Adjacent Borders, Countries with Most, All US States, Mean and Mean Absolute Deviation



How many US States are adjacent to Canada? How many US States are adjacent to Mexico? Which US State has the most adjacent US States? Which US State has the fewest adjacent US States? The answers were determined during class and are given on page two of the US States Adjacencies pdf.


In order to answer these questions, we first defined adjacency as “next to or adjoining something else.” Starting with the children, they had to name the adjacent child to where they were siting in the room. They all had two adjacencies; then I sat in the middle of the circle and the children could see that I now am adjacent to them all.


We then discussed strategic border protection for countries. Is a country more or less secure if it has a small number of adjacencies, a large number of adjacencies, or has natural borders such as mountains or bodies of water. We had a very thoughtful discussion where the children talked about friendly neighbors vs. enemies. They also talked about getting resources in and out of their country through other adjacent countries. Finally, they talked about the importance of having a port.


We expanded the definition of adjacent border to include only those borders which are shared; diagonal borders that only share a vertex are not considered adjacent (an example of this is Utah and New Mexico or Colorado and Arizona).


When we get into adjacent angles in geometry the definition requires having a common vertex and a common side. So the common vertex (corner) alone does not make them adjacent.


Before I gave the the US States challenge, where they would put the number of adjacencies in each state, I challenged them to find the number of adjacent countries in each of key countries world wide: China, Russia, Brazil, Democratic Republic of the Congo, Germany, Austria, France, and Serbia. The answers are on the US States pdf page 2 and at the bottom here. The may be disputes as to my answers and I would love the children to challenge. I did show them that Russia has a small land mass outside of its mainland called Kaliningrad that borders Poland and Lithuania. There are also several small countries that the children learned about including Liechtenstein, Luxembourg, Monaco, and Andorra.


I recommend that they check the answer key after completing all 50 states and correct their mistakes but redoing each state with an error. If they just erase their answer and replace the correct answer they learn nothing about “why.” I emphasized this during class.


I gave the children grades 3 and up a challenge to calculate the average (mean) adjacencies per US State [add the total of all 50 and divide by 50 = _______. They will get a fractional answer or a decimal. Answers are on page two of the pdf and at the bottom here. 


For grade 5, I gave the an additional challenge to calculate the Mean Absolute Deviation (MAD). This is very involved:


  1. Find the mean; 
  2. Find the absolute value of the difference between each data value and the mean: |data value – mean|;
  3. Find the sum of the absolute values of the differences;
  4. Divide the sum of the absolute values of the differences by the number of data values. = _______


The MAD helps determine whether the set's mean is a useful indicator of the values within the set. The larger the MAD, the less relevant is the mean as an indicator of the values within the set.













Mean Adjacencies = 4.2 or 4 1/5 for the 50 United States


MAD = 1.352 indicating not a lot of variability (Calculations on Pdf)

Countries_with_Most_Adjacencies.pdf3.98 MB
United_States_Adjacencies.pdf624.4 KB