Some of the students at Kahlo Middle School like to ride their bikes to and from school. They always ride unless it rains.
Let \(d\) be the distance in miles from a student's home to the school. Write two different expressions that represent how far a student travels by bike in a four week period if there is one rainy day each week.
The distance to school, and therefore home, is \(d\). Thus, the student rides \((d + d)\) miles in one day. Equivalently, she rides \((2d)\) miles in one day.
Repeatedly adding the distance traveled in one day for each school day of the week, we find that in one week the student travels \((2d + 2d + 2d + 2d + 2d)\) miles. Equivalently, she travels \(5(2d)\) or \((10d)\) miles in a normal, rain free week.
We know that she travels \((10d)\) miles in a normal rain free week. In a 4 week period she would normally ride \((10d + 10d + 10d + 10d)\) miles, but we need to subtract the miles for the rainy days. For each rain day we have to subtract \(2d\) miles. Therefore, she traveled \((10d+10d+10d+10d−2d−2d−2d−2d)\) or \((10d+10d+10d+10d−(2d+2d+2d+2d))\). Equivalently we can write \(4(10d) − 4(2d) = (40d − 8d)\).
If we decide to combine the rainy day miles with the weekly miles traveled ahead of time then the expression for one school week with one rain day looks like \((10d − 2d)\) or \((8d)\) and the four week total is \((8d + 8d + 8d + 8d)\). Equivalently we can write \(4(8d)\).
The equivalent expressions will vary greatly. Comparing the cases above we see that \((40d − 8d)\) and \(4(8d)\) represent the same distance traveled and therefore are equivalent expressions.
Commentary
This task asks students to find equivalent expressions by visualizing a familiar activity involving distance. The given solution shows some possible equivalent expressions, but there are many variations possible.