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INTERMEDIATE MATHEMATICS SUPPORT DOCUMENT Blending the TIPS4RM program With Math Makes Sense And Other resources Developed by Nancy Snyder & Kristina Somerton-Scott August 2007/Updated Sept. 08 NOTES: 1. The TIPS4RM lesson format is recommended, based on sound research: A. Minds On B. Action- a main Activity (pairs, groups, independent, or a combination) C. Consolidation/Debriefing session with the whole class D. some ‘practice’ The TIPS4RM document refers to Jazz Days – days where you can ‘do your own thing’. Practice skills & concepts, play a game, have a diagnostic assessment, etc. The front section of the TIPS4RM document is a great resource for Differentiation strategies, instructional strategies, learning styles of students, and more!! 2. Think/Pair/Share is an excellent strategy for students who are reluctant to share ideas orally as it gives them an opportunity to think first, then share their idea with one other person, which may give them confidence to share with the whole group. This strategy also gives the teacher reason to expect each person will have some idea of a solution/answer. For a more complete explanation of this strategy see the Think Literacy Mathematics – Subject Specific Examples Grades 7 – 9. 3. Modelling and Think Aloud are instructional strategies explained in the Resource section of this document 4. As a teacher circulates, interacting with groups, demonstrate positive ways to prompt or extend an idea or disagree with someone. (see Grade 8 TIPS4RM Introductory Unit 6.1 – Day 6, BLM 1) LESSON COMPONENTS A. MINDS ON (can be used for any lesson) These are used to introduce some mathematics for your lesson, as a quick diagnostic to assess ‘where’ your students are, or just to ‘wake up’ the students’ mind to mathematics in a fun way! B. ACTION Activities Depending on activity, students are in pairs, groups, work independently, or a combination is used. This is the main activity/purpose of the lesson, where concepts are investigated & learned. As a teacher circulates among student pairs or groups, use any and all opportunities to model and stress good mathematical vocabulary. When choosing or designing activities always think of what manipulatives you will make available for students to use. C. CONSOLIDATION/DE-BRIEFING Students share strategies/patterns/ideas they have learned during the ‘Action’. Teacher illustrates good listening techniques, how to paraphrase solutions, good communication of solutions. Use any and all opportunities to model and stress good mathematical vocabulary. D. PRACTICE/HOME ACTIVITY 1. Dice games and card games are great home activities 2. Activity sheets that help the student practice a skill 3. Any activity that assist with conceptual understanding 4. A journal entry ASSESSMENT Informal or formal observation of student’s learning skills, mathematical communication, and problem solving techniques. (Observation checklist/logs are attached at the end of this document or see John Van de Walle’s book Ch.5 on Assessment pg. 72/73 – An Observation Rubric) Formal assessments – quizzes & summatives are included in the TIPS4RM document and in the spiral-bound accompaniment to TIPS4RM. Also both the Math Makes Sense student text (chapter tests) and the MMS teacher resource binder offers assessment options. CONTENT AND REPORTING TARGETS Grade 8: Content and Reporting Targets * Strands for reporting purposes Term 1 – Content Targets Term 2 – Content Targets Term 3 – Content Targets Number Sense and Numeration* Number Sense and Numeration* Number Sense and Numeration* • integers • fractions • proportional reasoning • order of operations • powers and square roots • percents • rates • representations of numbers Measurement* Measurement Measurement* • surface area and volume of cylinders • relationships among units • circle measurement relationships Geometry and Spatial Sense* Geometry and Spatial Sense* Geometry and Spatial Sense • construction of a circle • properties of lines, angles, triangles, • similar figures Patterning and Algebra* and quadrilaterals • Pythagorean Relationship • algebraic expressions Patterning and Algebra • multiple representations of patterns • properties of polyhedra • use patterns to develop measurement • writing nth terms • transformations on the plane Data Management and Probability formulas • display and interpret data found in Data Management and Probability* Patterning and Algebra* patterns • experimental vs. theoretical • solve and verify linear equations probability Data Management and Probability* • complementary events • design and carry out an experiment TIPS4RM: Grade 8: Overview Grade 7: Content and Reporting Targets * Strands for reporting purposes Term 1 – Content Targets Term 2 – Content Targets Term 3 – Content Targets Number Sense and Numeration* Number Sense and Numeration* Number Sense and Numeration* • integers (Unit 2) • multiples and factors (Unit 7) • proportional relationships (ratios, • order of operations (Unit 4) • fractions (Unit 7) rates) (Unit 9) Measurement* • introduction to proportional • order of operations (Unit 10) • composite figures (Unit 4) relationships (percents) (Unit 7) Measurement • area of a trapezoid (Unit 4) • squares and square roots (Unit 4) • volume of right prisms (Unit 10) Patterning and Algebra* • order of operations (Units 4 and 5) Geometry and Spatial Sense* • describing patterns (Unit 2) Measurement* • similar and congruent triangles (Unit Data Management and Probability* • surface area of right prisms (Unit 4) 8) • collect, organize and analyse data • application of area measurements • transformations (Unit 8) (Unit 3) (Unit 4) Patterning and Algebra Geometry and Spatial Sense* • modelling and describing proportional • geometry of lines and angles (Unit 6) relationships (Unit 7) Patterning and Algebra* Data Management and Probability* • solving equations (Unit 5) • probability (Unit 7) • using variables (Unit 5) TIPS4RM: Grade 7: Overview Combined Grades 7 & 8: Refer to Overview of TIPS4RM http://www.edu.gov.on.ca/eng/studentsuccess/lms/tips4rm.html#grade78 RESOURCES to support this document & your Mathematics Program A. Web Resources TIPS4RM http://www.edu.gov.on.ca/eng/studentsuccess/lms/tips4rm.html Supports for the TIPS4RM http://www.curriculum.org/csc/library/tips/index.shtml The TIPS4RM site includes Impact Math activities: identified by strand Impact Math sites: NSN http://www.curriculum.org/csc/library/strategies/downloads/impact_number.pdf P&A http://www.curriculum.org/csc/library/strategies/downloads/impact_pattern.pdf GSS http://www.curriculum.org/csc/library/strategies/downloads/impact_geo.pdf M http://www.curriculum.org/csc/library/strategies/downloads/impact_measure.pdf DMP http://www.curriculum.org/csc/library/strategies/downloads/impact_data.pdf Think Literacy http://www.edu.gov.on.ca/eng/studentsuccess/thinkliteracy/library.html Leading Math Success http://www.edu.gov.on.ca/eng/studentsuccess/lms/library.html Virtual Manipulatives Library http://nlvm.usu.edu/en/nav/vlibrary.html OAME http://www.oame.on.ca/main/index1.php (Ontario Association of Mathematics Educators) OSAPAC http://www.osapac.org/bb.asp (Ontario Software licensing website with many teacher resources for GSP, TinkerPlots, TABS+, etc.) NCTM - many resources for intermediate http://www.nctm.org/resources/middle.aspx - Of special interest is their Illuminations section http://illuminations.nctm.org/ Number line – produce any number line you need http://incompetech.com/graphpaper/numberline/ Wired Math(UW) http://www.cemc.uwaterloo.ca/wired_math/english/index.shtml Teacher’s Guide to Data Discovery http://www.statcan.ca/english/freepub/12-593-XIE/12-593-XIE2007001.pdf A StatsCan teacher resource for data management which provides activities & lessons to address many concepts such as: primary data, secondary data, mean & median, types of graphs, analyzing data, etc. There are also many links to other sources and pre-made activities using TinkerPlots & Fathom. Figure This! http://www.figurethis.org/challenges/math_index.htm EQAO & TIMSS Math & Science (gr. 6 – 9) support document: http://www.eqao.com/pdf_e/00/00P009e.pdf B. Print Resources TIPS4RM document - binder and spiral notebook called Summative Tasks and Continuum & Connections (in schools Sept. 2007) Elementary and Middle School Mathematics: Teaching Developmentally by John D. Van De Walle (4th edition – orange & yellow) (Also a white book – grade 5 – 8 resource) (in school library) Math Makes Sense Text and Teacher Resource Binder by Pearson/Addison Wesley (in each school) Think Literacy Mathematics – Subject Specific Examples Grades 7 – 9 (in each school) The Super Source Activity Books – one per strand (in school library) Impact Math – one per strand (an old resource, but very good – maybe in school) Literacy Strategies for Improving Mathematics Instruction (Reference Library Ed Centre REF 510.71 LIT) C. Other Resources Intermediate manipulatives kit – refer to WCDSB website for contents list which should be at each school Geometer’s Sketchpad (GSP) - geometry software on the school network (TIPS4RM lessons offer ready- made GSP activities for teachers and students on its website - listed under Web Resources above) TinkerPlots – graphing software on the school network; useful for DMP; looking at patterns Note: Math Makes Sense refers to the Fathom software, which is the advanced version of TinkerPlots. Refer to the OSAPAC web address (listed above) for easy ‘how to use’ teacher resources for TinkerPlots. TABS+ - computer software on the school network that allows you to draw in 3D GRAPHERS – computer software on the school network used for DMP D. Instructional Strategies and strategies for students to use Some Mental Math strategies (i.e. not using a calculator or a ‘rule’) 1. When adding a column of numbers, have students look for groupings of 10 (or 100, etc.) e.g. Add: 22 37 55 the 2 + 8 sum to 10 So we know the units column sums to 25 18 the 3 + 7 sum to 10 Similarly, the 20, 30, & 50 in the tens column 23 sum to 100. 25 (units column) 100 (20 + 30 + 50) 30 (10 + 20 left in tens column) 155 Alternatively, the student could have written the 5 under the units column, carried the 2 to the tens column and then noticed there were three groups of 50 made in the tens column - from 20 & 30 at top, the carried 20 and 10, 20 at the bottom, and the 50 in the middle, thus making 150. Again arriving at an answer of 155. 2. When multiplying 27 x 82, model (use a Think Aloud) breaking this product into: i. 20 x 80 (like 2 x 8 with two zeros) 1600 ii. Plus 7 x 80 560 iii. Plus 2 ‘twenties’ plus 2 ‘seven’s 40 _ 14 2214 Note: The pattern of 20 x 80 being like 2 x 8 with two additional zeros should be reinforced in P&A (see TIPS4RM Unit 3 – Multiplying and Dividing Using Powers of 10, Day 2) 3. 49 x 5 can be calculated mentally by thinking of 50 x 5 – 5 x 1. Or, if a student is using a multiplication table up to 12 x 12 but their question is 13 x 5, help them to see it is the same as 12 x 5 + 1 x 5. 4. Give each group a number and ask them to represent it as many different ways as possible: a. 407 = 400 + 7 b. 407 = 4x100 + 7 c. 407 = 4 x102 + 7 Make it a challenge! d. 407 = 7 + 22 x 102 Can they do it using all 4 operations?! e. 407 = 4.07x100 f. 407 = four hundred and seven **This multiple representation is VERY g. 407 = 2x2x2x2x25 + 21 ÷ 3 important when doing ‘mental math’** h. 407 = 500 – 93 i. 407 = 4070/10 j. 407 = 4 centuries plus .07 of a century, etc. 5. For many other strategies, specific to different number concepts, see John A. Van De Walle’s book. Introduce, model, and use these strategies as the need arises with each student. (Differentiation) Inside/Outside Circle (a kinesthetic activity) The class numbers off “one, two, one, two, . . .” to form two groups – ‘ones’ form an Inside circle and ‘twos’ form an Outside circle. Students stand across from their ‘partner’ in the ‘other circle’ take turns sharing/communicating/explaining, as the activity indicates. (see TIPS Gr. 8 Introductory Unit Day 3) Think Aloud Strategy While the teacher is solving a problem or performing an operation, they ‘think out loud’, allowing students to listen to their thought process and giving the teacher an opportunity to use/model good mathematical terminology. During the think aloud it would be appropriate for the teacher to make a mistake, then acknowledge the error and revise the solution – all verbally. Using this technique, a teacher can illustrate a variety of strategies for mental math, performing operations, mathematical procedures, and problem-solving by using a different strategy for similar questions. Modelling Strategy Step 1: The teacher illustrates a technique, using an example. Step 2: Students are directed to solve a similar example, with a partner. Step 3: Students are directed to solve yet a similar example, independently. Then share solution with a partner. Then class shares results. The process can be repeated. Variations: I. Partners can change each time the process is repeated. After Step 2, two partners can pair up with another two partners, to compare/discuss before Step 3. II. You could insert a Step 1b where the teacher attempts another example, with assistance from students in the class. The teacher would circulate and observe during steps 2 & 3, focusing on students with higher need for assistance. This is a good strategy when first teaching a new concept. It is very effective when performed along with a Think Aloud. E. Other Resources Observation Logs 1. Refer to John Van De Walle’s book Chapter 5 (Assessment) for an Observation Checklist/Log 2. Sample Observation Templates attached at end of this document 3. Your own or those from a colleague Fermi Problems Setting a Context for Solving Fermi Problems: • Fermi problems at first might appear not to have an answer. Your initial response may be “I need more information or there is not enough information.” • There are many different ways to solve a Fermi problem. Be creative. Use any necessary tools. • You may use a variety of estimation strategies and need to take a few risks. Don’t be afraid to piggyback on someone else’s ideas. • You will be working as a team (a community of learners) and sharing strategies and ideas, encouraging and supporting each other using social skills that we will reinforce each day. What is a Fermi Problem? A Fermi problem is a multi-step problem that can be solved in a variety of ways, and whose solution requires the estimation of key pieces of information. Linking Assessment, p. 116 Why Start Grade 8 with Fermi Problems? The purpose of the first week of school is to set the tone for a positive academic environment and community of learners, which foster both mathematical processes and affective processes. The Fermi problems and social skills introduced in the first week of classes combine to “generate the kind of involvement and thinking processes that are at the root of quantitative literacy. Because important information is missing, students must ask themselves more questions about what they need to know and what they already know. Then they must construct a path of estimates that leads from the knowledge they have to the knowledge they need to acquire. The focus of this activity is on the process rather than the answer – a process that mirrors the ‘number sense’ we apply in everyday life when we make ‘ballpark’ estimates of our fuel consumption, our bank balances, or the time we’ll need to mark a class test.” Impact Math, Number Sense, p. 17 http://www.curriculum.org/csc/library/strategies/downloads/impact_number.pdf Who was Enrico Fermi? Fermi (1901-1954), a famous physicist, was known to mathematicians for his legendary estimation problems. He was able to answer impossible questions by mentally estimating large quantities for which there seemed to be insufficient information. “How many piano tuners are there in Chicago?” was one of his well-known problems. This seemingly unanswerable question often puzzled people. Fermi developed a series of subordinate questions leading to an estimate that was the right order of magnitude. The information in the table below is a summary of the sequence of questions, answers, and estimates listed in Impact Math, Number Sense. Example of a Fermi Question: How many piano tuners are there in Chicago? Process of questions to assist with answering this question: Question Answer 6 What is the population of Chicago? 3 × 10 To estimate pianos should we estimate people or Households rather than individuals tend to own households? pianos. Approximately how many There may be an average of 4 people per households are there in Chicago? household in Chicago, so the number of households is about 3 × 106 ÷ 4. What proportion of households in Maybe about 1 in 10 households has a piano. That Chicago has pianos? would suggest that there are about 3 × 106 ÷ 4 ÷ 10 or 7.5 × 104 pianos in Chicago. How many piano tuners are needed Assuming a piano is tuned once a year, then 75 to tune those pianos? 000 piano tunings are needed. If a piano tuner tunes approximately 3 pianos a day, and works 200 days a year, the number of tuners needed is about 75 000 ÷ 600 or 125. There are about 125 tuners in Chicago. Sample Fermi Problems 1. How many hours have you been alive ? 2. How many kilometers does your family drive in one year? 3. How many metres of spaghetti does your family eat in one year? 4. How many Rice Krispies would fit into our classroom? 5. How many times does a human heart beat in a lifetime? 6. How long would it take you to count to 1 000 000? Math Games, Puzzles, and Other Activity Suggestions 1. Card Games of any kind are great in math (War, Solitaire, Euchre, Gin Rummy, etc.) Other suggestions: a. Ten, Twenty, Thirty (from Family Math) • Groups of two to four • Deck of cards face down on table. Aces count 1 and face cards worth 10. • Turn up two cards. Place them in a row. • Take turns drawing a card to add to the ‘playing’ row – five cards in total. (always keep 5 in the ‘playing’ row, so adding to the row when necessary, on your turn) • Whenever there are 3 cards at either end of the row, or 2 at one end and 1 at the other, that add to 10, 20, or 30, you may take those cards – if it is your turn. You can also draw from the stack and use it with two cards from either end to obtain a 10, 20, or 30. Cards that are of no use are turned face down and re-used when the stack is done. • Play until all cards are used from the stack or there are no more plays. • Winner is the player with the most cards at the end of the game. • Variation: Play Jokers wild – they are worth whatever value a player wishes Play Aces worth 11 b. Make the Most of It! (from Family Math) • You will need the Aces through 9’s from all suits from a deck of cards & timer • partners or groups of three • Each player has a piece of paper and a pencil • Deal our 4 cards with the cards facing up. Using all four cards and a choice of multiplication, division, addition, and subtraction, have each player see how many different values a person can get in 5 minutes. () Brackets also allowed. • Players get one point for each answer. E.g. Suppose the 4 cards dealt facing up are: 4, 8, 9, 2. Some numbers that can be made are: 4+9+8+2 = 23 9x4x8x2 = 576 A great activity to practice Mental Math (9+8) x (4+2) = 102 Strategies! Calculator use also OK. Or try with and without using a calculator. Variation: See who can make the highest value 2. Dice Games a. Multiplication Max (from Family Math) (Partners or Groups of 3) Reject box Calculators permitted X Product • Each person makes a copy of this multiplication as shown • Take turns rolling a die or spinning a spinner (6 section spinner). Write the digit you roll or spin in the ‘box’ of your choice or the Reject box (reject box can only be used once) • The goal is to get the largest possible product, after 6 rolls • You cannot skip turns or move digits after they are recorded • After each player has completed six turns, compare the products to see which is the largest • Now work with other group members to help each person to rearrange their numbers to get the largest possible product. b. Diminishing Division Reject box (from Family Math) • Similar to Multiplication Max except your goal is to get the smallest quotient 835 • Extension: Does writing the division problem as a fraction (e.g. 835 ÷ 105 as ) 105 help you choose where to place the numbers? Why or why not? c. Mr. Mac’s Dice Game (Individual – no calculators. Encourages mental math!) • Decide on a number of rounds – 3 or 4 rounds is good. Each round consists of 8 rolls of two dice. • Choose one student to roll the dice (two dice needed) • All other students are players – they start the game by standing up. They need a piece of paper and a pencil. Across the top of the paper they write: Round 1, Round 2, etc. • The ‘roller’ rolls the two dice and announces the # on each die. Players write the total for the roll under Round 1. After each roll, players record the total of the two die on their paper. To continue to count rolls, a student must remain standing. After any roll of a round, a student may choose to sit down, which will keep their total point count ‘safe’. • The ‘snag’ is the following: During a round, if the roller rolls a 1 on one of the dice, the players left standing lose all of the points for that round. (any players who are sitting are safe) If the roller rolls double 1’s, the players left standing lose ALL points from the current round and from all previous rounds! (again, any player sitting are safe and can keep their points) Recall, after 8 rolls, another ‘round’ begins. At the beginning of EACH round, all players must start the round standing. • The player with the most points at the end of 3 or 4 rounds wins. (usually the winner and the roller get a prize) 3. Mathematical Puzzles The Extra Dollar Three business men rent a conference room from a hotel for thirty dollars. They decide to split the cost evenly so they each pay ten. Five minutes after they have paid, the hotel manager tells them that he forgot about the special deal currently going on, and informs them that the room is actually only twenty-five dollars instead of thirty. So each of the men take one dollar back from the five the hotel manager gave them back, leaving two dollars left over. Now they have each actually paid nine dollars. Three times nine is twenty-seven plus the two dollars left over equals twenty-nine. Where did the extra dollar go? Number Tricks (good for introducing algebraic expressions) Trick Algebraic modelling Pick any Whole Number. A number can be represented by x. Multiply it by 2. 2x Add 10 to it. 2x + 10 2 x + 10 Divide it by 2. (2x + 10)/5 or (2x + 10) ÷ 5 or 5 = x+5 Subtract 5. x + 5 – 5 which equals x You answer will always be the number you started with! Have students model this ‘trick’ using algebraic expressions to prove why the result always works. (see italics) There are lots of these number ‘tricks’ around – ask students to find some! Kaprekar’s Constant Take any four digit number, whose digits are not all the same, and do the following: • Rearrange the string of digits to form the largest and the smallest 4-digit numbers possible. (must be 4 digits!) • Take these two numbers find their difference – subtract the smaller from the larger. • Use the answer you obtain and repeat the above process. • See what happens as you repeat the process over and over – a pattern develops! E.g. Using 3142 4321 – 1234 = 3087 8730 – 3078 = 5652 6552 – 2556 = 3996 9963 – 3699 = 6264 \ 6642 – 2466 = 4176 7641 – 1467 = 6174 The process eventually hits 6174 and stays there! But more 7641 – 1467 = 6174 amazing is that EVERY 4-digit number (whose digits are not all the same) will eventually hit 6174, in at most 7 steps, and stay there!!!!! 4. Open Response Questions These are questions that can have more than one solution and more than one method for solving can be used. They can also have one solution, but many methods to solve can be employed. There are many benefits to Open Response Questions, some of which are: Enables students to learn through using the mathematical processes It empowers kids – they have a choice of method and answer IT IS A METHOD TO DIFFERENTIATE – students can use methods they are comfortable with (P & P, manipulatives, formulas) and allows for extension ‘on-the-spot’ A very ‘rich’ activity – ‘rich task’ and ‘rich talk’ It offers multiple entry points for students of all abilities Communication between students and during class sharing encourages use of good mathematical vocabulary Can often use it over again to reinforce a concept or illustrate another concept – reusable! Some Open Response Questions to use in an Intermediate Class: Context Example of Open Response Question Fractions I ate part of a pepperoni pizza and a different part of a veggie pizza. Which one did I eat more of? How much more? (students start with fractions they are comfortable with. The teacher can ‘take them further’ by asking “what if ” questions) Rate of Change Dan biked at a rate of 80 km in 4 hours. Annette said she can bike faster than that. What rate could Annette bike at? Use two methods to prove it is a faster rate than Dan’s. (Challenging students to use more than one method to prove an answer increases their inventory of strategies to solve problems) Estimation Find 5 items in the room that you think are between 10 cm and 25 cm. Check! Measurement How would you measure a puddle? (encourages good vocabulary – what could we measure? Volume? Area? SA? Perimeter?) The perimeter of an object is 33cm. What could the shape look like? Multiplication of 3 digit # x 1 digit # Pick a 3 digit number and a 1 digit number to multiply. Multiply them. Check using two methods. (we do not care what numbers students use, we care what process they use to multiply and whether they understand when the product is a 3 digit # and when it is a 4 digit #) OR Fill in the blanks with possible numbers: __ __ __ x __ __ __ __ __ Use of Equations The answer is 7. What could the equation be? (any equations, any grade) Factoring Two factors of a number are opposites. Find the number. (Answers could be -9 or -4, etc. since -3 x 3 are opposites) Integers Pick three different integers. At least one must be positive (use groups of 3) and one must be negative. Write three clues to describe/define your integers. All three clues must be necessary. Two of the three clues must involve addition and/or subtraction. Share your clues with another group and see if they work! Submit your work. Example: -5, 1, 5 are the integers picked Clue 1: the difference between my largest and smallest integer is 10. Clue 2: the sum of all my integers is 1 Clue 3: two of my integers are opposites NOTE: the order of my clues can make it more difficult to guess my integers OBSERVATION SHEET NAME:________________________ Date:__________ Notes: Focus: (circle focus areas) Communication (C) Attitude (LS) Collaboration (LS) Math Understanding (K & A) Reasoning (C/TI) Use of Tools & Equipment (K) C = Mathematical Communication LS = Learning skill K = Knowledge A = Application TI = Thinking/Inquiry OBSERVATION SHEET NAME:________________________ Date:__________ R 1 2 3 4 Communication Reasoning Math Understanding Use of Tools & Equipment Attitude Collaboration Comments: NOTE: You can modify this template by using any focus area you would like