the product

an equation showing that two ratios are equivalent

Example: 3 = 6 9

a solid whose base is a polygon and whose faces are triangles that meet at a common point (vertex)

pyramids

quadrilateral

quotient

a four-sided polygon

square rectangle parallelogram trapezoid

Q G quadrilaterals

quadrilateral nonregular rhombus

the result of dividing one number by another number

divisor)- quotient

dividend + divisor = Quotient

random sample a sample in which every person, object, or event in the population has the same chance of being selected for the sample

ratio a comparison of two numbers using division

Example: The ratio of two to five is 2:5 or$.

rational number a number that can be expressed as a ratio of two integers

Example: 34 can be written as i. 4.32 can be written as m. 3? can be written as

34 432 1

7

a part of a line that has one end point and extends infinitely in one direction . *

real number any rational or irrational number

reasonable within likely bounds; sensible

Example: A reasonable estimate is close to the actual answer; an answer of 2; cans is not reasonable, while 2 cans or 3 cans is reasonable.

47

rectangle

reflection

a parallelogram with right angles; a square is a special rectangle

a transformation of a figure by reflecting it over a line, creating a mirror image of the figure; the effect of a flip is a reflection

reflection i

reflection i

regular polygon. a polygon with equal sides and equal angles

remainder

regular polygons

the undivided part that is left after division; it is less than the divisor

Example:

48

represent

revise

rhombus

to present clearly; describe; show

to change or modify based on guess and check or on reflection and evaluation

a parallelogram with all four sides equal in length

right angle an angle whose measure is 90 degrees; see angle and triangle

right triangle

90"

a triangle having one right angle; see angle and triangle

rotation turning a figure around a given point

A

49

rule

sample

sample space

scalene

sequence

side

a procedure; a prescribed method; a way of describing the relationship between two sets of numbers

Example: In the fol/owing data. the rule is to add 3.

k t 3 6 5 8 9 72

a portion of a population or set used in statistics

Example: All boys under the age of ten constitute a sample of the population of a// male children.

a list of all possible outcomes of an activity

see triangle

a set of numbers arranged in a special order or pattern

a line segment connected to other segments to form the boundary of a polygon

similar figures having the same shape bot not necessarily the same siie (congruent corresponding angles and proportional corresponding sides)

similar triangles similar hexagons

50

simplest form

slide

solid

solution

solve

sphere

a fraction with a numerator and denominator having no common factor except 1

Example: The fraction 3 IS In simplest form, because the only number that evenly

divides 2 and 3 is 7; the fraction 6 is NOT in simplest form, becaose'4 and 10

both have a factor of 2.

2 . .

the effect of a slide is a translation; see translation

a geometric figure with three dimensions

result; answer; the process of finding the answer

to find an answer or solution to a problem

a closed surface consisting of all points in space that are the same distance from a given point (the center)

square a rectangle with congruent sides

standard form

standard units of measure

a number written with one digit for each place value

Example: The standard form for five hundred fony-six is 546. The standard form for three thousand six is 3.006.

units of measure commonly used, generally classified in the U.S. customary system or metric system

Example: feet, meters. acres, gallons. liters

51

stem-and-leaf plot

strategy

sum

survey

symbol

symmetrical

symmetry

table

a method of organizing data from least to greatest using the digits of the greatest place value to group data

Example: Ages of Adults af the Park

23 25 29 29 Data set Stem

2 3599 36 38 39 39 3 1 6 8 9 9 52 54 55 55 5 2 4 5 5

Key: 2/3 represents 23 years

a tool used in problem solving, such as looking for a pattern, drawing a diagram, working backward, etc.

the result of addition

addend + addend = sum

to get an overview by gathering data

a character used to represent operations, numbers. or relationships between numbers

having a line, plane, or point of symmetry such that for every point on the figure, there is a corresponding point that is the reflection of that point (see line of symmetry)

i symmetrical

the property of being balanced about a line, plane, or point

a method of displaying data in rows and columns

52

t-chart

three-

figure dimensional

transformatlon

translation

trapezoid

trend

a table of values; an input-output table

a shape having length, width, and height

one of three methods for moving a figure without changing its shape or size: translations (slides), reflections (flips), and rotations (turns)

a transformation of a figure by moving it without turning or flipping it in any direction: the effect of a slide is a translation

a quadrilateral that has 2 parallel sides; an alternate definition is a quadrilateral with at least 2 parallel sides (there is no common agreement on a definition of a trapezoid)

the general direction or tendency of a set of data

53

triangle

turn

two- dimensional figure

unknown

validate

variable

verify

vertex

a three-sided polygon

Triangles Classified Using Angle Sizes

triangle triangle acute equiangular obtuse right

triangle (all angles acute) (all angles equal) (has one obtuse angle) (has one 90' angle)

triangle

Triangles Classified Using Length of Sides

scalene equilateral triangle triangle

isosceles triangle (two sides equal)

(no sides equal) (all sides equal)

see rotation

a shape having length and width

in algebra, the quantity represented by a variable

Example: In the expression 4 * w = 24, the letter w is the unknown.

to substantiate, verify, or confirm

a quantity or symbol capable of assuming any of a set of values

Example: In the expression b < 1 0 0 , the variable b can be any number less than 100.

to establish as true by presentation of evidence

point at which two line segments. lines, or rays'meet to form an angle

Example:

vertex I L, 54

vertical

volume

whole number

extending straight up and down; perpendicular to the horizon

the number of cubic units it takes to fill a figure

any counting number or zero: 0, 1, 2, 3, . . .

word form

<

> - -

zero property of addition

zero property of multiplication

whole numbers (0. 1.2.3 .4 . 5. ... ) integers (...-3. -2. -1, 0, 1 . 2. 3, ... )

rational numbers (fractions) such as 2. 7 , ... I . .. . 1 2 Q

the expression of a number in words; reading the symbols

Example: 546 is "five hundred fofiysix"

symbol meaning "is less than"

symbol meaning "is greater than"

symbol meaning "equals" or "is equal to"

zero is the identity element of addition.(see identity property) adding zero to a number gives a sum identical to the original number;

Example: 4 + 0 = 4 56.89 i 0 = 56.89

the product of any number and zero is zero

Example: 4 x 0 = 0 0 X 456.7 = 0

55

a framework for achieving the...

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