MA.912.AR.5.5

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

• MA.912.F.1.6
• MA.912.F.1.8 
• MA.912.FL.3.4 
• MA.912.DP.2.9

Terms from the K-12 Glossary

• Exponential function

Vertical Alignment

Previous Benchmarks

• MA.912.NSO.1.2
• MA.912.AR.5.3
• MA.912.AR.5.4
• MA.912.F.1.8

Next Benchmarks

• MA.912.AR.5.7
• MA.912.AR.5.8
• MA.912.AR.5.9

Purpose and Instructional Strategies

• Students may need support with using the properties of exponents fluently. 
• Instruction includes connecting prior knowledge of constant rate of change, including linear functions and making comparisons as noted in MA.912.F.1.6. 
• When generating equivalent expressions to reveal the constant percent rate of change per unit interval, students should be encouraged to approach from different entry points and discuss how they are different but equivalent strategies (MTR.2.1). 
• Instruction includes interpreting percentages of growth/decay from exponential functions and see that ?? can be used to determine a percentage. 
  • For example, the function $f$($x$) = 500(1.16)^$x$ represents 16% growth of an initial value. 
  • Guide students to discuss the meaning of the number 1.16 as a percent. They should understand it represents 116%. Taking 116% of an initial value increases the magnitude of that value. (Students can test this in a calculator to confirm.) Taking this percentage repetitively leads to exponential growth. 
  • For example, the function $f$($x$) = 500(0.72)^$x$ represents 28% decay of an initial value. 
    • Guide students to discuss the meaning of the number 0.72 as a percent. They should understand it represents 72%. Taking 72% of an initial value decreases the magnitude of that value. (Students can test this in a calculator to confirm.) Taking this percentage repetitively leads to exponential decay. 
  • For example, the function $f$($x$) = 500(1)^$x$ represents an initial value that neither grows nor decays as $x$ increases. 
    • Guide students to discuss the meaning of the number 1 when it comes to growth/decay factors. They should understand it represents 100%. Taking 100% of an initial value causes the value to remain the same. (Students can test this in a calculator to confirm.) Taking this percentage repetitively leads to no change in the initial value (explaining the horizontal line that shows when $b$ =1 on the graph). 
• Instruction includes real-world problems on compound interest and continuously compounded interest in connection with MA.912.FL.3.4.

Common Misconceptions or Errors

• Students may not have fully mastered the Laws of Exponents and understand the mathematical connections between the bases and the exponents. 
• Students may struggle with representing the growth factor as a percent, rather than a whole number. 
• Students may represent the growth rate or decay rate within the formula instead of using the formula $f$($x$) = $a$(1 ± $r$)$x$. To address this misconception, students should record the values being used to showcase the growth or decay to see their mistake.

Instructional Tasks

• Four physicists describe the amount of a radioactive substance in grams, $Q$, left after $t$ years below. 
  
  Function 1: $Q$ = 300$e$^−0.0577$t$
  Function 2: $Q$ = 300($\frac{\text{1}}{\text{2}}$) ^{$\frac{\text{<math><mi>t</mi></math>}}{\text{12}}$}
  Function 3: $Q$ = 300(0.9439)^$t$
  Function 4: $Q$ = 252.290(0.9439)^$t$−3
  • Part A. Compare the four different functions describing the radioactive substance. 
  • Part B. Determine whether the described functions are equivalent. 
  • Part C. What is the constant percent rate of change annually of the radioactive substance? 
  • Part D. Why do you think each of the four physicists described the amount of radioactive substance differently? What information does each tell you? 

• A fisherman illegally introduces some fish into a lake, and they quickly propagate. The growth of the population of this new species (within a period of a few years) is modeled by $P$($x$) = 5$b$^$x$, where $x$ is the time in weeks following the introduction and b is a positive unknown base. 
  • Part A. Exactly how many fish did the fisherman release into the lake? 
  • Part B. Find $b$ if you know the lake contains 33 fish after eight weeks. 
  • Part C. Instead, now suppose that $P$($x$)= 5$b$$x$ and $b$ = 2. What is the weekly percent growth rate in this case? What does this mean in everyday language? 
  • Part D. How does the weekly percent growth rate compare in Part B and Part C?

• The equation $y$ = 42,500(0.9198)^$x$ represents the value of a car $x$ years after its initial purchase. The average rate of depreciation for vehicles is often measured in 5-year intervals. Write an equivalent expression to show the constant percent rate of change over 5 years. 

• The function $V$($t$)= 250,000(1.038)^$t$ represents the value ($V$) of a home $t$ years after its initial purchase. Interpret the annual percent rate of change in this context.

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