TEACHER NOTES Problem-solving objective
To use spatial visualisation, logical reasoning and measurement to solve problems.
Materials
counters in different colours, clocks
Focus
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These pages explore different ways of visualising the problems and analyse different possible solutions. Logical reasoning is needed, as well as an understanding of measurement (length, perimeter, area and direction). For each question, materials, diagrams or tables can be used to organise, sort and explore the data.
Discussion
Page 53 Using a diagram or counters to model the first two problems will help students understand the different pieces of information that the problems present. For Problem 1, Kevin needs to walk as far as the second, fourth, sixth (and so on) tree and back to the start. However, he can finish watering at the last tree and does not need to return to the tap. (Next time he can do the same in reverse, starting from the last tree.) Clearly, the answer to the second problem is not that the cyclists returned at the same time! A diagram or use of counters sorts out the difficulty posed by the problem. In the last problem, the information about ascent and descent of the balloon is not relevant. Rather, students need to understand that the path traces a rectangle because the directions are at right angles to each other and show that the final point is one side of the rectangle or 750 m from her starting point.
Possible difficulties
• Unable to visualise the paths taken by the person watering the garden, cyclists or balloonist • Using only 12-hour and not 24-hour time in considering the correct time • Thinking that the hands on a clock always line up on the hour • Including 12 o’clock as a time when the hands line up on a clock • Unable to draw or interpret diagrams to see the relationships among perimeter, side-length and area
Extension
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Teac he r
Page 55 The puzzle scrolls contain a number of different problems, all involving strategic thinking to find possible solutions. In most cases, students will find diagrams and lists are needed to manage the data while exploring the different possibilities. Concepts of space and measurement are explored in each scroll and students may use a number of different ways to find possible solutions, including the ‘try and adjust’ strategy.
could include more trees, less water per tree and © R. I . C.Pub l i ca t i on s hence more trees per bucket. • The cyclists could go from the clubhouse to the bay, •f orr evi ew pu r posesonl y• then go back to the clubhouse and finally return to the
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bay. • Students could write more complex problems involving balloons; clocks which lose or gain time; and area, side length and perimeters for a pentagon, hexagon or other polygon.
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Page 54 Each of these problems requires a good understanding of time, the hours and minutes in a day and how time is shown on a clock. Access to a clock with moveable hands will help students understand what happens when time is gained or lost on a clock, as well as the way an analog clock shows time over a 24 hour period. Using a list or table will help students keep track of the changes and possibilities.
• The problem involving watering trees from a bucket
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Problem-solving in mathematics
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