### The Situation

You room trying to buy a toy that is sold out in all of your local toy stores. You decide to usage the playthings R us website’s store locator to find the nearest keep location.

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### The Challenge(s)

Which stores are 5 miles or much less from you?Which stores room 10 miles or less from you?Which stores space 15 mile or less from you?Which stores space 25 miles or less from you?Which stores are 50 miles or much less from you?

These questions might be advantageous in help students down the problem solving path:

How go the toys R Us keep locator recognize how much away a store is?What are means of identify a place’s place on Earth?How can we determine the distance in between two places on Earth?What does the amount represent?

### Consider This

Here is a an easy overview for exactly how store locators work:

Customer enters their place as “(Street Address, City, State or ZIP Code)”Customer enters a “Search radius” of 5, 10, 15, 25, 50, or more miles.The website converts the customer’s location into GPS collaborates in (x, y) style where x is the latitude and y is the longitude.The website looks in ~ its database of keep locations and finds their general practitioners coordinates, additionally in in (x, y) style where x is the latitude and also y is the longitude.The website uses a formula (more on this later) to identify the distance between the GPS collaborates for the customer’s location and also the store’s location.Any stores that room within the find radius (i.e, “less than 5 miles away”) are displayed to the customer

After pointing out “The Situation” and also “The Challenges”, the way I intend this lesson come go would be to pose the question “How walk the playthings R Us store locator identify how far away a keep is?” college student will more than likely say something choose they take the 2 locations, watch them increase on a map, and also find out how far apart castle are. Have students check out how this would actually work and let them number out what details they need regarding where the customers and also stores space located.

I imagine students will ask for the street addresses. However, unless the students have outstanding geographical knowledge of Phoenix, Arizona, the addresses through themselves need to be useless come them without:

a paper map (which I would certainly not provide them as computers can’t use record maps)an online map (which i would likewise not give them as we are trying to discover how the store locator’s online map works fairly than actually usage them)

What we want students to establish is the GPS collaborates are the means to go. If students are not already suggesting that we use general practitioners coordinates, it would certainly be a good time come ask them, “What space other methods of identifying a place’s ar on Earth?”

Regarding gps coordinates, they can be written in three straightforward forms. Wikipedia list the forms as:

Coordinate containing degrees (integer), minute (integer), and seconds (integer, or genuine number) (DMS).Coordinate containing degrees (integer) and also minutes (real number) (MinDec).Coordinate comprise only levels (real number) (DegDec).

We desire to use choice three so the coordinates contain “only degrees” in a form like (40.446195, -79.948862). If students wonder what the an adverse and confident values mean, they assist define the Earth’s hemisphere (though the x and also y values room swapped). A an unfavorable latitude (x-value) means the southern Hemisphere if a optimistic latitude way the north Hemisphere. Similarly, a an unfavorable longitude way the western Hemisphere when a positive longitude method the eastern Hemisphere.

The display shot below is native a spreadsheet consisted of in the “Download files” attach that has actually the shopper’s info and every one of the stores’ general practitioners information. When they ask because that the general practitioners information, this is what friend can give them. I have mixed increase the stores’ order for this reason they room not noted from closest come farthest.

The following step is guiding them to realize exactly how to find the distance in between the gps coordinates. Looking at the data and how the selection between the GPS collaborates is much less than one degree, student may think that all of the stores are close together also though the street addresses are in different cities or zip codes. Eventually we desire students come realize the the latitude and also longitude, as an ordered pair, stand for a allude in a coordinate mechanism and shift the conversation to how to uncover the distance in between two clues in a coordinate system. This is a good place to rest from the problem, learn more about recipe to discover the distance between points, and then return to the problem to apply the new knowledge.

Regarding the formula used to determine the distance between the points in a name: coordinates system, there room two options:

The Pythagorean Theorem can be offered to find the distances between two points in a coordinate system. This will carry out a reasonably great approximation because that how far apart castle are and is what we use in this lesson. The problem originates from the fact that the earth is not flat and that we space actually recognize the distance in between two point out on a sphere.

Once students have actually learned just how to use the Pythagorean to organize to discover the distance between two point out in a coordinate system, the following step is to uncover the distance in between each of the stores and also “Your Address”. Below is an example of just how I would suppose it to look:

25 southern Arizona Place, Chandler AZ 85225(33.303385, -111.840829)

Chandler Store:

940 north 54th Street Chandler AZ 85226(33.316991, -111.968482)

Distance = √( (33.303385 – 33.316991)² + (-111.840829 – -111.968482)² ) ≈ 0.128376055575013

Make sure to ask college student to define what that number represents. College student may have no context because that what that number is. Is that the distance? If it is, what room the units? it is a great opportunity to exercise both Math exercise 2 (“attending come the an interpretation of quantities, no just just how to compute them” and also Math exercise 6 (“they room careful around specifying units of measure”) among others. Together it transforms out, that is the distance in between the 2 points measured in degrees. So the store is around 0.128 degrees away.

To price “The Challenge” questions, college student will require to transform it from levels to miles by multiplying by 69.11 as each each level is around 69.11 miles. As a result, student should end up through answers close come the people in the “Pythag Dist. In Miles” in the table below (click the image for a bigger size or download the spreadsheet):

With students having the distances in hand, have them list which shop should show up on the “Store Locator” search results page for 5, 10, 15, 25, and also 50 miles or much less away. Have actually them share what they mean to see on the really search results page, then display it to them and also discuss. Listed below are the yes, really search outcomes from the toys R Us keep Locator page along with my thoughts:

5 Mile Search outcomes – page 1

Students must expect no places to be provided on this results page. However, one save is detailed but is also provided as gift “Approx: 9 Miles” away. The is worth stating whether this need to be listed. Probably the keep wants to display at the very least one an outcome at all times therefore they appear to have a local presence. However, together we will view on the next image, an additional store is closer, so this appears to it is in a wrong on your site.

10 Mile Search results – web page 1

Students should have one or 2 of the stores listed on this page depending upon how strict they translate 10.36 miles. Keep in mind that a new store has appeared that is 7 miles away. I have the right to only guess the it is a mistake that prevented that from being presented on the 5 mile page rather of the “9 Miles” store. Also notice that the Mesa, AZ save is the just one to it is in excluded from ours list as result of error from utilizing the Pythagorean Theorem instead of the Haversine formula. Its street is calculated in ~ 12.27 miles rather of 10.76 miles.

15 Mile Search results – Pages 1 and 2

The mistake on the toys R united state website proceed to grow. Making use of both the Pythagorean theorem distance and also the Haversine Formula street measurements, only 3 stores space 15 mile or much less from ours location. Strangely, the website perform 8 shop of i m sorry the last 5 room all clearly listed as being farther than 15 miles away. This is a great opportunity to let students point out the mistakes, describe why they room wrong, and also “critique the thinking of others” (Math exercise 3). Students’ trust in being smarter 보다 a website need to be growing here.

25 Mile Search results – Pages 1 and 2

There room still mistakes with the results for 25 miles. Making use of both the Pythagorean organize distance and also the Haversine Formula distance measurements, only 5 stores are 25 mile or less from our location. Yet the website tho lists 8 shop of which the last 3 are all clearly listed as being farther 보다 25 mile away.

See more: Geometry Probability Worksheets, Geometric Probability Worksheet

50 Mile Search outcomes – Pages 1, 2, and also 3

The website finally gets it ideal in the end. All 10 stores must be presented in the list and also they are.