In the associated Algebra 1 lesson, students will need to be able to do some subtraction, multiplication, and distributive property with fractions in order to access the new learning about constructing an expression to represent exponential decay. For example, think about “The price of a new car is \$18,000 which depreciates by \(\frac13\) each year. What is its value after \(t\) years?” In order to do this, you need to know that decreasing 18,000 by \(\frac13\) (or \(18,\!000-\frac13 \boldcdot 18,\!000\) is the same as multiplying it by \(\frac23\) (or \(18,\!000\left(1-\frac13\right)\) which is \(18,\!000\left(\frac23\right)\)), and then that we are multiplying each new value by \(\frac23\) each year. So, the value after \(t\) years is \(18,\!000\left(\frac23\right)^t\).
The purpose of this support lesson is to remind students of what it means to subtract, multiply, and apply the distributive property when fractions are involved. In the first activity, students perform numerical calculations, and then encapsulate the process using a variable. This is an example of expressing regularity in repeated reasoning (MP8). In the second activity, students explain why some given pairs of expressions are equal, and then apply that reasoning to match some new pairs of expressions. This is an opportunity to notice and make use of structure (MP7).
- Use properties of operations to generate equivalent expressions.
- Let’s write equivalent expressions.