Traffic Jam

Task

You are stuck in a big traffic jam on the freeway and you are wondering how long it will take to get to the next exit, which is $1 \frac12$ miles away. You are timing your progress and find that you can travel $\frac23$ of a mile in one hour. If you continue to make progress at this rate, how long will it be until you reach the exit? Solve the problem with a diagram and explain your answer.

IM Commentary

It is much easier to visualize division of fraction problems with contexts where the quantities involved are continuous. It makes sense to talk about a fraction of an hour. The context suggests a linear diagram, so this is a good opportunity for students to draw a number line or a double number line to solve the problem. Linker cubes are also an appropriate tool to solve this problem. The linker cube solution suggests an algorithm for dividing fractions using a common denominator. The context of this problem would also work in the case where the dividend is smaller than the divisor, e.g. $\frac14 \div \frac23$.

Solutions

Solution: Number Line

Using a double number line where one line is measured in miles and the other one is measure in hours we get the following diagram.

In order to measure both $\frac12$ miles and $\frac13$ miles, we divide the 1 mile into $\frac16$ mile pieces. This way we can find $1 \frac12$ miles and $\frac23$ miles. Driving two $\frac23$ mile stretches takes two hours. That leaves $\frac16$ mile, which will take $\frac14$ hour to drive. Therefore, it takes $2 \frac14$ hours to drive $1 \frac12$ miles.

Since we are asking "How many $\frac23$ are there in $1\frac12$?" this is a "How many groups?" division problem: $$1\frac12\div\frac23 = ?$$ We have found that the answer to this division problem is $2\frac14$.

Solution: Linker Cubes

Using linker cubes we need at least 6 linker cubes to represent 1 mile in order to divide the mile into thirds and halves at the same time. (Note: Any multiple of 6 would also work.) So $1 \frac12$ miles is represented by 9 linker cubes and $\frac23$ of a mile is represented by 4 linker cubes. Now the question becomes: How many times do the 4 linker cubes ($\frac23$ mile) fit into the 9 linker cubes ($1 \frac12$ miles)? The answer is $1+1+\frac14 = 2 \frac14$. (see photo below)

Since we are asking "How many $\frac23$ are there in $1\frac12$?" this is a "How many groups?" division problem: $$1\frac12\div\frac23 = ?$$ We have found that the answer to this division problem is $2\frac14$.

Note: This problem could lead into a discovery of a “common denominator” procedure for dividing fractions: Find a common denominator of both fractions, then just divide the numerators: $1 \frac12 \div \frac23 = \frac96 \div \frac46 = 9 (\frac16) \div 4 (\frac16) = 9 \div 4 = \frac94 = 2 \frac14$.

Solution: Number line solution

Since $1\frac12=\frac96$ and it takes an hour to travel $\frac23 = \frac46$ miles, we can look at the number lines above and see that it will take $2\frac14$ hours to travel the distance to the exit.

Since we are asking "How many $\frac23$ are there in $1\frac12$?" this is a "How many groups?" division problem: $$1\frac12\div\frac23 = ?$$ We have found that the answer to this division problem is $2\frac14$.