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Asking the right questions

Asking the right questions. How to reach every student! November, 2007. What do we know about questioning in classrooms?. There are lots of teacher questions. It’s estimated they typically take up 30% or more of instructional time. Used primarily to check understanding

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Asking the right questions

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  1. Asking the right questions How to reach every student! November, 2007

  2. What do we know about questioning in classrooms? • There are lots of teacher questions. It’s estimated they typically take up 30% or more of instructional time. • Used primarily to check understanding • Primarily convergent and low level • They are often “rapid-fire”.

  3. Why would that be? • Maybe it is a management technique. • Maybe it is a way to ensure that the conversation doesn’t lead to misinformation. • Maybe it is because that’s all teachers are used to seeing.

  4. Some of the issues • We don’t pre-plan our questions. • We check rather than initiate. • We focus on rules, rather than underlying concepts.

  5. Some of the issues • We focus on details, rather than the big picture. • We over-scaffold. • We “bully”.

  6. Some of the issues • We don’t always make sure the question is broad enough to allow multiple entry points.

  7. Think about the difference between these • You saved $6.30 on a pair of jeans during a 15% off sale.How much did you pay for the jeans? OR • You saved $6.30 on a pair of jeans during a sale. What might the percent discount have been? How much might the jeans have originally cost?

  8. Possible discounts and payments • 15% off and a payment of $35.70 • 25% off and a payment of $18.50 • 10% off and a payment of $56.70 • 12% off and a payment of $46.20

  9. And even more open • You saved some money on a sale of jeans. • Choose your savings: $5, $7.50 or $8.20 and choose a percent as the discount. • What would the original price have been? • How much would you have paid?

  10. Possible solutions • Choose a $5 savings and a 20% discount. You would have paid $20 instead of $25. • Choose an $8.20 savings and a 10% discount. You would have paid $73.60 instead of $82 (very pricey jeans). • Choose a $5 savings and a 12% discount. You would have paid $36.67 instead of $41.67.

  11. Think about the difference between these • What is (-23)2 - (-22)3? OR • Write the number 128 as the sum or difference of powers of negative integers.

  12. 128 is • (-23)2 - (-22)3 • (-1)3- (-5)3 - (-2)2 - (-2)3 • [-8]2 - [-82] • (-10)2 + [(-2)2]3 - (-6)2 • - [ (-13) + (-13) + (-13) +….. + (-13) ]

  13. How might you open these up to a broader audience? • What is the sin of this angle ? • What is 1 ÷ 5–3? • What is the sum of the interior angle measures of a 5-sided polygon?

  14. We asked teachers: What sort of question would you choose to start a lesson? • Determining prerequisite knowledge • Determining if students already know what you’re planning to teach • Piquing curiosity (hooking them in)

  15. Starting a lesson • On the next slide I’ll list some questions that could be used to start a lesson on rate in grade 9 applied. • What would (or wouldn’t) each accomplish and which would you value most?

  16. Your choice • Write 3 sets of equivalent fractions for each: 3/4, 25/6, and 120/3. • Six cookies cost $3.99 at a bakery. How much would the bakery charge for 8 cookies? • The Olympic record for the men’s 100m butterfly is 51.25 s. The women’s 100m butterfly record is 56.61 s. What would you predict for the two records for the 200 m butterfly?

  17. And with a new topic… • On the next slide I’ll list some questions that could be used to start a lesson on adding and subtracting polynomials in grade 9. • What would (or wouldn’t) each accomplish and which would you value most?

  18. Your choice • Show me 3x2 + 2x + 1 and 2x2 + 3x +2 with your algebra tiles. • Here are some algebra tiles. How might you subtract 2x2 + x – 2 from 6x2 + x + 2? • When you calculate 3x2 + 2x + 1 – (2x2 – 3x +1), you start with an expression you can model with 6 tiles, subtract an expression you can model with 6 tiles, and end up with an expression you can model with 6 tiles. Does that usually happen?

  19. How can we use questions to focus on the important ideas?

  20. Linear Relations Consider 2x + 4 and 4x + 2 • For how many values are the expressions worth the same? How do you know? • Why is the value for which they are worth the same not far from 0? • What is that value? Why? • How do you know that both of these relations are linear?

  21. Exponential functions

  22. What about dividing rational numbers? • What do you think is the most important idea about dividing rational numbers? • What would you ask to get there?

  23. Questions to End the Lesson • The last thing you hear often sticks with you. • How could you end the class on dividing rational numbers we just talked about?

  24. How about these? • When you divide two rationals, how can you predict whether the answer will be positive or negative? How can you decide whether it will be greater than 1 or less than – 1? • Some people divide by a rational by multiplying by its reciprocal? Why does that make sense? What would you ask instead?

  25. How would you end these lessons? • A grade 9 applied lesson on the volume of pyramids • A grade 9 academic lesson on the meaning of m in y = mx + b • A grade 10 lesson introducing sine and cosine • A grade 11 university prep functions course lesson on annuities • A grade 12 data management lesson on combinations and permutations

  26. Creating engaging questions/tasks • Using interesting contexts Possible sources: • The student’s personal world • Facts and figures

  27. Make it personal • You are buying something on-line that costs $39 U.S. How much will you pay in Canadian dollars? • How can you use trig to figure out the height of the tree outside our classroom? • Estimate the number of meals you have eaten in your life. Use two different radical expressions to name that number.

  28. Make it personal • What is a fair price for car insurance for a 16 year old male? • You and three friends line up for a photo. What is the probability that you and your best friend end up standing next to each other? • Make up an arithmetic sequence where the 8th term is the sum of the ages of everyone who lives in your house.

  29. People and places The revolving restaurant in the CN tower completes 5/6 of a revolution every hour. If your dinner takes 2.5 hours, through how many radians have you rotated?

  30. People facts • Most people lose about 80 scalp hairs each day. • How long would it take to lose 1012 hairs?

  31. Records • The record for a person with the longest hair is a Chinese woman whose hair is 5.627m long. It took her 31 years, beginning at age 13, to grow it that long. • How many centimetres would her hair grow each day?

  32. Natural phenomena • Did you know that if you pour gravel into a pile, the shape will form a cylindrical cone with a slope of about 30°? • Suppose there is room for a pile that is 90 m wide. About how tall could the pile be? About how much gravel could be in the pile?

  33. For the curious

  34. For the curious

  35. For those with mathematical curiosity • Choose 3 consecutive numbers, square them and add. Divide by 3 and calculate the remainder. What happens? Why?

  36. Making connections • Consider the expressions x2 + 2x + 1. • Evaluate it for different values of x. • What do you notice? • What does it tell you about the expression?

  37. x2 + 2x + 1

  38. 2x2 - x - 1

  39. Questions for practice

  40. Inequalities • The common solution to both inequalities is x > 3. What could the values for the coefficients and constants be? ax + b > c dx2 + ex < f

  41. Possible solutions • x > 3 and – x2 < – 9 • 2x > 6 and 5 – x2 < – 4 • 6x + 7> 25 and – 2x2 – 6x < – 36

  42. Slope Place the digits 0-9 into the right spots. • A line with slope []/[] goes through (9,[]) and ([],1) • A line with slope 3/4 goes through ([],2) and ([],[]) • A line with slope 5/7 goes through ([],6) and ([],[])

  43. Slope Place the digits 0-9 into the right spots. • A line with slope 7/3 goes through (9,8) and (6,1) • A line with slope 3/4 goes through (0,2) and (4,5) • A line with slope 5/7 goes through (9,6) and (2,1)

  44. Creating questions

  45. Turn-it-around • One side of a right triangle is 5 cm long. What could the other side lengths be?

  46. Possibilities • 3 and 4 • 5 and 5√2 • 12 and 13

  47. Use blanks • The tenth term of an arithmetic sequence is 6[] (between 60 and 70). • What could the sequence be?

  48. Possible sequences • 55, 56, 57, 58, 59, 60, 61, 62, 63, 64 • 46, 48, 50, 52, 54, 56, 58, 60, 62, 64 • 37, 40, 43, 46, 49, 52, 55, 58, 61, 64 • 28, 32, 36, 40, 44, 48, 52, 56, 60, 64 • 19, 24, 29, 34, 39, 44, 49, 54, 59, 64 • 10, 16, 22, 28, 34, 40, 46, 52, 58, 64

  49. Relationships • The graph of y = a sin (k(x - d)) + c goes through (180°,9). What are possible values of a, k, d, and c?

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