8:00 -- Today this regular teacher is assigned morning yard duty, which I must complete.
8:15 -- This is the middle school where all classes rotate. Today the rotation starts with third. Also, at this school, homeroom is the same as third period.
As it turns out, this teacher has two classes that aren't English -- and it's just the luck of the rotation that these eighth grade electives are the first two classes of the day. Third period is Yearbook. These students visit the classrooms to gather photos of the other kids at the school.
9:15 -- Third period leaves and fourth period arrives. This is a computer class. The students are creating a Photoshop project where they put their favorite drawing on a brick background.
10:10 -- Fourth period leaves for snack break.
10:20 -- Fifth period arrives. This is the first of three English 7 classes.
These students are working on SBAC Prep. (Once again, it appears that my annual SBAC review here on the blog is timed perfectly for the middle school SBAC, not high school.) They have 28 multiple choice questions, on four pages of a packet, where they are given a sentence and must choose the correct definition of the underlined word. On page 5, they must read a passage on the Revolutionary War and answer four questions with one complete sentence each. And on page 6, they must read a passage on boxer Muhammad Ali and answer one question with a claim and evidence from the passage.
It's difficult to get through the entire packet, and I must rush to get to pages 5-6. I can see that the students know how to do the vocab questions on pages 1-4, but then I'm worried that the students don't understand how to do the comprehension questions on pages 5-6. This will hurt them when it's time for the real SBAC.
11:10 -- Fifth period leaves. Sixth period is conference period, which leads directly into lunch.
Every year, Teacher Appreciation Week is the first full week in May. I assume the week was chosen because it's just before the last day of school. Thus teachers can be appreciated for what they accomplished during the year that's about to end. Everyday this week, special foods are served for the teachers, including us subs, to enjoy. Today, a vegetarian lunch is served in the library.
12:50 -- First period arrives. This is the second of three English 7 classes.
The period after lunch at this school always starts with silent reading. Once again, the students have trouble with pages 5-6 of the packet.
2:00 -- First period leaves and second period arrives. This is the last of three English 7 classes.
This time, I decide to start with pages 5-6. I believe that this helps the students complete the short answer questions on page 5, but they still struggle with the paragraph response for page 6.
2:55 -- Second period leaves, thus ending my day.
The main classroom management issue is in first period. The regular teacher specifies not to give out restroom passes (except emergencies), but this is the class that should have given that rule. Once again, it's the class after lunch when students feel that they need to leave. On the other hand, she specifically states that these students need to remain in assigned seats. But one boy is sitting in the wrong seat -- and this ends up confusing me when it's time to take attendance.
Today on her Mathematics Calendar 2019, Theoni Pappas writes:
Find x.
(Once again, all the information is in an unlabeled diagram, so I must name the points. In Circle O, points A, B, C are on the circle and P is outside the circle. Tangent PA = 12 and on secant PBC, we also have PB = x and BC = 7.)
This is a power of a point problem, which fits Lesson 15-7 from last week:
12^2 = x(x + 7)
144 = x^2 + 7x
0 = x^2 + 7x - 144
0 = (x + 16)(x - 9)
x = -16, x = 9
The only valid length for PB is x = 9. Therefore the desired length is 9 -- and of course, today's date is the ninth.
This is what I wrote last year about today's lesson:
Question 5 of the SBAC Practice Exam is on solution steps:
A student solved 3/(x - 4) = x/7 in six steps, as shown.
Step 1: 3 = x(x - 4)/7
Step 2: 21 = x(x - 4)
Step 3: 21 = x^2 - 4x
Step 4: 0 = x^2 - 4x - 21
Step 5: 0 = (x - 7)(x + 3)
Step 6: x = -3, x = 7
Which statement is an accurate interpretation of the student's work?
A) The student solved the equation correctly.
B) The student made an error in step 2.
C) The student made an error in step 5.
D) Only x = 7 is a solution to the original equation.
We notice that this is definitely a second semester Algebra I question, since we obtain a quadratic equation in Step 4. The problem is asking us to check the solution.
As far as I can tell, there are no errors in the solution. We can double-check the specific steps mentioned in the answer choices. Step 2 is correct -- instead of dividing by 7, we multiply both sides by 7 -- and Step 5 correctly factors the quadratic polynomial. We can also check whether -3 is an extraneous root or not:
3/(-3 - 4) = -3/7
3/-7 = -3/7
-3/7 = -3/7
Therefore both values of x are correct solutions, and so the answer is A).
The girl from the Pre-Calc class correctly answers A). But the guy from the Pre-Calc class answers B) and not A). I'm not sure what error he finds in step 2 -- the cross multiplying in that step is correct.
Question 6 of the SBAC Practice Exam is on parallel lines in Geometry:
When a transversal intersects a pair of parallel lines it will create two pairs of alternate exterior angles.
Ricky claims the angles within each pair are congruent to each other, but not congruent to either angle in the other pair.
Part A
Draw a transversal through the point that supports Ricky's claim, or select NONE if there is not a situation to support the claim.
Part B
Draw a transversal through the point that refutes Ricky's claim, or select NONE if there is not a situation to refute the claim.
Well, we finally reach a Geometry question. But this question is tricky for Ricky. We notice that this question mentions alternate exterior angles, which don't appear in the Second Edition of the U of Chicago text. They do, however, appear in Lesson 5-4 of the modern Third Edition of the text.
Let's assume that Ricky knows that alternate exterior angles are congruent. But notice that Ricky is making two claims -- one is about which angles are congruent and the other about which angles are not congruent. In Part A, the transversal must satisfy both claims, while in Part B, the transversal must fail at least one of the claims. Since Ricky's first claim is always true, any counterexample for Part B must fail his second claim.
A good question to ask is, how are the angles in each pair related to each other? Well, it's easy to see that one angle in each pair forms a linear pair with an angle in the other pair. Thus the angles in the other pair are supplementary. If a counterexample to Ricky's second claim exists, the angles must be both congruent and supplementary. Such angles are called "right angles."
Therefore the transversal for Part B must be perpendicular to the parallel lines. It follows that the transversal for Part A must be oblique to the parallel lines.
The girl from the Pre-Calc class correctly answers an oblique line for Part A and a perpendicular line for Part B. But her transversals don't go through the point that is given on one of the lines. The question clearly states "draw a transversal through the point," and so the SBAC might not give her credit for this answer. I know that the guy from the Pre-Calc class certainly won't receive any credit for this question -- because he just leaves this question blank.
Hmm, this is the page on which both students start to struggle a little. I wonder whether this is the reason that today's lesson was my most popular post from the past 12 months -- so many students are having trouble with these two questions, and teachers are searching online for help. (And notice that 5 and 6 seem to be bad luck for SBAC review today -- the seventh grade English students can't complete pages 5-6, and the junior Pre-Calc students can't answer questions 5-6.)
SBAC Practice Exam Question 5
Common Core Standard:
Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
SBAC Practice Exam Question 6
Common Core Standard:
Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
Commentary: Solving quadratic equations by factoring appears in Lesson 12-8 of the U of Chicago Algebra I text, while proportions appear in Lesson 5-7 -- but notice that the steps above don't simply cross-multiply a proportion. Alternate interior angles appear in Lesson 5-6 of the U of Chicago Geometry text, but alternate exterior angles appear only in the modern Third Edition of the text. But students unfamiliar with alternate exterior angles might misread "exterior" as "interior" -- which would nonetheless lead them to the correct transversals for Parts A and B anyway. But even Pre-Calc students are having trouble with these questions, which suggests that students need to pay attention more to detail.
No comments:
Post a Comment