The goal of this lesson is to encounter two different growth patterns—one pattern is linear and the other is exponential, though students don't need to use those words, yet. Students think about and compare the patterns by performing calculations and using graphs. This lesson contains many opportunities for students to notice and make use of structure (MP7), for example, noticing that \(1,000 + 200 + 200 + 200 + 200\) can be expressed as \(1,000 + 200 \boldcdot 4\), which is useful when they must add on more and more 200's. There is also an opportunity to use appropriate tools strategically (MP5), for example, if students choose to use a spreadsheet to perform many iterations of such calculations.
They see that the pattern which grows by repeatedly doubling starts off slowly but eventually overtakes the other pattern, which increases by repeatedly adding the same amount. In fact, the first pattern eventually leaves the second pattern far behind. Throughout the unit, students will study exponential patterns systematically (eventually viewing them as functions) before returning to this comparison with linear functions toward the end of the unit.
Some technology is required for this lesson, but there are opportunities for students to select appropriate technology in the lesson. We recommend making scientific calculators and spreadsheet technology available.
- Compare linear and exponential relationships by performing calculations and by interpreting graphs that show two growth patterns.
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Acquire scientific calculators or devices that can run GeoGebra (recommended) or other spreadsheet technology. It is ideal if each student has their own device. (A GeoGebra Spreadsheet is available under Math Tools.)
- I can compare growth patterns using calculations and graphs.