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Overview OverviewOverview_Txt Unit 10 highlights a return to geometry from the point of view of transformations or “motions” of geometric figures. It focuses primarily ◆ Overview Text Bullet on reflections and the related topic of symmetry. Unit 10 also introduces formal operations with positive and negative numbers. Unit 10 has three main areas of focus: ◆ To guide the discovery of basic properties of reflections, involving 2-dimensional figures and the connection with line symmetry, ◆ To guide the application of reflections, rotations, and translations, and ◆ To introduce addition involving negative integers. 778 778 Unit 00 Unit Title Unit 10 Reflections and Symmetry Contents Lesson Objective 10◆ 1 Explorations with a Transparent Mirror 10◆ 2 Finding Lines of Reflection 10◆ 3 Properties of Reflections 10◆ 4 Line Symmetry 10◆ 5 Frieze Patterns 10◆ 6 Positive and Negative Numbers 10◆7 Progress Check 10 Page 794 To guide the exploration of reflections of 2-dimensional figures. 799 To guide the exploration of reflections; and to provide practice identifying lines of reflection. 805 To guide the discovery of basic properties of reflections. 810 To guide exploration of the connection between reflections and line symmetry. 816 To guide the application of reflections, rotations, and translations. 822 To introduce addition involving negative integers. 828 To assess students’ progress on mathematical content through the end of Unit 10. Unit Organizer 779 Learning In Perspective 780 Lesson Objectives Links to the Past Links to the Future 10◆1 To guide the exploration of reflections of 2-dimensional figures. Grade 3: Explore symmetry with geoboards and pattern blocks; complete symmetric figures; identify symmetric shapes and draw lines of symmetry. Grade 5: Applications and maintenance. Grade 6: Study rotations, rotational symmetry, and point symmetry. 10◆2 To guide the exploration of reflections; and to provide practice identifying lines of reflection. Grade 3: Explore symmetry with geoboards and pattern blocks; complete symmetric figures; identify symmetric shapes and draw lines of symmetry. Grade 5: Applications and maintenance. Grade 6: Study rotations, rotational symmetry, and point symmetry. 10◆3 To guide the discovery of basic properties of reflections. Grade 3: Explore symmetry with geoboards and pattern blocks; complete symmetric figures; identify symmetric shapes and draw lines of symmetry. Grade 5: Applications and maintenance. Grade 6: Study rotations, rotational symmetry, and point symmetry. 10◆4 To guide exploration of the connection between reflections and line symmetry. Grade 3: Explore symmetry with geoboards and pattern blocks; complete symmetric figures; identify symmetric shapes and draw lines of symmetry. Grade 5: Applications and maintenance. Grade 6: Study rotations, rotational symmetry, and point symmetry. 10◆5 To guide the application of reflections, rotations, and translations. Grades 1–3: Use straws, geoboards, and body turns to demonstrate rotations. Grade 3: Model polygons; change the shapes of constructed polygons; perform polygon calisthenics to form polygons and explore their properties. Grade 5: Transform ordered number pairs and explore the resulting transformations of geometric figures. Explore regular tessellations. Grade 6: Review regular tessellations; introduce notation for tessellations; find semiregular tessellations. Create translation tessellations. Explore topological transformations. 10◆6 To introduce addition involving negative integers. Grade 3: Review uses of positive and negative numbers to relate numbers to a zero point, as in temperatures and elevations, and to record change. Solve number stories about positive and negative numbers. Grade 4: Play Credits/Debits Game (Advanced Version) to practice subtraction of positive and negative integers. Grade 5: Solve addition/ subtraction stories with positive and negative numbers. Grade 6: Develop a rule for adding and subtracting positive and negative numbers; practice adding and subtracting positive and negative numbers. Unit 10 Reflections and Symmetry Key Concepts and Skills 10◆1 10◆2 10◆3 10◆4 10◆5 10◆6 Key Concepts and Skills Grade 4 Goals* Describe properties of congruent figures. Identify, describe, and sketch reflections of two-dimensional figures. Solve problems involving spatial visualization. Geometry Goal 2 Geometry Goal 3 Geometry Goal 3 Describe properties of congruent figures, right angles, and perpendicular lines. Explore lines of reflection and reflected images. Solve problems involving spatial visualization. Geometry Goal 2 Geometry Goal 3 Geometry Goal 3 Measure length. Draw and describe congruent figures. Explore basic properties of reflections. Solve problems involving spatial visualization. Measurement and Reference Frames Goal 1 Geometry Goal 2 Geometry Goal 3 Geometry Goal 3 Identify polygons and describe properties of regular polygons. Identify and draw lines of symmetry. Explore the connection between reflections and line symmetry. Solve problems involving spatial visualization. Describe rules for patterns and use them to solve problems. Geometry Goal 2 Geometry Goal 3 Geometry Goal 3 Geometry Goal 3 Patterns, Functions, and Algebra Goal 1 Identify and draw congruent figures. Identify, describe, and sketch reflections, rotations, and translations. Extend, describe, and create geometric patterns. Geometry Goal 2 Geometry Goal 3 Patterns, Functions, and Algebra Goal 1 Compare and order integers. Add signed numbers. Identify a line of reflection. Number and Numeration Goal 6 Operations and Computation Goal 2 Geometry Goal 3 * See the Appendix for a complete list of Grade 4 Goals. Unit Organizer 781 Ongoing Learning and Practice Math Boxes Math Boxes are paired across lessons as shown in the brackets below. This makes them useful as assessment tools. Math Boxes also preview content of the next unit. Mixed practice [10◆ 1, 10◆ 4 ], [10◆ 2, 10◆ 5], [10◆ 3, 10◆ 6] Mixed practice with multiple choice 10◆ 3, 10◆ 4, 10◆ 6 Mixed practice with writing/reasoning opportunity 10◆ 1, 10◆ 3, 10◆ 5 Practice through Games 1 2 4 3 Games are an essential component of practice in the Everyday Mathematics program. Games offer skills practice and promote strategic thinking. Lesson Game Skill Practiced 10◆ 1 Over and Up Squares Locating and plotting points on a coordinate grid Measurement and Reference Frames Goal 4 10◆ 2 Dart Game Experimenting with transparent mirrors and reflections Geometry Goal 3 10◆ 2 Pocket-Billiards Game Experimenting with transparent mirrors and reflections Geometry Goal 3 10◆ 2 Angle Tangle Estimating and measuring angles Measurement and Reference Frames Goal 1 10◆ 5 Polygon Pair-Up Identifying properties of polygons Geometry Goal 2 10◆ 6 Credits/Debits Game Adding positive and negative numbers Operations and Computation Goal 2 See the Differentiation Handbook for ways to adapt games to meet students’ needs. Home Communication ▲ Study Links provide homework and home communication. Home Connection Handbook provides more ideas to communicate effectively with parents. Unit 10 Family Letter provides families with an overview, Do-Anytime Activities, Building Skills Through Games, and a list of vocabulary. 782 Unit 10 Reflections and Symmetry Problem Solving Encourage students to use a variety of strategies to solve problems and to explain those strategies. Strategies that students might use in this unit: ◆ Drawing a picture ◆ Using a pattern ◆ Acting out the problem ◆ Using computation ◆ Using data in a table Lesson Activity 10 ◆ 1 Use the transparent mirror to “move” and draw reflected images of shapes. 10 ◆ 2 Play games that involve reflections. 10 ◆ 3 Fold paper to observe reflected images. 10 ◆ 4 Find lines of symmetry of polygons. 10 ◆ 5 Create and continue frieze patterns. 10 ◆ 6 Use credits and debits to practice addition of positive and negative numbers. Lesson s teach t that h problem rough s not jus olving, t about problem solving See Chapter 18 in the Teacher’s Reference Manual for more information about problem solving. Planning Tips Pacing Pacing depends on a number of factors, such as students’ individual needs and how long your school has been using Everyday Mathematics. At the beginning of Unit 10, review your Content by Strand Poster to help you set a monthly pace. MOST CLASSROOMS MARCH APRIL MAY NCTM Standards Unit 10 Lessons 10 ◆ 1 10 ◆ 2 NCTM 1, 3, 6–10 3, 5, 6–10 Standards 10 ◆ 3 10 ◆ 4 10 ◆ 5 10 ◆ 6 10 ◆ 7 1, 3–5, 6–10 3, 6–10 2, 3, 6–10 1, 2, 4, 6–10 6–10 Content Standards: 1 Number and Operations, 2 Algebra, 3 Geometry, 4 Measurement, 5 Data Analysis and Probability Process Standards: 6 Problem Solving, 7 Reasoning and Proof, 8 Communication, 9 Connections, 10 Representation Unit Organizer 783 Balanced Assessment Ongoing Assessment Recognizing Student Achievement Opportunities to assess students’ progress toward Grade 4 Goals: Lesson 10 ◆ 1 Content Assessed Plot points in the first quadrant of a coordinate grid. [Measurement and Reference Frames Goal 4] 10 ◆ 2 10 ◆ 3 Compare fractions with like numerators or like denominators; compare fractions to the benchmark 12. [Number and Numeration Goal 6] Use a transparent mirror to sketch and describe a reflection. [Geometry Goal 3] 10 ◆ 4 Describe a pattern and use it to solve problems. [Patterns, Functions, and Algebra Goal 1] 10 ◆ 5 Identify and sketch an example of a reflection and identify examples of translations and rotations. [Geometry Goal 3] 10 ◆ 6 Express the probability of an event as a fraction. [Data and Chance Goal 4] Use the Assessment Management System to collect and analyze data about students’ progress throughout the year. Informing Instruction To anticipate common student errors and to highlight problem-solving strategies: Lesson 10 ◆2 Find a common factor to use as intervals when labeling an axis Lesson 10 ◆4 Understand the difference between line symmetry and rotation symmetry Lesson 10 ◆6 Encourage strategies for adding positive and negative numbers to evolve over time 784 Unit 10 Reflections and Symmetry Periodic Assessment 10◆7 Progress Check 10 ASSESSMENT ITEMS CONTENT ASSESSED Self Oral/Slate Written Name equivalent fractions, decimals, and percents. [Number and Numeration Goal 5] ✔ ✔ ✔ Add signed numbers. [Operations and Computation Goal 2] ✔ ✔ ✔ ✔ ✔ Add and subtract fractions. [Operations and Computation Goal 5] Make reasonable estimates for multiplication and division problems. [Operations and Computation Goal 6] Open Response ✔ Measure an angle. [Measurement and Reference Frames Goal 1] ✔ ✔ Locate multiple lines of symmetry in a two-dimensional shape. [Geometry Goal 3] ✔ ✔ Identify and sketch examples of reflections; identify examples of translations and rotations. [Geometry Goal 3] ✔ ✔ ✔ Portfolio Opportunities Opportunities to gather samples of students’ mathematical writings, drawings, and creations to add balance to the assessment process: ◆ Inserting decimal points, Lesson 10◆ 1 ◆ Creating a paint reflection, Lesson 10◆ 2 ◆ Interpreting a cartoon, Lesson 10◆ 4 ◆ Calculating the mean, Lesson 10◆ 5 ◆ Creating frieze patterns to practice reflections, rotations, and translations, Lesson 10◆ 5 ◆ Solving problems involving pentominoes, Lesson 10◆ 7 Assessment Handbook Unit 10 Assessment Support ◆ Grade 4 Goals, pp. 37–50 ◆ Unit 10 Assessment Overview, pp. 126–133 ◆ Unit 10 Open Response • Detailed rubric, p. 130 • Sample student responses, pp. 131–133 Unit 10 Assessment Masters ◆ Unit 10 Self Assessment, p. 200 ◆ Unit 10 Written Assessment, pp. 201–203 ◆ Unit 10 Open Response, pp. 204 and 205 ◆ Unit 10 Class Checklist, pp. 284, 285, and 303 ◆ Unit 10 Individual Profile of Progress, pp. 282, 283, and 302 ◆ Exit Slip, p. 311 ◆ Math Logs, pp. 306–308 ◆ Other Student Assessment Forms, pp. 304, 305, 309, and 310 Unit Organizer 785 Differentiated Instruction Daily Lesson Support ENGLISH LANGUAGE LEARNERS EXTRA PRACTICE 10◆ 2 Building a Math Word Bank 10◆ 4 Creating a Line Symmetry Museum 10◆ 4 Creating shapes with line symmetry 10◆ 5 Creating frieze patterns 5-Minute Math 10◆ 3 Solving problems involving reflections READINESS ENRICHMENT 10◆ 1 Exploring reflections with paper folding 10◆ 2 Creating a paint reflection 10◆ 3 Creating reflections 10◆ 5 Exploring geometric patterns 10◆ 6 Using a calculator to skip count 10◆ 6 Using a number line to add positive and negative integers 10◆ 1 Exploring shadows and reflections 10◆ 2 Solving paper-folding puzzles 10◆ 2 Using technology to investigate the mirror as a virtual manipulative 10◆ 3 Constructing 3-dimensional buildings and their reflections 10◆ 4 Interpreting a cartoon involving line symmetry 10◆ 4 Exploring turn or rotation symmetry 10◆ 5 Exploring arrangements of four straws 10◆ 5 Exploring tessellations Adjusting the Activity 10◆2 Devising a scoring system for Pocket-Billiards Game 10◆3 Determining the latitude and longitude of a capital city 10◆4 Summarizing the discussion of line symmetry ELL A U D I T O R Y 䉬 K I N E S T H E T I C 10◆4 Having a volunteer cut out shapes ahead of time 10◆5 Sketching frieze patterns on centimeter grid paper ELL 10◆6 Experimenting with a calculator 䉬 T A C T I L E 䉬 V I S U A L Cross-Curricular Links Language Arts Technology Lesson 10◆ 1 Students discuss the use of the prefix pre-. Lesson 10◆ 2 Students investigate the use of a mirror as a virtual manipulative. Lesson 10◆ 5 Students use technology to explore tessellations. Art Lesson 10◆ 2 of reflection. Lesson 10◆ 5 their own. Lesson 10◆ 5 Lesson 10◆ 5 Students create a paint reflection to explore the concept Students complete three frieze patterns and design one of Social Studies Students create and continue patterns using pattern blocks. Students make frieze patterns by following directions. Lesson 10◆ 3 Students visit a second country in Region 4. Using the Projects Use Project 4, Making a Quilt, after Unit 10 to explore and apply ideas of pattern, symmetry, rotation, and reflection in the context of quilts. See the Differentiation Handbook for modifications to Project 4. Differentiation Handbook See the Differentiation Handbook for materials on Unit 10. 786 Unit 10 Reflections and Symmetry Language Support Everyday Mathematics provides lesson-specific suggestions to help all students, including non-native English speakers, to acquire, process, and express mathematical ideas. Connecting Math and Literacy Lesson 10◆ 1 Shadows and Reflections, by Tana Hoban, Greenwillow, 1990 Lesson 10◆ 3 Count Your Way through Japan, by Jim Haskins, Carolrhoda Books, 1988 Lesson 10◆ 3 Count Your Way through Russia, by Jim Haskins, Carolrhoda Books, 1987 Lesson 10◆ 3 Count Your Way through India, by Jim Haskins, Carolrhoda Books, 1992 Lesson 10◆ 3 Count Your Way through Israel, by Jim Haskins, Carolrhoda Books, 1992 How the Second Grade Got $8,205.50 to Visit the Statue of Liberty, by Nathan Zimelman, Albert Whitman & Company, 1992 Reflections, by Ann Jonas, Greenwillow, 1987 Unit 10 Vocabulary congruent credit debit frieze pattern image line of reflection line of symmetry opposite (of a number) preimage recessed reflection reflection (flip) rotation (turn) rotation (turn) symmetry symmetric translation (slide) transparent mirror Student Reference Book pp. 60, 108, 109, 230, 238, 257, and 258 Multiage Classroom ◆ Companion Lessons Companion Lessons from Grades 3 and 5 can help you meet instructional needs of a multiage classroom. The full Scope and Sequence can be found in the Appendix. Grade 3 6◆ 3, 6◆ 9 6◆ 9 6◆ 3, 6◆ 9 6◆ 3, 6◆ 9 1◆ 12, 6◆ 3 Grade 4 10◆1 10◆ 2 10◆ 3 10◆ 4 10◆ 5 10◆ 6 3 ◆8 3 ◆8 7 ◆ 7–7 ◆ 9, 7 ◆ 11 Grade 5 Professional Development Teacher’s Reference Manual Links Lesson Topic Section Lesson Topic Section 10 ◆ 1 Geometry Tools 3.2.4 10 ◆ 4 Line Symmetry 13.8.1 Transparent Mirrors 13.13.5 Polygons 13.4.2 Reflections, Rotations, and Translations 13.7.1 Congruence and Similarity 13.6.2 10 ◆ 2 10 ◆ 3 See 10 ◆ 2. 10 ◆ 5 See 10 ◆ 2. 10 ◆ 6 Positive and Negative Numbers 9.4 Operations with Positive and Negative Numbers 10.2.2 Unit Organizer 787 Materials Lesson 10◆ 1 10◆ 2 10◆ 3 10◆ 4 10◆ 5 10◆ 6 10◆ 7 Masters Manipulative Kit Items Other Items Teaching Masters, pp. 304 and 305 Study Link Master, p. 306 Game Master, p. 494 Teaching Aid Master, p. 389 slate 2 six-sided dice transparent mirror sheets of paper; colored pencils; crayons* ; scissors; Shadows and Reflections Study Link 10 ◆ 1 Study Link Master, p. 307 Game Master, p. 457 Teaching Master, p. 308 Teaching Aid Master, p. 389 slate Geometry Template transparent mirror ruler; large sheets of paper; paints, brushes, and dark marker; computer with Internet access (if available); scissors; protractor Study Link 10 ◆ 2 Teaching Master, p. 309 Study Link Master, p. 310 Teaching Aid Masters, pp. 389, 403, 419–421* , and 447 slate centimeter cubes pattern blocks transparent mirror ruler; blank paper Study Link 10 ◆ 3 Teaching Masters, pp. 311–314, 316, and 317 Teaching Aid Master, p. 389 Study Link Master, p. 315 slate pattern blocks Geometry Template transparent mirror scissors; magazines and newspapers; tape Study Link 10 ◆ 4 Teaching Aid Master, p. 389, 403* , and 437 Study Link Master, p. 318 Teaching Master, p. 319 Geometry Template straws pattern blocks slate transparent mirror straightedge; computer with Internet access; index cards; scissors; overhead or regular pattern blocks* ; Polygon Pair-Up Polygon Cards and Property Cards Study Link 10 ◆ 5 Teaching Master, p. 320 Game Master, p. 468 transparencies of Math Masters, pp. 318* and 321* Study Link Master, p. 322 deck of number cards calculator number line transparent mirror masking tape Study Link 10 ◆ 7 Assessment Masters, pp. 200–205 Study Link Masters, pp. 323–326 slate transparent mirror Geometry Template scissors * Denotes optional materials 788 Unit 10 Reflections and Symmetry Technology Assessment Management System, Unit 10 iTLG, Unit 10 Mathematical Background The discussion below highlights the major content ideas presented in Unit 10 and helps establish instructional priorities. Types of Geometry (Lessons 10◆ 1 and following) You may remember your high school geometry course as dealing with definitions, axioms, theorems (“Given ..., To Prove ...”), and perhaps straightedge-and-compass constructions. This form of synthetic geometry was first developed by Euclid about 300 B.C. and has been the model for teaching geometry ever since. 10 (3,9) 9 8 7 6 However, there are two modern geometries that cover the same topics: 5 Analytic geometry The study of figures in a coordinate plane. 3 (8,5) 4 2 Transformation geometry The study of certain operations on figures. These operations, or “transformations,” produce figures that are the same shape as (similar to) the original figures, or the same size and shape as (congruent to) the original figures. 1 0 0 (1,1) 1 2 3 4 5 6 7 8 9 10 Figure in a coordinate plane These two geometries are probably more useful than synthetic geometry. Both are featured in Grades 4–6 of Everyday Mathematics, with Unit 10 introducing the transformation approach. Learn more about types of geometry in Chapter 13 of the Teacher’s Reference Manual. “Isometric” or “Congruence” Transformations (Lessons 10◆ 1 and following) You may remember from your high school geometry course the emphasis on congruent figures, especially on proving triangles congruent by theorems called “side-angle-side” or “SAS,” and so on. This topic is handled in transformation geometry by rigid motions or isometric transformations, which do not change the size or shape of figures. These transformations— translations (slides), reflections (flips), and rotations (turns)—can duplicate any figure. (SAS and similar theorems of synthetic geometry apply only to triangles.) See the Teacher’s Reference Manual, Section 13.7, for more information about isometric transformations. Unit Organizer 789 Reflections (Flips) and Symmetry with Transparent Mirrors (Lessons 10◆ 1–10◆ 4) Everyone is familiar with mirrors and the exact—but reversed—images one sees in them. The device used in this unit, the transparent mirror, has an advantage over a regular mirror: It allows students to look through a mirror and reach behind it to touch or trace the mirror image (almost like Alice going “through the looking glass”). Transparent mirror As with any new tool, developing the skills for its use takes time, practice, and patience. For accurate placement of images, have students practice these skills: ◆ Lean down and look directly through the transparent mirror. ◆ Use the ends of the transparent mirror to keep it perpendicular to the paper. ◆ Use the inner part of the recessed edge to place the transparent mirror on points or lines or to draw mirror lines. ◆ Hold the transparent mirror firmly in position with one hand while drawing behind it or along its recessed edge. (This is one of the main skills to be learned.) It is probably a good idea to acquire or practice these skills yourself before teaching the lessons. Do the mirror exercises on journal pages and masters until you feel comfortable using the transparent mirror. To learn more about reflections and symmetry with a transparent mirror, refer to Section 13.8 in the Teacher’s Reference Manual. 790 Unit 10 Reflections and Symmetry Transformations as Design Tools (Lesson 10◆ 5) Geometric patterns are part of many designs—in arts and crafts from around the world, in architecture and engineering, and in paintings and other works of art (sometimes in disguised forms). Lesson 10-5, on frieze patterns, encourages students to explore reflections, symmetry, rotations, and translations in order to analyze and create designs. The authors believe that fourth graders will find these design tasks enjoyable. Some of their creations may be quite elegant. An interesting property of transformations is that two successive reflections across parallel mirror lines are equivalent to one translation. (The original image is reversed in the first reflection, but the mirror image of the first reflection can be a translated image.) Hence one can make friezes either by translating and tracing or by using transparent mirrors twice for each frieze copy. The authors hope that the principles learned here can be linked to teaching visual arts in your school. Geometry that is useful in both art and practical matters will be applied many times throughout Everyday Mathematics. For more about transformations as design tools, see Section 13.8 in the Teacher’s Reference Manual. 哬 哬 Unit Organizer 791 Operations with Positive and Negative Numbers (Lesson 10◆ 6) Since Kindergarten, Everyday Mathematics students have been using positive and negative numbers to identify locations on timelines, number lines, number grids, and thermometers. Since first grade, students have informally used addition and subtraction in going from one place to another and in finding distances. But Lesson 10-6 may be students’ first exposure to operations with positive and negative numbers. In this lesson, the numbers are limited to integers—whole numbers and their (negative) opposites. “Credits and debits” number stories are used to help make addition concrete. Single-digit numbers ensure that most problems can be done mentally. It is important in these “accounting” situations to name both the operation and the number. ◆ “Add +$3” is read “Add positive 3 dollars” (a credit transaction). ◆ “Add –$5” is read “Add negative 5 dollars” (a debit transaction). This distinguishes the addition operation from the numbers involved. Later lessons in Grade 4 introduce subtraction of positive and negative numbers and teach the use of the “change sign” key, which enables calculators to work with negative as well as positive numbers. Many practice and review exercises are included. Other operations with positive and negative numbers, as well as applications using them, are featured in Grades 5 and 6. See Section 10.2.2 of the Teacher’s Reference Manual for more information about operations with positive and negative numbers. 792 Unit 10 Reflections and Symmetry Confusing Notation for Positive and Negative Numbers (Lesson 10◆ 6) The use of the same notation with several meanings can be confusing. This is true of the symbol “”: ◆ The symbol “” attached to a numeral, as in 3, 0.5, or 37, is read “negative” and is used in naming numbers on the number line (“negative three,” “negative five-tenths,” “negative thirty-seven”). ◆ The symbol “” in a number model, preceding a positive or negative number, as in (3) or (17), is read “opposite of.” The opposite of a positive number is a negative number; the opposite of a negative number is a positive number. For example, the “opposite of positive 3” is negative 3, and the “opposite of negative 17” is positive 17. 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 Opposites ◆ The symbol “” in a number model, as in 17 3 14, is read “minus,” “subtract,” or “take away” and indicates the familiar subtraction operation. Example 17 3 14 is read “17 minus 3 equals 14.” 17 (3) 14 is read “17 minus 3 equals 14.” (17) 3 14 is read “the opposite of negative 17 minus 3 equals 14.” 17 (3) 14 is read “17 plus negative 3 equals 14.” 17 (3) 20 is read “17 minus negative 3 equals 20.” The meanings of the symbol “” can get quite tangled in number models like 17 3 = 20 (“Negative 17 minus 3 equals negative 20”) or 12 (4) = 8 (“12 take away the opposite of negative 4 equals 8”). Some mathematics programs of the past tried to reduce confusion by using “” only for subtraction. Positive and negative numbers were designated with small raised symbols (for example, 3, 7, 17), and opposites were indicated by “opp.” But everyday usage and nearly all algebra books continued to use the traditional notation, so students eventually had to reconcile the two notations. Given the problems associated with both notations, the authors have decided to use the traditional system. Help students sort it out when you read expressions by consistently saying “plus” or “minus” for addition or subtraction, and “positive,” “negative,” or “opposite” for numbers, as indicated by the context. Encourage students to do likewise when they eventually read expressions to each other and to themselves. Note Public or private speech is very helpful in dealing with complexities of meanings. It is verifiable and observable common sense for students to use speech, as well as sight, to sort out complicated symbolic expressions. (This is a key tenet of the learning theory of Lev S. Vygotsky. An excellent article on Vygotskian learning theory is “Why Children Talk to Themselves,” by Laura E. Berk, in the November 1994 issue of Scientific American.) To further explore confusing notation for positive and negative numbers, refer to Section 9.4.1 in the Teacher’s Reference Manual. Unit Organizer 793