Anyway, I began my day in an Integrated Math I class for freshmen, and I ended up in an Algebra I class for sophomores grandfathered on the traditionalist pathway. In my last post I mentioned some integrated math texts, and so of course I have more to say about that today. The freshmen were studying out of a text published by Core Connections. Here are its contents:
Volume 1:
Chapter 1: Functions
Chapter 2: Linear Functions
Chapter 3: Transformations and Solving
Chapter 4: Modeling Two-Variable Data
Chapter 5: Sequences
Volume 2:
Chapter 6: Systems of Equations
Chapter 7: Congruence and Coordinate Geometry
Chapter 8: Exponential Functions
Chapter 9: Inequalities
Chapter 10: Functions and Data
Chapter 11: Constructions and Closure
The students in this class were in Chapter 6, on Systems of Equations. Students had to determine whether to use substitution or elimination to solve certain systems, and then solve them.
Even though the other class was traditionalist Algebra I, there just happened to be some integrated materials in the classroom. One was a stack of orange MathLinks packets -- the ones that are labeled eighth grade yet are being used in freshmen classes. This packet, numbered 8-15, was titled, "Geometry Discoveries." Here are its contents of its four sections, with the contents of the first section in greater detail:
15.1 Similar Triangles
-- Establish the angle-angle criterion for similarity of triangles.
-- Apply the angle-angle criterion to solve problems.
-- Link concepts of parallel lines and similar triangles to slopes of lines.
-- Prove a famous theorem [the Pythagorean Theorem] using similar triangles.
15.2 Volume of Cylinders
15.3 Volume of Cones and Spheres
15.4 Skill Builders, Vocabulary and Review
There was also another integrated math text in this classroom, published by Carnegie Learning. I've already posted chapter lists for several texts in the past week, so I won't write out the chapter list for the Carnegie Learning text. The important thing to note is that Carnegie Learning, just like Core Connections and Pearson, has a paperback Integrated Math I text divided into two volumes. For some reason, this is apparently common among integrated texts.
Recall that the gold standards for integrated texts are usually considered to be Saxon and Singapore Math texts. I'm starting to change my mind about the Singapore Math text, only because the series that I discussed on the blog a few months ago -- the New Elementary Math text -- has four volumes for grades 7-10, but the ninth and tenth grade texts are out-of-print. It's awkward to keep discussing texts that no one has access to over and over again. Often when traditionalists recommend Singapore Math, they are referring to grades K-6, occasionally K-8, but never any grade higher than eighth.
So this leaves the Saxon texts as our sole remaining gold standard for integrated high school math. I have already mentioned that traditionalists want there to be a path for seniors to take Calculus. The Saxon texts are usually denoted by two grade levels, as in Saxon 87, which is for below-average eighth graders and above-average seventh graders. Obviously, if we want there to be a path to Calculus, we must assign this text to the lower grade, so Saxon 87 is for seventh graders. Recalling how the Saxon texts are labeled, we give the following integrated path from Saxon 87 to Calculus:
Grade 7: Math 87
Grade 8: Algebra 1/2 (half)
Grade 9: Algebra 1 (3rd edition, integrated)
Grade 10: Algebra 2 (3rd edition, integrated)
Grade 11: Advanced Math
Grade 12: Calculus
So in our discussion of the various integrated texts, we must see how the text assigned to eighth graders compares to Algebra 1/2, how the freshman text compares to Algebra 1, and so on.
I already concede that in order to get students to Calculus, the Common Core math classes labeled Integrated Math I, II, and III must be taken in grades 8-10. Then juniors can take Precalculus, and this allows the seniors to take Calculus. Notice that we are comparing three different course numbering systems -- the course to be taken by freshmen on the Calculus path is called Singapore New Elementary Math 3, Common Core Integrated Math 2, and Saxon Algebra 1.
I now want to discuss my ideal Integrated Math 1 course. Originally, I wanted to divide my class into twelve units. This was ideal because I've noticed that it takes about three weeks to cover one chapter of a math text. Since a typical school year is 38 weeks long (including partial weeks like Memorial Day week, but not vacation weeks like winter break), this allows each semester to span six chapters plus a finals week.
But now I'm considering having just ten units rather than twelve. When I was student teaching in an Algebra I class, one of the texts we used was Holt Explorations in Core Math. This text was divided into ten chapters, and I noticed that we covered Chapter 1 in September, Chapter 2 in October, Chapter 3 in November, all the way up to Chapter 10 in June. So the ten chapters roughly correspond to the months of the school year. This makes pacing much easier. Instead of counting out intervals of three weeks and trying to remember whether the current week was the first, second, or third week of the chapter, I could just look at the calendar. If it was around the 15th of the month, then the class should be halfway through the chapter, and when it's the last week of the month, the class should be getting reading for the test.
So now here's my ideal Integrated Math I course. My ideal course starts out with geometry -- and indeed, I'd start out my ideal Geometry course the same way:
Unit 1: Geometry Basics
Unit 2: Reflections
Unit 3: Rotations
Unit 4: Translations
Unit 5: Glide Reflections
Unit 6: Dilations
At this point Integrated Math I and Geometry diverge. Integrated Math I now goes into algebra:
Unit 7: Slopes and Lines
Unit 8: Inequalities
Unit 9: Systems
Unit 10: Functions
For Geometry, the remaining four units are:
Unit 7: Area
Unit 8: Volume
Unit 9: Trigonometry
Unit 10: Circles
Let's look at these units in detail. I would like the first six units of the course to be similar to what I posted here on the blog. In particular, Unit 1 corresponds roughly to the first three chapters of the U of Chicago Geometry text, while Units 2-5 correspond respectively to Chapters 4-7 of the text.
Now it may seem weird to spend roughly a semester just on transformations. But most of the other material in geometry ultimately go back to transformations -- this is why transformations are the cornerstone of Common Core Geometry. But I definitely want to make some changes to the way that I covered the material.
In particular, the congruence theorems SSS, SAS, and ASA are covered much too late in the U of Chicago text. The text uses transformations -- translations, reflections, and rotations -- to prove the three theorems SSS, SAS, and ASA, and since translations don't appear until Chapter 6, triangle congruence can't appear until Chapter 7. But this approach makes SSS, SAS, and ASA merely afterthoughts -- when we've seen several questions requiring SSS, SAS, or ASA on the PARCC.
Instead, we move SSS, SAS, and ASA up into Unit 2 with reflections. Notice that we can still prove SSS, SAS, and ASA using reflections only -- translations aren't needed at all. So we can cover much of the first half of Chapter 7 (U of Chicago) with Chapter 4 and make them into my Unit 2. I make these changes just as Dr. Franklin Mason has changed his curriculum after seeing what appears on the PARCC and SBAC exams.
Unit 3 can remain very much how it's posted here. It corresponds roughly to Chapter 5 of the U of Chicago, as we discuss polygons. Rotations will still be used to develop the properties of parallel lines, just as Dr. Hung-Hsi Wu does. A parallel postulate can appear right at the start of the second quarter of the course.
I was thinking about how I wanted to deal with the end of the first semester. I mentioned earlier that I wanted similarity to start the second semester, rather than end the first semester. My Unit 6 will be a tough unit, and I don't want to tantalize students into having a good grade for most of the semester, only to have it drop after covering the similarity chapter. Indeed, my idea is that the last unit before the end of the semester should be something that's easy to do, but hard to remember. So grades won't drop because the unit is easy, and though it's difficult to remember, the final occurs right afterward, before the students forget the material.
Furthermore, we consider the fact that if Unit 1 is in September, then Unit 4 is in December -- that is, just before winter break. On the Early Start Calendar, Unit 1 may be in August, which means that Unit 4 would be in November -- just before Thanksgiving break. So Unit 4 is likely to precede a holiday week no matter what. This is, of course, not necessarily the best time to introduce difficult material either. Instead, I want to put something more enjoyable here. Since many students enjoy drawing, this may be a good time (after teaching translations from Chapter 6) to introduce some of those constructions that appear on the PARCC but not in the U of Chicago text.
Unit 5 will contain glide reflections, as these are difficult to remember. Properties of parallelograms are also hard to remember, and so I can end the unit the same way I ended the semester here on the blog, with the last part of Chapter 7.
Then we reach Unit 6 -- the big chapter on dilations and similarity. This will cover material from Chapters 11 and 12 of the U of Chicago -- just as I began second semester here on the blog. Notice that there is so much to cover here -- the Dilation Distance Postulate, similarity using AA/SAS/SSS, the Pythagorean Theorem, the coordinate plane, the distance/midpoint formulas. transformations on the coordinate plane (barely covered in U of Chicago, strongly covered on the PARCC), and finally the slope formula.
By now you can surely figure out what I'm trying to accomplish here -- we teach the slope formula in Unit 6, and then segue right into slope and the linear equations of algebra in Unit 7. After all, this is the whole point of taking an integrated course -- to show the connections between the various branches of mathematics. But this won't work if Unit 6 becomes so unwieldy, with so much information packed in that it becomes unteachable.
Notice that we could begin Unit 6 with the Dilation Distance Postulate, and then move into similarity the same way that MathLinks packet 8-15 does it. Notice that we could get away with teaching only AA Similarity and not SAS or SSS. After all, the heavy-duty results of the similarity chapter -- the slope formula, Pythagorean Theorem, and eventually trigonometry all use AA Similarity only. The only theorem I saw in the U of Chicago text whose proof required SAS Similarity was the Converse to the Side-Splitter Theorem -- but of course the forward theorem used AA, and no theorem required SSS at all. By teaching AA only, there would be time to teach more of the bigger results.
As I mentioned in my last post, one thing that I like to do is have some lesson involving the number pi close to Pi Day. On a Labor Day Start calendar, March is Unit 7, while on an Early Start calendar, March would be Unit 8. In my traditionalist Geometry course, I made sure that Units 7 and 8 are on area and volume, so that both units teach formulas involving pi.
Now on my Integrated Math pathway, I delay area to Math II and volume to Math III. By doing so, both courses have lessons on pi that can be taught near Pi Day. Indeed, notice how we can continue this pattern into Precalculus, when trigonometry using radians is taught. The common angles whose trig values can be memorized are those of simple ratios of pi. And so that particular constant can be taught in Precalc near Pi Day as well.
But Integrated Math I, unfortunately, doesn't mention pi much at all. After all, most of the second semester is on Algebra I, and that class generally avoids mention of pi. The closest I can come is to mention the properties of real numbers during Units 7 and 8 -- and to explain what "real numbers" are, we can mention irrational numbers such as pi (and sqrt(2), since we just taught a lesson on the Distance Formula, which involves square roots). This is why I made Unit 8 on Inequalities -- this is another good place to sneak in pi, since we have the inequality 3 < pi < 22/7 involving pi.
To close out my Integrated Math I class, Unit 9 will be on systems -- in a Labor Day start class, Unit 9 will be in May, and the class I subbed in last week was learning systems. Unit 10, like Unit 5, is at the end of a semester, and June is certainly not the time when students want to see a very difficult math lesson! Here I decided to put in a short unit on functions. I've noticed that in many Algebra I texts, there is a chapter on functions just before the chapter on linear functions (and slope). The teacher often spends a day or two on the functions chapter, then goes straight into linear functions. I put my lesson on functions at the end, so that teachers can spend as little or as much time as is left at the end of the year before reviewing for the final.
By the way, since I'm so eager to show the connection between similarity and slope, one might wonder why I teach functions in Unit 10 rather than tie it to geometry transformations -- since transformations actually are functions. My fear is that in Unit 10, I wish to teach that the domain of a function y = f (x) consists of the input, or the set of x-values, while the range of a function consists of the output, or the set of y-values. But the input of a geometry transformation like T(x, y) has both an x- and a y-value -- and so does the output. So I avoid this confusion by placing functions far away from transformations -- although Math II or III could mention that transformations are functions.
Let's compare my Integrated Math I course to that from the Core Connections text -- which is supposed to be based on the PARCC Integrated Math I exam. One key omission in my text is the simple solving of linear equations. Here I'm assuming that solving at least one-step (and possibly two-step) equations was taught in the sixth- or seventh-grade courses, and unlike fractions, this is one thing that students do typically remember. Simple equations can also appear in some of the first semester geometry lessons. Yes, I know that I've said that I don't want to see too much algebra during the geometry lessons, but that was because I was worrying about students coming off of a C- or D- grade in Algebra I getting to Geometry and having so see so much algebra again -- but this is not a concern in an Integrated Math I class. And besides, my concern was more about having, say similarity questions that lead to quadratic equations. Of course, proportions will appear in the similarity unit, and some equation manipulation (such as solving for y) appears in the slope unit. All other types of linear equations (including equations with x on both sides, which often confuse students) can be taught or reviewed during the inequalities unit -- another reason that I placed that unit there.
But again, the gold standard for Integrated Math isn't Core Connections -- it's Saxon. So let's compare my Integrated Math I course to Saxon Algebra 1/2. The algebra side of my Integrated Math I course is on par with Algebra 1/2 -- mine is probably a little ahead, since I cover systems that Saxon doesn't cover until Algebra 1. Notice that both Core Connections and Saxon include lessons on collecting and graphing data, exponential functions, and other applied word problems. These can be incorporated into my Units 7-10 wherever they belong -- including exponential functions, which can be sneaked into Unit 10 if desired.
On the other hand, my geometry lessons differ from all of the other integrated texts. Similarity appears in both Core Connections and Saxon later than when I'd teach it -- but I want to show the connection between similarity and slope. In exchange, area and volume must appear later in my lessons than in Core Connections or Saxon.
There's one more thing that I must mention about my Integrated Math I course. My placement of Unit 5 was highly dependent on how students would feel about the semester final. But in order for students to reach calculus by senior year, Math I must be taken in the eighth grade -- a grade which, at many middle schools, has neither semesters nor finals. At middle schools where there are trimesters, it may be convenient to teach only nine units, as this would be three units per trimester. With three units per trimester, the first two trimesters would be geometry with only one trimester of algebra. There would be a tough Unit 6 at the end of the second trimester -- but once again, there are no finals, and the students already received a first trimester grade. (Of course, this is also an argument to go back to 12 units, as 12 is convenient for both semesters and trimesters.)
I owe the development of my Geometry and Integrated Math I courses to the mathematicians whom I regularly quote on the blog. The idea to teach SSS, SAS, and ASA early goes back to Dr. M, who changed his own course to reflect what appears on the PARCC and SBAC exams. From Dr. Wu, I keep the idea to use rotations to teach parallel lines. From Dr. David Joyce, I keep the idea to use similar triangles to teach slope. And one could argue that I first got the idea to teach only AA and no other similarity theorem from Danica McKellar, who gives only AA in her geometry for girls text.
Of course, neither my ideal Geometry nor my Integrated Math I course can become reality now, because both classes are bound by what appears on their respective PARCC exams. Furthermore, the fact that the PARCC exam is given so early in the year forces me to accelerate Units 7-9 so that they are completed before the PARCC PBA test. Unit 10 of my Geometry course, on circles, must remain the last chapter to be taught since questions from it appear on the PARCC EOY but not the PBA.
And one of those questions is today's featured PARCC problem. Question 29 of the PARCC practice exam is indeed on circles:
Point B is the center of a circle, and
Part A
Indicate which statements must be true.
Select all that apply.
(A) AD > CD
(B) Angle CBD =1/2 (Angle CAD)
(C) Angle ADC = 90 degrees
(D) Angle ABD = 2 (Angle ACD)
(E) Angle ABD = Angle DBC
Part B
If Angle BDA = 20 degrees, what is Angle CBD?
(A) 20 degrees
(B) 40 degrees
(C) 70 degrees
(D) 140 degrees
Like most other PARCC questions involving circles and angles, Section 15-3, the Inscribed Angle Theorem, is the only section of the U of Chicago text that really matters. For Part A, there are two correct answers. Angle ABD = 2 (Angle ACD) is correct because it gives the central angle as equal to double the inscribed angle, and Angle ADC = 90 degrees because it's inscribed in semicircle ADC. So the answers are (C) and (D).
For Part B, we must go back to isosceles triangles again, just like many other PARCC questions on inscribed angles. Triangle BDA is isosceles because two of its sides are radii. So Angle BAD, the same as CAD, must also be 20 degrees. Now CBD is the central angle for the inscribed angle CAD, so its measure must be double that of CAD, so it's 40 degrees. The correct answer is (B).
A common error for Part A is to include (B) as a correct choice. But we notice how (B) claims that the CBD is half of CAD, when it's actually double. Notice that if (B) had been correct, the answer to Part B would have been 10, not 40, degrees. Indeed, I'm actually surprised that 10 degrees wasn't one of the wrong choices for Part B.
There's nothing much more I can about this question that wouldn't be a repeat of my comments about the other circle-angle problems.
PARCC Practice EOY Exam Question 29
U of Chicago Correspondence: Section 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 the intercepted arc.
In a circle, the measure of an inscribed angle is one-half the measure of the intercepted arc.
Common Core Standard:
CCSS.MATH.CONTENT.HSG.C.A.2
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: I repeat my comments from Questions 8 and 24. Most of the inscribed angle problems in the U of Chicago text give the intercepted arc measure, but a few do require students to calculate the arc measure first before finding the inscribed angle measure. In the SPUR section, Objective B, including Questions 7 through 11, are good questions to consider.
CCSS.MATH.CONTENT.HSG.C.A.2
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: I repeat my comments from Questions 8 and 24. Most of the inscribed angle problems in the U of Chicago text give the intercepted arc measure, but a few do require students to calculate the arc measure first before finding the inscribed angle measure. In the SPUR section, Objective B, including Questions 7 through 11, are good questions to consider.
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