SBAC Practice Test Questions 23-24 (Day 175)

Today is the second day of training for my summer school Algebra I course. I met up with the other algebra teachers, and we're starting to reach a consensus on the grading.

First of all, the administrators agreed with the second semester cohort of teachers. The assignments don't count at all. Instead, the quizzes is 40%, the tests 50%, and the district final 10%. There are also minimum thresholds required to advance in Edgenuity. A score of 68% is required to advance beyond a quiz and 65% to advance beyond a test, and 60% is needed on the final.

But as I wrote last week, I still don't like how some of the opening lessons are taught in Edgenuity. In the first unit on solving equations (Chapter 2 of Glencoe), so many of the simplest examples contain decimals and fractions. And the students are taught to solve equations by graphing f (x) = LHS and g(x) = RHS and finding their point of intersection. As I wrote last week, the reason for Edgenuity teaching it this way is a pair of Common Core Standards:

Represent and solve equations and inequalities graphically.

CCSS.MATH.CONTENT.HSA.REI.D.10
Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

CCSS.MATH.CONTENT.HSA.REI.D.11
Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

But Glencoe teaches these standards in Lesson 3-2, after solving linear equations is covered. On Edgenuity, students are expected to graph linear equations in the plane first, and then they are taught to solve linear equations in one variable. This means that, for example, students aren't taught to convert a linear equation from standard form to slope-intercept form in order to graph it -- since at that point they don't know how to solve equations for y.

Of course, keep in mind that many of Edgenuity's equations have decimal or fractional solutions -- which aren't exactly easy to graph, much less locate as the intersection of two graphs.

In fact, there's only two situations when I'd teach this as the primary method of solving equations. If the equation is nonlinear, then an algebraic method is impossible (such as 2^x = x + 3.) The other situation is when a graphing calculator is available, so it's easy to graph y_1 = LHS and y_2 = RHS on the calculator. Otherwise, I prefer to do it like Glencoe's Lesson 3-2 -- after solving equations, so that students can see the connection between equations and graphs in preparation for solving equations by graphing in the two aforementioned cases (nonlinear equation or calculator available).

The Algebra II teachers (and probably the Algebra 1B teachers as well) watch the first few Edgenuity videos as well, and they don't necessarily like how these lessons are set up either. One of them remark that the teaching method in the video is how she'd set up her Honors Algebra II class -- not a summer class where two-thirds of the students received F's during the school year.

We fear that by the end of the first week of classes, the students will be completely lost unless we resort to supplementing Edgenuity with packets. That's right -- I might have to bring back that old "Packet Song" from last year in music break. I could make the packets out of worksheets published by Glencoe -- or I might create my own worksheets based on the U of Chicago Algebra I text!

Let's get back to Fawn Nguyen's Memorial Sunday post. She's describing how similar house cleaning is to lesson planning. Since we're still in the middle of lesson planning for summer school today, I'll post the rest of her list today.

http://fawnnguyen.com/house-cleaning-and-lesson-planning/

6. Swiffer products should be regarded as essentials like toothpaste and TP.
7. The person who did not put the TV remote control away in a designated spot shall be banished from the home (or get punched in the face).
8. Make your bed every morning.
9. Never go to bed unless the kitchen is clean. (If you dread this, then don’t cook.)
10. If you find the above 9 steps difficult to implement, then try step 1 again.

And here's how Nguyen compares these house cleaning steps to lesson planning. For #6, she points out that the essential products of the classroom are equity and access. She admits that she isn't perfect when it comes to equity and access in her class. We summer school teachers are also worried about the accessibility of the Edgenuity lessons. One Algebra II teacher warns us that there will probably be a few special ed students in each class -- will Edgenuity be understandable to them? And one Algebra I teacher -- the one who's currently student teaching -- is worried about her current students who will be taking her class again in the summer.

For Nguyen, step #7 is all about respect. In lesson planning, this means that teachers should respect other teachers by giving proper credit to the source of each lesson, and sharing good lessons. This is exactly what she and I are doing on our respective blogs.

Steps #8-9 are all about fresh starts or do-overs. Nguyen reminds us that just as we expect out students to persevere after a setback, we teachers need to pick ourselves up and move forward after lessons that don't work as well as they should have. And I've devoted numerous blog posts to unsuccessful lessons and how I'd improve upon them.

For the last step, Nguyen tells us that lesson planning can be a pain, especially on weekends. This is why we summer school teachers have been given these Wednesdays to do our lesson planning. I admit that Edgenuity has made the lesson planning easier, even though we don't necessarily like how the software presents some of the material.

I want to use some of these posts to revisit my most popular posts from the past twelve months -- and that includes my Lee Canter classroom management post. Last summer, I looked at his Succeeding with Difficult Students and wrote about his worksheets #4-5. I considered three hypothetical, but common, classroom situations and how I'd handle them.

I'd like to continue in this book especially in preparation for my summer class. But unfortunately, worksheets #4-5 are the only exercises I can do easily. Many of the worksheets require watching either a video that I can't access (the book was written a quarter-century ago, before YouTube) or students in our actual classrooms. I could go back to whom I remember of my class from last year (such as the "special scholar" from my January 6th post). I wish I'd thought about this book during the multi-day subbing assignments (such as the middle school Digital Video class from back in March, or the senior Economics class from about a month ago).

There's still some time to try out these exercises in these last few days of subbing before summer. I want to make sure that my management skills are much improved from last year.

Question 23 of the SBAC Practice Exam is on comparing rates:

Nina has some money saved for a vacation she has planned.

• The vacation will cost a total of $1600.
• She will put $150 every week into her account to help pay for the vacation.
• She will have enough money for the vacation in 8 weeks.

If Nina was able to save $200 a week instead of $150 a week, how many fewer weeks would it take her to save enough money for the vacation? Enter the result in the response box.

OK, I'd count this as a first semester Algebra problem that our students might be required to solve this summer. First of all, let's ignore the given value $1600 completely, since it has nothing to do with the solution of this problem. What we want to know is, how long will it take Nina to save the same amount of money at $200/wk. as she does at $150/wk. for eight weeks? Thus this is the equation:

200w = 150(8)

200w = 1200

w = 6

So it takes her six weeks to save the money as opposed to eight weeks. Therefore it will take her two fewer weeks (that is, 8 - 6) to save the money -- students should enter the number 2.

Sometimes students might forget that even though w = 6, they must enter the number 2, since the question is not how many weeks will she save, but how many fewer weeks. Sometimes I avoid this problem by letting x = 8 - w, so that as soon as I find x, I have the number I need to enter:

200(8 - x) = 150(8)

1600 - 200x = 1200

-200x = -400

x = 2

I often solve Pappas problems this way -- using the variable x only for the final value that I need to find rather than any intermediate values. Also, if there's a system of equation and Pappas asks us to solve for x, I might eliminate y even if x is easier to eliminate, so that I find the actual asked-for value more directly.

Question 24 of the SBAC Practice Exam is on quadrilaterals:

Consider parallelogram ABCD with point X at the intersection of diagonal segments AC and BD.

Evelyn claims that ABCD is a square. Select all the statements that must be true for Evelyn's claim to be true.

• AB = BD
• AD = AB
• AC = BX
• Angle ABC isn't 90
• Angle AXB = 90

Here's some Geometry -- yes, it's been some time since I posted Geometry on this Geometry blog. I'll look at the three length equations first. AD = AB is obviously true, since the sides of a square must indeed be congruent. On the other hand, AB = BD is definitely false -- the diagonal of a square is sqrt(2) times the length of a side, not the same length as a side. And AC = BX is false as well -- the diagonals of a square (or any rectangle) are congruent, and so AC can't be half of BD (which is what BX is, as the diagonals of a parallelogram bisect each other).

Now let's look at the angle statements. The statement that ABC isn't 90 is obviously false -- the angles of a square must be exactly 90. The other statement that AXB = 90 is true, since the diagonals of a square are also perpendicular. So students must select two correct answers, AB = AD and AXB = 90.

Notice that if all Evelyn knew about her parallelogram is AB = AD and AXB = 90, that would not be sufficient for her to conclude that it's a square. A rhombus, after all, also has congruent sides and perpendicular diagonals. (In fact, given that ABCD is a parallelogram, just one of these two is sufficient for her to conclude that ABCD is a rhombus.) On the other hand, adding the falsity of the claim that ABC isn't 90 (that is, the truth that ABC = 90) to either of the two selected statements is enough for Evelyn to conclude that ABCD is a square.

SBAC Practice Exam Question 23

Common Core Standard:

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

SBAC Practice Exam Question 24

Common Core Standard:

Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Commentary: The first equation I wrote for w can be solved as early as Lesson 4-4 of the U of Chicago Algebra I text, although the distributive property in the x equation must wait for Lesson 6-8 a little later on. Meanwhile, the properties of rectangles, rhombuses, and squares are covered in Lesson 5-4 of the Geometry text.