In this chapter, the authors encourage teachers to optimize the use of their textbooks by opening questions up for students. They identify four ways to do so:
1) Change the existing question - in particular, adding or omitting a bit to the question to make it more open ended to encourage students to explore the material further. They give many examples such as instead of asking students to plot 3 points on a coordinate grid and decide if it is an isoceles triangle, ask students to plot 6 points, and determine all the isoceles triangles they can.
2) Give the answer rather than the question - for example, tell students the hypotenuse of a right angle triangle is 17, and ask them to list as many triangles as they can.
3) Change the resources - Allowing, or not allowing technology changes the scope of the question. For example, coming up with fractions equal to 1/3 with or without a calculator.
4) Change the format - ask questions in different ways. The authors provide a whole page of different ways to express an equation like 4x+2=10.
These methods do not seem particularly innovative to me. Just about every textbook I use applies all of these quite regularly, so I'm not sure what new questions I can offer by analyzing three questions, but I will try...
I choose to work with the Grade 10 McGraw Hill Ryerson text I currently use.
Question 1: Earth has a diameter of approximately 8000 mi. Land forms 29% of the surface of the Earth. Assume Earth is a sphere. Estimate the area of the land on earth.
This question could be opened up by giving different aspects of the information - for example stating the area of the land on earth and the diameter and asking for the percent covered. Alternately, students could be asked to compare their result (assuming Earth is spherical) and research how far off they are from the reality, given that Earth is not an exact sphere.
Question 2: Evaluate without a calculator: (3^(1/6))(3^(5/6))
Students could be asked to follow up by listing a few other rational powers of 3 they could multiply (3^(1/6) by that they would be able to evaluate without a calculator.
Question 3: A circus recently had a sold-out performance. There were varying admission prices. The admission for premium seating was $250 for adults and $175 for students. The total revenue for premium seating was $29 125. The receipts showed that 130 premium seats were sold. Determine how many adults and how many students were in premium seats.
Students could be asked if they could find any other combination of adult and student tickets that had the same revenue, or they could be asked to model the situation using graphing technology, or they could be asked to come up with another scenario that could also be modelled with the equations they came up with for this question.
David, I don't teach high-school, so I appreciate the "real" questions with which I can practice the What-if-not approach from my reading with. Question 1 - first, students will break down the attributes into a list: surface area calculation; sphere; known diameter, etc. The teacher can then ask them to challenge a facet of the problem - how would the surface area of land on the earth change if it were growing like a balloon being blown up? Would the ratio of land to water stay the same? According to author of my readings, this method would open up possibilities for more discussion in the classroom. At the same time, I'm concerned with the challenge for students who are slow in adapting to the new approach. What do you think?
ReplyDeleteDavid, I don't teach high-school, so I appreciate the "real" questions with which I can practice the What-if-not approach from my reading with. Question 1 - first, students will break down the attributes into a list: surface area calculation; sphere; known diameter, etc. The teacher can then ask them to challenge a facet of the problem - how would the surface area of land on the earth change if it were growing like a balloon being blown up? Would the ratio of land to water stay the same? According to author of my readings, this method would open up possibilities for more discussion in the classroom. At the same time, I'm concerned with the challenge for students who are slow in adapting to the new approach. What do you think?
ReplyDeleteThanks David for the examples! For me, the second way: giving the answer rather than the question, is a useful strategy to learn. However, I'm wondering if this open-ended approach may be used well in all topics. Will there be scenarios where there is greater value with just one answer, rather than have students come up with multiple possibilities which may confuse rather than clarify?
ReplyDeleteFor question 3, two possible alternative ways that could vary the question could be to:
(i) vary the number of seats available since it's a sold-out performance, by having a larger venue; or even varying the ratio between the premium and ordinary seats. The impact on total revenue could be looked at.
(ii) bring in costs, and say the company still wants to get at least profits of $xxx (adding a constraint), and ask for possible combinations of ticket prices for 'adult' and 'children' seats. while holding ticket prices constant, students could also examine the ratio/breakdown between 'adult' and 'children' seats.
I like these suggestions very much!
ReplyDelete