## Reflection: Backwards Planning Identifying and Interpreting the Intercepts - Section 2: Another Problem: Scoring 30 Points

It often happens that my students will debate the "easiness" of one problem in relation to another.  They also debate the relative ease of different problem solving strategies or of algebraic methods vs. less formal methods.

Today when it came up, I allowed kids to spend a moment discussing which of these two problems was easier.  We all came to recognize that even though the date problem has thousands of solutions, it was easier than the basketball problem, which has just six.  So why was that?  I kind of hoped that this conversation would happen, and was glad that kids took the bait.  What would you say?  How would your kids answer this question?

I chose these two problems carefully, and deliberately presented them in this order.  Each problem has a discrete solution set, but the first set of data is (being informal here) "more continuous" than the other.  The first problem has a realistic output for every realistic input.  The second does not: for example, even though 5 3-point baskets would net me 15 points, there's no way to make another 15 points with a whole number of two point baskets.

Then there's the issue of slope.  For any amount of money that we take away from Xavier, we can just add that amount to Yadi's sum to stay in the solution set.  The same cannot be said for the 30 points problem.  If we look closely however, we'll see that two 3's can be replaced by three 2's.  How does slope help to express that?

Of course, each problem has a realistic pair of intercepts, because that's what the lesson was about.  I just don't hesitate to touch on all these other ideas, if they should come up.

Ease vs. Quantity
Backwards Planning: Ease vs. Quantity

# Identifying and Interpreting the Intercepts

Unit 7: Lines
Lesson 7 of 10

## Big Idea: The idea of an x-intercept sounds like a fiction to a lot of students, because it just makes too much SENSE! Here's a chance for students to see that such common sense abounds in algebra.

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Standards:
Subject(s):
Math, Algebra, Graphing (Algebra), linear functions, Linear and Nonlinear Equations, modeling, linear combinations, standard form of a linear equation, task selection and sequencing, backwards planning
43 minutes

### James Dunseith

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