Comparing Quadratic and Exponential Functions

5 min

Narrative

In this Warm-up, students compare the values of exponential expressions, by making use of their structure (MP7). The reasoning here prepares them to think about exponential growth later in the lesson.

Students should recognize that 92<1029^2 < 10^2 and 29<2102^9 < 2^{10}. Deciding whether 10210^2 or 292^9 is greater requires some estimation or further reasoning using properties of exponents.

For example, some students may recognize that 24=162^4 = 16 and 28=24⋅24=(24)22^8 = 2^4 \boldcdot 2^4 = \left(2^4\right)^2, so 28=1622^8 = 16^2, which is 256. Because 292^9 is greater than 282^8, it follows that 292^9 is greater than 256 and therefore greater than 10210^2.

As students discuss their thinking, listen for strategies that involve using properties of exponents or thinking about the structure of the expressions.

Launch

Arrange students in groups of 2. Give students a moment of quiet think time and then time to share their thinking with a partner.

In order to encourage students to rely on the structure of the expressions, they should not use a calculator to evaluate the expressions in this activity.

Student Task

List these quantities in order, from least to greatest, without evaluating each expression. Be prepared to explain your reasoning.

2102^{10}

10210^2

292^9

929^2

 

Sample Response

92,102,29,2109^2, 10^2, 2^9, 2^{10}

Activity Synthesis (Teacher Notes)

Select students to share their responses and reasoning. Highlight explanations that show that the expressions can be compared by analyzing their structure (as in the example in the Activity Narrative), and that it is not necessary to know their exact values to put the expressions in order.

Standards
Building On
  • 6.EE.1·Write and evaluate numerical expressions involving whole-number exponents.
  • 6.EE.A.1·Write and evaluate numerical expressions involving whole-number exponents.
Building Toward
  • F-LE.3·Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
  • F-LE.3·Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
  • F-LE.3·Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
  • F-LE.3·Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
  • HSF-LE.A.3·Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

15 min

15 min