The figure shows rectangle

*ABCD*in the coordinate plane with point

*A*at (0, 2.76),

*B*at (3.87, 2.76),

*C*at (3.87, 0), and

*D*at the origin. Rectangle

*ABCD*can be used to approximate the state of Colorado with the

*x*and

*y*scales representing hundreds of miles.

Part A

Based on the information given, how many miles is the perimeter of Colorado?

Part B

At the end of 2010, the population of Colorado was 5,029,196 people. Based on the information given, what was the population density at the end of 2010?

(A) 25 people per square mile

(B) 47 people per square mile

(C) 2,269 people per square mile

(D) 7,586 people per square mile

This question shouldn't be too difficult. For Part A, the perimeter of the rectangle is the sum of its side lengths -- we add 2.76 + 3.87 + 2.76 + 3.87 = 26.52. But notice that each unit is 100 miles, so this is actually a perimeter of 2,652 miles.

For part B, we notice that the area of the state is 276 * 387, or 106,812 square miles. Dividing the 2010 census figure of 5,029,196 by 106,812 gives about 47.1 people per square miles, which is (B).

Common errors will including confusing perimeter with area. I bet some students will try to enter the area of 106,812 in for Part A. If one makes the opposite mistake and use the perimeter for Part B, then one obtains a density of 1,896, which oddly enough isn't one of the choices.

Another common error will be to ignore the scale that one unit equals one

*hundred*miles. So some students might try to enter 26.12 in for Part A, or even make both mistakes and enter 10.6812. If we ignore the scale for Part B, we may obtain 470,846 for part B, or possibly 4,708 if a student thinks that one square unit equals 100 square miles rather than the correct 10,000 square miles. And I notice that 4,708 isn't one of the wrong choices either. I'm completely stumped where the three distractors come from for this problem. (I even tried dividing 5,029,196 by 2010, on the off chance that students might think that 2010 is something to divide by, but this gives 2,502, which isn't listed either.)

As for authenticity, to find the population density of a state or region can be a real-life question. I suppose that forcing the state onto the coordinate plane may be the only inauthentic part. Then again, many of the questions in Section 8-4 of the U of Chicago text use grids with different scales, but these were for irregular regions, not for simple rectangles. For rectilinear regions, such as those mentioned in Section 8-3, one unit is always one foot, one meter, or one of something. Notice that perimeter -- Part A of this question -- is covered in Section 8-1, but there are many polygons in this section, not just first rectangles. It may be better simply to list the dimensions of Colorado as 276 by 387 miles without using a coordinate plane.

Since we got through today's question so quickly, I can spend more of this post discussing the geography of Colorado. First, some may believe that 47 people per square mile is absurdly low, especially considering that there are often almost 47 students just in our classroom, which is a lot smaller than one square mile! But we must remember that population density isn't uniform -- people are concentrated in

*cities*. Because states contains wide-open areas with no population, such as Colorado's plains or mountain regions -- the density of a city is often a few orders of magnitude larger than the density of a state. For example, Denver -- Colorado's capital and largest city, had a population of 600,158 according to the same 2010 census and an area of 155 square miles, for a density of 3,872 people per square mile. The capital is 82 times as dense as the state as a whole.

In today's featured PARCC question, the southwest corner of Colorado is placed at the origin. This point is one of the most famous locations in the country -- Four Corners. Here's a link to a geography blog, Twelve Mile Circle, about the Four Corners:

http://www.howderfamily.com/blog/four-corners/

Notice that since the origin of our coordinate plane is Four Corners, the four quadrants (yes, again with the quadrants!) correspond to four different states. Quadrant I is of course Colorado, Quadrant II is Utah, Quadrant III is Arizona, and Quadrant IV is New Mexico.

Today's PARCC question is easy because we only had to find the perimeter and area of a rectangle. I now link to another page on the Twelve Mile Circle website about the shape of Colorado:

http://www.howderfamily.com/blog/colorado-not-rectangle/

If I can summarize for a moment, they maintain that a single 1879 survey of the Utah-Colorado boundary began at theFour Corners and marched straight up to Wyoming along the designated line of longitude. The survey party placed a marker at each mile along the way. They didn’t arrive where they expected on the Wyoming border and realized they’d made a mistake. A westward jog had to exist somewhere within the line but they knew not where.

Follow-up surveys revealed minor errors in 1885 and 1893. It didn’t matter. By then the boundary had already been fixed along the previous mile markers. Changing it would have required an agreement between both States and approval by Congress. Good luck with that!

So we see that the boundary of a state defined as the place where it is

*marked*, and not by an exact value such as a longitude. In the late 19th century GPS technology wasn't available, and so the boundary line is not perfectly straight. Therefore Colorado is

*not*a perfect rectangle -- but then again, PARCC did use the word "approximate."

But even if Colorado's perimeter were defined by latitude and longitude, the shape of Colorado would

*still*not be a rectangle. This is because the earth is not flat, but round, and in non-Euclidean spherical geometry, there are no rectangles. I've mentioned several time before that there exist Saccheri quadrilaterals -- figures that are both parallelograms and isosceles trapezoids (inclusive definition) and have two right angles, yet aren't rectangles (unless the geometry is Euclidean). But Colorado could not even be a Saccheri quadrilateral, since the sides of any quadrilateral must be line segments and the parallels of latitude aren't line segments (as they are arcs of

*small*, not

*great*, circles).

I've passed through the state of Colorado twice. The first time was on a long road trip, on the way from the Midwest back to California. I spent the night in Grand Junction, CO, which is the largest city in the western part of the state. Its population in the 2010 census was 58,566 and its area is about 38.6 square miles, for a population density of about 1,517 people per square mile. The other time was on a flight from Maryland to California. I could have flown back directly but I was able to switch to a flight that get me home earlier, but had a stop in Denver.

Here in Southern California, the state of Colorado has been in the news lately because its baseball team, the Rockies, is spending the week playing our local teams. First, last weekend the Dodgers played two games in the Mile High City -- it would have been three except the middle game was

*snowed*out. Then the Rockies came here to play two games against the Angels, and then they will stay to play four more games against the Dodgers.

Regarding academics and Common Core, Colorado is a PARCC state -- so it's unsurprising that there would be a question about the state on the PARCC. Like in most states, there is an anti-Core movement encouraging students to opt out of PARCC. Here is a link to Colorado's anti-Core page:

http://colohub.weebly.com/

I don't want to get into political arguments here, but this link does give a paper by Dr. James Milgram, regarding

*mathematical*problems with the Common Core Standards. Here is an overview of what Milgram finds problematic with the Core:

-- Uneven Mathematics Standards in the Primary Grades

-- Low Mathematics Expectations by the End of the Elementary Grades

-- Delayed Development of Pre-Algebra Skills

-- An Unproven Approach to Geometry (based on translations, reflections, etc.)

I have already mentioned many of these issues on the blog. Finally, I mentioned earlier that Colorado is the state that divides its school year into six "hexters." Actually, many schools around the country divides the year into six 6-week periods, but mainly Colorado calls them "hexters."

In today's worksheet, for a practice problem, I decided to use a different state, Wyoming, since this state, like Colorado, is rectangular. Actually, since the borders of Colorado aren't straight and Wyoming shares a border with Colorado, Wyoming can't be

*perfectly*rectangular either. PARCC could have used Wyoming instead of Colorado in today's featured practice question -- but of course the PARCC writers chose Colorado, because Wyoming is an SBAC state.

**PARCC Practice EOY Exam Question 22**

**U of Chicago Correspondence: Section 8-3, Fundamental Properties of Area**

**Key Theorem: Area Postulate**

**b. Rectangle Formula: The area of a rectangle with dimensions**

*l*and*w*is*lw*.

**Common Core Standard:**

CCSS.MATH.CONTENT.HSG.MG.A.2

Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

**Commentary: The areas of states appear several times in Chapter 8, but these are irregularly shaped states, not Colorado or Wyoming. In particular, the shape of Texas appears in Review Question 19 in Section 8-5, the trapezoidal shape of Nevada appears in Exploration Question 25 in Section 8-6, and the shape of Hawaii appears in Question 13 of the Progress Self-Test.**

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