CCSS.Math.Content.HSF-LE.A.1a
The standard
Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
Common Core State Standards for Mathematics · Construct and compare linear, quadratic, and exponential models and solve problems
What this standard means
Students need to show why a straight-line pattern adds the same amount each time the input changes by the same amount. They also need to show why an exponential pattern multiplies by the same amount over equal input steps.
Mastery means students can use tables, graphs, equations, and words to justify the pattern, not just name it. They often mix up adding and multiplying, especially when numbers rise fast. They also get stuck when input intervals are not 1, such as comparing every 2 hours or every 5 days.
Ways to teach it
- Give pairs two tables, one linear and one exponential, and have them mark equal input jumps, then calculate differences and ratios.
- Ask students to explain which pattern fits a savings account with weekly deposits and which fits interest growth, using complete sentences.
- Show a four-row table and ask students to label it linear, exponential, or neither, then justify with differences or ratios.
- Use phone battery drain versus bacteria growth to compare a repeated subtraction pattern with a repeated multiplication pattern.
Plan a lesson for CCSS.Math.Content.HSF-LE.A.1a
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Related standards
- CCSS.Math.Content.HSF-LE.A.1c
Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
- CCSS.Math.Content.HSF-LE.A.1
Distinguish between situations that can be modeled with linear functions and with exponential functions.
- CCSS.Math.Content.HSF-LE.A.3
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) a...
- 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...