# 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 $2\frac14$.

## Traffic Jam

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.