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Deerfield Community School Mathematics Curriculum August 2005 1 of 11 Deerfield Community School Mathematics Curriculum Table of Contents Contents Pages Philosophy and Essentials 1 through 11 Grade Level Curriculum Kindergarten K-1 through K-7 1st Grade 1-1 through 1-11 2nd Grade 2-1 through 2-13 3rd Grade 3-1 through 3-8 4th Grade 4-1 through 4-10 5th Grade 5-1 through 5-11 6th Grade 6-1 through 6-13 7th Grade 7-1 through 7-12 8th Grade 8-1 through 8-11 Resources Number and Operations NO-1 through NO-33 Geometry G-1 through G-58 Functions and Algebra FA-1 through FA-52 Data, Statistics, and DS-1 through DS-50 Probability Appendix A M(N&O)- 2-3 A1 Appendix B Measurement Benchmarks B1 through B2 2 of 11 THE EDUCATION OF DEERFIELD’S CHILDREN A SCHOOL PHILOSOPHY The purpose of the Deerfield Community School is to educate the young people of Deerfield by supporting goals which encourage growth in their personal lives, their homes, and community. Members of the staff are partners with parents and the larger community in supporting these goals. As partners we will meet, listen, and share knowledge. In this way we, as a community, share the commitment and the responsibility to provide for the growth of each child. We will offer each child the tools with which to live a healthy life, encouraging health as a value in itself and as an important element for social and intellectual functioning. We will foster the social development of each child, encouraging attention to others’ thoughts and a willingness to express one’s own as essential elements of learning. We will encourage the intellectual growth of each child. Each child is a ready learner. Teachers must provide children with developmentally appropriate learning opportunities which challenge and inspire. As adults, we are committed to preparing children for life as they become responsible and competent participants in the community. Children seek us as models and will listen to us. As we learn together, we will: read and write, work with concepts of number and shape, practice thinking scientifically, take an interest in the social goals of humanity, develop an involvement in the arts, and seek to live healthy lives. To do these things is basic to education. If we are successful, young people of Deerfield will grow up valuing themselves and others, and will contribute harmoniously in their own communities. 1 of 11 Mathematics Curriculum In developing the mathematics curriculum, we recognize that the Deerfield Community School is committed to five primary goals: • students will develop a firm grounding in essential computational skills; • students will develop strong mathematical problem solving and reasoning abilities; • students will develop positive attitudes about mathematics; • students will develop the ability to use appropriate technology to solve mathematical problems; and • students will develop the ability to effectively communicate their understanding of mathematics. Problem solving, while not a direct strand, should serve as the organizing feature of the mathematics curriculum as well as other areas of study and be applied to everyday activities. Problem solving must not be seen as a separate topic, but rather the centerpiece of the mathematics curriculum. Students should have many experiences in posing and solving problems from their world, from data that are meaningful to them, and from mathematical investigations. Students will use problem solving strategies to investigate and understand increasingly complex mathematical content. Students will communicate their understanding of mathematics. Reading, writing, talking, listening, and modeling provide students with the opportunity to integrate the language of mathematics into their world and help them to develop understanding. Actively exploring, investigating, describing, and explaining mathematical ideas promote communication which leads to a greater comprehension of mathematical concepts. 2 of 11 ESSENTIAL ENVIRONMENT The Essential Environment for the student working with mathematics is a setting which recognizes each person's abilities and looks forward to each person's contributions to the classroom community. The community supports individuals and sets standards which define what is expected of contributing members. As learning takes place, members of the learning community communicate their questions and understandings, reason their ways to solutions, connect what they have learned within mathematics and to other areas of learning, and undergo a process of problem solving in which mathematics is applied to everyday life. This Essential Environment includes these elements: Validation: The student is viewed directly and responded to directly by the teacher. The teacher does not label the student, but instead values his mathematical work and his actions in order to better understand his perspective. In this way appropriate work plans are determined in consultation with the student. The teacher's behavior in this regard becomes a model for students as they generate work plans in consultation with one another. There is an atmosphere of respect for the genuine efforts of each person. Evidence of this atmosphere appears in the actions of adults and students as they listen carefully and respond honestly to one another through constructive criticism and praise and help one another to revisit, extend, or deepen work. Involvement: Each student is encouraged to share particular mathematical interests and to develop expertise based upon these interests. Students realize that mathematics touches many places in people's lives and provides many points through which to develop interests and expertise. Students value one another's involvement in the various strands and seek each other out in order to share knowledge, understanding, and their excitement about learning. Expectation: The teacher expects the student to follow through on planned work. The student expects to be able to use the teacher and other students as resources to be called upon at appropriate times to help in the work. The teacher is open to consulting on expectations about student work, showing flexibility when this is important for student learning. Each student expects to refine some of his mathematical work for public use or display. Immersion: Mathematics is given real purpose. Students use calendars, clocks, tallies of lunch counts, order forms, and more as they go about the business of daily life in the classroom. Further, mathematics is linked with and used for other areas of study. Students gather data as they study the lives of others and as they investigate the natural world. Students create patterns as they undertake an art project. Students measure out quantities for a cooking project or a science experiment. Along with many other experiences, these uses of mathematics make it an essential part of everyday learning. 3 of 11 Wondering: A spirit of inquiry is encouraged. The teacher models this by openly acknowledging his own questions and his own mistakes. Students are given plenty of opportunity to share their own wonderings and to learn from their mistakes. Approximation is viewed as a valid tool of thinking in some situations. Problem solving, based on wonderings and careful thinking, becomes a part of life in the classroom. 4 of 11 ILLUSTRATING ESSENTIAL ENVIRONMENT We are a part of our environment, and the environment is a part of us. Illustrations of how the different elements of the Essential Environment work will show them working together, as in the following anecdotal record: When I arrived, the class was having a group meeting. The children had paired up and were conducting surveys among class members. Conversations I had had with the teacher prior to the visit established that the idea of surveys and data collection, along with representation of data through graphing, were understandings and actions which the teacher wanted the students to develop over a period of time. She does not want the students to merely learn how to read a graph. She wants them to develop reasoning skills related to data collection and interpretation. Today's steps in this ongoing process include communication. The intent of the meeting includes reviewing purpose and checking details: "Nicholas, who's your partner and what's your survey question?" Issues are fleshed out as a part of problem solving: "So how will someone answer your survey question?" Connections are made: "It'll be interesting to compare your data with the data we collected the other day." The teacher ends the group meeting by setting out the agenda for what will come next. Students who had finished compiling data based on their surveys would meet with her. Others would continue with their surveys, asking classmates the questions they had formulated. In this brief snapshot of a class meeting we see several elements described as essential to the classroom environment. The teacher is setting out the expectation, making certain that students understand their plans and have thought about how their plans will work. There is an element of wondering here. A question is asked which might prompt the student to put himself in the place of the person answering his question. A question is implied which refers back to previous work, to the immersion which has been established around this mathematical topic. There is validation. Students are addressed directly. The teacher has knowledge of the students' work. She also knows her purposes for the students. She has expertise in the area of mathematics and is directing students as they develop their plans and their own expertise. There is involvement. While the overall mathematical understandings and actions of the data collection project fits with the teacher's knowledge of what students need to learn, the content of the project is largely student determined. The teacher has modeled the process prior to this class meeting. All of the students have witnessed the development of a class graph based on a survey they participated in together. Now, they are off on their own, working with their own survey questions, developing their own graph models - pursuing their own interests and questions, following through on expectations, and growing in expertise. 5 of 11 ESSENTIAL ACTIONS The Essential Actions of mathematics form a web of strategies which give the person power to use mathematics for his own problem solving purposes. The Actions discussed below are common to most mathematical situations and many other learning situations, as a person organizes for problem solving. Recognizing: First, there must be recognition of a problem situation. Problem here is not meant in the negative sense. The problem to be solved may be a practical one, something in need of a solution before moving on; or it may be a source of wondering. As an example, we'll take a practical situation. It is a dry fall. Your dug well has been known to run out of water at this time of year. This is a problem. You need to have some idea of how much water you have available so that you can plan. The mathematical recognition of the problem situation spots the application of mathematics as a help toward planning for the lack of water. You figure that you will have to call upon your understanding of how to measure volume. This is also another aspect of recognition - a kind of classification - knowing what the situation entails and the understandings upon which you will have to call as you work toward a solution. Planning: After recognizing a problem situation and how math would apply to that situation, it becomes essential to develop a plan for working toward a solution. Planning involves mapping out the possibilities, then choosing from among them to develop a route which makes sense, given a person's mathematical understandings. Based upon your recognition of the problem situation and your mathematical understanding, you think that you need to find the volume of water and that volume can be measured by multiplying length times width times height. Finding volume will be step one. Next, you realize that you will have to find out your rate of use of the water in your well. In order to determine this you will have to compare the water levels at different times. Finally, you will need to use this comparison to figure out when you could expect to run out of water at your current rate of use. This will help you to know how much you will need to conserve water. 6 of 11 Researching: After recognizing a problem situation and developing a plan for coming to a solution there may be a need to gather more information. The initial need for research should become apparent as the problem situation is defined, but may come into play at any time during the problem solving process. To return to our example: You aren't sure just how much water you have now. Research here takes the form of the physical act of lifting the well cover and using some kind of measuring device to determine the depth of the water. Perhaps you have an idea of your own, but this may be a time to use another of the Essential Actions which may come into play at any time during the problem solving process - consulting. Your neighbor has an idea. Use a piece of rope which is weighted on the drop end, put it into the well and see how far up the wetness comes by measuring with a tape measure after bringing the rope up out of the well. Now that you have the height, you think that you need to find length and width. Suddenly, you are hit with a new problem. Your well is roughly circular. You will need to call upon another Essential Action. You will need to rethink your strategy. Rethinking: The need to think back upon a plan comes up frequently, the more frequently the more complicated the problem. Taking action, really working on something, may reveal gaps in planning which the initial thinking through did not reveal. Futhermore, assumptions may have been made which do not fit with the problem situation. The case of the dug well involves the latter. You suddenly realize that there is no way to measure width and length since the structure that you are measuring is not rectangular in shape. You are tempted to bring in another Essential Action, approximation, by inscribing a square on the top of your well (using fastened down string), allowing for a measurement of width and length which will help determine a large portion of the volume of the water. [You are already approximating (estimating) somewhat since you are unsure of how high the intake pipe is from the bottom of the well.] Or you could go back to researching, checking reference books for a formula which will help you determine volume for a cylindrical object. The wondering part of you wants to do this, but the physical act of measuring with the rope over a couple of days has brought you to a new realization. All you really need to know is how much the water goes down over a period of time. Length is the crucial factor, since distance across will remain the same over time (no matter how low the water gets - until it runs out!). You return to planning, having chosen a new strategy for problem solving, and in this case, having simplified the problem greatly. Solving: When the plan has been established, the research undertaken, and the rethinking accomplished, the problem may be solved. In the example of the dug well: You measured the length of wetness on the rope on Monday and find you have 8 feet. On Wednesday you have 7 feet six inches. On Friday you have 7 feet two inches. On Sunday you have 6 feet 7 inches. You convert to inches: Monday=96" Wednesday=90" Friday=86" Sunday=79", a loss of 6 inches, then 4 inches, then 7 inches. Your rate of loss is uneven, probably due to wash days being on some days and not on others. You rethink, realizing that the variables of the situation, including uneven use of water and the (much hoped for!) possibility of rain, make an exact calculation of the date of no water fairly useless. You could keep measuring and averaging loss per day to calculate this, but you realize that something else may be more useful. You go public with your information. 7 of 11 Communicating: When we communicate what we have discovered, we share our information and understandings. The form of expression becomes important. What is the best way to get my message across? Also important is the attention to appearance. What will someone else want to see and find easy to use? Mathematical expression may come out of the excitement of discovery and the desire to let the other person in on what has been found, seen, or understood. It may also serve a practical purpose, as in the case of the well. You call your family together around the dining room table. You have begun a graph entitled The Height of the Water in Our Well. You invite the family to help you color in the bars which show heights for the days you've tabulated so far. There is space left for information from future days to be entered. You also have a chart to take outside for recording height. Though the children are not to measure the length of wetness of the rope on their own (fortunately, the well cover is very heavy), you invite them to accompany you as you work on this project together. The chart will go on the kitchen wall, a reminder to conserve water, but also a possible reassurance that things aren't as bad as they seem. Then your daughter makes a suggestion. Extending: Communicating a mathematical discovery or mathematical information does bring a point of closure, but in many cases it may only be a pause along the way towards more work. Perhaps that piece of work will spark a new idea and be useful as a part of future work. Such was the case with the family facing drought. Recently, your daughter's class had set out a rain gauge at school to measure any rain that did fall during that dry season. She knows how to make a gauge and says she'd like to keep track of how much it rains at home. "Wouldn't it be interesting to see how much our well water changes after it rains?" Everyone agrees to help make and fill in a graph which will measure rainfall. Deepening: Deepening comes through reflection. After working through problems, a person may take stock of new understandings and new strategies. The person engages in self-assessment, considering what he knows and understands, and how to use what he has gained. The new learning will become a part of what the person brings to problem solving situations in the future. In the case of the well: You really knew it all along, but you didn't know that you knew it. Realizing that you just needed to measure the change in water height, you saw more clearly the logic of a situation involving a constant factor and a variable. You know that what really matters is how much water there is in the well, and that that has to do with a change in volume. But volume is a little harder to measure. Since length will change with volume and width will not, measuring length will be sufficient to indicate changes in the amount of water available. Hence, your new strategy has been to measure length only. 8 of 11 Additional Essential Actions embedded in the problem solving process: Consulting: Consulting may happen informally, or it may be a part of a strategy for problem solving. In some situations a person may consult the expertise of another person. Consulting is like researching, except that it implies working with another person to seek out information or ideas. It is natural to enter into consultations with others frequently during the problem solving process. Approximating: Approximating or estimating is useful for several purposes. First, estimation will give a good idea of the reasonableness of an answer. Thus, after a formal calculation or measurement takes place, comparing that answer with an estimate will be a good check. Second, some situations really require only an approximate answer. Since approximations can be done more easily in the head, they make sense when they are all that is required to fit a situation. Third, the world is rough. Building a house will require some close to exact measurements if the house is to stand right. This is true for many human made things. But if we are measuring some phenomenon, such as the depth of a dug well, an approximation will have to do. 9 of 11 ILLUSTRATING ESSENTIAL ACTIONS The Essential Actions of mathematics form the solid bed of purpose upon which mathematical knowledge and understandings can be built. In the home, in the work place, and in the school, problem solving becomes meaningful within the context of real action. In the example of the well, the setting for problem solving was the home, though there was a home/school connection when the rain gauge built in school was brought in to solve a problem at home. Home/school connections involving real action purposes for mathematics can work the other way as well. At the Deerfield Community School, one example of this is a project in which a grade level takes the perceived need for useful and informative calendars and meets this need through creating, designing, producing, and marketing calendars. Connections here exist not only between home and school, but also through the areas of learning and within mathematics. Students employ decision making, informational research skills, artistic skills, writing, measurement, patterning, data collection, graphing, data analysis, and more. During the fall of the year, students produce these calendars which will be sold to the general public in December. Students determine a theme for the calendar, develop a design both for the month grid and for facing pages, work through the specifics of layout, write and draw to provide content relative to the theme, then market the calendar, account for sales, and analyze sales data. All of these steps require problem solving strategies common to all learning, as well as problem solving strategies specific to mathematics. Students recognize problems, develop plans for solutions, consult with one another, research as needed, approximate on initial drafts, rethink plans, solve, then communicate solutions. Through this learning process, they extend and deepen their learning. The calendar project is an example of integrated learning in which mathematics plays a pivotal role. 10 of 11 ESSENTIAL UNDERSTANDINGS The section of the guide which follows is organized according to the state Grade Level Expectations (GLEs) which are broken down into the following four strands: Number and Operations; Geometry and Measurement; Functions and Algebra; Data, Statistics and Probability. Number and Operations: Students will develop number sense and an understanding of our numeration system. Students must understand numbers if they are to make sense of the ways numbers are used in their everyday world. Numbers are used to describe and interpret real-world phenomenon. Students need to use numbers to quantify, to identify location, to identify a specific object in a collection, to name, to measure, and to model real-world situations. They need to understand relative magnitude in order to make sense of everyday situations. Geometry and Measurement: Students will name, describe, model, classify, and compare geometric shapes and their properties with an emphasis on their wide applicability in human activity. Geometry helps students represent and describe the world in which they live. Students need to investigate, experiment, and explore geometric properties using both technology and hands on materials. Functions and Algebra: Students will recognize patterns and describe and represent relations and functions with tables, graphs, equations and rules, and analyze how a change in one element results in a change in another. One of the central themes of mathematics is the study of patterns, relations, and functions. This study requires students to recognize, describe, and generalize patterns and build mathematical models to predict the behavior of real-world phenomenon that exhibit the observed pattern. This study of patterns leads to an exploration of functions, a concept which is an important unifying idea in all aspects of mathematics. Data, Statistics, and Probability: Students will use data analysis, statistics and probability to analyze given situations and the outcomes of experiments. Collecting, organizing, displaying, and interpreting data, as well as using the information to make decisions and predictions, have become very important in our society. Statistical instruction should be carried out in a spirit of investigation and exploration so students can answer questions about data. Probability must be studied in familiar contexts encouraging students to model situations. Students need to investigate fairness, chances of winning, and uncertainty. Technology should be used as a tool throughout the investigation process. 11 of 11 Grade Level K Expectations by Strand Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(N&O)- Demonstrates conceptual understanding of rational o Rational number (N&O-1) Developing Number Concepts K-1 numbers with respect to: o Whole number (N&O-2) Book 1 – chapter 1 whole numbers from 0 to 12 through investigations o Fraction (N&O-3) that apply the concepts of equivalency in composing or o Equivalent numbers (N&O-14) Investigations in Number, Data decomposing numbers using models, explanations, o Composition of numbers and Space Series or other representations. (N&O-15) o Decomposition of numbers Demonstrates conceptual understanding of rational (N&O-16) Materials – unifix cubes, color numbers with respect to: positive fractional numbers (1/2) as “fair share” (i.e., o Area model to represent part to tiles, teddy bear counters etc. equal sized parts or sets) using models, explanations, whole relationship (N&O-17) or other representations. o Set model (N&O-18) These resources apply to all GLE’s in this section M(N&O)- Demonstrates understanding of the relative o Whole number (N&O-2) Developing Number Concepts K-2 magnitude of numbers from 0 to 20 through o Area model to represent part to Book 1 – chapters 1 & 3 investigations that demonstrate one-to-one whole relationship (N&O-17) correspondence. o Set model (N&O-18) Investigations in Number Data o Linear model (N&O-19) and Space Series Demonstrates understanding of the relative o Relative magnitude (N&O-20) magnitude of numbers from 0 to 20 through o Within number formats investigations that compare whole numbers to each (N&O-21) Materials – beans, unifix cubes, other or to benchmark whole numbers (5, 10). o Ordering (N&O-26) etc. Demonstrates understanding of the relative o Comparing (N&O-27) magnitude of numbers from 0 to 20 through o Number line (N&O-28) investigations that demonstrate an understanding of the relation of inequality when comparing whole numbers by using “1 more” or “1 less”. Demonstrates understanding of the relative magnitude of numbers from 0 to 20 through investigations that connect numbers orally and written as numerals to the quantities that they represent using models, representations, or number lines. These resources apply to all GLE’s in this section K-1 Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(N&O)- Demonstrates conceptual understanding of o Whole number (N&O-2) Developing Number Concepts – K-3 mathematical operations through investigations o Composition of numbers Book 2 involving addition and subtraction of whole numbers (N&O-15) (from 0 to 10) by solving problems involving joining o Decomposition of numbers actions, separating actions, part-part whole (N&O-16) Investigations in Number, Data relationships, and comparison situations. and Space Series Demonstrates conceptual understanding of mathematical operations through investigations involving addition of multiple one-digit whole numbers. Materials – teddy bear counters, (See Appendix A.) color tiles etc. These resources apply to all GLE’s in this section M(N&O)- None K-4 Demonstrates understanding of monetary value o Decimal (N&O-7) None through investigations involving knowing the names o Equivalent numbers (N&O-14) M(N&O)- and values for coins (penny, nickel and dime). o Composition of numbers K-5 (N&O-15) o Decomposition of numbers (N&O-16) M(N&O)- Mentally adds and subtracts whole numbers by All of this is referred to mentally! K-6 naming the number that is one more or one less than the original number. Developing Number Concepts – Book (IMPORTANT: The intent of this GLE is to embed 1 Chapter 3 and Book 2 Chapter 2 mental arithmetic throughout the instructional program, not to teach it as a separate unit.) K-2 Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(N&O)- Makes estimates of the number of objects in a set (up Developing Number Concepts – K-7 to 20) by making and revising estimates as objects are Book 1 Chapters 1 & 3 counted (e.g., A student estimates the number of pennies in a jar as 20. Then the student counts the first Investigations in Number Data 10 and makes another estimate based on those that and Space Series have been counted and those that remain in the jar.). (IMPORTANT: Estimation should be imbedded Materials – Pennies, beans, instructionally throughout all strands.) blocks etc. M(N&O)- None K-8 K-3 Geometry and Measurement GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(G&M)- Uses properties, attributes, composition, or o Attributes and properties K-1 decomposition to sort or classify polygons (triangles, (G&M-1) squares, rectangles, rhombi, trapezoids, and hexagons) o Uses composition and Investigations in Number Data or objects by using one non-measurable or measurable decomposition (G&M-5) and Space –“Making Shapes and attribute. o Polygons (G&M-2) Building Blocks” & “Collecting, o Non-measurable attribute Counting and Measuring” Uses properties, attributes, composition, or (G&M-8) decomposition to recognize, name, and build polygons o Measurable attribute (G&M-9) Also found in software form and circles in the environment. under school wide applications Materials – Pattern blocks, craft sticks, geoblocks etc. M(G&M)- None K-2 M(G&M)- None K-3 M(G&M)- None K-4 M(G&M)- None K-5 M(G&M)- None K-6 K-4 Geometry and Measurement GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(G&M)- Demonstrates conceptual understanding of o Measures and uses units of Developing Number Concepts – K-7 measurable attributes using comparative language to measure appropriately and Book 1 Chapters 1 & 3 describe and compare attributes of objects (length consistently (G&M-31) [longer, shorter], height [taller, shorter], weight [heavier , o Makes conversions within and lighter], temperature [warmer, cooler], and capacity across systems (G&M-32) [more, less]). Investigations in Number, Data and Space: “Collecting, Counting Demonstrates conceptual understanding of and Measuring” measurable attributes and compares objects visually and with direct comparison. Materials – ribbons, unifix cubes, blocks etc. M(G&M)- Determines elapsed and accrued time as it relates to Investigations in Number, Data K-8 calendar patterns (days of the week, yesterday, today, and Space: “Mathematical and tomorrow), the sequence of events in a day. Thinking in Kindergarten” Identifies a clock and calendar as measurement tools Developing Number Concepts – (days of week, months of the year). Book 1 Chapter 2 M(G&M)- Demonstrates understanding of spatial Investigation in Number, Data K-9 relationships by location and position using and Space – Series positional words to locate and describe where an object is found in the environment. Developing Number Concepts – Books 1 & 2 M(G&M)- None K-10 K-5 Functions and Algebra GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(F&A)– Identifies and extends to specific cases a variety of o Patterns (F&A-1) Investigations in Number, Data K–1 patterns (sequences of shapes, sounds, movement, o Extend a pattern (F&A-5) and Space: “Pattern Trains and colors, and letters) by extending the pattern to the next o Numeric patterns (F&A-3) Hopscotch Paths one, two or three elements, or by translating AB o Non-numeric patterns (F&A-4) patterns across formats (e.g., an abb pattern can be o Sequence (F&A-6) represented as snap, clap, clap or red, yellow, yellow) or o Pattern Summary Table by grade Developing Number Concepts – by identifying number patterns in the environment. level (F&A-9) Book 1 Chapter 2 Materials – pattern blocks, unifix cubes, teddy bear counters etc. M(F&A)– None K–2 M(F&A)– None K–3 M(F&A)– None o Equality (F&A-30) K–4 o Demonstrates conceptual understanding of equality by solving equivalence (F&A-31) o Number sentences (F&A-33) o Equation (F&A-32) o Algebraic equation notation (F&A-34) o Examples of forms of equations (F&A-35) K-6 Data, Statistics, and Probability GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(DSP) – Interprets a given representation created by the o Interprets a given representation Investigations in Number, Data K-1 class (models and tally charts) to answer questions (DSP-21) and Space: “Counting Ourselves related to the data, or to analyze the data to formulate o Representation (DSP-1) and Others” conclusions using words, diagrams, or verbal/scribed o Pictograph (DSP-2) responses to express answers. o Line plot (DSP-5) o Tally chart (DSP-3) (IMPORTANT: Analyzes data consistent with concepts o Frequency table (DSP-4) and skills in M(DSP)–K–2.) M(DSP) – Analyzes patterns, trends, or distributions in data in o Pattern (F&A-1) Investigations in Number, Data K-2 a variety of contexts by determining or using more, and Space: “Counting Ourselves less, or equal (e.g., Have there been more, less, or the and Others” same number of cloudy days compared to sunny days this week?). Developing Number Concepts Book 1 Chapter 2 M(DSP) – None K-3 M(DSP) – None K-4 M(DSP) – None o Combination (DSP-41) K-5 o Frequency table (DSP-4) o Tree diagram (DSP-28) o Solves problems using a variety of counting strategies (DSP-39) M(DSP) – None K-6 K-7 Grade Level 1 Expectations by Strand Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(N&O)- Demonstrates conceptual understanding of rational o Rational number (N&O-1) 1-1 numbers with respect to: o Whole number (N&O-2) Developing Number Concepts, whole numbers from 0 to 100 using place value, by o Fraction (N&O-3) Book 1:Counting, Comparing, & applying the concepts of equivalency in composing or o Proper fraction (N&O-4) Pattern decomposing numbers; and in expanded notation using o Improper fraction (N&O-5) models, explanations, or other representations. o Ratio (N&O-12) Developing Number Concepts, o Expanded Notation (N&O-13) Book :Addition & Subtraction Demonstrates conceptual understanding of rational o Equivalent numbers (N&O-14) numbers with respect to: positive fractional numbers (benchmark fractions: a/2, o Composition of numbers Developing Number Concepts, a/3, or a/4, where a is a whole number greater than 0 (N&O-15) Book 3: Place Value, and less than or equal to the denominator) as a part to o Decomposition of numbers Multiplication & Division whole relationship in area models where the (N&O-16) denominator is equal to the number of parts in the whole o Area model to represent part to Investigations In Number, Data, using models, explanations, or other whole relationship (N&O-17) Space: Mathematical Thinking At representations. o Set model (N&O-18) Grade 1 Investigations In Number, Data, Space: Building Number Sense Investigations: Quilt Squares & Block Towns Understanding Geometry (K. Richardson) 1-1 Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(N&O)- Demonstrates understanding of the relative o Whole number (N&O-2) 1-2 magnitude of numbers from 0 to 100 by ordering o Area model to represent part to Developing Number Concepts, whole numbers. whole relationship (N&O-17) Book 1:Counting, Comparing, & o Set model (N&O-18) Pattern Demonstrates understanding of the relative o Linear model (N&O-19) magnitude of numbers from 0 to 100 by comparing o Relative magnitude (N&O-20) Developing Number Concepts, whole numbers to each other or to benchmark whole o Within number formats Book 2: Addition & Subtraction numbers (5, 10, 25, 50, 75, 100). (N&O-21) Demonstrates understanding of the relative o Ordering (N&O-26) magnitude of numbers from 0 to 100 by demonstrating o Comparing (N&O-27) Investigations In Number, Data, an understanding of the relation of inequality when o Number line (N&O-28) Space: Mathematical Thinking At comparing whole numbers by using “1 more”, “1 less”, Grade 1 “5 more”, “5 less”, “10 more”, “10 less”. Investigations In Number, Data, Demonstrates understanding of the relative Space: Building Number Sense magnitude of numbers and by connecting number words (from 0 to 20) and numerals (from 0 to 100) to the Investigations In Number, Data, quantities and positions that they represent using Space: Number Games & Story investigations, models, representations, or number Problems lines. Developing Number Concepts, M(N&O)- Demonstrates conceptual understanding of o Whole number (N&O-2) Book 2: Addition & SuUbtraction 1-3 mathematical operations involving addition and o Composition of numbers subtraction of whole numbers (from 0 to 30) by solving (N&O-15) Investigations In Number, Data, problems involving joining actions, separating actions, o Decomposition of numbers Space: Mathematical Thinking At part-part whole relationships, and comparison (N&O-16) Grade 1 situations. Investigations In Number, Data, Demonstrates conceptual understanding of Space: Building Number Sense mathematical operations involving addition of multiple one-digit whole numbers. (See Appendix A.) Investigations In Number, Data, Space: Number Games & Story Problems M(N&O)- None 1-4 1-2 Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number Demonstrates understanding of monetary value by o Decimal (N&O-7) Investigations In Number, Data, knowing the names and values for coins (penny, nickel, o Expanded Notation (N&O-13) Space: Number Games & Story M(N&O)- dime, and quarter). o Equivalent numbers (N&O-14) Problems 1-5 o Composition of numbers Demonstrates understanding of monetary value by (N&O-15) adding collections of like coins together to a sum no o Decomposition of numbers greater than $1.00. (N&O-16) M(N&O)- Mentally adds and subtracts whole numbers by All of this is referred to mentally! 1-6 naming the number that is one or two more or less than the original number. Mentally adds and subtracts whole number facts to ten (e.g., 5 + 3 = 8; 8 – 5 = 3). (IMPORTANT: The intent of this GLE is to embed mental arithmetic throughout the instructional program, not to teach it as a separate unit.) Developing Number Concepts, M(N&O)- Makes estimates of the number of objects in a set (up Book 1:Counting, Comparing, & 1-7 to 30) and revises estimates as objects are counted Pattern (e.g., A student estimates the number of pennies in a jar as 28. Then the student counts the first 10 and makes Developing Number Concepts, another estimate based on those that have been Book :Addition & Subtraction counted and those that remain in the jar.). (IMPORTANT: Estimation should be embedded instructionally throughout all strands.) Investigations In Number, Data, Space: Mathematical Thinking At Grade 1 Investigations In Number, Data, Space: Building Number Sense 1-3 Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number Developing Number Concepts, M(N&O)- Applies properties of numbers (odd, even, Book 1:Counting, Comparing, & 1-8 composition, and decomposition [e.g., 5 is the same as Pattern 2 + 3]) and field properties (commutative and identity for addition) to solve problems and to simplify Developing Number Concepts, computations involving whole numbers. Book :Addition & Subtraction Investigations In Number, Data, Space: Mathematical Thinking At Grade 1 Investigations In Number, Data, Space: Building Number Sense 1-4 Geometry and Measurement GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number Understanding Geometry M(G&M)- Uses properties, attributes, composition, or o Attributes and properties (K.Richardson) 1-1 decomposition to sort or classify polygons (G&M-1) (triangles, squares, rectangles, rhombi, trapezoids, and o Uses composition and Investigations In Number, Data, hexagons) or objects by a combination of two decomposition (G&M-5) Space: Mathematical Thinking At nonmeasurable or measurable attributes. o Polygons (G&M-2) Grade 1 o Non-measurable attribute Uses properties, attributes, composition, or (G&M-8) Investigations In Number, Data, decomposition to recognize, name, build, and draw o Measurable attribute (G&M-9) Space: Quilt Squares & Block polygons and circles in the environment. Towns M(G&M)- None 1-2 M(G&M)- Given an example of a three-dimensional geometric Understanding Geometry 1-3 shape (rectangular prisms, cylinders, or spheres) finds (K.Richardson) examples of objects in the environment that are of the same geometric shape (e.g., show a wooden cylinder Investigations In Number, Data, and students identify common objects of the same Space: Quilt Squares & Block shape). Towns Understanding Geometry M(G&M)- Demonstrates conceptual understanding of (K.Richardson) 1-4 congruency by making mirror images and creating shapes that have line symmetry. Investigations In Number, Data, Space: Quilt Squares & Block Towns M(G&M)- None 1-5 Investigations In Number, Data, M(G&M)- Demonstrates conceptual understanding of the Space: Quilt Squares & Block 1-6 length/height of a two-dimensional object using non- Towns standard units (e.g. comparing objects to trains of small cubes, using iterations of a small unit to measure an Investigations In Number, Data, object). Space: Bigger, Taller, Heavier, Smaller 1-5 Geometry and Measurement GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(G&M)- Demonstrates conceptual understanding of o Measures and uses units of 1-7 measurable attributes using comparative language to measure appropriately and describe and compare attributes of objects (length consistently (G&M-31) [longer, shorter], height [taller, shorter], weight [heavier, o Makes conversions within and lighter], temperature [warmer, cooler], and capacity across systems (G&M-32) [more, less]). Investigations In Number, Data, Space: Bigger, Taller, Heavier, Demonstrates conceptual understanding of Smaller measurable attributes and compares objects visually, with direct comparison, and using non-standard units. M(G&M)- Determines elapsed and accrued time as it relates to Investigations In Number, Data, 1-8 calendar patterns (days of the week, months of the Space: Mathematical Thinking At year), the sequence of events in a day Grade 1 Recognizes an hour and “on the ½ hour”. Investigations In Number, Data, Space: Bigger, Taller, Heavier, Smaller Investigations In Number, Data, M(G&M)- Demonstrates understanding of spatial Space: Mathematical Thinking At 1-9 relationships using location and position by using Grade 1 positional words (e.g., close by, on the right, underneath, above, beyond) to describe one location in reference to another on a map, in a diagram, and in the Investigations In Number, Data, environment. Space: Quilt Squares & Block Towns Investigations In Number, Data, Space: Bigger, Taller, Heavier, Smaller 1-6 Geometry and Measurement GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(G&M)- None 1-10 1-7 Functions and Algebra GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number Developing Number Concepts, M(F&A)– Identifies and extends to specific cases a variety of o Patterns (F&A-1) Book 1:Counting, Comparing, & 1–1 patterns (repeating and growing [numeric and non- o Extend a pattern (F&A-5) Pattern numeric]) represented in models, tables, or sequences o Numeric patterns (F&A-3) by extending the pattern to the next one, two, or three o Non-numeric patterns (F&A-4) Developing Number Concepts, elements, by finding a missing element (e.g., 2, 4, 6, o Sequence (F&A-6) Book :Addition & Subtraction ___, 10), or by translating repeating patterns across o Pattern Summary Table by grade formats (e.g., an abb pattern can be represented as level (F&A-9) Investigations In Number, Data, snap, clap, clap; or red, yellow, yellow; or 1,2,2). Space: Mathematical Thinking At Grade 1 Investigations In Number, Data, Space: Building Number Sense Investigations In Number, Data, Space: Number Games & Story Problems M(F&A)– None 1–2 M(F&A)– None 1–3 1-8 Functions and Algebra GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(F&A)– Demonstrates conceptual understanding of equality o Equality (F&A-30) Developing Number Concepts, 1–4 by finding the value that will make an open sentence o Demonstrates conceptual Book 1:Counting, Comparing, & true (e.g., 2 + ��= 7) (limited to one operation and understanding of equality by Pattern limited to use of addition or subtraction) using models, solving equivalence (F&A-31) verbal explanations, or written equations. o Number sentences (F&A-33) Developing Number Concepts, o Equation (F&A-32) Book :Addition & Subtraction o Algebraic equation notation (F&A-34) Investigations In Number, Data, o Examples of forms of equations Space: Mathematical Thinking At (F&A-35) Grade 1 Investigations In Number, Data, Space: Building Number Sense Investigations In Number, Data, Space: Number Games & Story Problems 1-9 Data, Statistics, and Probability GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number Investigations In Number, Data, M(DSP) – Interprets a given representation created by the o Interprets a given representation Space: Mathematical Thinking At 1-1 class (models, tally charts, pictographs with one-to-one (DSP-21) Grade 1 correspondence, and tables) to answer questions o Representation (DSP-1) related to the data, or to analyze the data to formulate o Pictograph (DSP-2) Investigations In Number, Data, conclusions using words, diagrams, or verbal/scribed o Line plot (DSP-5) Space: Survey Questions & responses to express answers. o Tally chart (DSP-3) Secret Rules o Frequency table (DSP-4) (IMPORTANT: Analyzes data consistent with concepts Investigations In Number, Data, and skills in M( DSP)– 1– 2. ) Space: Number Games & Story Problems Developing Number Concepts, M(DSP) – Analyzes patterns, trends, or distributions in data in o Pattern (F&A-1) Book 1:Counting, Comparing, & 1-2 a variety of contexts by determining or using more, Pattern less, or equal. Developing Number Concepts, Book :Addition & Subtraction Investigations In Number, Data, Space: Mathematical Thinking At Grade 1 Investigations In Number, Data, Space: Building Number Sense Investigations In Number, Data, Space: Number Games & Story Problems Investigations In Number, Data, Space: Survey Questions & Secret Rules M(DSP) – None 1-3 M(DSP) – None 1-4 1 - 10 Data, Statistics, and Probability GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(DSP) – For a probability event in which the sample space o Combination (DSP-41) 1-5 may or may not contain equally likely outcomes, o Frequency table (DSP-4) groups use experiments to describes the likelihood or o Tree diagram (DSP-28) chance of an event (using “more likely,” “less likely”, or o Solves problems using a variety “equally likely”). of counting strategies (DSP-39) M(DSP) – None 1-6 1 - 11 Grade Level 2 Expectations by Strand Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number Developing Number Concepts, M(N&O)- Demonstrates conceptual understanding of rational o Rational number (N&O-1) Book 2: Addition & Subtraction 2-1 numbers with respect to: o Whole number (N&O-2) whole numbers from 0 to 199 using place value, by o Fraction (N&O-3) Developing Number Concepts, applying the concepts of equivalency in composing or o Proper fraction (N&O-4) Book 3: Place Value, decomposing numbers (e.g., 34 = 17 + 17; 34 = 29 + 5); o Improper fraction (N&O-5) Multiplication & Division and in expanded notation (e.g., 141 = 1 hundred + 4 o Ratio (N&O-12) tens + 1 one or 141 = 100 + 40 + 1) using models, o Expanded Notation (N&O-13) explanations, or other representations. o Equivalent numbers (N&O-14) Developing Number Concepts Demonstrates conceptual understanding of rational o Composition of numbers Using Unifix Cubes numbers with respect to: (N&O-15) positive fractional numbers (benchmark fractions: a/2, o Decomposition of numbers a/3, or a/4, where a is a whole number greater than 0 (N&O-16) Math Their Way and less than or equal to the denominator) as a part to o Area model to represent part to whole relationship in area and set models where the whole relationship (N&O-17) denominator is equal to the number of parts in the whole o Set model (N&O-18) Investigations: Mathematical using models, explanations, or other Thinking At Grade 2 representations. Investigations: Coins, Coupons & Combinations Investigations: Shapes, Halves & Symmetry Investigations: Putting Together & Taking Apart 2-1 Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number Developing Number Concepts, M(N&O)- Demonstrates understanding of the relative o Whole number (N&O-2) Book 2: Addition & Subtraction 2-2 magnitude of numbers from 0 to 199 by ordering o Area model to represent part to whole numbers. whole relationship (N&O-17) Developing Number Concepts, o Set model (N&O-18) Book 3: Place Value, Demonstrates understanding of the relative o Linear model (N&O-19) Multiplication & Division magnitude of numbers by comparing whole numbers o Relative magnitude (N&O-20) to each other or to benchmark whole numbers (10, 25, o Within number formats Math Their Way 50, 75, 100, 125, 150, or 175). (N&O-21) Demonstrates understanding of the relative o Ordering (N&O-26) Investigations: Mathematical magnitude of numbers by demonstrating an o Comparing (N&O-27) Thinking At Grade 2 understanding of the relation of inequality when o Number line (N&O-28) comparing whole numbers by using "1 more", "1 less", Investigations: Coins, Coupons & "10 more", "10 less", "100 more", or "100 less". Combinations Demonstrates understanding of the relative Investigations: Putting Together magnitude of numbers or by connecting number & Taking Apart words and numerals to the quantities they represent using models, number lines, or explanations. Developing Number Concepts Using Unifix Cubes 2-2 Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number Developing Number Concepts, M(N&O)- Demonstrates conceptual understanding of o Whole number (N&O-2) Book 2: Addition & Subtraction 2-3 mathematical operations involving addition and o Composition of numbers subtraction of whole numbers by solving problems (N&O-15) Developing Number Concepts, involving joining actions, separating actions, part-part o Decomposition of numbers Book 3: Place Value, whole relationships, and comparison situations. (N&O-16) Multiplication & Division Demonstrates conceptual understanding of Math Their Way mathematical operations involving addition of multiple one-digit whole numbers. (See Appendix A.) Investigations: Mathematical Thinking At Grade 2 Investigations: Coins, Coupons & Combinations Investigations: Putting Together & Taking Apart Developing Number Concepts Using Unifix Cubes M(N&O)- None 2-4 M(N&O)- Demonstrates understanding of monetary value by o Decimal (N&O-7) Math Their Way 2-5 adding coins together to a value no greater than $1.99 o Expanded Notation (N&O-13) and representing the result in dollar notation; o Equivalent numbers (N&O-14) Investigations: Coins, Coupons & o Composition of numbers Combinations Demonstrates understanding of monetary value (N&O-15) making change from $1.00 or less. o Decomposition of numbers Investigations: Putting Together (N&O-16) & Taking Apart Demonstrates understanding of monetary value by recognizing equivalent coin representations of the same value (values up to $1.99). 2-3 Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(N&O)- Mentally adds and subtracts whole number facts to Developing Number Concepts, 2-6 a sum of 20. Book 1:Counting, Comparing & Pattern Names the number that is 10 more or less than the original number, and mentally adds and subtracts two- Developing Number Concepts, digit multiplies of ten (e.g., 60 + 80, 90 – 30). Book 2: Addition & Subtraction (IMPORTANT: The intent of this GLE is to embed Developing Number Concepts, mental arithmetic throughout the instructional program, Book 3: Place Value, not to teach it as a separate unit.) Multiplication & Division Math Their Way Investigations: Mathematical Thinking At Grade 2 Investigations: Coins, Coupons & Combinations Investigations: Putting Together & Taking Apart Developing Number Concepts Using Unifix Cubes 2-4 Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number Developing Number Concepts, M(N&O)- Makes estimates of the number of objects in a set (up Book 2: Addition & Subtraction 2-7 to 50) by selecting an appropriate method of estimation. Developing Number Concepts, (IMPORTANT: The intent of this GLE is to embed Book 3: Place Value, estimation throughout the instructional program, not to Multiplication & Division teach it as a separate unit.) Math Their Way Investigations: Mathematical Thinking At Grade 2 Investigations: Putting Together & Taking Apart Developing Number Concepts Using Unifix Cubes Developing Number Concepts, M(N&O)- Applies properties of numbers (odd and even) and Book 2: Addition & Subtraction 2-8 field properties (commutative for addition, identity for addition, and associative for addition) to solve Developing Number Concepts, problems and to simplify computations involving Book 3: Place Value, whole numbers. Multiplication & Division Math Their Way Investigations: Mathematical Thinking At Grade 2 Investigations: Coins, Coupons & Combinations Investigations: Putting Together & Taking Apart 2-5 Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number 2-6 Geometry and Measurement GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number Understanding Geometry (K. M(G&M)- Uses properties, attributes, composition, or o Attributes and properties Richardson) 2-1 decomposition to sort or classify polygons or objects (G&M-1) by a combination of two or more non-measurable or o Uses composition and Investigations: Does It Walk, measurable attributes. decomposition (G&M-5) Crawl, or Swim? o Polygons (G&M-2) o Non-measurable attribute Investigations: How Long? How (G&M-8) Far? o Measurable attribute (G&M-9) Investigations: Shapes, Halves & Symmetry M(G&M)- None 2-2 M(G&M)- None 2-3 M(G&M)- Demonstrates conceptual understanding of 2-4 congruency by composing and decomposing two- dimensional objects using models or explanations (e.g., using triangular pattern blocks to construct a figure congruent to the hexagonal pattern block). Understanding Geometry (K. Richardson) Demonstrates conceptual understanding of congruency and uses line symmetry to demonstrate Investigations: How Long? How congruent parts within a shape. Far? Investigations: Shapes, Halves & Symmetry M(G&M)- 2-5 None 2-7 Geometry and Measurement GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(G&M)- Demonstrates conceptual understanding of o Polygons (G&M-9) Understanding Geometry (K. 2-6 perimeter and area by using models or manipulatives o Demonstrates conceptual Richardson) to surround and cover polygons. understanding of perimeter and area using models and Investigations: How Long? How manipulatives to surround and Far? cover polygons (G&M-26) Investigations: Shapes, Halves & Symmetry M(G&M)- Measures and uses units of measures appropriately o Measures and uses units of Investigations: How Long? How 2-7 and consistently, and makes conversions within measure appropriately and systems when solving problems across the content Far? consistently (G&M-31) strands. o Makes conversions within and Investigations: Does It Walk, across systems (G&M-32) Benchmarks in Appendix B. Crawl, or Swim? M(G&M)- 2-8 None M(G&M)- Demonstrates understanding of spatial 2-9 relationships using location and position by using positional language in two- and three-dimensional Understanding Geometry (K. situations to describe and interpret relative positions Richardson) (e.g., above the surface of the desk, below the triangle on the paper). Investigations: How Long? How Far? Demonstrates understanding of spatial relationships using location and position by creating Investigations: Shapes, Halves & and interpreting simple maps and naming locations on Symmetry simple coordinate grids. M(G&M)- None 2-10 2-8 Functions and Algebra GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number Developing Number Concepts, M(F&A)– Identifies and extends to specific cases a variety of o Patterns (F&A-1) Book 2: Addition & Subtraction 2–1 patterns (linear and non-numeric) represented in o Extend a pattern (F&A-5) models, tables, or sequences by extending the pattern o Numeric patterns (F&A-3) Developing Number Concepts, to the next element, or finding a missing element (e.g., o Non-numeric patterns (F&A-4) Book 3: Place Value, 2, 4, 6, ___, 10). o Sequence (F&A-6) Multiplication & Division o Pattern Summary Table by grade level (F&A-9) Math Their Way Investigations: Mathematical Thinking At Grade 2 Developing Number Concepts Using Unifix Cubes M(F&A)– None 2–2 M(F&A)– None 2–3 2-9 Functions and Algebra GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number Developing Number Concepts, M(F&A)– Demonstrates conceptual understanding of equality o Equality (F&A-30) Book 2: Addition & Subtraction 2–4 by finding the value that will make an open sentence o Demonstrates conceptual true (e.g. 2 + □ = 7) (limited to one operation and limited understanding of equality by Developing Number Concepts, to use of addition or subtraction). solving equivalence (F&A-31) Book 3: Place Value, o Number sentences (F&A-33) Multiplication & Division o Equation (F&A-32) o Algebraic equation notation Math Their Way (F&A-34) o Examples of forms of equations Investigations: Mathematical (F&A-35) Thinking At Grade 2 Investigations: Coins, Coupons & Combinations Investigations: Putting Together & Taking Apart 2 - 10 Data, Statistics, and Probability GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number Investigations: Does It Walk, M(DSP) – Interprets a given representation (pictographs with o Interprets a given representation Crawl, or Swim? 2-1 one-to-one correspondence, line plots, tally charts, or (DSP-21) tables) to answer questions related to the data, or to o Representation (DSP-1) Developing Number Concepts, analyze the data to formulate conclusions. o Pictograph (DSP-2) Book 2: Addition & Subtraction o Line plot (DSP-5) (IMPORTANT: Analyzes data consistent with concepts o Tally chart (DSP-3) Developing Number Concepts, and skills in M(DSP)-2-2.) o Frequency table (DSP-4) Book 3: Place Value, Multiplication & Division Math Their Way Investigations: Mathematical Thinking At Grade 2 Investigations: Coins, Coupons & Combinations Investigations: Shapes, Halves & Symmetry Investigations: Putting Together & Taking Apart Developing Number Concepts Using Unifix Cubes 2 - 11 Data, Statistics, and Probability GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number Developing Number Concepts, M(DSP) – Analyzes patterns, trends, or distributions in data in o Pattern (F&A-1) Book 2: Addition & Subtraction 2-2 a variety of contexts by determining or using more, less, or equal. Developing Number Concepts, Book 3: Place Value, Multiplication & Division Math Their Way Investigations: Mathematical Thinking At Grade 2 Investigations: Coins, Coupons & Combinations Developing Number Concepts Using Unifix Cubes M(DSP) – None 2-3 Developing Number Concepts, M(DSP) – Uses counting techniques to solve problems o Combination (DSP-41) Book 2: Addition & Subtraction 2-4 involving combinations using a variety of strategies o Frequency table (DSP-4) (e.g., student diagrams, organized lists, tables, tree o Tree diagram (DSP-28) Developing Number Concepts, diagrams, or others) o Solves problems using a variety Book 3: Place Value, (e.g., How many ways can you make 50 cents using of counting strategies (DSP-39) Multiplication & Division nickels, dimes, and quarters?). Math Their Way Investigations: Mathematical Thinking At Grade 2 2 - 12 Data, Statistics, and Probability GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number Investigations: Does It Walk, M(DSP) – For a probability event in which the sample space Crawl, or Swim? 2-5 may or may not contain equally likely outcomes, uses experiments to describe the likelihood or chance Math Their Way of an event using “more likely,” “less likely,” “equally likely,” certain or impossible. Investigations: How Many Pockets? How Many Teeth? M(DSP) – In response to a teacher or student generated Math Their Way 2-6 question or hypothesis, groups decide the most effective method (e.g., survey, observation, Investigations: Does It Walk, experimentation) to collect the data (numerical or Crawl, or Swim? categorical) necessary to answer the question. Investigations: Mathematical In response to a teacher or student generated Thinking At Grade 2 question or hypothesis, groups collect, organize, and appropriately display the data. In response to a teacher or student generated question or hypothesis, groups analyze the data to draw conclusions about the question or hypothesis being tested, and when appropriate make predictions. (IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP)–2–2.) 2 - 13 Grade Level 3 Expectations by Strand Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(N&O)- Demonstrates conceptual understanding of rational o Rational number (N&O-1) 3-1 numbers with respect to: o Whole number (N&O-2) whole numbers from 0 to 999 through equivalency, o Fraction (N&O-3) composition, decomposition, or place value using o Proper fraction (N&O-4) models, explanations, or other representations. o Improper fraction (N&O-5) o Decimal (N&O-7) Demonstrates conceptual understanding of rational o Ratio (N&O-12) numbers with respect to: o Equivalent numbers (N&O-14) positive fractional numbers (benchmark fractions: a/2, a/3, a/4, a/6, or a/8, where a is a whole number greater o Composition of numbers than 0 and less than or equal to the denominator) as a (N&O-15) part to whole relationship in area and set models where o Decomposition of numbers the number of parts in the whole is equal to the (N&O-16) denominator using models, explanations, or other o Area model to represent part to representations. whole relationship (N&O-17) o Set model (N&O-18) Demonstrates conceptual understanding of rational numbers with respect to: decimals (within a context of money) as a part of 100 using models, explanations, or other representations. o Whole number (N&O-2) M(N&O)- Demonstrates understanding of the relative o Fraction (N&O-3) 3-2 magnitude of numbers from 0 to 999 by ordering o Proper fraction (N&O-4) whole numbers or by comparing whole numbers to o Improper fraction (N&O-5) benchmark whole numbers (100, 250, 500, or 750). o Equivalent numbers (N&O-14) o Area model to represent part to Demonstrates understanding of the relative whole relationship (N&O-17) magnitude of numbers from 0-999 by comparing o Set model (N&O-18) whole numbers to each other. o Linear model (N&O-19) Demonstrates understanding of the relative o Relative magnitude (N&O-20) magnitude of numbers by comparing or identifying o Within number formats equivalent positive fractional numbers (a/2, a/3, a/4 (N&O-21) where a is a whole number greater than 0 and less than o Ordering (N&O-26) or equal to the denominator) using models, number o Comparing (N&O-27) lines, or explanations. o Number line (N&O-28) 3-1 Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(N&O)- Demonstrates conceptual understanding of o Whole number (N&O-2) 3-3 mathematical operations by describing or illustrating o Area model to represent part to the inverse relationship between addition and whole relationship (N&O-17) subtraction of whole numbers. o Set model (N&O-18) o Number line (N&O-28) Demonstrates conceptual understanding of o Linear model (N&O-19) mathematical operations by describing or illustrating o Inverse relationships in the relationship between repeated addition and operations (N&O-29) multiplication using models, number lines, or explanations. o Relationship between repeated addition and multiplication of whole numbers (N&O-30) o Concept of multiplication (N&O-37) M(N&O)- Accurately solves problems involving addition and o Decimal (N&O-7) 3-4 subtraction with regrouping to 999. o Relationship between repeated addition and multiplication of Accurately solves problems involving the concept of whole numbers (N&O-30) multiplication. o Relationship between repeated subtraction and division of whole Accurately solves problems involving addition or numbers (N&O-31) subtraction of decimals with regrouping in the context of o Accurately solves problems money. (N&O-35) o In and out of context (N&O-36) o Concept of multiplication (N&O-37) M(N&O)- None 3-5 3-2 Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(N&O)- Mentally adds whole number facts through 20; adds All of this is referred to mentally! 3-6 two-digit and one-digit whole numbers; adds combinations of two-digit and three-digit whole numbers that are multiples of ten (e.g., 60 +50, 300 + 400, 320 + 90). Mentally subtracts whole number facts through 20; a one-digit whole number from a two-digit whole number (e.g., 37 – 5); and subtracts two-digit whole numbers that are multiples of ten and three-digit whole numbers that are multiples one hundred (e.g., 50 – 20, 500 – 200). (IMPORTANT: The intent of this GLE is to embed mental arithmetic throughout the instructional program, not to teach it as a separate unit.) M(N&O)- Makes estimates in a given situation by identifying 3-7 when estimation is appropriate, selecting the appropriate method of estimation, and evaluating the reasonableness of solutions appropriate to grade level GLEs across content strands. (IMPORTANT: The intent of this GLE is to embed estimation throughout the instructional program, not to teach it as a separate unit.) M(N&O)- Applies properties of numbers (odd, even, and 3-8 multiplicative property of zero for single-digit whole numbers [6 x 0 = 0]) to solve problems and to simplify computations involving whole numbers. Applies field properties (commutative for addition, associative for addition, identity for multiplication, and commutative for multiplication for single digit whole numbers [e.g., 3 x 4 = 4 x 3]) to solve problems and to simplify computations involving whole numbers. 3-3 Geometry and Measurement GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(G&M)- Uses properties or attributes of angles (number of o Attributes and properties 3-1 angles) or sides (number of sides or length of sides) or (G&M-1) composition or decomposition of shapes to identify, o Non-measurable attribute describe, or distinguish among triangles, squares, (G&M-8) rectangles, rhombi, trapezoids, hexagons, or circles. o Measurable attribute (G&M-9) o Angles (G&M-10) o Triangle (G&M-3) o Quadrilaterals (G&M-4) o Polygons (G&M-2) o Uses composition and decomposition (G&M-5) M(G&M)- None 3-2 M(G&M)- None 3-3 M(G&M)- Demonstrates conceptual understanding of 3-4 congruency by matching congruent figures using reflections, translations, and rotations (flips, slides, and turns) (e.g., recognizing when pentominoes are reflections, translations and rotations of each other). Demonstrates conceptual understanding of congruency by composing and decomposing two and three-dimensional objects using models or explanations (e.g., Given a cube, students use blocks to construct a congruent cube.). Demonstrates conceptual understanding of congruency by using line symmetry to demonstrate congruent parts within a shape. M(G&M)- Demonstrates conceptual understanding of 3-5 similarity by identifying similar shapes. 3-4 Geometry and Measurement GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(G&M)- Demonstrates conceptual understanding of o Polygons (G&M-2) 3-6 perimeter of polygons on grids using a variety of o Demonstrates conceptual models or manipulatives. Expresses all measures using understanding of perimeter and appropriate units. area using models and manipulatives to surround and Demonstrates conceptual understanding of area of cover polygons (G&M-26) rectangles on grids using a variety of models or o Measures and uses units of manipulatives. Expresses all measures using measure appropriately and appropriate units. consistently (G&M-31) M(G&M)- Measures and uses units of measures appropriately o Measures and uses units of 3-7 and consistently, and makes conversions within measure appropriately and systems when solving problems across the content consistently (G&M-31) strands. o Makes conversions within and across systems (G&M-32) Benchmarks in Appendix B. M(G&M)- None 3-8 M(G&M)- Demonstrates understanding of spatial 3-9 relationships using location and position by interpreting and giving directions from one location to another (e.g., classroom to the gym, from school to home) using positional words. Demonstrates understanding of spatial relationships using location and position between locations on a map or coordinate grid (first quadrant) using positional words or compass directions. M(G&M)- Demonstrates conceptual understanding of spatial 3-10 reasoning and visualization by copying, comparing, and drawing models of triangles, squares, rectangles, rhombi, trapezoids, hexagons, and circles. Demonstrates conceptual understanding of spatial reasoning and visualization by building models of rectangular prisms from three-dimensional representations. 3-5 Functions and Algebra GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(F&A)– Identifies and extends to specific cases a variety of o Patterns (F&A-1) 3–1 patterns (linear and non-numeric) represented in o Extend a pattern (F&A-5) models, tables, or sequences by extending the pattern o Numeric patterns (F&A-3) to the next one, two, or three elements, or finding o Non-numeric patterns (F&A-4) missing elements. o Sequence (F&A-6) o Pattern Summary Table by grade level (F&A-9) M(F&A)– None 3–2 M(F&A)– None 3–3 M(F&A)– Demonstrates conceptual understanding showing o Equality (F&A-30) 3–4 equivalence between two expressions using o Demonstrates equality (F&A-31) representations of the expressions; or by finding open o Number sentences (F&A-33) sentence true (e.g., 2 + �� = 7). (Limited to one o Equation (F&A-32) operation and limited to use addition, subtraction, or o Algebraic equation notation multiplication.) (F&A-34) o Examples of forms of equations (F&A-35) 3-6 Data, Statistics, and Probability GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(DSP) - Interprets a given representation (line plots, tally o Interprets a given representation 3-1 charts, tables, or bar graphs) to answer questions (DSP-21) related to the data, to analyze the data to formulate o Representation (DSP-1) conclusions, or to make predictions. o Line plot (DSP-5) o Tally chart (DSP-3) (IMPORTANT: Analyzes data consistent with concepts o Frequency table (DSP-4) and skills in M(DSP)-3-2.) o Bar graph (DSP-6) M(DSP) - Analyzes patterns, trends, or distributions in data in o Pattern (F&A-1) 3-2 a variety of contexts by determining or using most o Mode (DSP-17) frequent (mode), least frequent, largest, or smallest. M(DSP) - Identifies or describes representations or elements o Representation (DSP-1) 3-3 of representations that best display a given set of o Identifies or describes data or situation, consistent with the representations representations or elements of required in M(DSP)–3–1. representations that best display a given set of data or situation Organizes and displays data using tables, tally (DSP-23) charts, and bar graphs, to answer questions related to the data, to analyze the data to formulate conclusions, to make predictions, or to solve problems. (IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP)–3–2.) M(DSP) - Uses counting techniques to solve problems o Combinations (DSP-41) 3-4 involving combinations and simple permutations using a o Simple Permutations (DSP-42) variety of strategies (e.g., student diagrams, organized lists, tables, tree diagrams, or others). 3-7 Data, Statistics, and Probability GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(DSP) - For a probability event in which the sample space o Sample space (DSP-30) 3-5 may or may not contain equally likely outcomes, o Event (DSP-31) determines the likelihood of the occurrence of an event (using “more likely”, “less likely”, or “equally likely”). For a probability event in which the sample space may or may not contain equally likely outcomes, predicts the likelihood of an event using “more likely,” “less likely,” “equally likely,” certain, or impossible and tests the prediction through experiments; and determines if a game is fair. M(DSP) - In response to a teacher or student generated 3-6 question or hypothesis, groups decide the most effective method (e.g., survey, observation, experimentation) to collect the data (numerical or categorical) necessary to answer the question; In response to a teacher or student generated question or hypothesis, groups collect, organize, and appropriately display the data. In response to a teacher or student generated question or hypothesis, groups analyze the data to draw conclusions about the question or hypothesis being tested, and when appropriate make predictions. (IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP)–3–2.) 3-8 Grade Level 4 Expectations by Strand Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number Demonstrates conceptual understanding of rational o Rational number (N&O-1) Fraction Hamburger: puzzle M(N&O)- numbers with respect to: o Whole number (N&O-2) Investigations 4-1 whole numbers from 0 to 999,999 through equivalency, o Fraction (N&O-3) Fraction Bars composition, decomposition, or place value using o Proper fraction (N&O-4) Graph Paper models, explanations, or other representations. o Improper fraction (N&O-5) Geoboards o Decimal (N&O-7) Demonstrates conceptual understanding of rational o Ratio (N&O-12) numbers with respect to: o Equivalent numbers (N&O-14) positive fractional numbers (benchmark fractions: a/2, o Composition of numbers a/3, a/4, a/5, a/6, a/8, or a/10, where a is a whole number greater than 0 and less than or equal to the (N&O-15) denominator) as a part to whole relationship in area, set, o Decomposition of numbers or linear models where the number of parts in the whole (N&O-16) are equal to, and a multiple or factor of the denominator o Area model to represent part to using models, explanations, or other whole relationship (N&O-17) representations. o Set model (N&O-18) o Linear model (N&O-19) Demonstrates conceptual understanding of rational o In and out of context (N&O-36) numbers with respect to: o Factor (N&O-38) decimals as hundredths within the context of money, or o Multiples (N&O-39) tenths within the context of metric measurements (e.g., 2.3 cm) using models, explanations, or other representations. o Fraction (N&O-3) M(N&O)- Demonstrates understanding of the relative o Proper fraction (N&O-4) 4-2 magnitude of numbers from 0 to 999,999 by ordering o Decimal (N&O-7) or comparing whole numbers using models, number o Equivalent numbers (N&O-14) lines, or explanations. o Area model to represent part to whole relationship (N&O-17) Demonstrates understanding of the relative o Set model (N&O-18) magnitude of numbers from 0 to 999,999 by ordering, o Linear model (N&O-19) comparing, or identifying equivalent proper positive o Relative magnitude (N&O-20) fractional numbers; or decimals using models, number lines, or explanations. o Within number formats (N&O-21) o Ordering (N&O-26) o Comparing (N&O-27) o Number line (N&O-28) 4-1 Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number Math Their Way: Beans and M(N&O)- Demonstrates conceptual understanding of o Fraction (N&O-3) Cups 4-3 mathematical operations by describing or illustrating o Proper fraction (N&O-4) Family Math the relationship between repeated subtraction and o Improper fraction (N&O-5) Unifix Cubes division (no remainders). o Area model to represent part to Math Bingo (all concepts) whole relationship (N&O-17) Power Blocks Demonstrates conceptual understanding of o Set model (N&O-18) mathematical operations the inverse relationship o Linear model (N&O-19) between multiplication and division of whole numbers; o Number line (N&O-28) or the addition or subtraction of positive fractional numbers with like denominators using models, number o Inverse relationships in lines, or explanations. operations (N&O-29) o Relationship between repeated addition and multiplication of whole numbers (N&O-30) o Concept of multiplication (N&O-37) M(N&O)- Accurately solves problems involving multiple o Whole number (N&O-2) 4-4 operations on whole numbers or the use of the o Fraction (N&O-3) properties of factors and multiples. o Proper fraction (N&O-4) o Decimal (N&O-7) Accurately solves problems involving addition or o Accurately solves problems subtraction of decimals and positive proper fractions (N&O-35) with like denominators. (Multiplication limited to 2 digits o Concept of multiplication by 2 digits, and division limited to 1 digit divisors.) (N&O-37) (IMPORTANT: Applies the conventions of order of o Factor (N&O-38) operations where the left to right computations are o Multiples (N&O-39) modified only by the use of parentheses.) M(N&O)- None 4-5 4-2 Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(N&O)- Mentally adds whole number facts through 20; adds All of this is referred to mentally! 4-6 two-digit whole numbers, combinations of two-digit and 3-digit whole numbers that are multiples of ten, and 4- digit whole numbers that are multiples of 100 (limited to two addends) (e.g., 67 + 24; 320 + 430; 320 + 90; 1,300 + 1,400) Mentally subtracts a one-digit whole number from a two-digit whole number (e.g., 67 – 9); and subtracts combinations of two digit and three-digit whole numbers that are multiples of ten (e.g., 50 – 20, 230 – 80, 520 – 200). Mentally multiplies whole number facts to a product of 100 and calculates related division facts. (IMPORTANT: The intent of this GLE is to embed mental arithmetic throughout the instructional program, not to teach it as a separate unit.) Unifix Cubes M(N&O)- Makes estimates in a given situation by identifying 4-7 when estimation is appropriate, selecting the appropriate method of estimation, and evaluating the reasonableness of solutions appropriate to grade level GLEs across content strands. (IMPORTANT: The intent of this GLE is to embed estimation throughout the instructional program, not to teach it as a separate unit.) M(N&O)- Applies properties of numbers (odd, even, 4-8 multiplicative property of zero, and remainders) to solve problems and to simplify computations. Applies field properties (commutative, associative, and identity) to solve problems and to simplify computations. 4-3 Geometry and Measurement GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(G&M)- Uses properties or attributes of angles (number of o Attributes and properties 4-1 angles) or sides (number of sides, length of sides, (G&M-1) Geoboards parallelism, or perpendicularity) to identify, describe, o Angles (G&M-10) Tangrams or distinguish among triangles, squares, rectangles, o Parallel lines (G&M-7) Human Bodies rhombi, trapezoids, hexagons, or octagons; or classify o Perpendicular (G&M-6) 3-D space figures o angles relative to 90 as more than, less than, or equal o Triangle (G&M-3) Power Blocks to. o Quadrilaterals (G&M-4) o Polygons (G&M-2) M(G&M)- None 4-2 M(G&M)- Uses properties or attributes (shape of bases or o Attributes and properties(G&M-3) 4-3 number of lateral faces) to identify, compare, or o Three-dimensional shapes describe three-dimensional shapes (rectangular (G&M-15) prisms, triangular prisms, cylinders, or spheres). M(G&M)- Demonstrates conceptual understanding of o Congruent (G&M-16) 4-4 congruency by matching congruent figures using o Reflection (G&M-17) reflections, translations, or rotations (flips, slides, or o Translation (G&M-18) turns), or as the result of composing or decomposing o Rotation (G&M-19) shapes using models or explanations. o Composing and decomposing shapes (G&M-21) o Matching congruent figures using reflections, translations, and rotations (G&M-20) Mirrors o Similar figures (G&M-23) o Describes the proportional effect on linear dimensions of polygons or circles when scaling up or down while preserving the angles of polygons (G&M-24) 4-4 Geometry and Measurement GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(G&M)- Demonstrates conceptual understanding of o Similar figures (G&M-23) 4-5 similarity by applying scales on maps, or applying o Describes the proportional effect characteristics of similar figures (same shape but not on linear dimensions of polygons Maps necessarily the same size) to identify similar figures, or or circles when scaling up or to solve problems involving similar figures. Describes down while preserving the angles relationships using models or explanations. of polygons (G&M-24) o Proportional reasoning (N&O-44) M(G&M)- Demonstrates conceptual understanding of o Polygons (G&M-2) 4-6 perimeter of polygons, on grids using a variety of o Demonstrates conceptual models, manipulatives, or formulas. understanding of perimeter and area of polygons and irregular Demonstrates conceptual understanding of area of figures on grids (G&M-26) rectangles, polygons or irregular shapes on grids using o Measures and uses units of a variety of models, manipulatives, or formulas. measure appropriately and Tiles consistently (G&M-31) Expresses all measures using appropriate units. o Whole number bases and whole number exponents, and fractional bases with whole number exponents (N&O-24) o Concept of multiplication (N&O-7) M(G&M)- Measures and uses units of measures appropriately o Measures and uses units of 4-7 and consistently, and makes conversions within measure appropriately and systems when solving problems across the content consistently (G&M-31) Tpe measure strands. o Makes conversions within and Scale across systems (G&M-32) rulers Benchmarks in Appendix B. yard sticks meter sticks M(G&M)- None 4-8 4-5 Geometry and Measurement GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(G&M)- Demonstrates understanding of spatial 4-9 relationships using location and position by interpreting and giving directions between locations on a map or coordinate grid (first quadrant). Demonstrates understanding of spatial relationships using location and position by plotting Computer CD points in the first quadrant in context (e.g., games, Maps mapping); Demonstrates understanding of spatial relationships using location and position by finding the horizontal and vertical distances between points on a coordinate grid in the first quadrant. M(G&M)- Demonstrates conceptual understanding of spatial 4-10 reasoning and visualization by copying, comparing, and drawing models of triangles, squares, rectangles, rhombi, trapezoids, hexagons, octagons, and circles. Demonstrates conceptual understanding of spatial reasoning and visualization by building models of rectangular prisms from two- or three dimensional representations. Geoboards Blocks 4-6 Functions and Algebra GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(F&A)– Identifies and extends to specific cases a variety of o Patterns (F&A-1) Unifix Cubes 4–1 patterns (linear and nonlinear) represented in models, o Extend a pattern (F&A-5) tables or sequences; and writes a rule in words or o Numeric patterns (F&A-3) symbols to find the next case. o Sequence (F&A-6) o Expresses generalization or rule using words or symbols (F&A-7) o Pattern Summary Table by grade level (F&A-9) M(F&A)– Demonstrates conceptual understanding of linear 4–2 relationships (y = kx) as a constant rate of change by identifying, describing, or comparing situations that represent constant rates of change. M(F&A)– Demonstrates conceptual understanding of o Whole number (N&O-2) 4–3 algebraic expressions by using letters or symbols to o Algebraic expressions (F&A-25) represent unknown quantities to write simple linear o Evaluating algebraic expressions algebraic expressions involving any one of the four (F&A-26) operations; or by evaluating simple linear algebraic o Linear relationships (F&A-10) expressions using whole numbers. o Proportional linear relationships (y = kx) (F&A-11) o Non-proportional linear relationships (y = mx + b) (F&A-12) o Number sentences (F&A-33) o Equation (F&A-32) o Algebraic equation notation (F&A-34) o Examples of forms of equations (F&A-35) 4-7 Functions and Algebra GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(F&A)– Demonstrates conceptual understanding of equality o Equality (F&A-30) 4–4 by showing equivalence between two expressions using o Demonstrates equality (F&A-31) models or different representations of the expressions, o Number sentences (F&A-33) by simplifying numerical expressions where left to right o Equation (F&A-32) computations may be modified only by the use of o Algebraic equation notation parentheses [e.g., 14 - (2 × 5)] (F&A-34) o Examples of forms of equations and by solving one-step linear equations of the form (F&A-35) ax = c and/or x±b=c where a, b, and c are whole numbers with a≠ 0. 4-8 Data, Statistics, and Probability GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(DSP) - Interprets a given representation (line plots, tables, o Interprets a given representation Graphing 4-1 bar graphs, pictographs, or circle graphs) to answer (DSP-21) Time Liner questions related to the data, to analyze the data to o Representation (DSP-1) formulate or justify conclusions, to make predictions, or o Line plot (DSP-5) to solve problems. o Frequency table (DSP-4) o Bar graph (DSP-6) (IMPORTANT: Analyzes data consistent with concepts o Pictograph (DSP-2) and skills in M(DSP)-4-2.) o Circle graph (DSP-8) M(DSP) - Analyzes patterns, trends, or distributions in data in o Pattern (F&A-1) 4-2 a variety of contexts by determining or using o Median (DSP-16) measures of central tendency (median or mode), or o Mode (DSP-17) range. o Range (DSP-18) M(DSP) - Organizes and displays data using tables, line plots, 4-3 bar graphs, and pictographs to answer questions related to the data, to analyze the data to formulate or justify conclusions, to make predictions, or to solve problems. (IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP)–4–2.) M(DSP) - Uses counting techniques to solve problems in o Solves problems using a variety 4-4 context involving combinations or simple permutations of counting strategies (DSP-25) (e.g., Given a map - Determine the number of paths o Combination (DSP-41) from point A to point B.) using a variety of strategies o Permutation (DSP-42) (e.g., organized lists, tables, tree diagrams, or others). o Frequency table (DSP-4) o Tree diagram (DSP-28) 4-9 Data, Statistics, and Probability GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(DSP) - For a probability event in which the sample space o Sample space (DSP-30) Tiles – different colored 4-5 may or may not contain equally likely outcomes, o Event (DSP-31) Dice determines the theoretical probability of an event and o Theoretical probability (DSP-32) Red beans expresses the result as part to whole (e.g., two out of o Ratio (N&O-12) Playing cards five). For a probability event in which the sample space may or may not contain equally likely outcomes predicts the likelihood of an event as a part to whole relationship (e.g., two out of five, zero out of five, five out of five) and tests the prediction through experiments; and determines if a game is fair. M(DSP) - In response to a teacher or student generated 4-6 question or hypothesis, groups decide the most effective method (e.g., survey, observation, experimentation) to collect the data (numerical or categorical) necessary to answer the question. In response to a teacher or student generated question or hypothesis, groups collect, organize, and appropriately display the data. In response to a teacher or student generated question or hypothesis, groups analyze the data to draw conclusions about the question or hypothesis being tested, and when appropriate make predictions. In response to a teacher or student generated question or hypothesis, groups ask new questions and make connections to real world situations. (IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP)–4–2.) 4 - 10 Grade Level 5 Expectations by Strand Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(N&O)- Demonstrates conceptual understanding of rational o Rational number (N&O-1) From the series Investigations 5-1 numbers with respect to: o Whole number (N&O-2) of Number, Data and Space whole numbers from 0 to 9,999,999 through o Fraction (N&O-3) equivalency, composition, decomposition, or place value Proper fraction (N&O-4) o Mathematical Thinking at Grade 5 using models, explanations, or other representations. o Improper fraction (N&O-5) o Mixed number (N&O-6) Factors, multiples, square Demonstrates conceptual understanding of rational o Decimal (N&O-7) number, prime number, addition numbers with respect to: o Percent (N&O-8) and subtraction games, landmark positive fractional numbers (proper, mixed number, and improper) (halves, fourths, eighths, thirds, sixths, o Ratio (N&O-12) numbers –100, 1000, 10,000, twelfths, fifths, or powers of ten (10, 100, 1000)), o Equivalent numbers (N&O-14) solving larger number operations decimals (to thousandths), or benchmark percents o Composition of numbers using cluster patterns includes (10%, 25%, 50%, 75% or 100%) as a part to whole (N&O-15) games, mental math, worksheets relationship in area, set, or linear models using models, o Decomposition of numbers (tiles and graph paper) explanations, or other representations*. (N&O-16) o Area model to represent part to Building on Numbers You Know whole relationship (N&O-17) o Set model (N&O-18) Strategies for estimation and o Linear model (N&O-19) computation, algebraic patterns o * Specifications for Area, Set, includes games and worksheets and Linear Models (Pg. NO-12) Name That Portion Fractions, decimals and percents includes several games and worksheets 5-1 Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(N&O)- Demonstrates understanding of the relative o Rational number (N&O-1) From the series Investigations 5-2 magnitude of numbers by ordering, comparing, or o Whole number (N&O-2) of Number, Data and Space identifying equivalent positive fractional numbers, o Fraction (N&O-3) decimals, or benchmark percents within number formats o Proper fraction (N&O-4) Name That Portion (fractions to fractions, decimals to decimals, or percents o Improper fraction (N&O-5) to percents); or integers in context using models or o Mixed number (N&O-6) Fractions, decimals and percents number lines. o Decimal (N&O-7) includes several games and o Percent (N&O-8) worksheets o Ratio (N&O-12) o Equivalent numbers (N&O-14) Patterns of Change o Composition of numbers (N&O-15) Tables and graphs includes o Decomposition of numbers software (N&O-16) o Area model to represent part to whole relationship (N&O-17) o Set model (N&O-18) o Linear model (N&O-19) M(N&O)- Demonstrates conceptual understanding of o Whole number (N&O-2) Building on Numbers You Know 5-3 mathematical operations by describing or illustrating o Meaning of remainders with the meaning of a remainder with respect to division of respect to division of whole Strategies for estimation and whole numbers using models, explanations, or solving numbers (N&O-32) computation, algebraic patterns problems. o Accurately solves problems includes games and worksheets (N&O-35) Demonstrates conceptual understanding of mathematical operations with respect to addition and subtraction of decimals and positive proper fractions with unlike denominators. 5-2 Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number From the series Investigations M(N&O)- Accurately solves problems involving multiple o Whole number (N&O-2) of Number, Data and Space 5-4 operations on whole numbers or the use of the o Fraction (N&O-3) properties of factors, multiples, prime, or composite o Proper fraction (N&O-4) Building on Numbers You Know numbers. o Decimal (N&O-7) o Accurately solves problems Strategies for estimation and Accurately solves problems involving addition or (N&O-35) computation, algebraic patterns subtraction of fractions (proper) and decimals to the o Factor (N&O-38) includes games and worksheets hundredths place. (Division of whole numbers by a one- o Multiples (N&O-39) or two-digit divisor.) o Prime numbers (N&O-40) Name That Portion (IMPORTANT: Applies the conventions of order of o Composite numbers (N&O-41) operations with and without parentheses.) Fractions, decimals and percents includes several games and worksheets M(N&O)- None 5-5 M(N&O)- Mentally calculates change back from $1.00, $5.00, All of this is referred to mentally! 5-6 and $10.00; Mentally calculates multiplication and related division Mathematical Thinking at Grade 5 facts to a product of 144; multiplies a two-digit whole number by a one-digit whole number (e.g., 45 x 5), two- Factors, multiples, square digit whole numbers that are multiples of ten (e.g., 50 x number, prime number, addition 60), a three-digit whole number that is a multiple of 100 and subtraction games, landmark by a two- or three-digit number which is a multiple of 10 numbers –100, 1000, 10,000, or 100, respectively (e.g., 400 x 50, 400 x 600). solving larger number operations using cluster patterns includes Mentally calculates division of 3- and 4-digit multiples of powers of ten by their compatible factors (e.g., games, mental math, worksheets 360 ÷ 6; 360 ÷ 60; 3600 ÷ 6; 3600 ÷ 60; (tiles and graph paper) 3600 ÷ 600; 360 ÷ 12; 360 ÷ 120; 3600 ÷ 12; 3600 ÷ 120; 3600 ÷ 1200). (IMPORTANT: The intent of this GLE is to embed mental arithmetic throughout the instructional program, not to teach it as a separate unit.) 5-3 Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number From the series Investigations M(N&O)- Makes estimates in a given situation by identifying of Number, Data and Space 5-7 when estimation is appropriate, selecting the appropriate method of estimation, determining the level Mathematical Thinking at Grade 5 of accuracy needed given the situation, analyzing the effect of the estimation method on the accuracy of Factors, multiples, square results, and evaluating the reasonableness of solutions number, prime number, addition appropriate to grade level GLEs across content strands. and subtraction games, landmark numbers –100, 1000, 10,000, (IMPORTANT: The intent of this GLE is to embed estimation throughout the instructional program, not to solving larger number operations teach it as a separate unit.) using cluster patterns includes games, mental math, worksheets (tiles and graph paper) Mathematical Thinking at Grade 5 M(N&O)- Applies properties of numbers (odd, even, and 5-8 divisibility) to solve problems and to simplify Factors, multiples, square computations. number, prime number, addition and subtraction games, landmark Applies field properties (commutative, associative, numbers –100, 1000, 10,000, identity, and distributive) to solve problems and to solving larger number operations simplify computations. using cluster patterns includes games, mental math, worksheets (tiles and graph paper) Building on Numbers You Know Strategies for estimation and computation, algebraic patterns includes games and worksheets 5-4 Geometry and Measurement GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(G&M)- Uses properties or attributes of angles (right, acute, o Attributes and properties From the series Investigations 5-1 or obtuse) or sides (number of congruent sides, (G&M-1) of Number, Data and Space parallelism, or perpendicularity) to identify, describe, o Angles (G&M-10) classify, or distinguish among different types of o Parallel lines (G&M-7) Picturing Polygons triangles (right, acute, obtuse, equiangular, or o Perpendicular (G&M-6) equilateral) or quadrilaterals (rectangles, squares, o Triangle (G&M-3) 2-D geometry includes software rhombi, trapezoids, or parallelograms). o Quadrilaterals (G&M-4) and manipulatives (Power Polygons) M(G&M)- None 5-2 M(G&M)- Uses properties or attributes (shape of bases, o Attributes and properties 5-3 number of lateral faces, or number of bases) to (G&M-1) identify, compare, or describe three-dimensional o Three-dimensional shapes Containers and Cubes shapes (rectangular prisms, triangular prisms, (G&M-15) cylinders, spheres, pyramids, or cones). 3-D geometry and volume includes manipulatives for volume M(G&M)- None 5-4 M(G&M)- Demonstrates conceptual understanding of 5-5 similarity by describing the proportional effect on the linear dimensions of triangles and rectangles when scaling up or down while preserving angle measures, or Picturing Polygons by solving related problems (including applying scales on maps). Describes effects using models or 2-D geometry includes software explanations. and manipulatives (Power Polygons) 5-5 Geometry and Measurement GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(G&M)- Demonstrates conceptual understanding of o Triangle (G&M-3) 5-6 perimeter of polygons, and the area of rectangles or o Polygons (G&M-2) right triangles through models, manipulatives, or o Three-dimensional shapes formulas. (G&M-15) From the series Investigations o Demonstrates conceptual Demonstrates conceptual understanding of area of of Number, Data and Space understanding of perimeter and polygons or irregular figures on grids, and volume of area of polygons and irregular rectangular prisms (cubes) using a variety of models, Picturing Polygons figures on grids (G&M-26) manipulatives, or formulas. o Demonstrates conceptual 2-D geometry includes software understanding of perimeter, Expresses all measures using appropriate units. and manipulatives (Power area, volume, or surface area Polygons) using models and manipulatives (G&M-28) Containers and Cubes o Measures and uses units of 3-D geometry and volume measure appropriately and includes manipulatives for consistently (G&M-31) volume o Whole number bases and whole number exponents, and fractional bases with whole number exponents (N&O-24) M(G&M)- Measures and uses units of measures appropriately o Measures and uses units of 5-7 and consistently, and makes conversions within measure appropriately and systems when solving problems across the content consistently (G&M-31) Measurement Benchmarks strands. o Makes conversions within and across systems (G&M-32) Estimating and measuring Benchmarks in Appendix B. standard and metric linear, weight, capacity and time (scales, metric and standard weights, metric and standard measuring tapes, metric and standard rulers, stop watches) 5-6 Geometry and Measurement GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(G&M)- None 5-8 M(G&M)- Demonstrates understanding of spatial 5-9 relationships using location and position by interpreting and giving directions between locations on a map or coordinate grid (all four From the series Investigations quadrants). of Number, Data and Space Demonstrates understanding of spatial Picturing Polygons relationships using location and position by plotting points in four quadrants in context (e.g., 2-D geometry includes software games, mapping, identifying the vertices of and manipulatives (Power polygons as they are reflected, rotated, and Polygons) translated). Patterns of Change Demonstrates understanding of spatial relationships using location and position by Tables and graphs includes determining horizontal and vertical distances software between points on a coordinate grid in the first quadrant. Math A Way of Thinking Coordinate graphing section M(G&M)- Demonstrates conceptual understanding of 5-10 spatial reasoning and visualization by building models of rectangular and triangular Containers and Cubes prisms, cones, cylinders, and pyramids from two- or three dimensional representations. 3-D geometry and volume includes manipulatives for volume 5-7 Functions and Algebra GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number From the series Investigations M(F&A)– Identifies and extends to specific cases a variety of o Numeric patterns (F&A-3) of Number, Data and Space 5–1 patterns (linear and nonlinear) represented in models, o Extend a pattern (F&A-5) tables, sequences, or in problem situations; and writes a o Sequence (F&A-6) Patterns of Change rule in words or symbols for finding specific cases of a o Linear relationships (F&A-10) linear relationship. o Proportional linear relationships Tables and graphs includes (y = kx) (F&A-11) software o Non-proportional linear relationships (y = mx + b) Building Blocks for Algebra K- (F&A-12) 8 o Expresses generalization or rule using words or symbols (F&A-7) patterns and functions includes o Concrete situations (F&A-20) games o Pattern Summary Table by grade level (F&A-9) Patterns of Change M(F&A)– Demonstrates conceptual understanding of linear 5–2 relationships (y = kx) as a constant rate of change by Tables and graphs includes identifying, describing, or comparing situations that software represent constant rates of change (e.g., tell a story given a line graph about a trip). 5-8 Functions and Algebra GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number Patterns of Change M(F&A)– Demonstrates conceptual understanding of o Whole number (N&O-2) 5–3 algebraic expressions by using letters to represent o Algebraic expression (F&A-25) Tables and graphs includes unknown quantities to write linear algebraic expressions o Evaluating algebraic expressions software involving any two of the four operations; or by evaluating (F&A-26) linear algebraic expressions using whole numbers. o Linear relationships (F&A-10) o Proportional linear relationships Building Blocks for Algebra K- (y = kx) (F&A-11) 8 o Non-proportional linear relationships (y = mx =b) patterns and functions includes (F&A-12) games o Number sentences (F&A-33) o Equation (F&A-32) o Examples of forms of equations (F&A-35) o Algebraic equation notation (F&A-34) Patterns of Change M(F&A)– Demonstrates conceptual understanding of equality o Whole number (N&O-2) 5–4 by showing equivalence between two expressions o Equality (F&A-30) Tables and graphs includes using models or different representations of the o Demonstrates equality (F&A-31) software expressions, o Number sentences (F&A-33) by solving one-step linear equations of the form o Equation (F&A-32) Building Blocks for Algebra K- ax = c and/or o Algebraic equation notation 8 x ± b = c and/or (F&A-34) x/a = c, o Examples of forms of equations patterns and functions includes where a, b, and c are whole numbers with a≠ 0; or by determining which values of a replacement set (F&A-35) games make the equation (multi-step of the form ax ± b = c o Linear relationships (F&A-10) where a, b, and c are whole numbers with a≠ 0) a true o Proportional linear relationships statement (e.g., 2x + 3 = 11, {x: x = 2, 3, 4, 5}). (y = kx) (F&A-11) o Non-proportional linear relationships (y = mx+b) (F&A-12) 5-9 Data, Statistics, and Probability GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number From the series Investigations M(DSP) – Interprets a given representation (tables, bar graphs, o Interprets a given representation of Number, Data and Space 5-1 circle graphs, or line graphs) to answer questions (DSP-21) related to the data, to analyze the data to formulate or o Representation (DSP-1) Data: Kids, Cats and Ads justify conclusions, to make predictions, or to solve o Frequency table (DSP-4) problems. o Bar graph (DSP-6) Statistics (stop watches) o Circle graph (DSP-8) (IMPORTANT: Analyzes data consistent with concepts o Line graph (DSP-12) and skills in M(DSP)-5-2.) Data: Kids, Cats and Ads M(DSP) – Analyzes patterns, trends, or distributions in data in o Pattern (F&A-1) 5-2 a variety of contexts by determining or using o Mean (DSP-15) Statistics (stop watches) measures of central tendency (mean, median, or mode) o Median (DSP-16) or range to analyze situations, or to solve problems. o Mode (DSP-17) o Range (DSP-18) Data: Kids, Cats and Ads M(DSP) – Identifies or describes representations or elements o Representation (DSP-1) 5-3 of representations that best display a given set of o Identifies or describes Statistics (stop watches) data or situation, consistent with the representations representations or elements of required in M(DSP)-5-1. representations that best display a given set of data or situation Organizes and displays data using tables, bar (DSP-23) graphs, or line graphs to answer questions related to the data, to analyze the data to formulate or justify conclusions, to make predictions, or to solve problems. (IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP)-5-2.) M(DSP) – None 5-4 5 - 10 Data, Statistics, and Probability GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number Between Never and Always M(DSP) – For a probability event in which the sample space o Sample space (DSP-30) 5-5 may or may not contain equally likely outcomes, o Experimental probability Probability includes games determines the experimental or theoretical probability (DSP-33) (spinners, colored cubes) of an event and expresses the result as a fraction. o Theoretical probability (DSP-32) o Event (DSP-31) For a probability event in which the sample space o Fraction (N&O-3) may or may not contain equally likely outcomes, o Ratio (N&O-12) predicts the likelihood of an event as a fraction and tests the prediction through experiments; and determines if a game is fair. Between Never and Always M(DSP) – In response to a teacher or student generated 5-6 question or hypothesis decides the most effective Probability includes games method (e.g., survey, observation, experimentation) to (spinners, colored cubes) collect the data (numerical or categorical) necessary to answer the question. In response to a teacher or student generated question or hypothesis collects, organizes, and appropriately displays the data. In response to a teacher or student generated question or hypothesis analyzes the data to draw conclusions about the question or hypothesis being tested, and when appropriate makes predictions. In response to a teacher or student generated question or hypothesis asks new questions and makes connections to real world situations. (IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP)–5–2.) 5 - 11 Grade Level 6 Expectations by Strand Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(N&O)- Demonstrates conceptual understanding of rational o Rational number (N&O-1) Glencoe Mathematics Application 6-1 numbers with respect to ratios (comparison of two o Whole number (N&O-2) and Concepts course 1 2004 whole numbers by division a/b, a : b, and a ÷ b , where o Proper fraction (N&O-4) edition plus corresponding b ≠ 0); and rates (e.g., a out of b, 25%) using models, o Improper fraction (N&O-5) Resource Masters explanations, or other representations*. o Mixed number (N&O-6) o Percent (N&O-8) Calculators (classroom set) o Ratio (N&O-12) o Equivalent numbers (N&O-14) o Area model to represent part to whole relationship (N&O-17) o Set model (N&O-18) o Linear model (N&O-19) o Proportional reasoning (N&O-44) o *Specifications for Area, Set, and Linear Models (Pg. NO-12) 6-1 Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(N&O)- Demonstrates understanding of the relative o Rational number (N&O-1) Glencoe Mathematics Application 6-2 magnitude of numbers by ordering or comparing o Fraction (N&O-3) and Concepts course 1 2004 numbers with whole number bases and whole number 3 3 o Proper fraction (N&O-4) edition plus corresponding exponents (e.g.,3 , 4 ), integers, or rational numbers o Improper fraction (N&O-5) Resource Masters within and across number formats (fractions, decimals, o Mixed number (N&O-6) or whole number percents from 1- 100) using number o Decimal (N&O-7) Calculators (classroom set) lines or equality and inequality symbols. o Percent (N&O-8) o Integer (N&O-9) o Equivalent numbers (N&O-14) o Relative magnitude (N&O-20) o Within number formats (N&O-21) o Across number formats (N&O-22) o Whole number bases and whole number exponents, and fractional bases with whole number exponents (N&O-24) o Ordering (N&O-26) o Comparing (N&O-27) o Number line (N&O-28) o Describing or illustrating the meaning of a power (N&O-33) 6-2 Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(N&O)- Demonstrates conceptual understanding of o Proper fraction (N&O-4) Glencoe Mathematics Application 6-3 mathematical operations by describing or illustrating o Improper fraction (N&O-5) and Concepts course 1 2004 the meaning of a power by representing the relationship o Decimal (N&O-7) edition plus corresponding between the base (whole number) and the exponent o Whole number bases and whole Resource Masters 3 3 (whole number) (e.g.,3 , 4 ). number exponents, and fractional bases with whole Calculators (classroom set) Demonstrates conceptual understanding of number exponents (N&O-24) mathematical operations and the effect on the o Relationship between repeated Fraction Bars magnitude of a whole number when multiplying or dividing it by a whole number, decimal, or fraction. addition and multiplication of whole numbers (N&O-30) o Relationship between repeated Demonstrates conceptual understanding of subtraction and division of whole mathematical operations of addition and subtraction of numbers (N&O-31) positive fractions and integers; and multiplication and o Effect on magnitude of a whole division of fractions and decimals. number when multiplying or dividing by a whole number, fraction, or decimal (N&O-34) o Describing or illustrating the meaning of a power (N&O-33) M(N&O)- Accurately solves problems involving single or o Fraction (N&O-3) Glencoe Mathematics Application 6-4 multiple operations on fractions (proper, improper, and o Proper fraction (N&O-4) and Concepts course 1 2004 mixed), or decimals; o Improper fraction (N&O-5) edition plus corresponding o Mixed number (N&O-6) Resource Masters Accurately solves problems involving addition or o Decimal (N&O-7) subtraction of integers; percent of a whole; or problems o Percent (N&O-8) Calculators (classroom set) involving greatest common factor or least common o Accurately solves problems multiple. (N&O-35) Fraction Bars (IMPORTANT: Applies the conventions of order of o GCF (N&O-42) operations with and without parentheses.) o LCM (N&O-43) Inter Chips Connected Mathematics Series M(N&O)- None 6-5 6-3 Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(N&O)- Mentally calculates change back from $5.00, $10.00, 6-6 $20.00, $50.00, and $100.00. All of this is referred to mentally! Mentally multiplies a two-digit whole number by a one- Glencoe Mathematics Application digit whole number (e.g., 45 x 5), two-digit whole and Concepts course 1 2004 numbers that are multiples of ten (e.g., 50 x 60), a edition plus corresponding three-digit whole number that is a multiple of 100 by a Resource Masters two- or three-digit number which is a multiple of 10 or 100, respectively (e.g., 400 x 50, 400 x 600). Calculators (classroom set) Mentally divides 3- and 4-digit multiples of powers of ten by their compatible factors (e.g., 360 ÷ 6; 360 ÷ 60; 3600 ÷ 6; 3600 ÷ 60; 3600 ÷ 600; 360 ÷ 12; 360 ÷ 120; 3600 ÷ 12; 3600 ÷ 120; 3600 ÷ 1200). Mentally determines the part of a whole number using benchmark percents (1%, 10%, 25%, 50%, and 75%). (IMPORTANT: The intent of this GLE is to embed mental arithmetic throughout the instructional program, not to teach it as a separate unit.) 6-4 Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(N&O)- Makes estimates in a given situation by identifying Glencoe Mathematics Application 6-7 when estimation is appropriate, selecting the and Concepts course 1 2004 appropriate method of estimation, determining the level edition plus corresponding of accuracy needed given the situation, analyzing the Resource Masters effect of the estimation method on the accuracy of results, and evaluating the reasonableness of solutions Calculators (classroom set) appropriate to grade level GLEs across content strands. (IMPORTANT: The intent of this GLE is to embed estimation throughout the instructional program, not to teach it as a separate unit.) M(N&O)- Applies properties of numbers (odd, even, Glencoe Mathematics Application 6-8 remainders, divisibility, and prime factorization) to and Concepts course 1 2004 solve problems and to simplify computations. edition plus corresponding Resource Masters Applies field properties (commutative, associative, identity [including the multiplicative property Calculators (classroom set) of one, e.g., 1 = 2/2 and 2/2 x ¾ = 6/8, so ¾ = 6/8], distributive, and additive inverses) to solve problems and to simplify computations. 6-5 Geometry and Measurement GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(G&M)- Uses properties or attributes of angles (right, acute, o Attributes and properties Glencoe Mathematics Application 6-1 or obtuse) or sides (number of congruent sides, (G&M-1) and Concepts course 1 2004 parallelism, or perpendicularity) to identify, describe, o Angles (G&M-10) edition plus corresponding classify, or distinguish among different types of o Parallel lines (G&M-7) Resource Masters triangles (right, acute, obtuse, equiangular, scalene, o Perpendicular (G&M-6) isosceles, or equilateral) or quadrilaterals (rectangles, o Triangle (G&M-3) squares, rhombi, trapezoids, or parallelograms). Calculators (classroom set) o Quadrilaterals (G&M-4) Connected Mathematics Series Geo Boards M(G&M)- None 6-2 Glencoe Mathematics Application M(G&M)- Uses properties or attributes (shape of bases, o Attributes and properties and Concepts course 1 2004 6-3 number of lateral faces, number of bases, number of (G&M-1) edition plus corresponding edges, or number of vertices) to identify, compare, or o Three-dimensional shapes Resource Masters describe three-dimensional shapes (rectangular (G&M-15) prisms, triangular prisms, cylinders, spheres, pyramids, Calculators (classroom set) or cones). Connected Mathematics Series Geo Boards M(G&M)- Demonstrates conceptual understanding of Glencoe Mathematics Application 6-4 congruency by predicting and describing the and Concepts course 1 2004 transformational steps (reflections, translations, and edition plus corresponding rotations) needed to show congruence (including the Resource Masters degree of rotation) and as the result of composing and decomposing two- and three-dimensional objects using Calculators (classroom set) models or explanations. Connected Mathematics Series Demonstrates conceptual understanding of congruency by using line and rotational symmetry to Geo Boards demonstrate congruent parts within a shape. 6-6 Geometry and Measurement GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(G&M)- Demonstrates conceptual understanding of o Similar figures (G&M-23) Glencoe Mathematics Application 6-5 similarity by describing the proportional effect on the o Polygons (G&M-2) and Concepts course 1 2004 linear dimensions of polygons or circles when scaling up o Sum of the measures of interior edition plus corresponding or down while preserving the angles of polygons, or by angles of polygons (G&M-13) Resource Masters solving related problems (including applying scales on o Describes the proportional effect maps). Describes effects using models or explanations. on linear dimensions of polygons Calculators (classroom set) or circles when scaling up or down while preserving the angles Connected Mathematics Series of polygons (G&M-24) o Solves problems involving scaling Geo Boards up or down and their impact on angle measure, linear dimensions, and areas of polygons and circles when the linear dimensions are multiplied by a constant factor (G&M-25) o Proportional reasoning (N&O-44) 6-7 Geometry and Measurement GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number o Polygons (G&M-2) M(G&M)- Demonstrates conceptual understanding of o Triangle (G&M-3) Glencoe Mathematics Application 6-6 perimeter of polygons, the area of quadrilaterals or o Three-dimensional shapes and Concepts course 1 2004 triangles, and the volume of rectangular prisms by (G&M-15) edition plus corresponding using models, formulas, or by solving problems. o Demonstrates conceptual Resource Masters understanding of perimeter, Demonstrates conceptual understanding of the area, volume, or surface area Calculators (classroom set) relationships of circle measures (radius to diameter using models and manipulatives and diameter to circumference) by solving related (G&M-28) Connected Mathematics Series problems. Expresses all measures using appropriate units. o Demonstrates conceptual understanding of perimeter, Geo Boards area, volume, or surface area by solving problems (G&M-29) o Demonstrates conceptual understanding of the relationships of circle measures by solving related problems (G&M-30) o Measures and uses units of measure appropriately and consistently (G&M-31) o Irrational number (N&O-10) o Whole number bases and whole number exponents, and fractional bases with whole number exponents (N&O-24) o Measures and uses units of M(G&M)- Measures and uses units of measures appropriately measure appropriately and Glencoe Mathematics Application 6-7 and consistently, and makes conversions within consistently and Concepts course 1 2004 systems when solving problems across the content o Makes conversions within and edition plus corresponding strands. across systems Resource Masters o Irrational number Calculators (classroom set) Benchmarks in Appendix B. Connected Mathematics Series Geo Boards M(G&M)- 6-8 None 6-8 Geometry and Measurement GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(G&M)- None 6-9 M(G&M)- None 6-10 6-9 Functions and Algebra GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(F&A)– Identifies and extends to specific cases a variety of o Patterns (F&A-1) Glencoe Mathematics Application 6–1 patterns (linear and nonlinear) represented in models, o Numeric patterns (F&A-3) and Concepts course 1 2004 tables, sequences, graphs, or in problem situations; or o Non-numeric patterns (F&A-4) edition plus corresponding writes a rule in words or symbols for finding specific o Extend a pattern (F&A-5) Resource Masters cases of a linear relationship; or writes a rule in words or o Sequence (F&A-6) symbols for finding specific cases of a nonlinear o Linear relationships (F&A-10) Calculators (classroom set) relationship. o Proportional linear relationships (y = kx) (F&A-11) Identifies and extends to specific cases a variety of patterns and writes an expression or equation using o Non-proportional linear words or symbols to express the generalization of a relationships (y = mx + b) linear relationship (e.g., twice the term number plus 1 or (F&A-12) 2n + 1). o Non-linear relationships (F&A-18) o Expresses generalization or rule using words or symbols (F&A-7) o Concrete situations (F&A-20) o Pattern Summary Table by grade level (F&A-9) M(F&A)– Demonstrates conceptual understanding of linear o Ratio (N&O-12) Glencoe Mathematics Application 6–2 relationships (y = kx; y = mx + b) as a constant rate o Proportional reasoning (N&O-44) and Concepts course 1 2004 of change by constructing or interpreting graphs of real o Linear relationships (F&A-10) edition plus corresponding occurrences and describing the slope of linear o Proportional linear relationships Resource Masters relationships (faster, slower, greater, or smaller) in a (y = kx) (F&A-11) variety of problem situations. o Non-proportional linear Calculators (classroom set) relationships (y = mx + b) Demonstrates conceptual understanding of linear (F&A-12) relationships (y = kx; y = mx + b) as a constant rate of change and describes how change in the value of o Distinguishes between constant one variable relates to change in the value of a and varying rates (F&A-19) second variable in problem situations with constant o Slope (F&A-14) rates of change. o Describes the meaning of slope and intercept in concrete situations (F&A-15) o Concrete situations (F&A-20) 6 - 10 Functions and Algebra GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(F&A)– Demonstrates conceptual understanding of o Algebraic expression (F&A-25) Glencoe Mathematics Application 6–3 algebraic expressions by using letters to represent o Evaluating algebraic expressions and Concepts course 1 2004 unknown quantities to write linear algebraic expressions (F&A-26) edition plus corresponding involving two or more of the four operations; or by o Number sentences (F&A-33) Resource Masters evaluating linear algebraic expressions (including those o Equation (F&A-32) with more than one variable); or by evaluating an o Algebraic equation notation Calculators (classroom set) expression within an equation (e.g., determine the value (F&A-34) of y when x = 4 given y = 3x - 2). o Examples of forms of equations (F&A-35) o Linear relationships (F&A-10) o Proportional linear relationships (y = kx) (F&A-11) o Non-proportional linear relationships (y = mx + b) (F&A-12) M(F&A)– Demonstrates conceptual understanding of equality o Whole number (N&O-2) Glencoe Mathematics Application 6–4 by showing equivalence between two expressions using o Equality (F&A-30) and Concepts course 1 2004 models or different representations of the expressions, o Demonstrates equality edition plus corresponding solving multi-step linear equations of the form (F&A-31) Resource Masters ax ± b = c, where a, b, and c are whole numbers with o Linear relationships (F&A-10) a≠ 0. o Proportional linear relationships Calculators (classroom set) (y = kx) (F&A-11) o Non-proportional linear relationships (y = mx =b) (F&A-12) o Number sentences (F&A-33) o Equation (F&A-32) o Algebraic equation notation (F&A-34) o Examples of forms of equations (F&A-35) 6 - 11 Data, Statistics, and Probability GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(DSP) – Interprets a given representation (circle graphs, line o Interprets a given representation Glencoe Mathematics Application 6-1 graphs, or stem-and-leaf plots) to answer questions (DSP-21) and Concepts course 1 2004 related to the data, to analyze the data to formulate or o Representation (DSP-1) edition plus corresponding justify conclusions, to make predictions, or to solve o Circle graph (DSP-8) Resource Masters problems. o Line graph (DSP-12) o Stem-and-leaf plot (DSP-9) Calculators (classroom set) (IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP)-6-2). Connected Mathematics Series M(DSP) – Analyzes patterns, trends or distributions in data in o Pattern (F&A-1) Glencoe Mathematics Application 6-2 a variety of contexts by determining or using o Mean (DSP-15) and Concepts course 1 2004 measures of central tendency (mean, median, or mode) o Median (DSP-16) edition plus corresponding or dispersion (range) to analyze situations, or to solve o Mode (DSP-17) Resource Masters problems. o Dispersion (DSP-19) o Range (DSP-18) Calculators (classroom set) Connected Mathematics Series M(DSP) – Organizes and displays data using tables, line Glencoe Mathematics Application 6-3 graphs, or stem-and-leaf plots to answer questions and Concepts course 1 2004 related to the data, to analyze the data to formulate or edition plus corresponding justify conclusions, to make predictions, or to solve Resource Masters problems. Calculators (classroom set) (IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP)–6–2). Connected Mathematics Series M(DSP) – Uses counting techniques to solve problems in o Solves problems using a variety Glencoe Mathematics Application 6-4 context involving combinations or simple permutations of counting strategies (DSP-25) and Concepts course 1 2004 using a variety of strategies (e.g., organized lists, tables, o Combination (DSP-26) edition plus corresponding tree diagrams, models, Fundamental Counting Principle, o Permutation (DSP-27) Resource Masters or others). o Frequency table (DSP-4) o Tree diagram (DSP-28) Calculators (classroom set) o Fundamental Counting Principle (DSP-29) Connected Mathematics Series 6 - 12 Data, Statistics, and Probability GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(DSP) – For a probability event in which the sample space o Sample space (DSP-30) Glencoe Mathematics Application 6-5 may or may not contain equally likely outcomes, o Experimental probability and Concepts course 1 2004 determines the experimental or theoretical probability (DSP-33) edition plus corresponding of an event in a problem-solving situation. o Theoretical probability (DSP-32) Resource Masters o Event (DSP-31) For a probability event in which the sample space Calculators (classroom set) may or may not contain equally likely outcomes, predicts the theoretical probability of an event and tests Connected Mathematics Series the prediction through experiments and simulations; and designs fair games. M(DSP) – In response to a teacher or student generated Glencoe Mathematics Application 6-6 question or hypothesis decides the most effective and Concepts course 1 2004 method (e.g., survey, observation, experimentation) to edition plus corresponding collect the data (numerical or categorical) necessary to Resource Masters answer the question. Calculators (classroom set) In response to a teacher or student generated question or hypothesis collects, organizes, and Connected Mathematics Series appropriately displays the data. In response to a teacher or student generated question or hypothesis analyzes the data to draw conclusions about the question or hypothesis being tested, and when appropriate makes predictions. In response to a teacher or student generated question or hypothesis asks new questions and makes connections to real world situations. (IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP)–6–2.) 6 - 13 Grade Level 7 Expectations by Strand Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(N&O)- Demonstrates conceptual understanding of rational o Rational number (N&O-1) Glencoe Applications and 7-1 numbers with respect to percents as a means of o Percent (N&O-8) Concepts Course 2 (2004) comparing the same or different parts of the whole when o Ratio (N&O-12) the wholes vary in magnitude (e. g., 8 girls in a o Equivalent numbers (N&O-14) Glencoe Applications and classroom of 16 students compared to 8 girls in a o Area model to represent part to Concepts Course 2 (2004) classroom of 20 students, or 20% of 400 compared to whole relationship (N&O-17) Resource Kit 50% of 100) using models, explanations, or other o Set model (N&O-18) representations*. o Linear model (N&O-19) Mathimagination Series Demonstrates conceptual understanding of rational o Whole number bases and whole (Steve and Janis Marcy) numbers with respect to percents as a way of number exponents, and expressing multiples of a number (e. g., 200% of 50) fractional bases with whole Mathematics – A Way of Thinking using models, explanations, or other number exponents (N&O-24) (Robert Baratta-Lorton) representations*. o Comparing (N&O-27) o Number line (N&O-28) Guiding Children’s Learning of Demonstrates conceptual understanding of square roots o Multiples (N&O-39) Mathematics of perfect squares, rates, and proportional reasoning. o Proportional reasoning (Kennedy and Tipps) (N&O-44) o * Specifications for Area, Set, and Linear Models (Pg. NO-12) M(N&O)- Demonstrates understanding of the relative o Rational number (N&O-1) Pre-Algebra with Pizzazz Series 7-2 magnitude of numbers by ordering, comparing, or o Integer (N&O-9) (Steve and Janis Marcy) identifying equivalent rational numbers across number o Equivalent numbers (N&O-14) formats, numbers with whole number bases and whole o Relative magnitude (N&O-20) Glencoe Applications and 3 3 number exponents (e.g., 3 , 4 ), integers, absolute o Across number formats Concepts Course 2 (2004) values, or numbers represented in scientific notation (N&O-21) using number lines or equality and inequality symbols. o Absolute value (N&O-23) Glencoe Applications and o Scientific notation (N&O-25) Concepts Course 2 (2004) o Ordering (N&O-26) Resource Kit o Comparing (N&O-27) o Describing or illustrating the meaning of a power (N&O-33) 7-1 Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(N&O)- Demonstrates conceptual understanding of 7-3 operations with integers and whole number exponents (where the base is a whole number) using models, diagrams, or explanations. M(N&O)- Accurately solves problems involving proportional o Percent (N&O-8) Glencoe Applications and 7-4 reasoning, percents involving discounts, tax or tips, and o Ratio (N&O-12) Concepts Course 2 rates. o Accurately solves problems (N&O-35) Glencoe Applications and Accurately solves problems involving addition or o Proportional reasoning Concepts Course 2 (2004) subtraction of integers, raising numbers to whole (N&O-44) Resource Kit number powers, and determining square roots of perfect square numbers and non- perfect square numbers. Pre-Algebra with Pizzazz Series (IMPORTANT: Applies the conventions of order of (Steve and Janis Marcy) operations, including parentheses, brackets, or exponents). M(N&O)- None 7-5 M(N&O)- Mentally calculates benchmark perfect squares and All of this is referred to mentally! 2 2 2 2 2 2 7-6 related square roots (e. g., 1 , 2 … 12 , 15 , 20 , 25 , 2 2 100 , 1000 ). Mentally determines the part of a number using benchmark percents and related fractions (1%, 10%, 25%, 33⅓ %, 50%, 66⅔%, 75%, and 100%) (e. g., 25% of 16; 33⅓% of 330). (IMPORTANT: Mental arithmetic should be imbedded instructionally throughout all strands .) 7-2 Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(N&O)- Makes estimates in a given situation (including tips, 7-7 discounts, and tax) by identifying when estimation is appropriate, selecting the appropriate method of estimation, determining the level of accuracy needed given the situation, analyzing the effect of the estimation method on the accuracy of results, and evaluating the reasonableness of solutions appropriate to grade level GLEs across content strands. (IMPORTANT: The intent of this GLE is to embed estimation throughout the instructional program, not to teach it as a separate unit.) M(N&O)- Applies properties of numbers (odd, even, Glencoe Applications and 7-8 remainders, divisibility, and prime factorization) to solve Concepts Course 2 (2004) problems and to simplify computations. Glencoe Applications and Applies field properties (commutative, associative, Concepts Course 2 (2004) identity, distributive, inverses) to solve problems and Resource Kit to simplify computations. Pre-Algebra with Pizzazz Demonstrates conceptual understanding of field properties as they apply to subsets of the real numbers (e. g., the set of whole numbers does not have additive inverses, the set of integers does not have multiplicative inverses). 7-3 Geometry and Measurement GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number Glencoe Applications and M(G&M)- Uses properties of angle relationships resulting from o Attributes and properties Concepts Course 2 7-1 two or three intersecting lines (adjacent angles, vertical (G&M-1) angles, straight angles, or angle relationships formed by o Angle (G&M-10) Glencoe Applications and two non-parallel lines cut by a transversal), or two o Angle relationships formed by Concepts Course 2 (2004) parallel lines cut by a transversal to solve problems. two or more lines cut by a Resource Kit transversal (G&M-11) o Parallel lines (G&M-7) Mathimagination Series Math – A Way of Thinking Moving On with Geoboards (Creative Publications) M(G&M)- 7-2 Applies theorems or relationships (triangle inequality o Triangle (G&M-3) Glencoe Applications and or sum of the measures of interior angles of polygons) o Triangle inequality (G&M-12) Concepts Course 2 to solve problems. o Sum of the measures of interior angles of polygons (G&M-13) Glencoe Applications and o Angles (G&M-10) Concepts Course 2 (2004) o Polygons (G&M-2) Resource Kit M(G&M)- None 7-3 M(G&M)- Applies the concepts of congruency by solving o Congruent (G&M-16) Glencoe Applications and 7-4 problems on a coordinate plane involving reflections, o Reflection (G&M-17) Concepts Course 2 translations, or rotations. o Translation (G&M-18) o Rotation (G&M-19) Glencoe Applications and o Solving problems on a coordinate Concepts Course 2 (2004) plane using reflections, Resource Kit translations, or rotations (G&M-22) 7-4 Geometry and Measurement GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(G&M)- Applies concepts of similarity by solving problems o Similar figures (G&M-23) Glencoe Applications and 7-5 involving scaling up or down and their impact on angle o Angles (G&M-10) Concepts Course 2 (2004) measures, linear dimensions and areas of polygons, o Polygons (G&M-2) and circles when the linear dimensions are multiplied by o Sum of the measures of interior Mathimagination Series a constant factor. Describes effects using models or angles of polygons (G&M-13) explanations. o Describes the proportional effect Math – A Way of Thinking on linear dimensions of polygons or circles when scaling up or Moving On with Geoboards down while preserving the angles (Creative Publications) of polygons (G&M-24) o Solves problems involving scaling Geoboards up or down and their impact on angle measure, linear dimensions, and areas of polygons and circles when the linear dimensions are multiplied by a constant factor (G&M-25) o Factor (N&O-38) o Proportional reasoning (G&M-44) 7-5 Geometry and Measurement GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(G&M)- Demonstrates conceptual understanding of the area o Quadrilaterals (G&M-4) Glencoe Applications and 7-6 of circles or the area or perimeter of composite figures o Triangle (G&M-3) Concepts Course 2 (2004) (quadrilaterals, triangles, or parts of circles), and the o Three-dimensional shapes surface area of rectangular prisms, or volume of (G&M-15) Mathimagination Series rectangular prisms, triangular prisms, or cylinders using o Demonstrates conceptual models, formulas, or by solving related problems. understanding of perimeter, Math – A Way of Thinking Expresses all measures using appropriate units. area, volume, or surface area using models and manipulatives Moving On with Geoboards (G&M-28) (Creative Publications) o Demonstrates conceptual understanding of perimeter, Geoboards area, volume, or surface area by solving problems (G&M-29) Tessellations – The Geometry of o Demonstrates conceptual Patterns understanding of the (Creative Publications) relationships of circle measures by solving related problems (G&M-30) o Measures and uses units of measure appropriately and consistently (G&M-31) o Irrational number (N&O-10) o Whole number bases and whole number exponents, and fractional bases with whole number exponents (N&O-24) M(G&M)- 7-7 None M(G&M)- None 7-8 M(G&M)- 7-9 None Demonstrates conceptual understanding of spatial M(G&M)- reasoning and visualization by sketching three- 7-10 dimensional solids; and draws nets of rectangular and triangular prisms, cylinders, and pyramids and uses the nets as a technique for finding surface area. 7-6 Functions and Algebra GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(F&A)– Identifies and extends to specific cases a variety of o Patterns (F&A-1) Glencoe Applications and 7–1 patterns (linear and nonlinear) represented in models, o Numeric patterns (F&A-3) Concepts Course 2 tables, sequences, graphs, or in problem situations. o Non-numeric patterns (F&A-4) o Extend a pattern (F&A-5) Glencoe Applications and Identifies and extends to specific cases a variety of o Sequence (F&A-6) Concepts Course 2 (2004) patterns and generalizes a linear relationship using o Linear relationships (F&A-10) Resource Kit words and symbols. o Proportional linear relationships (y = kx) (F&A-11) Identifies and extends to specific cases a variety of patterns and generalizes a linear relationship to find a o Non-proportional linear Pre-Algebra with Pizzazz specific case; or writes an expression or equation using relationships (y = mx + b) words or symbols to express the generalization of a (F&A-12) nonlinear relationship. o Non-linear relationships (F&A-18) o Equation (F&A-32) o Expresses generalization or rule using words or symbols (F&A-7) o Generalizes a pattern to find a specific case (F&A-8) o Concrete situations (F&A-20) o Pattern Summary Table by grade level (F&A-9) 7-7 Functions and Algebra GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(F&A)– Demonstrates conceptual understanding of linear o Ratio (F&A-12) Glencoe Applications and 7–2 relationships (y = kx; y = mx + b) as a constant rate o Proportional reasoning (N&O-44) Concepts Course 2 of change by solving problems involving the o Linear relationships (F&A-10) relationship between slope and rate of change, by o Proportional linear relationships Glencoe Applications and describing the meaning of slope in concrete situations, (y = kx) (F&A-11) Concepts Course 2 (2004) or informally determining the slope of a line from a table o Non-proportional linear Resource Kit or graph. relationships (y = mx + b) (F&A-12) Pre-Algebra with Pizzazz Demonstrates conceptual understanding of linear relationships and distinguishes between constant o Slope (F&A-14) and varying rates of change in concrete situations o Intercept (F&A-17) represented in tables or graphs; or describes how o Non-linear relationships change in the value of one variable relates to (F&A-18) change in the value of a second variable in problem o Informally determines slope situations with constant rates of change. (F&A-16) o Distinguishes between constant and varying rates (F&A-19) o Solves problems involving linear relationships (F&A-13) o Concrete situations (F&A-20) M(F&A)– Demonstrates conceptual understanding of o Whole number bases and whole Glencoe Applications and 7–3 algebraic expressions by using letters to represent number exponents, and Concepts Course 2 unknown quantities to write algebraic expressions fractional bases with whole (including those with whole number exponents or more number exponents (N&O-24) Glencoe Applications and than one variable); or by evaluating algebraic o Algebraic expression (F&A-25) Concepts Course 2 (2004) expressions (including those with whole number o Evaluating algebraic expressions Resource Kit exponents or more than one variable); or by evaluating (F&A-26) an expression within an equation (e.g., determine the 3 o Number sentences (F&A-33) Pre-Algebra with Pizzazz value of y when x = 4 given y = 5x - 2). o Equation (F&A-32) o Examples of forms of equations (F&A-35) o Algebraic equation notation (F&A-34) 7-8 Functions and Algebra GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(F&A)– Demonstrates conceptual understanding of equality o Equality (F&A-30) Glencoe Applications and 7–4 by showing equivalence between two expressions o Demonstrates equality Concepts Course 2 (expressions consistent with the parameters of the left- (F&A-31) and right-hand sides of the equations being solved at o Linear relationships (F&A-10) Glencoe Applications and this grade level) using models or different o Proportional linear relationships Concepts Course 2 (2004) representations of the expressions, solving multi-step (y = kx) (F&A-11) Resource Kit linear equations of the form ax ± b = c with a ≠ 0, o Non-proportional linear ax ± b = cx ± d with a, c≠ 0, and (x/a) ± b = c with a ≠ 0, relationships (y = mx + b) Pre-Algebra with Pizzazz where a, b, c and d are whole numbers; or by translating a problem-solving situation into an equation consistent (F&A-12) with the parameters of the type of equations being o Number sentences (F&A-33) solved for this grade level. o Equation (F&A-32) o Examples of forms of equations (F&A-35) o Algebraic equation notation (F&A-34) 7-9 Data, Statistics, and Probability GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(DSP) – Interprets a given representation (circle graphs, o Interprets a given representation Glencoe Applications and 7-1 scatter plots that represent discrete linear (DSP-21) Concepts Course 2 relationships, or histograms) to analyze the data to o Representation (DSP-1) formulate or justify conclusions, to make o Circle graph (DSP-8) Glencoe Applications and predictions, or to solve problems. o Scatter plot (DSP-13) Concepts Course 2 (2004) o Histogram (DSP-7) Resource Kit (IMPORTANT: Analyzes data consistent with concepts o Linear relationships (F&A-10) and skills in M(DSP)-7-2.) o Proportional linear relationship (F&A-11) o Non-proportional linear relationship (F&A-12) M(DSP) – Analyzes patterns, trends, or distributions in data in o Pattern (DSP-1) Glencoe Applications and 7-2 a variety of contexts by solving problems using o Mean (DSP-15) Concepts Course 2 measures of central tendency (mean, median, or mode), o Median (DSP-16) dispersion (range or variation), or outliers to analyze o Mode (DSP-17) Glencoe Applications and situations to determine their effect on mean, median, or o Dispersion (DSP-19) Concepts Course 2 (2004) mode; o Range (DSP-18) Resource Kit o Outlier (DSP-20) Analyzes patterns, trends, or distributions in data in o Analyzes the impact of outliers a variety of contexts by evaluating the sample from which the statistics were developed (bias). on the mean, median and mode (DSP-22) o Evaluates samples from which the statistics were developed (bias) (DSP-24) M(DSP) – Identifies or describes representations or elements o Representation (DSP-1) Glencoe Applications and 7-3 of representations that best display a given set of o Identifies or describes Concepts Course 2 data or situation, and organizes and displays data representations or elements of using tables, line graphs, scatter plots, and circle representations that best display Glencoe Applications and graphs to answer questions related to the data, to a given set of data or situation Concepts Course 2 (2004) analyze the data to formulate or justify conclusions, to (DSP-23) Resource Kit make predictions, or to solve problems. (IMPORTANT: Analyzes data consistent with concepts and skills in M( DSP)– 7– 2 .) 7 - 10 Data, Statistics, and Probability GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(DSP) – Uses counting techniques to solve problems in Glencoe Applications and 7-4 context involving combinations or permutations (e.g., Concepts Course 2 How many different ways can eight students place first, second, and third in a race?) using a variety of Glencoe Applications and strategies (e.g., organized lists, tables, tree diagrams, Concepts Course 2 (2004) models, Fundamental Counting Principle, or others). Resource Kit What Are My Chances Book A, Book B (Creative Publications) M(DSP) – For a probability event in which the sample space o Sample space (DSP-30) Glencoe Applications and 7-5 may or may not contain equally likely outcomes, o Experimental probability Concepts Course 2 determines the experimental or theoretical probability (DSP-33) of an event in a problem- solving situation. o Theoretical probability (DSP-32) Glencoe Applications and o Event (DSP-31) Concepts Course 2 (2004) For a probability event in which the sample space Resource Kit may or may not contain equally likely outcomes, predicts the theoretical probability of an event and tests the prediction through experiments and simulations For a probability event in which the sample space may or may not contain equally likely outcomes compares and contrasts theoretical and experimental probabilities. 7 - 11 Data, Statistics, and Probability GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(DSP) – In response to a teacher or student generated Glencoe Applications and 7-6 question or hypothesis decides the most effective Concepts Course 2 method (e. g., survey, observation, experimentation) to collect the data (numerical or categorical) necessary to Glencoe Applications and answer the question. Concepts Course 2 (2004) Resource Kit In response to a teacher or student generated question or hypothesis collects, organizes, and appropriately displays the data. In response to a teacher or student generated question or hypothesis analyzes the data to draw conclusions about the question or hypothesis being tested while considering the limitations that could affect interpretations. In response to a teacher or student generated question or hypothesis, makes predictions when appropriate. In response to a teacher or student generated question or hypothesis asks new questions and makes connections to real world situations. (IMPORTANT: Analyzes data consistent with concepts and skills in M( DSP)– 7– 2. ) 7 - 12 Grade Level 8 Expectations by Strand Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(N&O)- Demonstrates conceptual understanding of rational o Rational number (N&O-1) Glencoe, Pre-Algebra (2003) 8-1 numbers with respect to percents as a way of o Percent (N&O-8) Textbook describing change (percent increase and decrease) o Ratio (N&O-12) using explanations, models, or other o Area model to represent part to Glencoe, Pre-Algebra (2003) representations*. whole relationship (N&O-17) Resource Kit o Set model (N&O-18) o Linear model (N&O-19) Glencoe, Applications and o Proportional reasoning Connections (1995) (N&O-44) Resource Kit o * Specifications for Area, Set, and Linear Models (Pg. NO-12) Algebra In the Real World – Futures Channel M(N&O)- Demonstrates understanding of the relative o Rational number (N&O-1) Glencoe, Algebra I (2003) 8-2 magnitude of numbers by ordering or comparing o Integer (N&O-9) rational numbers, common irrational numbers (e.g. o Irrational number (N&O-10) Creative Publications, Middle , , ), numbers with whole number or fractional bases o Real numbers (N&O-11) School Math with Pizzazz (1989) and whole number exponents, square roots, absolute o Equivalent numbers (N&O-14) values, integers, or numbers represented in scientific o Relative magnitude (N&O-20) notation using number lines or equality and inequality o Across number formats symbols. (N&O-21) o Absolute value (N&O-23) o Whole number bases and whole number exponents, and fractional bases with whole number exponents (N&O-24) o Scientific notation (N&O-25) o Ordering (N&O-26) o Comparing (N&O-27) o Number line (N&O-28) o Describing or illustrating the meaning of a power (N&O-33) M(N&O)- 8-3 None 8-1 Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(N&O)- Accurately solves problems involving proportional o Percent (N&O-8) Glencoe, Pre-Algebra (2003) 8-4 reasoning (percent increase or decrease, interest rates, o Integer (N&O-9) Textbook markups, or rates). o Ratio (N&O-12) o Whole number bases and whole Glencoe, Pre-Algebra (2003) Accurately solves problems involving number exponents, and Resource Kit multiplication or division of integers. fractional bases with whole number exponents (N&O-24) Glencoe, Applications and Accurately solves problems involving squares, o Accurately solves problems Connections (1995) cubes, and taking square or cube roots. (N&O-35) Resource Kit (IMPORTANT: Applies the conventions of order of o Proportional reasoning operations.) (N&O-44) Algebra In the Real World – Futures Channel Glencoe, Algebra I (2003) Creative Publications, Middle M(N&O)- None School Math with Pizzazz (1989) 8-5 M(N&O)- Mentally calculates benchmark perfect squares and All of this is referred to mentally! 2 2 2 2 2 2 2 8-6 square roots (e.g., 1 , 2 … 12 , 15 , 20 , 25 , 100 , 2 1000 ). Mentally determines the part of a number using benchmark percents and related fractions (1%, 10%, 25%,33⅓ %, 50%,66 ⅔%, 75%, and 100%) (e.g., 25% of 16; 33⅓ % of 330). (IMPORTANT: Mental arithmetic should be imbedded instructionally throughout all strands.) 8-2 Number and Operations GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(N&O)- Makes estimates in a given situation (including tips, Glencoe, Pre-Algebra (2003) 8-7 discounts, tax, and the value of a non-perfect square Textbook root as between two whole numbers) by identifying when estimation is appropriate, selecting the Glencoe, Pre-Algebra (2003) appropriate method of estimation; determining the level Resource Kit of accuracy needed given the situation; analyzing the effect of the estimation method on the accuracy of Glencoe, Applications and results; and evaluating the reasonableness of solutions Connections (1995) appropriate to grade level GLEs across content strands. Resource Kit (IMPORTANT: The intent of this GLE is to embed estimation throughout the instructional program, not to Algebra In the Real World – teach it as a separate unit.) Futures Channel Glencoe, Algebra I (2003) Creative Publications, Middle School Math with Pizzazz (1989) M(N&O)- Applies properties of numbers (odd, even, 8-8 remainders, divisibility, and prime factorization) to solve Glencoe, Pre-Algebra (2003) problems and to simplify computations. Textbook . Applies field properties (commutative, associative, Glencoe, Pre-Algebra (2003) identity [including the multiplicative property of one, e.g. Resource Kit 0 3 0+3 3 0 2 x 2 = 2 = 2 , so 2 = 1], distributive, inverses) to solve problems and to simplify computations. Glencoe, Applications and Demonstrates conceptual understanding of Connections (1995) field properties as they apply to subsets of real Resource Kit numbers when addition and multiplication are not defined in the traditional ways (e.g., If a ∆ b = a + b - 1, Glencoe, Algebra I (2003) is ∆ a commutative operation?). 8-3 Geometry and Measurement GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(G&M)- None 8-1 Glencoe, Applications and M(G&M)- Applies the Pythagorean Theorem to find a missing o Pythagorean Theorem Connections (1995) 8-2 side of a right triangle, or in problem solving situations. (G&M-14) Resource Kit o Triangle (G&M-3) M(G&M)- None 8-3 M(G&M)- None 8-4 M(G&M)- Applies concepts of similarity to determine the impact o Three-dimensional shapes Glencoe, Applications and 8-5 of scaling on the volume or surface area of three- (G&M-15) Connections (1995) dimensional figures when linear dimensions are o Similar figures (G&M-23) Resource Kit multiplied by a constant factor; o Triangle (G&M-3) o Describes the proportional effect Native American Geometry Applies concepts of similarity to determine the length on linear dimensions of polygons Replacement Unit - of sides of similar triangles or circles when scaling up or http://www.earthmeasure.com down while preserving the angles Applies concepts of similarity to solve problems involving growth and rate. of polygons (G&M-24) Compasses, Protractors, o Solves problems involving scaling Straightedges up or down and their impact on angle measure, linear dimensions, and areas of Glencoe, Algebra I (2003) polygons and circles when the linear dimensions are multiplied by a constant factor (G&M-25) o Ratio (N&O-12) o Proportional reasoning (N&O-44) 8-4 Geometry and Measurement GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(G&M)- Demonstrates conceptual understanding of surface o Demonstrates conceptual Glencoe, Applications and 8-6 area or volume by solving problems involving understanding of perimeter, Connections (1995) surface area and volume of rectangular prisms, area, volume, or surface area Resource Kit triangular prisms, cylinders, or pyramids. Expresses all using models and manipulatives measures using appropriate units. (G&M-28) Centimeter Cubes o Demonstrates conceptual understanding of perimeter, area, volume, or surface area by solving problems (G&M-29) Glencoe, Algebra I (2003) o Three-dimensional shapes (G&M-15) o Measures and uses units of measure appropriately and consistently (G&M-31) o Irrational number (N&O-10) o Whole number bases and whole number exponents, and fractional bases with whole number exponents (N&O-24) M(G&M)- None 8-7 M(G&M)- None 8-8 M(G&M)- None 8-9 M(G&M)- None 8-10 8-5 Functions and Algebra GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number o Patterns (F&A-1) M(F&A)– Identifies and extends to specific cases a variety of o Numeric patterns (F&A-3) Glencoe, Pre-Algebra (2003) 8–1 patterns (linear and nonlinear) represented in models, o Non-numeric patterns (F&A-4) Textbook tables, sequences, graphs, or in problem situations. o Extend a pattern (F&A-5) o Sequence (F&A-6) Glencoe, Pre-Algebra (2003) Identifies and extends to specific cases a variety of o Expresses generalization or rule Resource Kit patterns and generalizes a linear relationship (non- using words or symbols (F&A-7) recursive explicit equation) o Pattern Summary Table by grade Glencoe, Applications and level (F&A-9) Connections (1995) Identifies and extends to specific cases a variety of patterns and generalizes a linear relationship to find a o Linear relationships (F&A-10) Resource Kit specific case. o Proportional linear relationships (y = kx) (F&A-11) Algebra In the Real World – Identifies and extends to specific cases a variety of o Non-proportional linear Futures Channel patterns and generalizes a nonlinear relationship relationships (y = mx + b) using words or symbols, or generalizes a common (F&A-12) nonlinear relationship to find a specific case. o Non-linear relationships (F&A-18) Glencoe, Algebra I (2003) o Concrete situations (F&A-20) o Generalizes a pattern to find a specific case (F&A-8) 8-6 Functions and Algebra GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(F&A)– Demonstrates conceptual understanding of linear o Ratio (N&O-12) Glencoe, Pre-Algebra (2003) 8–2 relationships (y = kx; y = mx + b) as a constant rate o Proportional reasoning (N&O-44) Textbook of change by solving problems involving the o Linear relationships (F&A-10) relationship between slope and rate of change. o Proportional linear relationships Glencoe, Pre-Algebra (2003) Demonstrates conceptual understanding of linear (y = kx) (F&A-11) Resource Kit relationships (y = kx; y = mx + b) as a constant rate o Non-proportional linear of change by informally and formally determining relationships (y = mx + b) Glencoe, Applications and slopes and intercepts represented in graphs, tables, or (F&A-12) Connections (1995) problem situations. o Non-linear relationships (F&A-18) Resource Kit Demonstrates conceptual understanding of linear o Intercept (F&A-17) relationships (y = kx; y = mx + b) as a constant rate o Slope (F&A-14) Algebra In the Real World – of change by describing the meaning of slope and o Informally determines slope Futures Channel intercept in context (F&A-16) o Solves problems involving linear Glencoe, Algebra I (2003) Distinguishes between linear relationships relationships (F&A-13) (constant rates of change) and nonlinear o Describes the meaning of slope relationships (varying rates of change) represented and intercept in concrete in tables, graphs, equations, or problem situations. situations (F&A-15) Describes how change in the value of one variable o Distinguishes between constant relates to change in the value of a second variable in and varying rates (F&A-19) problem situations with constant and varying rates of change. 8-7 Functions and Algebra GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number o Rational number (N&0-1) M(F&A)– Demonstrates conceptual understanding of o Whole number bases and whole Glencoe, Pre-Algebra (2003) 8–3 algebraic expressions by evaluating and simplifying number exponents, and Textbook algebraic expressions (including those with square fractional bases with whole roots, whole number exponents, or rational numbers); or number exponents (N&O-24) Glencoe, Pre-Algebra (2003) demonstrates conceptual understanding of algebraic o Algebraic expression (F&A-25) Resource Kit expressions by evaluating an expression within an o Evaluating algebraic expressions equation (e.g., determine the value of y when x = 4 (F&A-26) Glencoe, Applications and given y 7 x 2 x ). o Simplifying algebraic equations Connections (1995) (F&A-27) Resource Kit o Number sentences (F&A-33) o Equation (F&A-32) Algebra In the Real World – o Examples of forms of equations Futures Channel (F&A-35) o Algebraic equation notation (F&A-34) M(F&A)– Demonstrates conceptual understanding of equality o Equality (F&A-30) Glencoe, Pre-Algebra (2003) 8–4 by showing equivalence between two expressions o Demonstrates equality (F&A-31) Textbook (expressions consistent with the parameters of the left- o Linear relationships (F&A-10) and right-hand sides of the equations being solved at o Proportional linear relationships Glencoe, Pre-Algebra (2003) this grade level) using models or different (y = kx) (F&A-11) Resource Kit representations of the expressions, solving formulas for o Non-proportional linear a variable requiring one transformation (e.g., d = rt; relationships (y = mx =b) Glencoe, Applications and d/r = t). (F&A-12) Connections (1995) Demonstrates conceptual understanding of equality o Number sentences (F&A-33) Resource Kit by solving multi-step linear equations with integer o Formula (F&A-28) coefficients. o Write equivalent forms of Algebra In the Real World – formulas (F&A-29) Futures Channel Demonstrates conceptual understanding of equality o Equation (F&A-32) by showing that two expressions are or are not o Examples of forms of equations Glencoe, Algebra I (2003) equivalent by applying commutative, associative, or (F&A-35) distributive properties, order of operations, or o Simplifying algebraic equations substitution. (F&A-27) o Algebraic equation notation Demonstrates conceptual understanding of equality (F&A-34) by informally solving problems involving systems of linear equations in a context. 8-8 Data, Statistics, and Probability GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(DSP) – Interprets a given representation (line graphs, scatter o Interprets a given representation Glencoe, Pre-Algebra (2003) 8-1 plots, histograms, or box-and-whisker plots) to analyze (DSP-21) Textbook the data to formulate or justify conclusions, to make o Representation (DSP-1) predictions, or to solve problems. o Histogram (DSP-7) Glencoe, Pre-Algebra (2003) o Scatter plot (DSP-13) Resource Kit (IMPORTANT: Analyzes data consistent with concepts o Box-and-whisker plot (DSP-21) and skills in M(DSP)-8-2.) Glencoe, Applications and Connections (1995) Resource Kit M(DSP) – Analyzes patterns, trends, or distributions in data in o Pattern (DSP-1) Glencoe, Pre-Algebra (2003) 8-2 a variety of contexts by determining or using o Mean (DSP-15) Textbook measures of central tendency (mean, median, or mode), o Median (DSP-16) dispersion (range or variation), outliers, quartile values, o Mode (DSP-17) Glencoe, Pre-Algebra (2003) or estimated line of best fit to analyze situations, or to o Dispersion (DSP-19) Resource Kit solve problems. o Range (DSP-19) o Outlier (DSP-20) Glencoe, Applications and Analyzes patterns, trends, or distributions in data in o Quartiles (DSP-11) Connections (1995) a variety of contexts by evaluating the sample from which the statistics were developed (bias, random, or o Estimated line of best fit Resource Kit non-random). (DSP-14) o Evaluates samples from which Glencoe, Algebra I (2003) the statistics were developed (bias) (DSP-24) o Analyzes the impact of outliers on the mean, median and mode (DSP-22) Glencoe, Pre-Algebra (2003) M(DSP) – Organizes and displays data using scatter plots to o Scatter plot (DSP-13) Textbook 8-3 answer questions related to the data, to analyze the o Representation (DSP-1) data to formulate or justify conclusions, to make o Identifies or describes Glencoe, Pre-Algebra (2003) predictions, or to solve problems; or identifies representations or elements of Resource Kit representations or elements of representations that best representations that best display display a given set of data or situation, consistent with a given set of data or situation Glencoe, Applications and the representations required in M(DSP)-8-1. (DSP-23) Connections (1995) Resource Kit (IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP)-8-2.) Glencoe, Algebra I (2003) 8-9 Data, Statistics, and Probability GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(DSP) – Uses counting techniques to solve problems in o Solves problems using a variety Glencoe, Pre-Algebra (2003) 8-4 context involving combinations or permutations using a of counting strategies (DSP-25) Textbook variety of strategies (e.g., organized lists, tables, tree o Permutation (DSP-27) diagrams, models, Fundamental Counting Principle, or o Frequency table (DSP-4) Glencoe, Pre-Algebra (2003) others). o Tree diagram (DSP-28) Resource Kit o Fundamental Counting Principle (DSP-29) Glencoe, Applications and Connections (1995) Resource Kit Glencoe, Algebra I (2003) M(DSP) – For a probability event in which the sample space Glencoe, Pre-Algebra (2003) 8-5 may or may not contain equally likely outcomes, Textbook determines the experimental or theoretical probability of an event in a problem-solving situation. Glencoe, Pre-Algebra (2003) Resource Kit For a probability event in which the sample space may or may not contain equally likely outcomes, Glencoe, Applications and predicts the theoretical probability of an event and tests Connections (1995) the prediction through experiments and simulations. Resource Kit For a probability event in which the sample space may or may not contain equally likely outcomes, Glencoe, Algebra I (2003) compares and contrasts theoretical and experimental probabilities. 8 - 10 Data, Statistics, and Probability GLE Grade Level Expectations Terms Defined (Definition Number) Notes and Resources Number M(DSP) – In response to a teacher or student generated Glencoe, Pre-Algebra (2003) 8-6 question or hypothesis decides the most effective Textbook method (e. g., survey, observation, experimentation) to collect the data (numerical or categorical) necessary to Glencoe, Pre-Algebra (2003) answer the question. Resource Kit In response to a teacher or student generated Glencoe, Applications and question or hypothesis collects, organizes, and Connections (1995) appropriately displays the data. Resource Kit In response to a teacher or student generated question or hypothesis analyzes the data to draw Glencoe, Algebra I (2003) conclusions about the question or hypothesis being tested while considering the limitations that could affect interpretations. In response to a teacher or student generated question or hypothesis makes predictions when appropriate. In response to a teacher or student generated question or hypothesis asks new questions and makes connections to real world situations. (IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP)–8–2.) 8 - 11

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