Some people may notice that the word "integers" doesn't appear in the Common Core Standards. Instead, the standards refer to the set of "rational numbers":

#### Apply and extend previous understandings of numbers to the system of rational numbers.

CCSS.Math.Content.6.NS.C.5

Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

If we think about it, we notice that the students first learn about natural numbers, then about fractions (of which the natural numbers are a subset), then we suddenly want to jump to integers -- of which the set of fractions is

*not*a subset. The students already know about fractions, so as soon as they learn about positive and negative signs, they technically already have the entire field

**Q**of rational numbers. Of course, the simplest examples -- including all the examples that they saw today -- were integers.

Students learned about adding and subtracting integers today -- next week they will learn how to multiply

and divide them. Today's lesson is actually a seventh grade topic:

CCSS.Math.Content.7.NS.A.1

Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

Then again, this is the end of the year. Most sixth graders have already completed the SBAC (yet some students are still taking it), and so this lesson is a preview of next year.

With all these visits to middle school math classes this week, angles and parallels are still on my mind. Recall that I'm still trying to come up with a plan to prove the Corresponding Angles Test as a theorem without using Playfair's Parallel Postulate.

Of course, you may ask, why can't I just accept the Corresponding Angles Test as a postulate, just as Dr. Franklin Mason is now doing? I guess that the words of Dr. David Joyce still haunt me:

**Chapter 7**[of the Prentice-Hall text -- dw] is on the theory of parallel lines. It begins by postulating that corresponding angles made by a transversal cutting two parallel lines are equal. One postulate is enough, but for some reason two others are also given: the converse to the first postulate, and Euclid's parallel postulate (actually Playfair's postulate). A proliferation of unnecessary postulates is not a good thing. One postulate should be selected, and the others made into theorems.

Dr. M's original presentation did select only one postulate -- Playfair's. Then he used Playfair and the Parallel Tests to prove the Parallel Consequences. The Parallel Tests are themselves proved using the Triangle Exterior Angle Inequality, which in turn follows from SAS. It was only later on when Dr. M added a Corresponding Angles Test Postulate to avoid the TEAI proof -- which may be a bit tricky for high school students to understand.

My concern is mainly that I want Corresponding Angles to be first, and then derive the theorems for Alternate Interior and Same-Side Interior from Corresponding Angles. It's possible to modify the TEAI proof so that it refers to corresponding rather than alternate interior angles, but a natural reading of other proofs -- such as Dr. Hung-Hsi Wu's rotation proof -- uses alternate interior angles.

I was just reading the geometry text of the mathematician Adrien Legendre -- which, if you recall, was the first text to challenge the dominance of Euclid's

*Elements*. Now Legendre's Proposition 60 appears to give priority to a Same-Side Interior Angle Test Theorem. Here's is Legendre's proof, converted to two-column format (and replacing the Euclidean phrase "sum to two right angles" with "are supplementary angles"):

Given: Angles

*CAB*,

*ABD*are supplementary

Prove: Lines

*AC*| |

*BD*

*Proof:*

Statements Reasons

1.

*CAB*,

*ABD*supplementary 1. Given

2. Let G be the midpoint of

3. Draw

*EG*perpendicular to 3. Through point there's line perp. a given line

*AC*intersecting

*BD*at

*F*

4.

*GBF*,

*GBD*supplementary 4. Linear Pair Theorem

5.

*GAE*,

*GBD*supplementary 5.

*GAE*,

*GBD*are just

*CAB*,

*ABD*renamed

6. Angle

*GAE*=

*GBF*6. Angles suppl. to same angle are congruent

7.

*AG*=

*BG*7. Definition of midpoint

8. Angle

*AGE*=

*BGF*8. Vertical Angles Theorem

9. Triangle

*AGE*=

*BGF*9. ASA Congruence Postulate

10. Angle

*AEG*=

*BFG*10. CPCTC

11.

*BD*,

*EG*perpendicular 11. All right angles are congruent.

12.

*AC*| |

*BD*12. Two Perpendiculars Theorem

Notice that just like Legendre, I on the blog have already proved the Two Perpendiculars Theorem independently of any previous theorem. So we both can use the Two Perpendiculars Theorem to prove this Same-Side Interior Angle Test.

But then we notice that this proof works better as an Alternate Interior Angles Test -- we notice that Step 6 of this proof ultimately shows that the alternate interior angles are congruent, so we might as well have this be given and drop Steps 4-5 altogether. I mentioned earlier that Wu's rotation proof can be converted to a proof using triangle congruence. Well, this is exactly the proof -- notice that the two triangles proved congruent (

*AGE*and

*BGF*) can be mapped to each other via exactly the rotation that appears in the Wu proof.

So if Legendre could change a natural Alternate Interior Angles Test into a Same-Side Interior Angles Test, then I can change it into a Corresponding Angles Test instead. We simply give that the corresponding angles are congruent and use the Vertical Angle Theorem for the alternate interior angles necessary to produce the congruent right triangles.

Then again, I wonder whether we can just take the original Wu rotation proof and attempt to adjust it so that it gives a Corresponding Angles Test instead. Notice that for every pair of corresponding angles, one is exterior and the other is interior. So we simply rotate the exterior angle to produce what one could call its "alternate

*exterior*angle." We then use Vertical Angle Theorem to obtain the angle that is corresponding with the original preimage angle, without anyone wondering why we don't apply Vertical Angles to the

*preimage*to give alternate

*interior*angles.

Speaking of corresponding angles, they appear in the final PARCC question. Question 32 of the PARCC Practice Test is about constructions:

Part A:

The figure shows line

*r*, points

*P*and

*T*on line

*r*, and point

*Q*not on line

*r*. Also shown is ray

*PQ*.

(A standard construction for parallel lines appears. Point

*W*is constructed on line

*r*, and point

*S*is constructed not on line

*r*.)

Consider the partial construction of a line parallel to

*r*through point

*Q*. What would be the final step in the construction?

(A) draw a line through

*P*and

*S*

(B) draw a line through

*Q*and

*S*

(C) draw a line through

*T*and

*S*

(D) draw a line through

*W*and

*S*

*Part B:*

Once the construction is complete, which of the reasons listed contribute to proving the validity of the construction?

(A) When two lines are cut by a transversal and corresponding angles are congruent, the lines are parallel.

(B) When two lines are cut by a transversal and vertical angles are congruent, the lines are parallel.

(C) definition of segment bisector

(D) definition of angle bisector

For Part A, even if one isn't familiar with this particular construction, we notice that the question asks for the final step. Since we are trying to construct a line parallel to

*r*through

*Q*, the final step ought to be -- to draw the line parallel to

*r*through

*Q*, of course! Thus drawing a line through

*P*,

*T*, or

*W*make no sense as all of these points lie on

*r*, so any line passing through these points must intersect

*r*, not be parallel to

*r*. The only possible line parallel to

*r*is line

*QS*. So choice (B) is correct.

For Part B, we want to prove that lines

*r*and

*QS*are parallel. We are copying the angle

*QPW*, and this angle and the copied angle are corresponding angles. So by the Corresponding Angles Test, we conclude that the lines are parallel. So choice (A) is correct.

Notice that the U of Chicago text doesn't even give the construction of copying an angle -- much less the construction of a line parallel to a given line through a point not on the line, which depends on the former construction. For all its faults, the Prentice-Hall text mentioned by Joyce gives copying an angle as the second construction in its Section 1-1, and we've already mentioned that the Pearson Integrated Math II test gives the same construction in its own first section as well. The U of Chicago text focuses mainly on constructing perpendicular bisectors -- even its angle bisector construction is really just a form of its perpendicular bisector construction.

Indeed, on the blog, I gave a parallel line construction based on the perpendicular bisector construction (which is probably how the U of Chicago would present it if it were to do so). This construction is justified by the Two Perpendiculars Theorem.

But the construction on the PARCC exam is justified by the Corresponding Angles Test. So once again, we want Corresponding Angles to have priority -- there's nothing wrong with performing the construction to use alternate interior rather than corresponding angles, but it's just not done. I wonder whether a student who saw Alternate Interior Angles given priority and Corresponding Angles proved using Alternate Interior and Vertical Angles might even be tricked into selecting choice (B) rather than (A) for Part B above.

Today is an activity day. I decided that the students need more practice with constructions, and so I have them draw a simple design that requires them to construct parallel lines. This is similar to the tic-tac-toe design that I had them draw earlier -- except there are no perpendicular lines. So students will have to use the parallel line construction based on corresponding angles that is covered on today's worksheet.

This is the last of the PARCC Practice Test questions. According to the school district whose calendar I'm using on the blog, there are three days of school left -- finals week. And so in my next post, I will post a final exam based on both U of Chicago and PARCC practice questions. I don't provide a review worksheet since all 32 PARCC questions actually are review questions. On Monday I post only the test itself.

**PARCC Practice EOY Exam Question 32**

**U of Chicago Correspondence:**

**Section 3-4. Parallel Lines**

**Key Theorem: Corresponding Angles Postulate:**

**If two coplanar lines are cut by a transversal so that two corresponding angles have the same measure, then the lines are parallel.**

**Common Core Standard:**

CCSS.MATH.CONTENT.HSG.CO.D.12

Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).

*Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line*.

**Commentary: The construction of parallel lines doesn't appear anywhere in the U of Chicago text at all. One can use the perpendicular bisector twice to construct parallel lines, but this is never directly stated in the text.**

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