CCSS.Math.Content.HSF-LE.A.3
The standard
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
Common Core State Standards for Mathematics
What this standard means
Students need to compare growth patterns using tables and graphs, not just equations. They should see that repeated multiplication starts slowly, but later passes repeated addition or polynomial growth. They need to explain when one function is larger than another and use evidence from values or a graph.
Mastery looks like students making a table, graphing both functions, finding where the exponential model passes the other model, and describing the pattern in words. Students often get stuck because the exponential values may be smaller at first, so they assume it will stay that way. They also mix up constant difference and constant ratio.
Ways to teach it
- Have students graph y=2^x, y=10x, and y=x^2 on grid paper for x=0 to 10, then mark each crossing point.
- Ask students to write which job offer they would choose: $100 per day or $1 doubled daily for 10 days, with evidence.
- Give a table with one linear, one quadratic, and one exponential pattern, then ask students to identify which grows fastest by x=8.
- Compare viral video shares to saving a fixed amount weekly, using a spreadsheet to show when the share count passes the savings total.
Plan a lesson for CCSS.Math.Content.HSF-LE.A.3
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Related standards
- CCSS.Math.Content.HSF-LE.A.1a
Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
- CCSS.Math.Content.HSF-LE.A.1
Distinguish between situations that can be modeled with linear functions and with exponential functions.
- CCSS.Math.Content.8.F.B.5
Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or ...
- CCSS.Math.Content.HSF-LE.A.2
Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pa...