Chukat- Two Approaches to Distance Traveled as Related to Rate of Speed

This week I would like to follow-up on my post from Parshat Chukat last year. In last year's post, I looked at the distance that the Children of Israel traveled in Bamidbar Chapters 20-21. Using an estimated route, I calculated that they traveled approximately 312.5 miles in this final section of their wandering in the desert.

"The Children of Israel, the entire assembly, arrived at the Wilderness of Zin in the first month and the people settled in Kadesh. Miriam died there and she was buried there." ~Numbers 20:1

From Midrash, we learn that these two chapters take place in the final year of the wandering. We learn in this first statement that we are beginning this section from the first month of this final year of wandering. Throughout the course of these 2 chapters, we have Miriam's death with 30 days of mourning and Aaron's death with 30 days of mourning. This means that out of the final full year of wandering, the Children of Israel had at most 10 months of travel time. While we know that there were periods of traveling and periods of encampment, we can still calculate an average rate of travel in terms of miles traveled per month. We could break it down further to calculate average miles traveled per day or even per hour. Let's remember that when you look at an average rate, it doesn't mean that you are traveling consistently over the whole time period, but if you broke up your entire travel so that you were traveling equal distances each hour, day, week, or month, this is how much you would have traveled in that space of time.

Back in Parshat Vayetzei, I first looked at Rate of Speed, where you express how fast or slow something or someones covers a distance by comparing the distance covered to the amount of time that it takes to cover the distance. This comparison is expressed as a ratio. So, if I travel 30 miles over the course of 1 hour, then my rate of speed would be 30 miles per hour (30mi/1hr or 30mph). If I travel 100 miles in 2 hours, then my rate could be expressed as 100mi/2hr or reduced as 50mph (since 100/2 reduces to 50/1).

Using this understanding of rate of speed, how could we look at the travel speed of the Children of Israel during the end of their wandering in the desert?

Rate of travel per month: 

Based on the estimates that I calculated last year, the distance traveled was 312.5 miles, and using my timeline logic above, they had 10 months of potential travel time.

Therefore, the rate per month would have been: 

312.5mi/10mo

This reduces to an average of 31.25mi/1mo or 31.25 miles per month

Follow-up:

Following this thinking-

*you can compare how many days would have been in the 10 months of travel time, and then calculate the estimated rate of miles per day

*how many hours would have been in those 10 months? what would the average rate of miles per hour have been?

*would the miles per hour calculation be a realistic estimate? Think about what this would mean for travel time and rest time during each day or week. How might you adjust this calculation to account for resting time each day (don't forget stopping for Shabbat)? How might you adjust the original calculations to allow for estimated periods of encampment based on the timeline that we're given in the parsha?

An alternate approach:
Looking at it from a different angle, if we estimate a reasonable rate of speed for their travel, how many hours/days/weeks/months would they have needed to be traveling and how much time would they have had for their periods of encampment?

*What does this mean? What would be a reasonable estimate for the amount of distance they were able to travel in an hour? a day? a week? a month? By quantifying some of these rates, how might their travel have lined up with the information that is provided in the parsha? How long would it have taken them to get between their different stops where they requested and were denied permission to cross through other lands? How might you create an estimated timeline of their travel using all of this information?

*Personal connection: Think about it similar to planning a road trip- how much mileage do you have to cover? What stops or activity locations might you be planning, and what are the distances between the stops? What is a reasonable rate of speed for your travel? How long will it take you between each stop? Can you fit your trip within a given timeframe or do you need more time to reasonably make your full trip?

Note: It is important to remember that this information is all hypothetical, since we don't have day by day or month by month itemizations of travel time or encampment periods. Additionally, my calculations from last year were merely estimates as well. That being said, thinking about the situation within this context can help students create a semblance of understanding around what the Children of Israel might have actually been going through in their travels- offering a realistic link to this "story".

Chukat- Measuring Using Scale on a Map to Compare Distance

In this week's parsha, through chapters 20 and 21, the Jews are trying to make there way through the desert up to Israel. Along the way, they make requests to Edom, Amor, and Moab to be able to cut through their land, but their requests are denied. They are left with no choice but to travel around the outer perimeter of the Edomite, Amorite, and Moabite lands, where they are eventually able to overcome the Amorites and they end up settling in the land of the Amorites and the Moabites (which was overtaken by the Amorites). By looking at a map of their route, we can see that it appears as though their route took them much longer than if they had been able to travel in a straight path up from the Red Sea up to the Dead Sea. Is this accurate? How much farther did they travel due to their not being allowed to peacefully travel through these other lands?

We know that the Jews were at Mount Hor, where we learn of Aaron's death (Numbers 20;24-29). Let's use this stop at Mount Hor as a starting point for our calculations. So, to clarify exactly what we're investigating in order to answer our questions:

1) What is the distance between Mount Hor and Jericho (the city where Joshua first brought the Jews when they entered Israel)? I should note here that if the Jews took a more straightforward route, it's possible that their point of entry would not have been Jericho. For the purpose of calculation and comparison, however, it seemed the most logical point to use in order to maintain the same start point and end point.

2) What is the distance that was traveled instead, by going around these other nations? (For consistency, we will base our answer on the estimated route given on the map linked above.)

3) What is the difference between the distance of the straightforward route and the distance of the circuitous route?

The map that we're using conveniently gives us a scale to use to estimate the mileage. Using this scale, we can compare the distances.

Answer 1: If we draw a straight line between Mount Hor and Jericho and then measure this distance, we measure a distance of just less than 125 miles.

Answer 2: If we measure the combined distances along the estimated route given on the map, we measure a distance of approximately 312.5 miles. 

Answer 3: To find the difference between the two distances, we take the longer distance (#2) and subtract from it the shorter distance (#1) to find how far the extra distance traveled was. 
312.5 - 125 = 187.5
So, from our calculations, the Jews traveled an extra distance of 187.5 miles due to the circuitous route that they needed to take. Another way to look at this is that they traveled more than twice the distance that they actually needed to in order to reach their destination.

Mathematical Connection:
Using a map with scale measurements to measure distances allows students an opportunity to practice many important math skills which can be differentiated based on level:
*measuring with a ruler (either using a ruler to measure the distance and comparing it to the scale given, or using the scale measurement as a ruler replacement)
*using scale factor (a standard measure = a represented distance on the map)
*adding measured distances together- finding a successful technique for accurately measuring distances that may or may not be in a straight line (using string on the map to measure, marking locations on a marking paper to add distances, measuring with a ruler and adding measures together, etc.)

Additionally, such an activity also offers integration with and reinforcement of Geography skills.

Everyday Connection:
Have you ever needed to compare driving routes- whether just trying to get around town or when planning a road trip? Don't just rely on an app or GPS to tell you where to go. Check it out for yourself. Is your GPS really giving you the shortest route? (Of course, you'd have to take into account the speed limits on different roadways and other road restrictions that might keep you from really taking the most ideal route just based on looking at a map). How much farther do you need to travel if you choose to pass by a specific landmark or stop along the way?