Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

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Patterns, Functions, and Algebra
Session 5 Part A Part B Part C Part D Part E Homework
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Notes for Session 5, Part C

Note 11

The exercises in this part of the session link the concept of direct variation, covered in Session 4, to the idea of problem solving, which will be covered in Session 6.

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Note 12

Groups: Work in pairs on Problems C1 and C2.

Consider sketching the situations or drawing tables to help come up with the equations for Achilles and the tortoise. After generating graphs for these situations, compare the graphs with the toothpick graph developed in Part A. In the case of Achilles and the tortoise, non-integer values make sense, so there is no problem with drawing connected lines between points.

The graph should look like this:

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Note 13

Problem C6 addresses the idea of parallel lines. You might want to refer to the graphs drawn in Part B. Ask if lines with the same slope ever intersect.

Students and teachers often miss the essential connection between solving linear equations and finding the intersection of lines. The intersection of two lines happens at an (x, y) pair that satisfies both linear equations. At this point in each equation, the xs are equal and the ys are equal. To find where this happens, set the ys equal to each other and solve each equation for x.

When trying to solve an equation like 5x = 5x + 25 by the usual method of "doing the same thing to both sides," students end up with the equation 0 = 25 and don't know how to interpret it. The point is that the two lines are parallel, so they never intersect, which means there is no solution to the equation. And the equation 0 = 25 is never true. Solving equations is covered more completely in the next session, but if there is time, think about some of these ideas now.

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