A trig student told me he was hanging out with a group of friends, some who’d had me, some who hadn’t. One was bitching about his four page test.

My students snorted. “Ed’s tests are double sided single pages. Once we had a three page test, but only for the space.”

A debate ensued, and those with the widest range of math teacher experience agreed: My tests are shortest, and hardest.

I’m not sure what this means. I don’t try to make my tests difficult. But periodically I’ve perused other teachers’ tests off the copier, and…wow. They are four or five pages. The questions are straightforward. They are typically of what I would call rote difficulty–they could have peeled off a few pages of one of these tests. When math teachers snort about regurgitating algorithms, these are the tests they have in mind.

I used to have more traditional *looking* tests, but even back then I wasn’t an exact match for typical. Once I started down the multiple answer path, it became even easier to wander miles off the reservation. But without question, multiple answer tests make it easier to assess understanding on multiple topics—thus shorter tests.

This semester, I finally decided to start my class with a functions unit. Regular readers know that I’ve been beefing up my functions curriculum, after initially (as a new teacher) giving it a perfunctory treatment. But I still began the year with linear equations. This last semester start, though, I went back to the textbooks. Why do they always start with functions? I finally started to grasp the logic: beginning with functions allows the teacher to work with transformations, parent functions, mapping, as well as challenging algebra (solving for x in a square root or quadratic function, etc).

So I mapped out a basic plan:

- Function definition: domain, range, criteria
- Function notation
- Transforming functions
- Four parent functions (line, quadratic, square root, absolute value). I told them we’d be introducing lines to ignore them until the next unit.
- Transforming parent functions.
- Solving for input and output

I originally planned to introduce inverses, but the kids were maxed out. This was a much tougher first unit than linear equations, and a good chunk of the lower ability students were struggling with the abstractions. Generally, I was pleased.

Some new questions from my first functions unit test–which was a single page, double-sided.

Notice I slipped in a couple function notation questions? That’s how I save space.

Here’s a mapping question:

Again with the function notation! Am I the only math teacher whose kids simply can’t compute the difference between “f(3) = ” and “If f(a) = 3, a=”???? I do my best to beat it into their heads.

Here’s another way I use space effectively, I think:

So a graph, some free-response algebra, and conceptual understanding. (Most of them DO NOT understand how to read graphs, and missed d.) Time and again, I had to show the students how to write the equation, but they are learning how to isolate. Relatively few order of operations errors.

I didn’t ask them to graph this next one, but again, practice at setting up an equation to find the input given the output. Another plus of doing functions early is an introduction to quadratics, which is a tremendously tough Algebra 2 unit.

Hands down, this next question had the weakest response. The strongest students understood it, but many of the same students who were able to graph the square root were flummoxed by this one. Go figure. But again, notice that I assess several different knowledge areas with the same question.

**A New Quiz**

I don’t usually discuss my quizzes, which are often relatively straightforward compared to my multiple answer assessments. But I created a quiz on Thursday that I’m really pleased with. It’s my second quiz for linear functions. The students have learned the three different linear forms. The first quiz covers slope intercept and standard form, which are the forms for modeling situations. This one focuses on point slope and creating equations from points, as well as parallel and perpendicular points. We actually did much more modeling of real-life situations than this quiz shows. Usually, my quizzes are a very reliable guide to what the students have done in the previous week, but this was an attempt to see how well they could transfer knowledge and work several concepts in combination.

The quiz itself, I think was cool. I stole the nuggets of two ideas from textbooks, but the presentation and questions are mine own.

I’ve seen this crickets question in both Pearson and Holt Harcourt books. I built the graph on Desmos, and was dismayed that a number of kids counted the barely visible lines, rather than use the points. But most of them didn’t.

Notice that this is a relatively easy question. I didn’t want to focus on the algebra needed to find the y-intercept. I wanted them instead to look at the patterns (the 120 chirps is exactly halfway between 0 and 240), and think about what graphs *say* vs. what they *mean*. Most kids confused question c and d, explaining that the temperature was too cold for chirping, or that the crickets died. But after a few pushes, they go…”negative chirps?” which is fun.

Here, I’m just testing their fluency:

Lots of room for self-correction. One student asked me why all her solutions were “Neither”, and I suggested that perhaps she should check her algebra, where she’d handled a negative value incorrectly. Other students plotted the points incorrectly and, because they were only able to find slopes from the graphs, couldn’t catch their mistake–thus giving me an opportunity to reiterate the importance of using different methods to validate and self-correct.

As part of the work leading up to this quiz, they’d derived the Celsius to Fahrenheit conversion algorithm, given two points. I decided to give them the formula to see if they could recognize the errors in verbal description and work a solution using fractions.

And then my favorite:

I got the basic idea from my new favorite textbook series, Big Ideas Math, then played with the goals a bit. Big Ideas has wonderful scenarios.

As always, if you spot any errors or ambiguities, let me know.

March 5th, 2016 at 11:00 pm

In the beacon problem it states that the beacon is along the shoreline. This could be taken to mean “on the shoreline”. That is how I read it. Then plotting the shore line from the instructions becomes difficult if not impossible.

March 6th, 2016 at 12:38 am

Good point. I’ll change that in newer versions. I went through this test verbally with the kids, as I always do with a new test.

March 5th, 2016 at 11:23 pm

[…] Source: Education Realist […]

March 6th, 2016 at 6:58 am

In the one about two pairs of points and equations of lines I have often wondered what happened to the mathematically elegant form of the equation (y-y1)/(y2-y1)=(x-x1)/(x2-x1) and variations. Not only can you just shove the numbers in, but it has built in meaning in terms of slopes.

March 6th, 2016 at 2:08 pm

Not sure what that is. I do derive point slope from the slope formula.

March 17th, 2016 at 1:45 am

Just found your reply. Ok, let point A be (x1,y1) and point B be (x2,y2) Then for any point on the line joining A and B, we’ll call it P, with coordinates (x,y) the slpe from A to P is the same as the slope from P to B. That is what the formula says.

March 30th, 2016 at 2:00 am

not so elegant when y1=y2, but sure, that’s intuitive. i consider parametric form the most intuitive for line segments at least

March 7th, 2016 at 9:00 pm

I’ve been getting caught up on reading many of your posts today. I’m teaching algebra this year for the first time in awhile. This time I’ve become much more aware of the fact that how I say and write things affects student understanding. I admire how you reflect so much on assessing this understanding in an efficient way.

All that being said, I’ve found myself becoming much more careful about the language of math for my audience. In particular, being more precise, but not to the point of absurdity. For example, in the last example I believe it should be noted that the shoreline is straight (linear), as I don’t consider that an automatic assumption here. Also, I don’t think the verb ‘evaluate’ is best for the problem with the two lines. In class I’m sure it makes sense due to your examples, but I see it as ambiguous as an outsider. And RULE seems to have two meanings in the earlier problem.

I don’t mean to come off as a nitpicking obsessive. I have just been on a heavy sel-reflection kick lately, and have taken to heart the CC practice of speaking with meaning for both me and my students. Of course, it would be absurd to always state every little detail, but I am finding that it helps my students, and me, to understand math problems better when I avoid too much hand waving and you-know-what-I-means. As with improving writing, I’ve found it’s helpful to have someone else look at my work. Or if that’s not possible, to at least hear myself through more critical ears. Which is exactly what you’re doing.

Sorry for the longwindedness. And keep up the good work. I wish all teachers could get off automatic pilot and reflect upon their craft like you do.

March 8th, 2016 at 3:20 am

Precision is my downfall. But evaluate is meant in sense of “evaluate the linear equation at x=whatever”.

Thanks for the kind words. And yes, I take feedback here and change it. I also look at my quizzes and change them to make them clearer over time.

March 13th, 2016 at 4:27 am

Three unclear points about question 4:

1a) Is the shoreline a straight line, with the beacon either out to sea, or inland?

1b) Or is the shoreline kinked, so that it runs at least 100 miles due north from beacon, and at least 250 miles due east of the beacon, with the beacon at the corner of the shoreline?

2) Is the student supposed to assume that the helicopter starts at a point 200 miles east of the beacon, and 150 miles north of the beacon?

3) The phrase “homing in on the beacon” implies that the helicopter will travel to the beacon. Will the helicopter’s flight path start by travelling perpendicular to the shoreline, reach the shoreline, and then turn to go toward the beacon?

March 13th, 2016 at 5:01 am

1a) beacon is in lower left. 2) no idea, and the kids weren’t confused on that point. 3) No. 4) Yes, I realized that and will fix it in future versions.

March 13th, 2016 at 4:41 am

Question 3 has a formatting error. Some formatting tools convert open paren – capital C – close paren (C) into the copyright symbol ©.

This question is about converting temperatures, so it would be correct to use degrees Fahrenheit (°F) and degrees Celsius (°C). This would probably prevent the formatting tool from generating the copyright symbol.

As an aside:

The results of subtracting temperatures are in Fahrenheit degrees (F°) and Celsius degrees (C°). The conversion formula can be expressed as:

(Fahrenheit temperature) = 32°F + (Celsius temperature – 0°C) * (1.8 F°) / (1 C°)

or

(Fahrenheit temperature – 32°F) = (Celsius temperature – 0°C) * (1.8 F°) / (1 C°)

March 13th, 2016 at 4:56 am

Regarding test question 4’s true-false question 4a:

I was taught that if r(a) = 1, then a is an element of the set {1, -3}. In Windows, the element symbol is available using the Character Map’s Symbol font, as character 0xCE.

This is not the same as saying that a is the set {1, -3}, which is implied by the equality symbol used on the test.

March 13th, 2016 at 5:02 am

Yeah, I worried about that. Thanks for the info.

March 30th, 2016 at 1:56 am

You have a vacuously-true problem with choice d: “RULE(Jolene) = 2 => Jolene = B”. The trouble is that it’s impossible for RULE(x) to be 2. So the statement is vacuously true according to the math-logic definition of “false => P” being true for all P.

March 30th, 2016 at 1:58 am

“The trouble is that it’s impossible for RULE(x) to be 2. ”

That’s the point. The students tend to read it backwards.

And that last sentence is why I can’t deal with formal logic.

March 30th, 2016 at 2:01 am

I know that’s the point, and I consider it a good question for a student *not* knowledgable about the precise mathematical meaning of “if X then Y”. A perfected version of your answer would be “it’s possible for RULE(x) to be 2, and if it is, x is ..”

March 30th, 2016 at 2:04 am

OK, I’ll change it.

March 30th, 2016 at 2:04 am

this is the “if pigs could fly, then …” issue