Ce qui est important dans l'enseignement des mathématiques. Marian Small novembre 2017

Transcription

1 Ce qui est important dans l'enseignement des mathématiques Marian Small novembre 2017

2 Playing with math Uae your linking cubes. Show that the mean of 4, 7 and 7 is 6.

3 Playing with math Uae your linking cubes. Show that the mean of 4, 7 and 7 is 6.

4 Playing with math Uae your linking cubes. Now create 5 numbers so that the median is half of the mean.

5 Playing with math Use your linking cubes. Now create 5 numbers so that the median is half of the mean.

6 Playing with math Use your linking cubes. Build two rectangular prisms with the same volume but very different heights.

7 Ordre du jour Part 1: Grade 6 Course Plan Why is it the way it is? Part 2: Open Questions

8 Agenda Part 3: Comprehension Questions Part 4: Scaffolding Part 5: À Pas de Géant 5-6 Part 6: Questions

9 Part 1: Gr. 6 Course Plan

10 Part 1 There is NO perfect plan, but plans do help.

11 Part 1 It is important to teach to bigger idea or essential understandings throughout the grades to make them very visible. Here is an example or two and what they mean.

12 Part 1 There are many ways to compare numbers. Using benchmark numbers is one very useful way. E.g. when comparing 20 to 200, relations involve 10 x, 1/10 of, 180 more, 180 less E.g. when comparing 1/8 to 4/7, one is about 1/10 ish and the other is about ½ ish

13 Part 1 Any calculation describes a variety of situations. E.g. What might be the problem you are solving by multiplying 25 x 350?

14 Part 1 Sometimes known measurements can be used to calculate unknown measurements. If you know the radius of a circle, what other measurements of the circle do you know?

15 Part 1 Composing and decomposing shapes or objects often helps us calculate their measures. How could breaking up this shape help you get its area?

16 Part 1 Algebra is a way to generalize number relationships. E.g. You choose two numbers. If you add them, the answer is triple the smallest number. What could the numbers be?

17 Part 1 You choose two numbers. If you add them, the answer is triple the smallest number. What could the numbers be? x + [] = 3x

18 Part 1 Any equation can represent many different situations. E.g. What might the equation 3x 2 = 10 represent?

19 Part 1 Probabilities are always predictions that can never be certain, since they are based on random events. E.g. You rolled a pair of die and got totals of 6, 10, 4, 12 and 2. What will happen next?

20 Part 1 There are also essential understandings- not quite as big, but important. For example: Every terminating decimal can be thought of as a whole number of tenths, hundredths, thousandths, etc. E.g is 234 thousandths

21 Part 1 We sometimes use letters (variables) to describe a quantity more generally but other times we use them as a placeholder to describe something we don t know yet. E.g. compare x in these two situations: 5 + x = 37 and x + x = 2x

22 Part 1 The big ideas and the essential understandings that emerge in each topic the curriculum requires are suggested to you.

23 Part 1 It is important to revisit strands and sometimes ideas or topics several times during a year. The proposed sequence does that. The reasons for the sequence are explained. There are other okay sequences, but the provided one does suit lots of kids.

24 Part 1 There is discussion of 3-part lessons, what they mean and how I think they need to look. Let s talk about one lesson as an example here.

25 So it might look like this Learning goal: I can describe different real-life situations an equation might describe.

26 Flyer $9 $20 $12 $6 13 $10

27 Main Problem Choose one, two or three items from the picture. Decide how many of each item you want to buy and calculate the total cost. Write an equation to describe what you just did. Use a different variable for the price of each type of item you choose to buy.

28 Main Problem For example, if you bought 2 boxes of markers and a ball, you would write: 2 x m + b = 21

29 Main Problem Have a classmate figure out which items you had in mind based on your equation.

30 Consolidation After posting some of the equations students created: Look at some of the equations your classmates wrote. How could you tell how many different types of items they bought? How could you tell how many of each type they bought?

31 Consolidation What part of the equation told you how much they spent? What operation signs were in your equations? Why those?

32 Part 2: Open Questions

33 Part 2 Open questions are a useful important way to differentiate instruction, as well as to make better and richer math talk.

34 Compare these two Question 1: Write 2/5 as a percent. Question 2: You have to write []/[] as a percent. Do you care more about the numerator or the denominator? What values would you like for the blanks?

35 Compare these two Question 1: Read this number: Question 2: You read a number and you say exactly four words. What might the number be if it s big? If it s little?

36 Compare these two Question 1: Estimate and calculate Question 2: You divide a 4-digit number by a 2-digit number. The answer has 2-digits. What numbers might you have divided?

37 Compare these two Question 1: What is the missing side length if the perimeter is 40 cm? 16 cm 14 cm

38 Compare these two Question 2: Draw different triangles with a perimeter of 40 cm where one side length is missing. Make the missing length sort of big.

39 Compare these two Question 1: What is the volume? 13 cm 6 cm 8 cm

40 Compare these two Question 2: Choose a length, width and height for a rectangular prism so the volume is a bit more than 600 cm 3.

41 Let s try Start with a regular question and change it. For example: An isosceles triangle has two equal angles of 40. What is the measure of the third angle? COULD BECOME: An isosceles triangle has one angle of 40. What could the other angles measure?

42 Let s try OR Choose the measure for the equal angles of an isosceles triangle. What could the other angle be? OR An isosceles triangle has two equal angles of less than 50. What could be the measure of the third angle?

43 Let s try Or you might start with: Solve a + 5 = 53 and change it to: E.g. You solve an equation by subtracting 5. What might the equation be? You solve an equation and the solution is 48. What might the equation be?

44 Let s try The solution to an equation is a little less than 50. What could the equation be? OR You solve an equation by subtracting. What could the equation be?

45 Maybe Two fractions with different numerators are really close together. What could they be? Which is more? Let s use our fraction strips.

46 Now it s your turn. Take some standard questions and open them up.

47 Part 3: Comprehension Questions

48 Part 3 Too often, we ask questions focused on connaissance and not compréhension. What is the difference and how could we do better?

49 Compare these questions Question 1: Put these fractions in order: 2/3, 4/7, 8/5, 9/2 and 2/9. Question 2: Without getting any decimal values or equivalent fractions, how do you know that 2/3 > 4/9?

50 Compare these questions Question 1: Plot the points (4,8) and (5, 10) on a grid. Question 2: You plot points on a coordinate grid. One point is WAY higher and a little to the right of (100,50). What could it be?

51 Compare these questions Question 1: Add Question 2: You add two decimal thousandth numbers. The answer is just a little bit more than the first number. What could the second number be?

52 Compare these questions Question 1: What is the area of a triangle if the base is 10 cm and the height is 8 cm? Question 2: A triangle and a parallelogram have the same area. If the triangle is just a little taller than the parallelogram, what else do you know about the shapes?

53 Compare these questions Question 1: What are the factors of 48? Question 2: You can make 8 different rectangles with a certain set of linking cubes. How many cubes might it be?

54 Or Question 1: What is the 100 th number of 4, 7, 10, 13,? Question 2: How does this picture help you figure out the terms in the pattern 4, 7, 10,? How does it help you get the 100th term?

55 Or

56 Compare these questions Question 1: What is the value of 4x 3 when x = 10? Question 2: The value of an algebraic expression is 37 when x = 10. What could the expression be?

57 It is important That you ask more compréhension questions of your students.

58 Part 4: Scaffolding

59 Part 4 When students struggle, we try to rescue them by leading them. Maybe this doesn t really help--- they learn to wait for help instead of trying to work things out themselves. They need help, but we need to help without telling them too much.

60 Part 4 For example, the exercise was: Put these fractions in order from least to greatest: 3/5 8/9 11/352 17/5 2/17 91/32 Someone writes: 2/17 3/5 8/9 11/352 17/5 91/32 What do you say?

61 Part 4 For example, the exercise was: You multiply a fraction by a whole number and the answer is close to 30, but a little less. The student says 3 x 2/5. What do you say?

62 Part 4 For example, the exercise was: Create a triangle with the same area as this parallelogram. The child draws: 5 cm 4 cm What do you say? 5 cm 8 cm

63 Part 4 For example, the problem was: The area of one parallelogram was reduced by 4 cm 2 when the base was decreased by 1 cm and the height increased by 1 cm. What were the two parallelograms bases and heights? A student does now know where to start. What do you say?

64 Part 5: À Pas de Géant

65 Part 5 Do you have questions?

66 Part 6: Your Questions

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