Tuesday, May 17, 2016

PARCC Practice Test Question 20 (Day 163)

Today I subbed in a middle school history class. Normally I don't announce non-math subbing assignments here on the blog, but this one is significant because:

-- It's the first time I subbed at a middle school since the announcement that I will be teaching at a middle school full-time next year.
-- As SBAC testing continues, the middle school is on a block schedule. Wednesdays are Common Planning days at the elementary and middle schools in our district, and so the blocks are set up with odd periods on Mondays and Thursdays and even periods on Tuesdays and Fridays -- just like the middle school I'll be at in the fall.

So this is another chance to look critically at the block schedule and see how teachers plan for block periods, even in subjects other than my own. In this class, the eighth graders have four tasks to complete during the 1:50 block:

-- Finish the movie Glory. California eighth graders study U.S. History up to around the Civil War.
-- Take a 10-minute break.
-- Listen to a lecture about the Battle of Vicksburg.
-- Work on the Civil War project that is due next week.

Traditionalists often complain about watching movies when they occur in lieu of a task with more academic rigor. It's hard to know whether the students would have watched the video had there not been block periods, since the Civil War is always taught at the end of the year, right around the time of the SBAC. (Even before the SBAC, eighth graders always must take the NCLB science tests, which here in California are given to fifth, eighth, and tenth graders.)

But it's that 10-minute break that will annoy traditionalists the most. The savings in passing time are offset by the need to give the students a break during the period. The class is now divided essentially into two 50-minute "periods."

The project mentioned above may sound perfect for a block lesson, except that it's one of those multimedia projects that is best completed using technology at home. So if the students have no research to do, then it ultimately turns into study hall. So again, we see that there are many parts of this block period for traditionalists to dislike. Of course, I point out that even though I want to see what block periods look like, these block periods are biased by the fact that these are lesson plans written for a sub. For example, the "study hall" could occur because the teacher can't think about how to fill 1:50 for a sub, not because the teacher always has study time every block period. The same is true about the video -- there might be no video today had the regular teacher been present.

Chapter 20 of Morris Kline's Mathematics and the Physical World is "Old Foes With New Faces." In this chapter, the "old foes" are electricity and magnetism.

"The magnet's name the observing Grecians drew/From the magnetic region where it grew." -- Lucretius

Kline begins:

"The world of nature is vaster than what the hand can touch, the eye see, or the ear hear. Beyond the senses lies a world that has been effectively explored only within the last hundred years or so."

It is the world of electricity and magnetism. Kline's quote from the poet Lucretius tells us the origin of the word magnet, and Kline repeats this origin story:

"For example, Thales knew that iron ores containing lodestone, such as those found near Magnesia in Asia Minor, attract iron. He is also supposed to have known that after amber is rubbed it attracts light particles of matter."

And indeed, the word electricity comes from the Greek word for amber, although Kline does not point this out.

Now the key formula for this chapter was discovered by the 18th century French physicist Augustin Coulomb, and it describes the electrical force between two charged particles:

F = k q_1 q_2 / (r^2)

Kline points out that this is very similar to the equation for the gravitational force. Kline proceeds by pointing out that electricity and magnetism are closely related --the electromagnetic force.

Question 20 of the PARCC Practice Test is on inscribed angles:

20. In circle O, points A, B, C, and D lie on the circle; arc AD is congruent to BC; and the measure of arc AB is twice the measure of arc BC.

[Moving clockwise, the points lie on the circle in the order D, A, B, C.]

Part A
The measure of angle ACD is Choose... (a third, half, equal to, twice, three times) the measure of angle ADC.

Part B
The measure of angle ADC is Choose... (a third, half, equal to, twice, three times) the measure of angle BCD.

This question clearly involves the Inscribed Angle Theorem of Lesson 15-3, since all of the angles mentioned are inscribed angles. Of course, it may be tricky to look at the diagram and determine which angles subtend which arcs. An easy way to tell is to note that the endpoints of the arc are the two points named in the inscribed angle other than the vertex. Since the inscribed angle is half the measure of the arc, we substitute into the original problem:

Part A
Half the measure of arc AD is Choose... (a third, half, equal to, twice, three times) half the measure of arc AC.

Part B
Half the measure of arc AC is Choose... (a third, half, equal to, twice, three times) half the measure of arc BD.

Of course, we can multiply everything by two to get rid of the "half":

Part A
The measure of arc AD is Choose... (a third, half, equal to, twice, three times) the measure of arc AC.

Part B
The measure of arc AC is Choose... (a third, half, equal to, twice, three times) the measure of arc BD.

Both of these require us to calculate the measure of arc AC. We see that arc AC is the sum of arcs AB and BC, and the former is said to be twice the latter. Therefore AC has thrice the measure of BC, which in turn has the same measure as AD. So AD must have a third of the measure of AC -- that is, for Part A, the measure of angle ACD is a third of the measure of angle ADC.

For Part B, we see that arc BD is the sum of arcs AB and AD, and once again, the former is said to be twice the latter (as again, BC and AD have the same measure). So both AC and BD have thrice the measure of BC -- that is, AC and BD are congruent. Therefore the measure of angle ADC is equal to the measure of angle BCD.

Notice that we must be careful when just blindly substituting in the arc measures. If the angle is obtuse, then it must subtend a major arc, so we can't just substitute a minor arc measure (fortunately all the arcs needed for the problem are minor and all the angles are acute). Also, we can't generally ignore the factor of 1/2 -- here I did because all of the angles were inscribed. If we had a mix of inscribed and central angles then the factor of 1/2 becomes significant.

PARCC Practice EOY Question 20
U of Chicago Correspondence: Lesson 15-3, The Inscribed Angle Theorem

Key Theorem: Inscribed Angle Theorem

In a circle, the measure of an inscribed angle is one-half the measure of its intercepted arc.

Common Core Standard:
Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

Commentary: This question may be slightly more abstract than a typical problem from the U of Chicago text. Question 9 from Lesson 15-3 and Question 33 from the SPUR Review for Chapter 15 are two questions where the arc lengths and angle measures are unknown.

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