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M ATHEMATICS

M ATHEMATICS. as a T eachable M oment. P M eaning P C hoice P D iversity P T rust P T ime. 1. C reate M eaning . S tudent P rojects S tudent W ritten P roblems and S olutions S ports / P ets / C ooking D ate / S pecial D ays / Season / W eather

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M ATHEMATICS

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  1. MATHEMATICS as a Teachable Moment PMeaning PChoice PDiversity PTrust PTime 1

  2. CreateMeaning • Student Projects • Student Written Problems andSolutions • Sports /Pets /Cooking • Date /Special Days / Season /Weather • Place (Home / Community / School) • Games • Discussion in pairs, small groups and as a class 2

  3. Control Commitment Challenge Projects, Student Written Problems, “How Many Ways” sheets, and Discussion are all embedded with choice. GiveChoices Choices provide meaning through a sense of: 3

  4. Encourage alternative strategies. Ask “How?” not “Why?” Give students choice in the order, methods, strategies and topics. Make sure all students are involved in creating rules and sharing strategies. ValueDiversity Diversity should be treated as a positive factor in the classroom. We need to: 4

  5. Ask open-ended questions and value diverse strategies for solving problems. Ask students to explain, discuss and show – especially when their answers are correct. Value errors as opportunities to investigate conceptual understanding and create new understandings. Create aClimate of Trust 5

  6. Teach the curriculum in an integrated manner so that there are opportunities to review every major theme or skill set. Value accuracy over speed. Avoid: Races Contests and Strictly Timed Basic Fact Tests Integrate the intended learning outcomes (ILOs) into major themes and evaluate over the whole year. Ensure There is Adequate Time 6

  7. Review ofSilent Mouthing Use the “silent mouthing technique: ä ä Student Feedback to give: to give: When students make errors give them hints, suggest that they are close, acknowledge that they are a step ahead or say, “That is the answer to a different question.” Slower processors and complex thinkers the time they need to do the question. 7

  8. Review ofPlace Value Place value should be taught at least once a week but preferably a place value connection should be made almost every day. The connections to algebraic thinking should be made (collecting like terms) as this will pay off when doing operations with fractions and algebraic expressions. 8

  9. Organization of theCURRICULUM All four strands (ŸNumber Sense, ŸSpatial Sense, ŸProbability and Data Sense and ŸPattern and Relationship Sense) should be covered every month (every week in Primary). Problem solving often embeds three of the strands depending on whether the problem has a focus on spatial relationships or data relationships. It is usually preferable to introduce a new topic through a problem. The Japanese teachers use this technique effectively. 9

  10. Graphing is a tool for making meaning if the data is collected from the students. Eventually the “Weekly Graph” becomes a day for teaching proportional thinking, decimals, fractions, percents, graphing, patterns and relations, and probability. The “Weekly Graph” is intended to be student driven by the fourth week at the latest. MakingMeaningwith theWEEKLY GRAPH 10

  11. Watch for the “big ideas” in the video. • What teaching techniques are effective? • What Mathematical concepts are covered? 11

  12. How do we find the time to teach this way? How many ILOS were covered in the previous video clip? 12

  13. T Intermediate eaching Students NEW Strategies for OLD Ideas Where do we find the time to teach this way? If students are taught this way, how will they do on the FSA tests? 13

  14. Multi-step Division and Decimal Fractions Placement of the decimal in the quotient should be done by asking, “Where does it make sense to put the decimal so that the answer makes sense?” The first few times multi-step division is taught it should be done as a whole class. The errors made should be used as opportunities to investigate conceptual understanding. 14

  15. Process for Teaching 1 ÷ 9 If possible, do multi-step division on grid paper (cm graph paper works well). If grid paper is not available, use lined paper turned sideways so that the lines become grids for keeping the numerals in the correct position. 15

  16. 1 1 0 ) 1 0 0 9 . -0 1 0 - 9 1 1 0 1 9 9 = 1 1 0 1 1 1 1 . 1 - 9 ) 1 9 0 0 0 0 0 0 0 0 . Ÿ 0.1 0.1 = or . 1 ÷ 9 16

  17. The multi-digit regrouping system we use for subtracting is based on the principle of equivalence and is done differently in parts of Europe. Some Europeans use a system that depends on the principles of balance and equivalence. Many algorithms are culture specific time savers that create accuracy. 17

  18. The algorithm we use for multi-digit multiplication has changed considerably over the years. In the fifties we moved the second product over one space which paralleled the way we multiplied using adding machines. Now we add a zero for the second product. Many algorithms are culture specific time savers that create accuracy. In the middle ages we used a box or window method. 18

  19. Algorithms in the 21stCentury Algorithms should be developed through discussion with learners because the purpose of teaching algorithms is to develop understanding. The focus should be on accuracy, then on efficiency. The most efficient algorithm today is always based on today’s technology. The most efficient algorithm today is the calculator or the computer but we do need to understand the underlying concept or we don’t know if the answer makes sense. 19

  20. 5 7 7 7 6 7 8 7 3 7 4 7 2 7 1 7 , , , , , , , , How many remainders did it take before you achieved a repeating decimal pattern? Do you notice any patterns? FRACTIONS are RICH in PATTERNS Working at your table or in your group, assign different members of the group to find the decimal fraction for: 20

  21. WEEKONE of the Weekly Graph 40 100 1 4 2 8 4 16 3 12 5 20 = = = = = Memorable Fractions and Their Equivalent Buddies Common Fractions Simplest Form Decimal Equivalent Percentage Equivalent For example: (the first fraction illustrated in the video) On your “Memorable Fractions” sheet please write in all the fractions studied in the video. Include some of the equivalent fractions for these. There were three other fractions in the problem. There was one fraction from the graph. 21

  22. WEEKONE of the Weekly Graph Have the students draw a bar graph of the results. In the end, the class will have developed assessment criteria from a meaningful context by having students notice what makes a graph a good communication tool. Self-evaluation is often the most effective. Do not give students criteria for creating a good graph. Discuss the results and focus on the fact that graphs are supposed to give you a lot of information at a glance. This means that the graph should be neat, have a title and a legend (if necessary). 22

  23. WEEKONE of the Weekly Graph Have students discuss (write) what they know about the class by analyzing the data (graph). Can they think of any questions or extensions? Use these for further research. Use the think/pair/share method to create discussion, then share as a group (valuing diversity, creating trust and developing meaning through choice). 23

  24. WEEKTWO of the Weekly Graph Collect data. Decide which fractions (decimals and percents) you wish to study. If you are worried about coloring in the hundreds squares for a tricky fraction, leave this part until the next day and try it yourself. Enter the fractions on the Memorable Fraction sheet. Draw a circle graph of the data. Review the criteria. 24

  25. Have the students find the prime factorization of 360 and the prime factorization of the number who voted (e.g. 30). Write the equation in fractional form: 360 30 2 x 2 x 2 x 3 x 3 x 5 = 2 x 3 x 5 CIRCLE GRAPHS Can be rich in CURRICULUM Connections If the number of voters in the class is: 12 15 18 20 24 30 36 or Do the following: 25

  26. 360 30 2 x 2 x 2 x 3 x 3 x = 2 x 3 x 5 5 2 x 2 x 5 = = 40 100 x 20 20 5 20 = 2 5 1 4 PRINCIPLE of ONE Find the ones. 5 This principle was used in the video to make equivalent fractions – in particular: 26

  27. PRINCIPLEof EQUIVALENCE Throughout the video and on the “Memorable Fractions” sheet, the students have been making equivalent fractions and have learned that every fraction can be expressed as an infinite number of common fractions, exactly one decimal fraction and one percentage fraction. It can also be expressed as a ratio. 27

  28. x = = x 5 20 1 4 5 5 PRINCIPLEof BALANCE In the video one student noticed that when equivalent fractions are generated, both the numerator and denominator have to be multiplied by the same number. This is also an example of the Principle of One as: 28

  29. 2x = 26 2 = 2 Please solve the following: Don’t forget to show your steps. 2x + 5 = 31 2x + 5 – 5 = 31 – 5 2x = 26 x = 13 29

  30. PRINCIPLEof ZERO This step is necessary for equation solving and is the only principle that is not generated in doing the “Weekly Graph”. It should have been generated much earlier in the primary grades when doing the “How Many Ways Can You Make a Number” activity during Calendar Time. 30

  31. How Many Ways? 31

  32. How Many Ways? 32

  33. P How Many Different Ways Can You Make a Number? Criteria Criteria Mark Mark 1 1 Where any sentence shows knowledge of the power of zero (e.g. 6 – 6 + 10 = 10 or 10 + 0 = 10) Where any sentence uses doubling and halving to generate new questions (e.g. 4 x 6 = 24, 2 x 12 = 24, 1 x 24 = 24) Where any sentence shows knowledge of the power of one (e.g. 6 ÷ 6 + 9 = 10 or 10 x 1 = 10) Where any sentence shows knowledge of the commutative principle (e.g. 6 + 4 = 10 and 4 + 6 = 10) Where any sentence shows knowledge of the numberNote: this applies only for numbers greater than 10, such as 24. In upper intermediate grades, award marks for exponential notation also. (e.g. 20 + 4 = 24 and 2 x 10 + 4 = 24) Where any sentence contains brackets, such as: (3 + 2) + (3 + 2) + (3 + 2) + (3 + 2) + 4 = 24 Where any sentence contains exponents, square roots, factorials, or fractions. Note: there should be no expectation of the demonstration of exponents, square roots or factorials before grade six, but their use should be acknowledged and rewarded where a student chooses to employ such operations in earlier grades. Where any sentence contains the Addition operation Where any sentence contains the Subtraction operation Where any sentence contains the Multiplication operation Where any sentence contains the Division operation Where any sentence contains more than two terms (e.g. 2 x 3 + 5 = 10) Where any sentence contains more than two operations (e.g. 2 x 3 + 4 = 10) Where any sentence contains a number more than the goal number (in this case 10) Where any sentence contains a number substantially greater than the goal number (in this case 50 or 100) Where any group of sentences shows evidence of a pattern (e.g. 1 + 9, 2 + 8, 3 + 7) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 33

  34. These four principles should be generated by and attributed to students. They are all you need to solve most equations and work with rational expressions throughout high school. PRINCIPLESof EQUATION SOLVING PrincipleofZero PrincipleofOne PrincipleofEquivalence PrincipleofBalance 34

  35. ALGEBRAIC THINKING Connections The principle of one and the principle of balance are used in rationalizing radical expressions. The principles of one and balance can be used to generate an easy to remember algorithm for dividing fractions. to 35

  36. x x = = = 1 x We refer to this as Invert and Multiply which has no other application in mathematics. x The Principle of One has many applications. 14 15 5 7 7 5 7 5 2 3 5 7 2 3 2 3 5 7 PRINCIPLE of ONE 36

  37. ALGEBRAIC THINKING Connections All four principles are used in equation solving. to Equivalenceis used in all facets of mathematics. Balanceis used in equation solving as well as multiplication and division of rational expressions. The Principle of Zerois extensively in simplifying rational expressions. 37

  38. Probability Makes Meaning Probability can be introduced during the “Weekly Graph” process. Probability was introduced in the first session when playing “hangman” which is an activity students love to play. Probability sense is an important skill we use in everyday life. 38

  39. Place the decks face down side-by-side. Predict the sum if you were to turn over the top two cards. Collect the predictions from the whole group. Ten-Frame Probability In your group, have one person shuffle the red deck (cards numbered 1 to 10) and a different person shuffle the blue deck. 39

  40. How many did you get? Check with other groups to see how many they got. Discuss the reason for your answers until you come to a consensus. What would the probability be for getting 6? 5? It often is. What is the probability of turning over two cards whose sum is 7? Now watch the video. Ten-Frame Probability Turn over the two decks and find all the combinations that equal 7. Was the most common prediction a 7? 40

  41. T Intermediate eaching Students NEW Strategies for OLD Ideas Which ILOs were covered in the activity? What are some connected or follow-up activities that you could use? 41

  42. ADDING FACTS Connections to Introducing the ten-frame cards this way allows grade four to eight students to look at numbers in a new way and learn to add visually without counting. The games shown in the video are called “Solitaire 10” and “Concentration 10”. Some students in intermediate grades have difficulty adding, and this is a new way to learn an old concept of making tens. 42

  43. Connections “ALL THE to FACTS” Sheet 43

  44. Visual tools are powerful. After just this one lesson, which may take two or three days to complete, most students when asked to visualize how to make ’15’ with the cards will say, “Get a ten and a five”. Connections Now ask them to cover up or take away nine. When they say, “Six”, ask them how they see the six. They should say, “One and Five”. This tool works for subtracting 9, 8 and 5, which is almost half of the subtracting facts. SUBTRACTION to FACTS 44

  45. with PROBABILITY Connections All of the fractions generated in the video were for ‘what you would expect to get’. This is called the “Expected Probability”. What we are really interested in is the “Experimental Probability”. The next step is to have each pair or students do 100 trials each and compare the Expected Probability to the Experimental Probability. The difference explains why people gamble. 45

  46. with TECHNOLOGY Connections If each student in the class does 100 trials and then the data is put on a spreadsheet, it is clear that while some students will win if they pick their favourite number, others will lose. However, the experimental results for the whole class will usually mirror the expected probability. Gambling then is a tax on the under-educated, often the poor. Government figures the odds, pays less than the expected probability, and makes lots of money. 46

  47. with TECHNOLOGY Connections 47

  48. with TECHNOLOGY Probability of a new student in class wanting a specific kind of pizza, liking a certain pop star, or wearing a certain kind of clothing. Connections Do the same activity with six-sided, ten-sided, or twelve-sided dice. Probability of getting a specific number or color of SmartiesTM or other candies on Halloween or Valentines Day. 48

  49. DECIMAL FRACTIONS PROJECT to the Connections Do you see any patterns? Take out the Decimal Fractions Project sheet. Enter all the fractions and decimals collected so far. Find the prime factorization of the denominator for each fraction (use fractions in their lowest terms only.) 49

  50. 5 20 1 4 STUDENTFRACTIONDECIMALINVESTIGATIONSHEET 5 = 5 so 4 = 2 x 2 = 10 = 2 x 5 9 = 3 x 3 50

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