Kindergarten Math Pacing Guide SEPTEMBER *Everyday Kindergarten Math (EKM) Unit 1: Activities 1-1 thru 1-10; 1-12,1-14 FEBRUARY Unit 1: 1-11 Unit 2: 2-8 Unit 3: 3-8; 3-11 Unit 6: 6-1; 6-2; 6-7 Unit 7: 7-1; 7-5 Unit 8: 8-7 OCTOBER Unit 2: Activities 2-1 thru 2-7; 2-9; 2-13 thru 2-16 Unit 3: Activity 3-1 (numeral writing may vary in pacing sequence depending on class make-up and pre- requisite);3-14 Unit 4: Activities 4-10 *Developing Number Concepts – Chapter 1 MARCH Unit 3: 3-7 Unit 5: 5-7; 5-11; 5-12 Unit 6: 6-3; 6-6; 6-9 thru 6-12; 6-14; 6-16 *Developing Number Concepts – Chapter 3 NOVEMBER Unit 3: Activities 3-3 thru 3-6; 3-9 thru 3-12; 3-15 Unit 4: 4-13; 4-14 Unit 5: 5-3; 5-1 *Developing Number Concepts – Chapter 2 APRIL Unit 7: 7-6 thru 7-7 thru 15 Unit 8: 8-5; 8-6 *Developing Number Concepts – Chapter 2 DECEMBER Unit 2: Activities 2-10 thru 2-12 Unit 3: 3-16 Unit 4: 4-2; 4-3; 4-5;4-6 *Developing Number Concepts – Chapter 2 MAY Unit 6: 6-4; 6-13 Unit 8: 8-2; 8-3; 8-9 thru 8-14 JANUARY Unit 4: Activities 4-1; 4-4; 4-7; 4-8; 4-11; 4-12; 4-15 4-16 Unit 5: 5-2; 5-4; 5-5; 5-8; 5-9; 5-15; 5-16 Unit 8: 8-4*Developing Number Concepts – Chapter 2 JUNE Review of concepts where needed or “catch-up” month
Grade: Kindergarten Standard 4.1 Number and Numerical Operations Essential Question: How can problems in the real world be solved with mathematics? How can estimation be useful to us? How do numbers help us reason out solutions to problems? How do basic operations help us understand numbers? Knowledge/Skills/Understandings Assessments Learning Experiences Resources A. Number Sense 1. Use real life experiences, physical materials, and technology to construct meaning for numbers. Observation of students exploration with manipulatives Oral assessment Mid Year/End of Year Assessments "Look and Find" (Numbers 1 -10) Egg-Carton Mathematics Everyday Math: Teacher's Guide * Demonstrates understanding of one-one correspondence. (e.g., places one placement at each place, gives each child one cookie, places one animal in each trunk, hands out manipulatives to be shared with a friend saying "One for you, one for me.") (Preschool) Listen and Count Everyday Math: Teacher's Guide *Verbally counts 20 or more objects in a random arrangement. Mid-Year and Final: Correspondence to 20 Teacher number cards Act. Making Towers Grow and Shrink Creations Unifix Cubes Digi-Blocks Everyday Math: Teacher's Guide Developing Number Concepts: Counting, Comparing, and Pattern, Kathy Richardson *Identifies small numbers without counting. Final: Subitizing with Finger Patterns Finger Counting Say-It Fast Cards High Roller game Everyday Math: Teacher's Guide Developing Number Concepts: Counting, Comparing, and Patterns, Kathy Richardson
*Writes numerals 0 -10. *Reads and writes numbers, 100 or less. Final: Writes numerals 0 - 10, randomly. Oral / Slate Assessments Slate writing activities Number Books: Writing 0 Number Hunt and 100 Chart Preparation for 100 Day 100 Number Grids Double Digit Dice Game Class Collection Project (100 poster) Writing on Backs Spin a number (1 -10) Interrupted Skip Counts (0 Number Grid Say the Next Number by 10s Everyday Math: Teacher's Guide Developing Number Concepts: Counting, Comparing and Pattern, Kathy Richardson Teacher Number Cards Activities *Counts backwards from 10. *Counts backwards from 22. Mid-Year and Final: Counting Backwards 10 - 0 Countdown Counting Backwards with / without calculators Say the Next Number Everyday Math: Teacher's Guide *Counts by 10's to 100. *Counts by 5's. *Counts by 2's. Final: Skip Counts by 10's Skip count by 10's Counting on calculators Say the Next Number by 10s Introduction to Skip Counting by 2's Count Fingers by 5's Count using tally marks Count to 70 by 10s Everyday Math: Teacher's Guide *Introduce concept of zero Oral discussion Everyday Math: Teacher's Guide
*Developing the basic meaning of addition and subtraction in real situations, in children's own number stories, oral problems, concrete objects and number lines. Teacher Observation Joining Objects Change to Less Pocket Game Disappearing Train High Roller What Number Am I Thinking of? Everyday Math: Teacher's Guide Developing Number Concepts: Counting, Comparing and Pattern, Kathy Richardson *Participates in solving oral number stories. Oral assessment Student sharing of own stories and solutions Number Stories throughout the Year Everyday Math: Teacher's Guide *Divide even groups in half Observation of student while exploring task with manipulatives Divide Groups into Half Groups use manipulatives Everyday Math: Teacher's Guide 3. Develop proficiency with basic addition and subtraction number facts using a variety of strategies. *Develop proficiency with basic addition using the strategy of counting on. High Roller Top It Everyday Math: Teacher's Guide *Explore counting one more or one less) Oral / Slate Assessment One More or One less Developing Number Concepts: Counting, Comparing, and Pattern, Kathy Richardson C. Estimation 1. Judge without counting whether a set of objects has less than, more than, or the same number of objects as a reference set. *Determines which of two groups is more and which is less. Final: Weather Graph More or Less Activities (DNC) Stack, Tell, Spin and Win Everyday Math: Teacher's Guide Developing Number Concepts: Counting, Comparing, and Pattern, Kathy Richardson
Grade: Kindergarten Standard 4.4 Data Analysis, Probability, and Discrete Mathematics Essential Questions: How can classifying help me in organizing data to solve problems? How can statistics help us to understand real world situations? How can the study of real world data help us understand and make accurate predictions? Knowledge/Skills/Understandings Assessments Learning Experiences Resources A. Data Analysis 1. Collect, generate record, and organize data in response to questions, claims or curiosity. Oral Assessment Attendance Routine Recording Daily Temperature Everyday Math: Teacher's Guide *Data collected from students' everyday experiences. Student participation in collection and recording of data Favorite Colors graph Everyday Math: Teacher's Guide *Data generated from chance devices, such as spinners and dice. Observation of students during play Spinners, Dice games Graph sums of dice throws Everyday Math: Teacher's Guide 2. Read, interpret, construct and analyze displays of data. *Pictures, tally chart, pictograph, bar graph, Venn Diagram Smallest to largest, most frequent (mode) Calendar activities Reminder for Tally Marks Everyday Math: Teacher's Guide *Arrange pictures of events in temporal order. (Preschool) Teacher Observation
*Describe how he/she solved mathematical problems in his/her own way. (Preschool) Student Discussion Teacher supplement B. Communication 1. Use communication to organize and clarify their mathematical thinking. Teacher observation What's My Rule? Everyday Math: Teacher's Guide 4. Use the language of mathematics to express mathematical ideas precisely. Teacher observation Birthday Graph Dismantling the Monthly Calendar Everyday Math: Teacher's Guide *Uses mathematical terms when conversing with others (e.g., "Which car is faster?" "My building is taller than yours." "I have more sand in my bucket."). (Preschool) Teacher observation Model appropriate math vocabulary Calendar activities C. Connections 2. Use connections among mathematical ideas to explain concepts. Slow and Fast Compare activities with variable speeds Everyday Math: Teacher's Guide The Tortoise and the Hare * Making connections between two dimensional and three dimensional forms (circle-sphere, square-cube, triangle-pyramid) (Preschool) Student -teacher interview 2-D and 3-D Objects Ask questions what is same and different between a square and a cube, circle and cylinder Teacher supplement with questions 3. Recognize that mathematics is used in a variety of contexts outside of mathematics. Teacher observation Noticing Numbers in the environment Patterns All Around Simple Hopscotch Telephone Book 100th school day Everyday Math: Teacher's Guide
D. Reasoning 2. Use reasoning to support their mathematical conclusions and problem solutions. Observation of partner play "Feely" Bag or Box Partners Everyday Math: Teacher's Guide *Use emergent mathematical knowledge as a problem-solving tool. (Preschool) Teacher observation Engage student in play that facilitates simple problem solving Teacher supplement Use literature books 6. Evaluate examples of mathematical reasoning and determine whether they are valid. Teacher observation Everyday Math: Teacher's Guide E. Representations 1. Create and use representations to organize, record, and communicate mathematical ideas. -concrete representations -pictorial representations -symbolic representations Observation of student while interacting with manipulatives as they explore math ideas Provide opportunities for students to model mathematical ideas using manipulatives and other models Everyday Math: Teacher's Guide NJ Mathematics Curriculum Framework, selected activities grades K-2 *Identifying the meaning of common signs and symbols. (Preschool) Teacher observation / Student responses Classroom symbols, school symbols, bathroom symbols, street light 2. Select, apply and translate among mathematical representations to solve problems. Teacher observation Classroom Playing Cards Everyday Math: Teacher's Guide F. Technology
Unit Number of Days (approximate) Unit Number of Days (approximate) 1 – Establishing Routines 18 6 – Developing Fact Power 17 2 – Everyday Uses of Numbers 18 7 – Geometry & Attributes 12 3 – Visual Patterns, Number Patterns, & Counting 19 8 – Mental Arithmetic, Money and Fractions 14 4 – Measurement & Basic Facts 17 9 – Place Value and Fractions 13 5 – Place Value, Number Stories & Basic Facts 18 10 – Year-End Review 12 Mid-Year Benchmark Assessment 2 End-of-the-Year Benchmark Assessment 2 Grade: 1 Standard 4.1 Number and Numerical Operations Essential Question: How can problems in the real world be solved with mathematics? How do numbers help us reason out solutions to problems?
How do basic operations help us understand numbers? Knowledge/Skills/Understandings Assessments Learning Experiences Resources A. Number Sense 1. Use real-life experiences, physical materials, and technology to construct meanings for numbers. *Whole numbers through hundreds *Ordinals *Proper fractions (denominators of 2, 3, 4, 8, 10) Oral / Slate Assessment Unit Assessments Do Now Teacher Observation Discussion Open Ended Writing Tasks Recognizing Student Achievement sections in Everyday Math 3 Direct Instruction Collaborative and Cooperative Learning Open Ended Writing Tasks Place Value Mat Cuisenaire Rods Digi-blocks Base Ten Blocks Calendar skills Number line Slides Giant Floor Number Line EM Games and Activities Auntie Pasta’s Fraction Game Coins Geometric shapes Number grid Pattern Blocks Calculator EM Skills Link Everyday Mathematics, Lesson 1.2, 5.6, 8.3, 10.7 *Count by 2's to 100; Count by 5's to 100 Choral skip counting, Stop and start counting Tallies & Nickels & Pennies Everyday Mathematics Lesson 1.2, 1.12. 1.13, 2.7, 2.9, 2.10, 3.2, 3.3 ongoing throughout the year *Count up and back by 1's on the number grid to 100 Everyday Mathematics, Lesson 1.5, 1.7, 2.1, 9.1 EM Skills Link EM teacher resource for grid
* Writes any number 100 or less EM Project 5: Calendar for the New Year EM Project 6: Celebrate the Hundredth School day Everyday Mathematics, Lesson 1.2, 1.3, 1.4, 1.9, 4.10 Fish Eye: A Book You Can Count On, Lois Ehlert 1, 2, 3, Tana Hoban Can You Count Ten Toes?: Count to 10 in 10 Different Languages, Leslie Evans City by Numbers, Stephen T, Johnson *Count up and back by 10's starting with any given number up to and including 100 Try to develop mental math for 10s Everyday Mathematics, Lesson 6.6, 9.2 *Practice counting on a number line Everyday Mathematics, Lesson 3.2, 3.5, 3.6, 3.9, 5.4 *Write and count tallies Everyday Mathematics, Lesson 1.7, 2.9 *Find equivalent names for a number Everyday Mathematics, Lesson 6.2 *Explore parts and total relationships Everyday Mathematics, Lesson 3.14, 4.6, 5.8, 9.4 Fraction Action, Loreen Leedy
*Measuring to the nearest inch and half inch Everyday Mathematics, Lesson 4.5 Tom Thumb (story about inchlings), Margaret Read MacDonald Inch By Inch, Leo Lionni Jack and the Beanstalk, Traditional Folktale *Introducing the yard Everyday Mathematics, Lesson 4.2 *Use a tape measure to measure curved and flat objects in inches Everyday Mathematics, Lesson 4.6 *Explore area by counting units Everyday Mathematics, Lesson 5.4 * Introduce and tell time using the analog clock *Tell time to the nearest hour and half hour Everyday Mathematics, Lesson 2.5, 2.6, 3.7, 10.2 Teacher supplement with clock songs *Telling time on the quarter hour Everyday Mathematics, Lesson 4.8 *Telling time to five minutes Everyday Mathematics, Lesson 10.2 Big Book: “What Time Is It?” * Understand digital notation for time Everyday Mathematics, Lesson 6.10 *Introduce the use of the second hand for timing tasks Everyday Mathematics, Lesson 2.5
C. Connections 1. Recognize recurring themes across mathematical domains. Everyday Mathematics, Lesson 1.2, 1.9 Teacher supplement 2. Use connections among mathematical ideas to explain concepts. Project: 5 Apple Math Project 2 Amaryllis Plant 3. Recognize that mathematics is used in a variety of contexts outside of mathematics. Project 1 Geometric Gift Wrap and Greeting Cards Everyday Mathematics, Lesson 1.9, 2.2 Link to Social Studies: discuss museum, community helpers 4. Apply mathematics in practical situations in other disciplines. Seed Unit, Climate Zones Science Curriculum, Social Studies Curriculum 5. Trace the development of mathematical concepts over time and across cultures (cf. world languages and social studies standards). Everyday Mathematics, Lesson 4.9 See Science, Social Studies curriculum for resources and connections D. Reasoning 1. Recognize that mathematical facts, procedures, and claims must be justified. Everyday Mathematics, Lesson 7.4 2. Use reasoning to support their mathematical conclusions and problem solutions. Everyday Mathematics, Lesson 1.10, 7.4 4. Rely on reasoning, rather than answer keys, teachers, or peers, to check the correctness of their problem solutions. Teacher supplement
4. Construct, use and explain procedures for performing addition and subtraction calculations with: *pencil and paper *mental math *calculator Everyday Mathematics, Lesson 4.8, 4.9 EM Skills Book Website Resources for practice *Add and subtract multiples of 10 Everyday Mathematics, Lesson 1.5, 1.8, 1.10, 3.1, 3.4, 4.5, 6.5, 7.1, 10.9 * Know compliments of 10. (using two digit numbers) Everyday Mathematics, Lesson 7.2 Illumination web-site (Ten Frames) Website resources *Find missing addends for the next multiple of 10. Everyday Mathematics, Lesson 7.2, selected activities Elementary School Mathematics with Pizzazz, Creative Publications 5. Use efficient and accurate pencil and paper procedures for computation with whole numbers. *Addition of 2-digit numbers *Subtraction of 2-digit numbers Everyday Mathematics, activities, ongoing throughout the year Website resources Elementary School with Pizzazz, Creative Publications * Devise and use strategies for finding sums and differences of 2-digit numbers. Everyday Mathematics, Lesson 4.6, 4.8 Teacher Supplement Website resources for practice EM Skills Book 4.9,
C. Estimation 1. Judge without counting whether a set of objects has less than, more than or the same number of objects as a reference set. Teacher supplement 2. Determine the reasonableness of an answer by estimating the result of computations (e.g., 15 + 16 is not 211). Everyday Mathematics, Lesson 4.5 and ongoing throughout the year 3. Explore a variety of strategies for estimating both quantities (e.g., the number of marbles in a jar) and results of computation. *Example - Estimating the number of marbles in a jar. Teacher supplement
3. Use graphing calculators and computer software to investigate properties of functions and their graphs. Teacher supplement Website resources Texas Instruments resources 4. Use calculators as problem-solving tools (e.g., to explore patterns, to validate solutions). TI-108 5. Use computer software to make and verify conjectures about geometric objects. Activities in conjunction with technology teacher Website resources National Virtual Manipulatives Library website 6. Use computer-bases laboratory technology for mathematical applications in the sciences (cf. science standards). Activities in conjunction with technology teacher and or science lessons
Third Grade Math Pacing Guide Unit Number of Days (approximate) Unit Number of Days (approximate) 1 – Routines/Review & Assessment 16 7 – Multiplication & Division 15 2 – Adding & Subtracting Whole Numbers 15 8 – Fractions 11 3 – Linear Measures & Area 14 9 – Multiplication & Division 17 4 – Multiplication & Division 14 10 – Measurement & Data 13 5 – Place Value, Whole Numbers & Decimals 16 11 – Probability 8 6 – Geometry 15 End-of-the-Year Benchmark Assessment 1
Marking Period Packets: MP 1, 2, 3, 4 Addition separate, subtraction separate;; Multiplication and division facts separate 1. Develop the meaning of the four basic arithmetic operations by modeling and discussing a large variety of problems. *Addition and subtraction: joining, separating, comparing *Multiplication: repeated addition, area/array *Division: repeated subtraction, sharing Amanda Bean's Amazing Dream, Cindy Neuschwander Everyday Mathematics, Lessons 2.1, 2.2, 4.1 - 4.8, Groundworks-Reasoning with Numbers, Creative Publications Teaching Student-Centered Mathematics, Grades 3-5, Lovin and Van de Walle *Know basic addition and subtraction facts up to 20 Teacher Supplement Everyday Mathematics, Unit 2 Website resources *Complete fact and number families Everyday Mathematics, Lesson 2.1, 4.6, 4. 7, 4.8, 7.1, 7.3 *Solve addition and subtraction multi-digit number stories Everyday Mathematics, Lesson 2.4 - 2.6, 2.9 2. Develop proficiency with basic multiplication and division number facts using a variety of fact strategies (such as "skip counting" and "repeated subtraction") Teacher Supplement Everyday Mathematics, Unit 4; Lessons 7.1, 9.1 Elementary School Mathematics with Pizzazz, Creative Publications *Know multiplication facts having 2, 5, or 10 as a factor Teacher supplements Everyday Mathematics, Lesson 4.4 - 4.6 The Best of Times, Greg Tang *Use basic facts to solve fact extensions Everyday Mathematics, Unit 2, 7
5. Count and perform simple computations with money *use cents notation (¢) Everyday Mathematics, assorted Math Boxes in Units 1, 2, 4, 5, 7, 9 Teacher supplement Website resources *Count combinations of bills and coins and write the total using dollars and cents notation Everyday Mathematics Lesson 1.9, 1.10, 7.7, 9.5, 9.7 26 Letters and 99 Cents, Tana Hoban 6. Select pencil-and-paper, mental math, or a calculator as the appropriate computational method in a given situation depending on the context and numbers Everyday Mathematics, Unit 2, 7, 9 Elementary School Mathematics with Pizzazz, Creative Publications Best of Math I and II, Exemplars CD *Solve number stories involving equal groups by using multiplication Everyday Mathematics, Lessons 4.1, 4.2, 7.1, 7.2, 7.3, 7.8, 9.1 - 9.6, 11.1 Each Orange Had Eight Slices: A Counting Book, Paul Giganti Sea Squares, Joy Hulme One Hundred Hungry Ants, Elinor J. Pinczes *Solve number stories involving equal sharing and equal grouping Everyday Mathematics, Unit 4 *Solve number stories involving positive and negative numbers Everyday Mathematics, Lesson 9.13 Teacher supplements 7. Check the reasonableness of results of computations Everyday Mathematics, Units 2, 7, 9 Website resources Exemplars, Best of Math I and II CD Elementary School Mathematics with Pizzazz, Creative Publications
C. Estimation 1. Judge without counting whether a set of objects has less than, more than, or the same number of objects as a reference set. Everyday Mathematics, Unit 1 and ongoing activities throughout the year Teaching Student-Centered Mathematics, Grades 3-5, Lovin and Van DeWalle 2. Construct and use a variety of estimation strategies (e.g., rounding and mental math) for estimating both quantities and the result of computations. Everyday Mathematics Units 2, 7, 9 *Estimating Costs Everyday Mathematics, Lesson 7.7 3. Recognize when an estimate is appropriate, and understand the usefulness of an estimate as distinct from an exact answer. Everyday Mathematics, Unit 7 Exemplars, Best of Math I and II CD 4. Use estimation to determine whether the result of a computation (either by calculator or by hand) is reasonable. Everyday Mathematics, Unit 2, 7, 9 Teacher supplement Exemplars, Best of Math I and II CD *Estimate answers to multi-digit addition and subtraction problems Everyday Mathematics, Unit 2
D. Discrete Mathematics- Vertex-Edge Graphs and Algorithms 1. Follow, devise, and describe practical sets of directions (e.g., to add two 2-digit numbers). Everyday Mathematics, all units 2. Explore vertex-edge graphs *vertex, edge *path District Teacher-Made Worksheets Website resources 3. Find the smallest number of colors needed to color a map. District Teacher-Made Map Coloring worksheets Website resources
3. Select and use various types of reasoning and methods of proof. Best of Math I and II, Exemplars CD Sample open-ended problems from various Website resources NJASK State rubric 4. Rely on reasoning, rather than answer keys, teachers, or peers, the check the correctness of their problem solutions Elementary School Mathematics with Pizzazz, Creative Publications Best of Math I and II, Exemplars CD 5. Make and investigate mathematical conjectures Best of Math I and II, Exemplars CD Navigation Series, NCTM selected problems Sample NJASK Open-ended problems * Counterexamples as a means of displaying conjectures Teacher supplement with website resources Best of Math I and II, Exemplars CD * Verifying conjectures using informal reasoning or proofs. Teacher supplement Sample open-ended problems from website resources NJ Mathematics Curriculum Frameworks, selected activities for grades 3 - 4 6. Evaluate examples of mathematical reasoning and determine whether they are valid Best of Math I and II, Exemplars CD Teacher supplement with website resources E. Representations 1. Create and use representations to organize, record, and communicate mathematical ideas. Everyday Mathematics activities ongoing throughout the year Best of Math I and II, Exemplars CD Teacher supplement with problems from various resources
* Concrete representations (e.g., base-ten blocks or algebra tiles) Everyday Mathematics activities ongoing throughout the year Best of Math I and II, Exemplars CD Navigation Series, NCTM selected problems * Pictorial representations (e.g., diagrams, charts, or tables) Everyday Mathematics activities ongoing throughout the year Best of Math I and II, Exemplars CD NJ Mathematics Curriculum Framework, selected activities grades 3-4 Everyday Mathematics, Lesson 7.9 * Symbolic representations (e.g., a formula) Everyday Mathematics activities ongoing throughout the year Sample open-ended problems from NJASK and other states * Graphical representations (e.g., a line graph) Everyday Mathematics activities ongoing throughout the year Teacher supplement with website resources Everyday Mathematics, Lesson 5.12, 11.5 2. Select, apply, and translate among mathematical representations to solve problems Teacher supplement Best of Math I and II, Exemplars CD NavigationSeries, NCTM selected problems Website resources for Open-ended problems NJ Mathematics Curriculum Frameworks, selected activities for grades 3-4 3. Use representations to model and interpret physical, social, and mathematical phenomena Everyday Mathematics activities, ongoing throughout the year Website resources: e.g., National Library of Virtual Manipulatives Elementary School Mathematics with Pizzazz, Creative Publications Navigation Series, NCTM selected problems
F. Technology 1. Use technology to gather, analyze, and communicate mathematical information. Everyday Mathematics, calculator activities ongoing throughout the year Teacher made materials in conjunction with Technology teacher Website resources 2. Use computer spreadsheets, software, and graphing utilities to organize and display quantitative information (cf. workplace readiness standard 8.4-D). Microsoft Office tools such as Word, Excel, PowerPoint United Streaming, videos Website resources Consult with technology teacher to supplement 3. Use graphing calculators and computer software to investigate properties of functions and their graphs. Teacher supplement Website resources Texas Instruments resources 4. Use calculators as problem-solving tools (e.g., to explore patterns, to validate solutions). Best of Math I and II, Exemplars CD Everyday Mathematics, activities ongoing throughout the year Sample open-ended problems using website resources *Introduce memory keys on a calculator Everyday Mathematics, Lesson 10.8 5. Use computer software to make and verify conjectures about geometric objects. NCTM's Illuminations website National Library of Virtual Manipulatives website
6. Use computer-based laboratory technology for mathematical applications in the sciences (cf.science standards). Teacher supplement with websites in conjunction with Technology teacher Texas Instrument website lessons Fourth Grade Math Pacing Guide Unit Number of Days (approximate) Unit Number of Days (approximate) 1- Geometric Figures 12 7 - Fractions & Probability 16 2 - Numbers and Data 13 8 - Perimeter & Area 12
*Multiplication of 2-digit numbers Everyday Mathematics, Lesson 5.5, 5.6, 5.7 *Division of 3-digit numbers by 1-digit numbers Everyday Mathematics, Lesson 6.1, 6.2, 6.3, 6.4 5. Construct and use procedures for performing decimal addition and subtraction. Everyday Mathematics, Lesson 4.6 Money *Construct and use procedures for performing decimal multiplication and division Everyday Mathematics, Lesson 9.8, 9.9 6. Count and perform simple computations with money. Everyday Mathematics, Lesson 4.6 Exemplar: Skating Party *Standard dollars and cents notation Everyday Mathematics, Lesson 4.6 The Lunch Line (Hello Math Reader level 3) 7. Select pencil-and-paper, mental math, or a calculator as the appropriate computational method in a given situation depending on the context and numbers. Everyday Mathematics, Lesson 3.7, Exemplars, Best of Math I and II CD “Shopping for Shoes” 8. Check the reasonableness of results of computations. Everyday Mathematics, Activities ongoing throughout the year Lesson 3.8 *Calculate unit prices to determine which product is the "better buy." Everyday Mathematics, Lesson 12.4, 12.5 9. Use concrete models to explore addition and subtraction with fractions. Everyday Mathematics, Lesson 7.1 – 7.6
How can coordinate grid systems help in understanding locations? How does the mathematics of geometry enable us to wonder and understand our natural and physical world? Knowledge/Skills/Understandings Assessments Learning Experiences Resources A. Geometric Properties 1. Identify and describe spatial relationships of two or more objects in space. Unit Assessments Do Now Teacher Observation Discussion Open Ended Writing Tasks Recognizing Student Achievement sections in Everyday Math 3 Direct Instruction Collaborative and Cooperative Learning Open Ended Writing Tasks EM Games Polygon Tiles Centimeter cubes Geometric solids Geometry Riddles, “I have, who has” Match up game Polygon tiles, tangrams, pattern blocks Mira/ Transparent Mirror Angelegs Protractor Master Rulers MIRA –transparent mirrors Everyday Mathematics, activities ongoing throughout the year Groundworks: Reasoning with Geometry *Direction, orientation, and perspectives (e.g., which object is on your left when you are standing here?) Project 6: Building & Viewing Structures Teacher-made supplements Teaching Student-Centered Mathematics, Grades 3-5, Lovin and VanDeWalle, page 246 Directions using a map * Relative shapes and sizes Teacher-made supplements NJ Mathematics Curriculum Framework for grades 3-4, selected activities *Shadows (projections) of everyday objects Teacher-made supplements
2. Describe and use geometric transformations (slide, flip, turn). Slide (translation) Flip (reflection) Turn (rotation) Everyday Mathematics Lessons Unit 10 Elementary School Mathematics with Pizzazz, Creative Publications Grandfather's Tang Story, Ann Tompert * Use a transparent mirror to draw reflection of a figure.) Everyday Mathematics Lessons Unit 10 *Relate turns and angles Everyday Mathematics, Lesson 6.5 3. Investigate the occurrence of geometry in nature and art. Project 4: Making a Quilt NJ Mathematics Curriculum Framework, selected activities for grades 3-4 Shapes, Shapes, Shapes, Jon Scieszka EM Teacher's Guide, page 895 Everyday Mathematics, Unit:1,10,11 Greedy Triangle, Marilyn Burns C. Coordinate Geometry 1. Locate and name points in the first quadrant on a coordinate grid. Hurkle Everyday Mathematics, Lesson 6.8, 6.9 Teacher supplement Fly on the Ceiling, Julie Glass 2. Use coordinates to give or follow directions from one point to another on a map or grid. Everyday Mathematics, Lesson 6.9 Teacher supplement
* Use the statistical landmarks of maximum and minimum number and range in a set of data Everyday Mathematics, Lesson 2.5 B. Probability 1. Use everyday events and chance devices, such as dice, coins, and unevenly divided spinners, to explore concepts of probability PIG, The Two-Dice Sum Game Everyday Mathematics, Lesson 7.11 District Teacher-Made Packet of Probability worksheets About Teaching Mathematics, Marilyn Burns, page 70, 71, 73, 74 Groundworks-Reasoning with Data and Probability, Creative Publications *Likely, unlikely, certain, impossible, improbable, fair, unfair Everyday Mathematics, Lesson 7.11, 7.12 District Teacher-Made Probability worksheets *More likely, less likely, equally likely Teaching Student-Centered Mathematics, Grades 3-5, Lovin and Van De Walle, Chapter 12 *Probability of tossing "heads" does not depend on outcomes of previous tosses Teacher-made supplements teacher supplement 2. Determine probabilities of simple events based on equally likely outcomes and express them as fractions. District Teacher-Made Probability worksheets Everyday Mathematics, Lesson 7.11, 7.12 3. Predict probabilities in a variety of situations (e.g., given the number of items of each color in a bag, what is the probability that an item picked will have a particular color). Everyday Mathematics, Lesson 7.11, 7.12 District Teacher-Made Probability worksheets Math by All Means, Probability 3-4, Marilyn Burns *What students think will happen (intuitive) Everyday Mathematics, Lesson 7.11, 7.12
*Organized lists, charts, tree diagrams Teacher-made supplements Best of Math I and II, Exemplars CD *Dividing into categories (e.g., to find the total number of rectangles in a grid, find the number of rectangles of each size and add the results) Teacher-made supplements D. Discrete Mathematics - Vertex - Edge - Graphs and Algorithms 1. Follow, devise, and describe practical sets of directions (e.g., to add two 2-digit numbers). Everyday Mathematics, Units 2, 5, 7, 9 Teacher supplement 2. Play two-person games and devise strategies for winning the games (e.g., "make 5" where players alternately add 1 or 2 and the person who reaches 5, or another designated number, is the winner.) Game of Nim Game of Pig Odd Number Wins Math By All Means Probability 3-4, Marilyn Burns 3. Explore vertex-edge graphs and tree diagrams. Teaching Student-Centered Mathematics, Grades 3-5, Lovin and VanDeWalle, page 348 *Vertex, edge, neighboring/adjacent, number of neighbors District teacher-made worksheets NJ Mathematics Curriculum Framework, selected activities for grades 3-4 *Path, circuit (i.e., path that ends at its starting point) District teacher-made worksheets 4. Find the least number of colors needed to color a map or a graph. Use US state maps, NJ county maps and abstract drawings Everyday Mathematics, Lesson 9.7 Map Coloring District Teacher Made worksheets
Number Theory 12 7 – Exponents and Negative Numbers 14 2 – Estimation and Computation 13 8 – Fractions and Ratios 15 3 – Geometry Explorations & The American Tour 13 9 – Coordinates, Area, Volume & Capacity 13 4 - Division 10 10 – Using Data; Algebra Concepts and Skills 12 5 – Fractions, Decimals, and Percents 15 11 - Volume 10 6 – Using Data; Addition & Subtraction of Fractions 13 12 – Probability, Ratios, and Rates 10 Mid-Year Benchmark Assessment 1 End-of-the-Year Benchmark Assessment 1 Fifth Grade Math Pacing Guide Unit Number of Days (approximate) Unit Number of Days (approximate)
* Explore rates Provide opportunities to solve number stories using rates Everyday Mathematics, Lesson 12.1 Teacher supplement Website resources Teaching Student-Centered Mathematics in grades 3-5, 5-8, Lovin and Van de Walle * Explore uses of ratios and way of expressing ratios Provide opportunities to explore ratios in everyday life; model and solve problems involving ratios of part to whole relationship and number stories EM American Tour data EM Activity: Musical Ratios. Everyday Mathematics, Lesson 12.3, 12.4, 12.5 Teacher supplement Middle School Mathematics with Pizzazz, Creative Publications NJ Mathematics Curriculum Framework in grades 5-6, selected activities *Solve rate story problems EM, Unit 12 checking progress EM Activity: Find your Heart Rate. Provide multiple opportunities to model using manipulatives, discuss rate tables, and explore relationships between numbers using rate stories Everyday Mathematics, Lesson 12.6, 12.7, 12.8 Teacher supplement Best of Math I and II, Exemplars CD Website resources *Introduce cross multiplication for solving ratio problems (proportions) Practice setting up equivalent ratios that yield the same solutions. Everyday Mathematics, Lesson 12.5 Teacher supplement Middle School Mathematics with Pizzazz, Creative Publications Website resources 6. Compare and order numbers. EM Activity: Ranking States by Their Populations Everyday Mathematics, Lesson 3.2, 4.6 *Order and compare positive and negative numbers Explore practical uses for negative numbers: golf scores, football gains and losses, having and owing money, above and below sea level, stock market * Use number line Everyday Mathematics, Lesson 2.10, 5.3, 7.7 Sketchpad: Jump Along.
*Add and subtract positive and negative numbers EM Unit 7 checking progress Use two color counters activities Student should be able to demonstrate add / subtraction with counters or using the number line. EM Game: Top It, Credit/Debits Game Calculators should not be used for computation Everyday Mathematics, Lesson 7.8, 7.9, 7.10 (NO calculator) 1. Recognize the appropriate use of each arithmetic operation in problem situations. EM Unit 2, checking progress EM Project 3: Ancient Multiplication Algorithm Everyday Mathematics, Units 2.2, 2.3, 2.4, 2.8, 2.9, 4.2, 4.6 2. Construct, use, and explain procedures for performing addition and subtraction with fractions and decimals EM Unit 2, 5, 8 checking progress Students should have a successful strategy for computation. Students should be able to show understanding of computation method they are using. Encourage students to use appropriate and efficient procedures as time and problem warrants. Everyday Mathematics, Lesson 2.2, 2.3, 2.4, 5.3, 6.8, 6.9, 6.10, 8.1, 8.2, 8.3, 8.4 • Pencil-and-paper Provide practice appropriate to student need • Mental math Provide class competitions Teacher supplement • Calculator Use appropriate numbers and monitor student use Discuss answer interpretations EM Game: Beat the Calculator Website resources
C. Estimation 1. Use a variety of estimation strategies for both number and computation. EM Unit 2 checking progress Provide ongoing opportunities for student guesses in estimation jars filled with a variety of objects. Everyday Mathematics, Lesson 2.1, 2.5, 2.7 *Make magnitude estimates for product of multi-digit numbers, including decimals EM Unit 2 checking progress Encourage students always to make estimates whenever doing computation whether on a calculator or by hand EM Game: Multiplication Bull's Eye Everyday Mathematics, Lesson 2.7 2. Recognize when an estimate is appropriate, and understand the usefulness of an estimate as distinct from an exact answer. EM Activity: Estimation EM Activity: Estimate Reaction Time EM Activity: American Tour: Population estimates Literature Link Everyday Mathematics, Lesson 2.1, 2.5. Counting on Frank, Rusty Bresser * Use sampling to make an estimate Explore the enormity of such large numbers through class discussions EM Activity: Making Time Estimates for 1 Billion and 1 Trillion Literature Link EM Activity: Earth's Water Surface and School's Land area. Everyday Mathematics, Lesson 2.10, 9.7 Website resources for real data numbers How Much is a Million?, David Schwartz 3. Determine the reasonableness of an answer by estimating the result of operations. EM Unit 2 checking progress Provide opportunities to practice estimation prior to computation Encourage good estimates and define key numbers to Everyday Mathematics, Lesson 2.1, 2.5, 2.7
* Know the properties of geometric solids Use or create geosolid models EM Game: 3-D Shape Sort, EM Activity: polyhedral Dice, EM Activity: Cube pattern, EM Activity: rectangular Prism Pattern, Everyday Mathematics, Lesson 11.1, 11.2 Website resources *Identify faces, vertices and edges (Grade 4 secure) Use geosolids models Literature Link Everyday Mathematics, Lesson 11.2 The Boy Who Reversed Himself, Flatland 4. Understand and apply the concepts of congruence and symmetry (line and rotational). EM Study Link 9.3 (reflection) Look for congruent and similar shapes in wallpaper, fabric, art Everyday Mathematics, Lesson 3.6, 9.3 Teacher supplement B. Transforming Shapes 1. Use a translation, a reflection, or a rotation to map one figure onto another congruent figure. EM Game: Robot Activity:" Shape Cutter" This is new vocabulary for students: slide, flip and turn are previous vocabulary used Provide projects in which student must use one shape and show all three transformations Everyday Mathematics, Lesson 3.6, 3.8, 9.3 Illuminations website Teaching Student-Centered Mathematics, Grades 3-5, Lovin and Van de Walle Teacher supplement Middle School Mathematics with Pizzazz, selected activities *Define and create tessellations EM Activity: Tessellation Museum. Everyday Mathematics, Lesson 3.8 Teaching Student-Centered Mathematics, Grades 3-5, Lovin and Van de Walle. The Art of MC Escher, "Tessellmania" Sketchpad: Making Kaleidoscopes, Sliding Shapes, Mosaic Tile Designs.
* Reading a ruler (previous grade secure) Provide multiple opportunities for students to measure and draw lengths using both centimeter and inch ruler. Teacher supplement Website resources Transparency Rulers - Master Rulers *Select reasonable measurement units , US and metric EM Activity: Finding Personal Measures for One's Body Allow students to create own multiple choice problems using both US customary and metric units for length, weight and capacity Everyday Mathematics, Lesson 6.2 Math Masters Sample multiple choice assessment like items from NJ state and other state assessments 2. Convert measurement units within a system (e.g., 3 feet = ___ inches). Activity: "Measuring Up" EM Activity: Finding Measurement Equivalencies Illuminations website Teacher supplement Middle School Mathematics with Pizzazz, selected activities Everyday Mathematics, Lesson 9.10 3. Know approximate equivalents between the standard and metric systems (e.g., one kilometer is approximately 6/10 of a mile). Reference Page of Journal 2 (back cover) Everyday Mathematics, Lesson 6.2, 11.6 *Examine the relationship among liter, milliliter, and cubic centimeter Use models to demonstrate equivalencies Everyday Mathematics, Lesson 9.10 4. Use measurements and estimates to describe and compare phenomena. Project 8- Pendulums Provide some informal indirect measurement examples Everyday Mathematics, Lesson 2.5, 10.5 * Understand the concept of capacity and how to calculate it Solve problems involving units of weight, see Teacher's guide for examples Everyday Mathematics Lesson 11.6 Teacher supplement with Open-ended problems from various sources
5. Solve problems with elapsed time (Grade 4 secure) Provide students with opportunities to write and solve number stories involving elapsed time: include fractions, decimals and whole numbers, e.g., daily schedule, cooking, appointments, getting ready for school EM teacher's guide Teacher supplement Website resources E. Measuring Geometric Objects 1. Use a protractor to measure angles. Geometry template, Have students explore using both half and circle protractors EM Activity: Pattern Block Angles Everyday Mathematics, Lesson 3.3 3.4 Middle School Mathematics with Pizzazz, selected activities *Estimate the measure of an angle Cooperative group exploration activities Everyday Mathematics, Lesson 3.6, 3.8 Sketchpad: Angle Estimation. * Measure an angle within 2 degrees Review ways to name angles Everyday Mathematics, Lesson 3.4, 3.9 Math Masters Sketchpad: Introducing Protractors. 2. Develop and apply strategies and formulas for finding perimeter and area. EM Project 6: Sports Areas EM Math Masters EM Project 7: Polygon Areas and Pick's Theorem Use geoboard activities Everyday Mathematics, Lessons 9.3, 9.4, 9.5, 10.8, and 10.9 Sketchpad: Rectangles with Same Area, Rectangles with Same Perimeter. *Square Use models to explore formulas Teacher supplement • Rectangle Graph paper, geoboard Explore to find formula Everyday Mathematics, Lesson 9.4
3. Recognize that rectangles with the same perimeter do not necessarily have the same area and vice versa. Explore rectangles using geoboard activities and other models Everyday Mathematics Lesson 9.4 4. Develop informal ways of approximating the measures of familiar objects (e.g., use a grid to approximate the area of the bottom of one's foot). Everyday Mathematics Journal Teacher supplement Estimation Challenge Grade: 5 Standard 4.3 Patterns and Algebra Essential Questions: How can patterns help in problem solving? How can symbols be used to help us in problem solving? How does the study of algebra help us understand mathematical patterns as the patterns found in nature & the real world? Knowledge/Skills/Understandings Assessments Learning Experiences Resources A. Patterns
1. Recognize, describe, extend, and create patterns involving whole numbers. Teacher Observation/RSA (Recognizing Student Achievement) Test/Quiz Exit Ticket EM Unit 10 checking progress EM Activity: Number Patterns using dots, Math Masters. "Patterns that Grow" Provide multiple opportunities for exploring patterns involving manipulatives, numbers, models and other objects Everyday Mathematics Units 1, 2, 7, 10 Teacher supplement Illuminations websites Sketchpad: The Envelope • Descriptions using tables, verbal rules, simple equations, and graphs Everyday Mathematics, Lesson 10.4, 10.5, 10.7 Website resources Navigations in Algebra, grades 3-5, NCTM, selected activities *Explore Fibonacci Sequence Provide opportunities for students to explore pattern and find patterns in the Fibonacci sequence, especially those found in nature: sunflower seeds, nautilus shell, pine cones Teacher supplement Website and Library resources B. Functions & Relationships 1. Describe arithmetic operations as functions, including combining operations and reversing them. Allow student to explore ways to evaluate expressions Everyday Mathematics Lesson 7.4, 7.5 Groundworks for Algebra, Creative Publications Sketchpad: Function Machines. *Translate word descriptions into algebraic expression Discuss common symbols for words, e.g., twice, half Everyday Mathematics, Lesson 10.3 *Complete "What's My Rule?" EM Unit 10 checking progress Everyday Mathematics, Lesson 10.3, 10.4 2. Graph points satisfying a function from T-charts, from verbal rules, and from simple equations. EM Unit 10 checking progress assessment Explore linear relationships Everyday Mathematics Lesson 10.4,10.5, 10.6 C. Modeling
•Write and solve number sentences with variables for division number stories Allow students to use manipulatives and other models to assist in solving stories Everyday Mathematics, Lesson 4.5, 4.6 2. Draw freehand sketches of graphs that model real phenomena and use such graphs to predict and interpret events. EM Activity: Predicting Old Faithful's Next Eruption EM Project 8: Pendulum Everyday Mathematics, Lesson 10.5, 10.7 *Changes over time Explore line graph data Website resources *Rates of change (e.g., when is plant growing slowly/rapidly, when is temperature dropping most rapidly/slowly) Explore: temperature over a month time, growth of plants, height of growth from child to adult Teacher supplement Website resources D. Procedures 1. Solve simple linear equations with manipulatives and informally *Whole-number coefficients only, answers also whole numbers * Variables on one side of equation EM Unit 9, 10 checking progress and assessment. Use pan-balance, geoboard, manipulatives and other models to solve Everyday Mathematics, Lesson 9.6, 10.3, 10.4, 10.5, 10.7. Middle School Mathematics with Pizzazz, selected activities Teaching Student Centered Mathematics, Grade 3-5 & 5-8, Lovin & Van de Walle, selected activities *Solve one-step pan-balance problems Demonstrate using manipulatives and models, then use symbols before using variables Students should explore equation solving in cooperative groups Everyday Mathematics, Lesson 10.1. *Solve two-step pan-balance problems Begin with manipulatives and models Everyday Mathematics, Lesson 10.2.
2. Understand and use the concepts of equals, less than, and greater than in simple number sentences. (Grade 4 secure) *Symbols (<, >, =) EM Unit 7 checking progress Incorporate use of these symbols when comparing all numbers including fractions and decimals SRB, American Tour Section Website or library book resources 1. Understand, name and apply the properties of operations and numbers. (Grade 4 secure) *Commutative *Identity element for multiplication *Associative *Division by zero *Any number multiplied by zero is zero Apply these properties when simplifying numerical expression including whole numbers, decimals, fractions and integers. Teacher supplement Middle School Mathematics with Pizzazz, Creative Publications Grade: 5 Standard 4.4 Data Analysis, Probability, and Discrete Mathematics Essential Questions: How can classifying help me in organizing data to solve problems? How can statistics help us to understand real world situations? How can the study of real world data help us understand and make accurate predictions? Knowledge/Skills/Understandings Assessments Learning Experiences Resources
A. Data Analysis 1. Collect, use, generate, organize, and display data. Teacher Observation/RSA (Recognizing Student Achievement) Unit 6 Checking Progress/Quiz Exit Ticket Evaluation of student completed projects using a rubric EM Activity: Class Data for number of States students / adults have visited EM Activity: Collecting Exercise Data Everyday Mathematics, Lesson 6.1, 12.7 Best of Math I & II, Exemplars CD • Data generated from surveys Have student generate surveys to collect data about themselves Introduce the American Tour Census Data EM Journal EM Activity: Interpret data in the news Everyday Mathematics, Lesson 3.1, 3.2, 6.1, 6.6 2. Read, interpret, select, construct, analyze, generate questions about, and draw inferences from displays of data. Activity: "Information Represented Graphically" EM Activity: Education and Earnings EM Activity: Estimating Colonial Populations EM: American Tour, School Days, EM Journal EM Activity: reading Graphs, Mystery Graphs. Illuminations website Everyday Mathematics, Lesson 3.1, 3.2, 5.12, 10.7 Teacher supplement with various graphs from newspapers, periodicals, encyclopedias, and other related documents Sketchpad: Target Mean Game.
• Bar graph, line graph, circle graph, table EM Unit 5 checking progress Discuss differences of each type of graph and why one over another is appropriate for different types of data EM activity: Interpreting Data EM Activity: Create circle graphs with Class Snack Survey EM Activity: Acting out the Construction of a Circle graph Teacher supplement Website resources Everyday Mathematics, Lesson 5.9, 5.10, 5.11, 6.1, 6.4, 6.6 Best of Math I and II, Exemplars *Use a Percent Circle to find percents of circle graphs Everyday Mathematics, Lesson 5.10, 5.11 EM Math Masters *Measures pieces of a circle graph; interpret a circle graph Find and use circle graphs Use protractor to measure sectors Everyday Mathematics, Lesson 6.5 Teacher supplement *Construct circle graph Everyday Mathematics, Lesson 5.11Math Masters *Construct, read and interpret plots (B/D) Use various sample data sets to create stem and leaf plots EM Journal activity; Measuring the Great Span EM Math Masters EM Activity: Reaching and Jumping Website sources for data Everyday Mathematics, Lesson 6.3, 6.4 • range, median, and mean Provide multiple opportunities to explore these measures and their value in different contexts Discuss meaning for average and different uses of Teacher supplement Best of Math I & II, Exemplars CD Sketchpad: Target Mean Game *Identify the statistical landmarks for maximum, minimum, median, and mode EM Unit 2 checking progress EM Activity: The Great Span EM Activity: Estimate Reaction Time EM Activity: States visited by Adults and Students Illuminations website and others Middle School Mathematics with Pizzazz, Creative Publications Everyday Mathematics, Lessons 2. 5, 6.1, 6.4, 6.5, 6.6
B. Probability 1. Determine probabilities of events. Teacher Observation/RSA (Recognizing Student Achievement) EM Unit 12 Checking Progress/Quiz Exit Ticket EM Journal: Describing Chance Probability Meter Poster EM Activity: Thumbtack Experiment Everyday Mathematics, Lesson 2.6, 12.2 Jumanji, Chris Van Allsburg Teacher supplement Navigations in Data Analysis and Probability, grades 3-5, 5-8, NCTM, selected activities Best of Math I and II, Exemplars CD • Event, probability of an event EM Activity: US Census results on the Probability Meter Everyday Mathematics, Lesson 2.6 and 12.2 Middle School Mathematics with Pizzazz, Creative Publications • Probability of certain event is 1 and of impossible event is 0 Whole class discussion about real events and their certainty or uncertainty e.g., day/ night, drive to Europe from NY Everyday Mathematics, Lesson 2.6 Teacher supplement Probability Meter Poster *Compute the probability of outcomes when choices are equally likely EM Activity: Probability Investigations Everyday Mathematics. Lesson 12.2 Teacher supplement 2. Determine probability using intuitive, experimental, and theoretical methods (e.g., using model of picking items of different colors from a bag). Provide multiple opportunities for students to find probability through simulations using coins, dice, cubes in a bag, etc. Have students explore experimental versus theoretical probabilities and discuss why they may be different Everyday Mathematics Unit 2, 6, and 12, selected activities Teacher supplement Website resources Sketchpad: Red Light, Green Light
• Given numbers of various types of items in a bag, what is the probability that an item of one type will be picked Provide multiple opportunities for students to find probability through simulations using coins, dice, cubes in a bag, etc. EM Activity: Taking a Small Sample of Candy Colors Everyday Mathematics, Lesson 2.6, 6.5, and 12.5 Navigation in Data and Analysis in Grades 3-5, NCTM, selected activities • Given data obtained experimentally, what is the likely distribution of items in the bag Perform probability simulations and give results of experiment Everyday Mathematics, Lesson 12.2 Teacher supplement 3. Model situations involving probability using simulations (with spinners, dice) and theoretical models. Provide students with opportunities to model probability situations using manipulatives. EM Activity: Making Spinners Everyday Mathematics, Lesson 2.6, 12.2 Jumanji, Chris Van Allsburg Teacher supplement C. Discrete Mathematics-Systematic Listing and Counting 1. Solve counting problems and justify that all possibilities have been enumerated without duplication. EM Activity: Counting License Plates "The Handshake Problem" Website resources Best of Math I and II, Exemplars CD Sketchpad: Hide the Spinner • Organized lists, charts, tree diagrams, tables EM Activity: Venn Diagrams Everyday Mathematics, Lesson 11.2, 12.2 Sample Open-ended problems form various sources *Use tree diagrams to find all possible ways a sequence of choices can be made (D/S) Use Multiplication Counting Principle and provide multiple opportunities to write out all possible outcomes Everyday Mathematics, Lesson 12.2 Teacher supplement
2. Explore the multiplication principle of counting in simple situations by representing all possibilities in an organized way (e.g., you can make 3 x 4 = 12 outfits using 3 shirts and 4 skirts). Fourth Grade Extension Activity: "Four-by-Four Block" Everyday Mathematics, Lesson 12.2 Explain It! Selected problems Website resources Sketchpad: How Many Ways *Use the Multiplication Counting Principle to find the total number of possible outcomes of a sequence of choices EM Activity: Probability Investigations Everyday Mathematics, Lesson 12.2 Middle School Mathematics with Pizzazz, Creative Publications D. Discrete Mathematics-Vertex-Edge Graphs and Algorithms 1. Devise strategies for winning simple games EM Game: Finish First Design games to play and find strategies for winning: dice Everyday Mathematics, Lesson 6.2 Teacher supplement (e.g., start with two piles of objects, each of two players in turn removes any number of objects from a single pile, and the person to take the last group of objects wins) and express those strategies as sets of directions. EM Activity: Making Spinners Everyday Mathematics, Lesson 12.2 * Explore vertex-edge graphs and tree diagrams (Grade 4) Use models and design communication chains: office manager can talk with every employee, each employee can talk to office manager and one other employee Everyday Mathematics, Lesson 12.2 Teacher supplement Website resources * Vertex, edge, neighboring/adjacent number of neighbors (Grade 4) Use manipulatives and other models Teacher supplement *Path, circuit (Paths that end at its starting point) (Grade 4) Use museum, zoo, or school floor plans Teacher supplement NCTM website Question Quest Level A
Grade: 5 Standard 4.5 Mathematical Processes Big Idea: Mathematical understandings are an essential part of our lives in and out of school and as such all children need to have an instinctive sense of mathematical resources that they can rely on to help them progress through life. Essential Questions: How will learning to "think" mathematically enable us to make a life, make a living, and make a difference? How does the use of technology enable us to have a deeper understanding of mathematics? Knowledge/Skills/Understandings Assessments Learning Experiences Resources A. Problem Solving Note - Since Everyday Mathematics is a spiraling program, NJ standard 4.5 occurs throughout daily lessons. Additional resources that are used in conjunction with the program are listed below. 1. Learn mathematics through problem solving, inquiry, and discovery. Teacher Observation/RSA (Recognizing Student Achievement) Test/Quiz Exit Ticket Provide multiple opportunities with partner/small group/independent investigations of problem solving using manipulatives and other models. Problem Solving Diagram Best of Math I and II, Exemplars CD Everyday Mathematics Projects Everyday Mathematics 2. Solve problems that arise in mathematics and in other contexts (cf. workplace readiness standard 8.3). Provide examples of math in "real life" : shopping, discounts, counting and making change, scheduling for after school events, birthday and party planning Everyday Mathematics Lessons 5.8, 5.9, and 5.10 Best of Math I and II, Exemplars CD • Open-ended problems Evaluation of student solution using a rubric Provide students with opportunities to score other students solutions using a rubric Sample open-ended questions from NJ and other states Best of Math I and II Exemplars CD
5. Monitor their progress and reflect on the process of their problem solving activity. Provide student with opportunities to tell what they know and how they know and what they might not completely understand or are unclear on. Teacher supplement B. Communication 1. Use communication to organize and clarify their mathematical thinking. EM Project 5: How Would you Spend $1,000, 000? (rubric provided) Best of Math I and II, Exemplars CD Teacher supplement • Reading and writing • Discussion, listening, and questioning Provide opportunities for students to discuss, listen and critique each other's work Teacher supplement 2. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others, both orally and in writing. Use cooperative groups and partners to share mathematically thinking Best of Math I and II, Exemplars CD 3. Analyze and evaluate the mathematical thinking and strategies of others. Teacher Observation Group presentations of winning strategies for games EM Game: First to 21
3. Use graphing calculators and computer software to investigate properties of functions and their graphs. Explore Microsoft Excel activities using formulas and charts Graphing calculators Texas Instruments resources Teacher supplement 4. Use calculators as problem-solving tools (e.g., to explore patterns, to validate solutions). Student created tessellation Explore tessellations Everyday Mathematics Lesson 3.8 5. Use computer software to make and verify conjectures about geometric objects. Observation of students while interacting with software/ Provide appropriate opportunities for students to interact with software Geometer's Sketchpad Best of Math I and II Exemplars CD Use additional website resources as needed 6. Use computer-based laboratory technology for mathematical applications in the sciences (cf. science standards). Journal writing Geometer's Sketchpad Geometer's Sketchpad
Unit 1 – Data and Landmarks 15 Unit 8 – Rates and Ratios 13 Unit 2 – Operations with Whole Numbers and Decimals 20 Unit 5 – 2 Dimensional Geometry 15 Metric and Customary Measurement 8 3 Dimensional Geometry 7 Unit 3 – Variables, Formulas and Graphs 20 Unit 6 - Number Systems and Algebra Concepts 14 Prime Time- Factors, Multiples, and Prime Numbers 10 Unit 7 - Probability 5 Unit 4 – Rational Number Uses and Operations 20 Unit 9 - Variables, Formulas and Graphs 10 Mid-Year Assessment 1 End-of-the-Year Assessment 1 Unit Number of Days (approximate) Unit Number of Days (approximate) Sixth Grade Math Pacing Guide
Grade: 6 Topic 4.3 Patterns and Algebra Essential Questions: How can patterns help in problem solving? How can symbols be used to help us in problem solving? How does the study of algebra help us understand mathematical patterns as the patterns found in nature & the real world? Knowledge/Skills/Understandings Assessments Learning Experiences Resources A. Patterns 1. Recognize, describe, extend, and create patterns involving whole numbers, and rational numbers: descriptions using tables, verbal simple equations and graphs; formal iterative formulas; recursive patterns, including Pascal's Triangle and the Fibonacci Sequence; Generating sequences by using calculators to repeatedly apply a formula. Unit Assessments Do Now Teacher Observation Class Discussion Slate Assessments Open Ended Writing Tasks RSA Sections of EM3 Direct Instruction Cooperative – Collaborative Learning Experiences Written Explanations Open Ended Writing Tasks Games Geometer’s Sketch Pad Links Groundworks Series – Algebraic Reasoning Van De Walle Chapters 22-25 Everyday Mathematics, Book 6- – Units, 3, 6, 9 Geometer’s Sketch Pad Lesson Links B. Functions and Relationships 1. Describe the general behavior of functions given by formulas or verbal rules Van De Walle Chapter 23 C. Modeling 1. Use patterns, relations, and linear functions to model situations: Using variables to represent unknown quantities; Using concrete materials (manipulatives), tables, graphs, verbal rules, algebraic expressions/equations/ inequalities. Van De Walle Chapter 23
5. Monitor their progress and reflect on the process of their problem solving activity B. Communication 1. Use communication to organize and clarify their mathematical thinking: Reading and writing; Discussion, listening and questioning. Everyday Mathematics, Book 6 New Jersey Math Frameworks 1-4 Explain It, Creative Publications, Creative Publications Question Quest – Level B 8 Step Model Drawing, Char Forsten 2. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others, both orally and in writing. 3. Analyze and evaluate the mathematical thinking and strategies of others. 4. Use the language of mathematics to express mathematical ideas precisely. C. Connections 1. Recognize recurring themes across mathematical domains (e.g., two linear equations have a unique solution because the lines they represent intersect at a single point). Everyday Mathematics, Book 6- New Jersey Math Frameworks 1-4 Explain It, Creative Publications Question Quest – Level B 8 Step Model Drawing, Char Forsten
Grade: 6th Algebra 1A Topic 4.3 Patterns and Algebra Essential Questions: How can patterns help in problem solving? How can symbols be used to help us in problem solving? How does the study of algebra help us understand mathematical patterns as the patterns found in nature & the real world? Knowledge/Skills/Understandings Assessments Learning Experiences Resources A. Patterns Recognize, describe, extend, and create patterns involving whole numbers, rational numbers, and integers: Descriptions using tables, verbal and symbolic rules, graphs, simple equation or expressions; Finite and infinite sequences; Arithmetic sequences; Geometric sequences; Generating sequences by using calculators to repeatedly apply to a formula. Unit Assessments Do Now Teacher Observation Discussion Investigation Results Open Ended Writing Tasks Direct Instruction Lesson Investigation 4.3, 4.4, 4.6 Modeling Graphing Calculators Discovering Algebra Chapter 4 B. Functions and Relationships 1. Graph functions, and understand and describe their general behavior: Equations involving two variables; Rates of change (informal notion of slope). Lesson Investigations 4.4, 4.6, 4.7, 5.1, 5.1, 5.2, 5.3 Discovering Algebra Chapters 4 and 5 C. Modeling 1. Analyze functional relationships to explain how a change in one quantity can result in a change in another, using pictures, graphs, charts, and equations. Lesson Investigations 3.2, 4.3, 4.4, 4.6, 4.7, 5.1, 5.2, 5.3 Discovering Algebra Chapters 3, 4, 5
D. Discrete Mathematics - Vertex- Edge Graphs and Algorithms 1. Use vertex-edge graphs to represent and find solutions to practical problems: finding the shortest network connecting specified sites; Finding the shortest route on a map form one site to another: Finding the shortest circuit on a map that makes a tour of specified sites. What Do You Expect? New Jersey Math Frameworks 12 Groundworks Series Data and Probability NJ Ask Coach
1. Use technology to gather, analyze, and communicate mathematical information. Geometer’s Sketch Pad TinkerPlots Geometer’s Sketch Pad Lesson Links Brain Pop 2. Use computer spreadsheets, software, and graphing utilities to organize and display quantitative information Discovering Algebra Chapter 1 Geometer’s Sketchpad Lesson Links Geometer’s Sketchpad Tinker Plots 3. Use graphing calculators and computer software to investigate properties of functions and their graphs. TI 73 and TI 84 Calculators Discovering Algebra – Chapters 1-5 4. Use calculators as problem-solving tools (e.g., to explore patterns, to validate solutions). Discovering Algebra – Chapters 1-5 5. Use computer software to make and verify conjectures about geometric objects. Geometer’s Sketch Pad Geometer’s Sketch Pad Lesson Links Tinker Plots 6. Use computer-based laboratory technology for mathematical applications in the sciences Discovering Algebra – Chapters 1-5 Geometer’s Sketch Pad Geometer’s Sketch Pad Lesson Links Tinker Plots
3rd Grade Gifted and Talented Math Pacing Guide Unit Number of Days (Approximate) Unit Number of Days (Approximate) 1 – Naming & Constructing Geometric Figures 12 7 – Fractions & Their Uses, Probability 17 2 – Using Numbers & Organizing Data 13 8 – Perimeter & Area 11 3 – Multiplication & Division Number Sentences & Algebra 19 9 – Fractions, Decimals, & Percents 12 4 – Decimals & Their Uses 14 10 – Reflections & Symmetry 10 5 – Big Numbers; Estimation & Computation 14 11 – 3-D Shapes, Weight, Volume, & Capacity 10 6 – Division, Map Reference, Measure of Angles 13 12 - Rates 9 Hands on Algebra 5
Authentic Performance Students use color tiles and record solutions on graph paper Literature Link: Spaghetti and Meatballs for All Math by All Means: Perimeter and Area Grades 5-6, mathsolutions.com 3. Measure and compare the volume of objects using materials such as rice or cubes. Ticket out the Door Everyday Mathematics 11.4, 11.5, 11.7 Folding Geometric set (3D -- 2D) View Thru geometric solids set Completion of worksheet How Many Cubes? NCTM 2003 Yearbook, p. 136
● Input/output table, T-charts Everyday Mathematics What's my rule? JP 59 & 64 (Mbox) Rate Tables 12.2 ● Combining two function machines ● Reversing a function machine Completion of Blackline Masters Tiling a Patio Growing Patterns Equilateral Triangles Triangle-Rule Machine Navigations through Algebra 3-5 C. Modeling 1. Recognize and describe change in quantities. NFC Class graph Everyday Mathematics 2.8 9.6 ● Graphs representing change over time (e.g., temperature, height) Grow and measure Amaryllis Bulb. Help students focus on the shape of the data: range, maximum, minimum, outlier ● How change in one physical quantity can produce a corresponding change in another (e.g., pitch of a sound depends on the rate of vibration) 2. Construct and solve simple open sentences involving any one operation (e.g., 3 x 6 = _____, n = 15 ÷ 3, 3 x ____ = 0, 16 - c = 7). Whole class discussion manual p. 191 Number Balance Everyday Mathematics 3.10 Different uses of variables Variables as unknowns Van De Walle, p. 428 Class discussion: Solve b + b + b - 20 = 16 Number sentences involving relational thinking, p.31 Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary Schools, Carpenter, Franke, Levi Heinemann, 2003
4. Pose problems of various types and levels of difficulty 5. Monitor their progress and reflect on the process of their problem solving activity B. Communication 1. Use communication to organize and clarify their mathematical thinking Completion of tally sheet Count on Mathematics for Number Sense Lesson: Every Breath You Take NCTM.org.Illuminations lesson ● Reading and writing Completion of worksheet Literature Link: A Tale of Two Stories: Pigging Out Mathematics and Children's Literature: Getting the Facts: Counting on Frank ● Discussion, listening and questioning Collect and respond with post-it "nudge note" Explain to a student in 2nd grade what you learned about decimals today Van De Walle p. 70 Class discussion Even/Odd Class Discussion Classroom Discussions: Using Math Talk to Help Students Learn, Chapin O'Connor, Canavan/Anderson Math Solutions, 2003 2. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others, both orally and in writing. Class discussion Everyday Mathematics lesson 3.7-3.10 Writing Prompt: What about the work we did today was easy? What was hard? What do you still have questions about? 3. Analyze and evaluate the mathematical thinking and strategies of others. Math Congress: Teacher uses scoring rubric for feedback to students Parent Volunteers Best of Math Exemplars II CD Rom
E. Representations 1. Create and use representations to organize, record, and communicate mathematical ideas. Students share at overhead Problem: What happens to the area of a rectangle if the lengths of its sides are doubled? Principles and Standards for School Mathematics, NCTM, 2000, p. 205 ● Concrete representations (e.g., base-ten blocks or algebra tiles) Students create graphs using spreadsheet Collecting, Representing, and Interpreting Data Principles and Standards for School Mathematics, NCTM, 2000, E - example 5.5 (members only) ● Pictorial representations (e.g., diagrams, charts, or tables) whole class discussion Are there more even or odd products in the multiplication table? Principles and Standards for School Mathematics, NCTM, 2000, pp. 208-9 ● Symbolic representations (e.g., a formula) math congress Problem Prompts adaptations NCTM 2001 Yearbook, pp. 77-89 ● Graphical representations (e.g., a line graph) Writing Prompt: Reflection on playing the game Chip Trading 2. Select, apply, and translate among mathematical representations to solve problems 3. Use representations to model and interpret physical, social, and mathematical phenomena. F. Technology 1. Use technology to gather, analyze, and communicate mathematical information. Teacher reflection after observing students (see last page of article) Virtual Tile Turning Applets: Patterns Shape Tool (Illuminations) NCTM On-Math Journal, Spring 2003, (members only), Printed copy of article attached. Math Congress/class discussion Problem of the week Math Arena http://mathforum.org
2. Demonstrate an understanding of place value concepts. State assessment "Capture Decimals" Mental Math & Reflexes, Everyday Mathematics 2.5 TERC, Name That Portion Everyday Mathematics - 5 Units 2,4,5 3. Recognize the decimal nature of United States currency and compute with money. 4. Demonstrate a sense of the relative magnitudes of numbers. Observation of students playing game Do Now: Mental Math & Reflexes (Everyday Mathematics 2.10) First to 100 "Number Names" Everyday Mathematics - 5 Units 2,4,7 Problem Parade 5. Use whole numbers, fractions, and decimals to represent equivalent forms of the same number. Everyday Mathematics Unit 5 Assessment PM 5-59 Pizzazz C-25, C-74 Everyday Mathematics - 5 Unit 5 Open Court (1991) Middle School with Pizzazz Everyday Mathematics Unit 1 Assessment 6. Develop and apply number theory concepts in problem solving situations. Authentic Performance Task Project Factor Capture Game Candy Box Research Ryan's Primes PM 6-52 Variety of Problems "The Sieve of Eratosthenes" "Deficient, Abundant, & Perfect Numbers" Everyday Mathematics - 5 Unit 1 Multiplication - Grade 3, M. Burns, 1991 Teaching Children Mathematics, May 1997 Practice Masters, Open Court, 1991 Logic Number Problems Everyday Mathematics - 5, Project 1,2 • Primes, factors, multiples
4. Develop informal ways of approximating the measures of familiar objects (e.g., use a grid to approximate the area of the bottom of one's foot). Authentic Performance Task "Foot Area Perimeter" About Teaching Mathematics, Marilyn Burns Everyday Mathematics Units 6 & 9
Grade: 4 - Gifted and Talented Topic 4.3 Patterns and Algebra Essential Questions: How can patterns help in problem solving? How can symbols be used to help us in problem solving? How does the study of algebra help us understand mathematical patterns as the patterns found in nature & the real world? Knowledge/Skills/Understandings Assessments Learning Experiences Resources A. Patterns 1. Recognize, describe, extend, and create patterns involving whole numbers. "Patterns that Grow" (5 lessons) Everyday Mathematics - 5 Units 1, 2, 7, 10 Illuminations - "Lessons" • Descriptions using tables, verbal rules, simple equations, and graphs Unit 10 Assessment B. Functions & Relationships 1. Describe arithmetic operations as functions, including combining operations and reversing them. Unit 7 Assessment Everyday Mathematics - The Four 4's Project Everyday Mathematics - 5 Unit 7 Everyday Mathematics - 5 Unit 9 2. Graph points satisfying a function from T-charts, from verbal rules, and from simple equations. Unit 10 Assessment Everyday Mathematics - 5 Unit 10 (Lessons 4 & 6) C. Modeling 1. Use number sentences to model situations. • Using variables to represent unknown quantities • Using concrete materials, tables, graphs, verbal rules, algebraic expressions/equations Student completion of booklet. Student demonstration First to 100 - Everyday Mathematics Hands on Algebra Everyday Mathematics - Units 3, 4, 10 Hands on Algebra, Borenson & Assoc., 1994 2. Draw freehand sketches of graphs that model real phenomena and use such graphs to predict and interpret events. Circulate, complete, and share Journal p. 360 Everyday Mathematics 10.5 - Predicting Old Faithful's Next Eruption Everyday Mathematics - Unit 10 Lesson 5 • Changes over time
D. Discrete Mathematics-Vertex-Edge Graphs and Algorithms 1. Devise strategies for winning simple games (e.g., start with two piles of objects, each of two players in turn removes any number of objects from a single pile, and the person to take the last group of objects wins) and express those strategies as sets of directions. Circulate, complete, and share "Vertex coding" Discrete Mathematics Workshop - Lisa Ryden, September 2000
Grade: 5th Gifted and Talented Topic 4.1 Number and Numerical Operations Essential Question: How can problems in the real world be solved with mathematics? How can estimation be useful to us? How do numbers help us reason out solutions to problems? How do basic operations help us understand numbers? Knowledge/Skills/Understandings Assessments Learning Experiences Resources A. Number Sense 1. Use real-life experiences, physical materials, and technology to construct meanings for numbers (unless otherwise noted, all indicators for grade 6 pertain to these sets of numbers as well). Everyday Mathematics Assessments 2 4 6 and 9 Teacher made tests/quizzes White Boards Student sharing Exit Slips Direct Instruction * collaborative learning * Written explanations Land Fractions Open Ended Tinkerplots SketchPad Everyday Math Game CMLs Menu of Problems Cuisenaire Rods A Collection of Math Lessons, Marilyn Burns Problem Parade Doug Monteath & Don Volle Continental Math League monthly problems Mathematics Teaching in Middle School Middle School with Pizzazz; "Mathematics Teaching in Middle School" (May 1997) ● All integers Van de Walle books Everyday Math Books 1 and 2 Hand on Equations About Teaching Mathematics, Marilyn Burns Developing Number Sense Series, ● All fractions as part of a whole, as subset of a set, as a location on a number line, and as divisions of whole numbers ● All decimals ● Percents ● Whole numbers with exponents
1. Recognize, describe, extend, and create patterns involving whole numbers, and rational numbers and integers Everyday Mathematics Assessments 4,6 8 &9 Teacher made tests/quizzes White Boards Student sharing Exit Slips Mental Math Modeling/Collaborative Activity Direct Instruction * collaborative learning * Written explanations Various puzzles Open Ended Tinkerplots SketchPad Everyday Math Game CMLs Menu of Problems Cuisenaire Rods Group Investigation Everyday Mathematics unit 3, 6, 8, 9; NCTM Addendum Grades 5-8, Patterns & Functions Van de Walle pg 417-435 Chap. 22-25; ● Descriptions using tables, verbal and symbolic rules, expressions, simple equations or graphs. Journaling Groundworks Hands on Algebra Do Nows In the Balance, Creative Publications (Grades 4-6) Problem Parade About Teaching Mathematics, M. Burns, pages 112-124; Hands On Algebra Book III ● Finite and infinite sequences ● Formal iterative formulas ● Recursive patterns, including Pascal's Triangle and the Fibonacci Sequence: 1, 1, 2, 3, 5, 8 ● Generating sequences by using calculators to repeatedly apply a formula B. Functions and Relationships 1. Describe the general behavior of functions given by formulas or verbal rules ● Graph Functions C. Modeling
Knowledge/Skills/Understandings Assessments Learning Experiences Resources A. Data Analysis 1.Collect, generate, organize, and display data. Graph display ● Data generated from surveys Everyday Mathematics Assessments 1 & 7 Teacher made tests/quizzes White Boards Student sharing Exit Slips Mental Math Class Sharing Teacher Observation Modeling/Collaborative Activity Direct Instruction * collaborative learning * Written explanations Various puzzles Open Ended Tinkerplots SketchPad Everyday Math Game CMLs Menu of Problems Snap Cubes Group Investigation Everyday Mathematics units 1 & 7 Van de Walle Used Numbers, Dale Seymour Publications (1992);Navigations through Probability Math by All Means, Marilyn Burns NCTM Illuminations; Problem Parade 2.Read, interpret, select, construct, analyze, generate questions about, and draw inferences from displays of data. Journaling Groundworks Partner work/ Disease Z Do Nows Partner work/ "Tiles in the Bag" Spinner Puzzles; Shake-n-Spill Pascals Triangle Discrete Math packet ● Bar graph, line graph, circle graph, table, histogram Shop-Rite Problem Game of Pig ●Range, median, and mean ● Calculators and computers used to record and process information 3. Respond to questions about data, generate their own questions and hypotheses, and formulate strategies for answering their questions and testing their hypotheses.
B. Probability 1. Determine probabilities of events. ● Event, complementary event, probability of an event ● Multiplication rule for probabilities ● Probability of certain event is 1 and of impossible event is 0 ● Probabilities of event and complementary event add up to 1. 2. Determine probability using intuitive, experimental, and theoretical methods (e.g., using model of picking items of different colors from a bag. ● Given numbers of various types of items in a bag, what is the probability that an item of one type will be picked ● Given data obtained experimentally, what is the likely distribution of items in the bag. 3. Explore compound events. 4. Model situations involving probability using simulations (with spinners, dice) and theoretical models. 5. Recognize and understand the connections among the concepts of independent outcomes, picking at random, and fairness.
A. Problem Solving 1. Learn mathematics through problem solving inquiry and discovery Exit slips Teacher observation with feedback Mental Math Reflexes Estimation Making Conjectures Showing use of efficient math strategies Effective use of technology Various open-ended questions Monthly CML's Monthly menu problems Portfolio evaluation Various activities Sharing problem-solving strategies CMLs Exemplars Teaching Children Mathematics, NCTM -Menu of Problems KSK attachments # 17a & 17b Van de Walle Math Curse Sir Circumference 2. Solve problems that arise in mathematics and in other contexts Journaling; reflexive writing; class discussions Division with Fractions attachment #18 "Fat Content in Foods" "Golden Ratio" Everyday Mathematics, Unit 8 Lesson 5 Everyday Mathematics, Unit 8 Lesson12 Everyday Mathematics, embedded throughout Hands-On Algebra, Book III Everyday Mathematics, Unit 4 Lesson 10 About Teaching Mathematics, M. Burns, pages 102 and 105 ● Open ended problems Group discussion; Examining & use student Reference Book Strategy Share Student Discourse NCTM illuminations grades 6-8 NCTM illuminations; Navigations through Geometry Geometer's Sketch Pad Tinkerplots Game of Set ● Non-routine problems Partner work; Journal pages; "Graphing Garbage" King Arthur Problem; Prison Problem Get-it-Together Family Math Book Kagan Cooperative Learning Activities Super Source ● Problems with multiple solutions ● Problems that can be solved in several ways