Distance to School

Task

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.

IM 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.

Solution

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.

Expression 1

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 $4(10d) − 4(2d) = (40d − 8d)$.

Expression 2

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.