Almost Variation with Inequalities

I‘ve started thinking about Modules 0 for Grade 6. And I’ve written my first sequence for inequalities, which I’ll show below. Although I tried to design the sequence using ideas from variation theory, I found that the specific goal I had for this sequence—writing inequalities of the form x < c and c < x from number line models—did not make it easy to think of a boatload of questions I could ask, each slightly different from the previous one. Plus, I had some slightly more robust instructional goals in mind. Still, I found that it paid off to even just try thinking about variation.

So, I start with the video below, which serves as the first (and only) instructional worked example in the sequence.


I use the Silent Teacher method, wherein I essentially show the worked example twice, the second time with my voice annotating what I’m seeing, doing, and thinking as I write the inequality to represent the two models. In the lesson, I include a brief reminder to students above the video what the inequality symbols mean and what the equals sign means.

My assumptions with regard to this content are that students have seen and used inequality symbols for a long time before they get to Grade 6, though primarily with positive numbers and not variables or negatives. So, this represents a kind of “start-again” topic, which is one reason why I include the block models along with the number line model. It is a compromise between extending the concept and reviewing it: so I do a bit of both.

Another reason I include the block models is because they make a solid, albeit abstract, connection to the use of inequalities with algebraic expressions to express relative values in situations where we don’t know one of the values. We know that q above represents a number greater than x, but we can’t mark q on the number line because we don’t know its exact value. This is what the thinking question below the video is hopefully getting at. It’s numbered in case an instructor wants to assign the sequence to a student.

The Sequence

After the video, there is a sequence of a mere 8 questions. The first of these, shown at the right, is not a typical “Your Turn” type of question, where the student tries out a technique on a very similar problem. Here we unpack the other ways to express the inequalities shown in the video—it’s important to constantly make the point that there is almost always a few different ways of looking at mathematical relationships—and we include the equation, in part because research tells us that comparing the equals sign with other relational operators reinforces the correct relational view of the equals sign.

Next up is a more typical Your Turn, with a block model and number line model both closely mirroring the models shown in the video.

Students can write n or 1 to represent the single block (or the point labeled with both n and 1 on the number line). Doing so helpfully reinforces a slightly better meaning of “variable,” which is a letter that represents any quantity, known or unknown.

And here, for the first time (in a thinking question), I ask students to relate the number line model to the blocks model.

The next question in the sequence is an example of some minimal variation. What’s different here is that the m and n block towers switch sides in the illustration, and the inequality model on the number line shifts to the right. Everything else stays the way it was.

We could continue in this way, adding or subtracting blocks, switching sides, etc., but this kind of model has limitations that don’t allow for examining more of the variation space. But we can hint at the fact that adding the same number to both sides of an inequality doesn’t change the direction of the inequality.

And that’s what we do in the next exercise in the sequence. Here also, the known number is moved along the number line. The thinking question I ask here is:

Would adding 1 block to each tower change the direction of the inequality? Why or why not?

I phrase the question as a hypothetical because, strictly speaking, it’s not evident from the diagram that I added exactly 1 block to tower m.

And Now for a Big Change

Now we see how this isn’t really a sequence of minimal variation. One reason for the change-up is that I realized too late that the model I started with could only show the greater quantity as the unknown quantity. I thought about changing to a different model, one which could show the full range of variation, but I couldn’t think of a situation that worked.

This example, in which the larger quantity (the greater height) is the known, was too good to pass up. And it gave me a context to foreshadow subtracting both sides of an inequality by the same number, which is what (kind of) happens in the next exercise.

Here, though—and again—it was not plausible to hit this balance of operations idea directly (plus, it’s outside of the scope anyway). We only hint at it. But we still ask the thinking question—again, as a hypothetical—about whether subtracting the same value from both quantities changes the direction of the inequality.

The height examples, and perhaps all of the items in the sequence, lie somewhere between minimal variation and maximal variation. At some point while designing it, I had to stop searching for more perfect examples and just run with it.

The final two items in the sequence present two more (more or less abstract) situations where inequalities seem to fit.

The first, shown at the right, is the “swarm,” which contains too many items to count, though we can know for sure that the number is a greater value than 6. Here too is an example situation that better fits with the idea of a larger unknown that couldn’t be handled by the earlier block models.

In this example, I’ve switched up the labels on the number line for a small taste of minimal variation within all the macro variation going on.

Finally, there’s temperature and a quick example showing negative numbers.

What we get at here, also, is that we haven’t left the universe of comparing numbers just because we’re introducing a little algebra. Plus, I’ve eliminated the number line model here, just for a little flavor—and it’s too close in appearance to the thermometer levels. I didn’t want that confusion creeping in.

How to Use Guzinta Math

  • Adults and students work together to complete modules—at school, at home, or both.
  • Students check in regularly. When their Practice Meter for a lesson is in the red, they should complete one or more modules in that lesson to get their Practice Meter to blue (or at least out of red).
  • Adults check in 1 day, 1 week, and 1 month after first going through a lesson module and require that all practice levels be out of the red or in the blue on that day.
  • PLEASE read the recommendations for adults at the end of this post.
  • Your Practice Meter levels are saved, even if you uninstall and then reinstall the application. To reset a Practice Meter for a lesson, click on the Guzinta Math logo at the center of the lesson homepage.

First things first: head over to the download page and install the application. When you start the application each time, you start at the homepage, which shows all 15 lessons for Grade 6.

After you have completed at least one module in one of the 15 lessons, this will activate your Practice Meter (explained below) for that lesson, and you will see a meter level to the right of the lesson on the homepage. This will allow you to, at a glance, see what concepts need your attention. Above, you can see that I’ve completed at least one module in the Ratio Names lesson (a green bar is shown to the right of the lesson). The Instructor Notes link at the bottom right allows you the ability to download the complete PDF of all the Instructor Notes for the grade level (279 pp).

Lesson Structure

Click on one of the lessons to be taken to the lesson homepage. Every lesson (with the exception of Plotting Ratios) in Grade 6 contains five modules. At the right, the lesson homepage for Ratio Names is shown. Notice the green Practice Meter level in the center icon. This mirrors the level shown on the main home page. Hover over this icon to see the numeric Practice Meter level (the one shown is at 36 currently).

Notice also the Home button in the bottom right corner of the lesson homepage. This can be found on all lesson homepages and will take you back to the main home page.

The 3 main modules, located in the center quad-panel are numbered 1 to 3 (Algebraic Expressions shown at right). The fourth square in the quad-panel is a link to the Instructor Notes for the lesson. Click on that to download a PDF of these notes.

The modules do not necessarily have to completed in any particular order. However, completing them in the order given is recommended.

Adult Interaction Is a Must

Each of the first modules in every lesson is labeled with Guided Practice (Equations and Inequalities shown at left). This means that if it is used in a classroom, it should be used as an activity centerpiece involving teachers and students. If it is used at home, the first module in particular should be the focus of both parent(s) and student.

The Instructor Notes provide an outline for interactive teaching and learning discussions with this material. It is recommended that every module—when first completed—be done together with adult and student.

To use Guzinta Math at home, a teacher may assign a module for homework or practice. Parent and student then discuss and complete the module together and, if it is requested, return the student to school the next day with the completion certificate or email it. (Parents can follow the Instructor Notes for each module as well.) The material can be used also at school or solely at school. In that case, different modules can be completed as a class and others may be assigned for homework. It is not the case that adults should be doing all or even most of the work during these interactions, but they should attend to them, rather than plop students down in front of a monitor to complete these activities alone.

Practice Meter

At 25 and below, the Practice Meter color is red. Between 26 and 79, the color is green. And, a level of 80 or above makes the color blue. Once a module has been completed—either at school with a teacher or at home with a parent or caregiver, the time is recorded for that lesson, along with the Practice Meter level. As time passes, the Practice Meter level decreases to represent a forgetting of the content.

In the first 7 days, the meter decreases at a rate of about 54% each day. That is, it loses a little more than half its value each day. The Practice Meter level of 36 (green) mentioned above would be about 16 or 17 (red) a day later, if no work is done in the lesson. From 7 days to 28 days, the meter only loses about 17% of its value each day. From 28 to 90 days, only 6% is lost each day. And from 90 days on, only 1% is lost each day.

The purpose of the Practice Meter is to provide a visual indication of forgetting and to alert students, teachers, and parents when it is time to revisit a lesson. Forgetting is very useful for learning, so it is important to allow time for the Practice Meter level to decrease before recharging it. A good schedule to keep would be to use a module together as a class or with parents and students as homework and then check in 1 day, then 1 week, and then each month after this first start. Have students get their meters out of the red or in the blue, at least for these check-ins. The goal is to keep this content alive throughout the year—yes, even if students are repeating the same questions. Repetition is excellent for novice learners!

Time-Released Practice Questions

Because the application timestamps the beginning of a student’s work in a lesson, this allows it to reveal new practice questions over time. The table below shows the number of questions (excluding Module 0) asked in each lesson, starting on Day 1 and then the extra questions revealed on Days 4, 9, and 22. These days are measured separately for each lesson, and the timer doesn’t start until after adults and students together complete at least one module in the lesson.

Lesson Day 1 Day 4+ Day 9+ Day 22+
Ratio Names 31 +5 +8 +6
Ratio Tables 28 +10 +7 +5
Comparing Ratios 45 +0 +0 +0
Plotting Ratios 12 +0 +0 +0
Measure Conversions 26 +6 +12 +6
Fraction Division 28 +6 +12 +4
Long Division 22 +6 +10 +6
GCF and LCM 33 +6 +6 +5
Negative Numbers 31 +0 +0 +0
Quadrants in the Plane 31 +7 +6 +6
Order and Absolute Value 34 +7 +6 +3
Numeric Expressions 28 +8 +8 +6
Algebraic Expressions 46 +0 +0 +0
Equations & Inequalities 29 +0 +0 +0

Across the entire Grade 6, students have 424 practice questions on Day 1. Then, over time, this number grows to 651 total practice questions. The zeros in the table show 5 lessons which do not time-release new questions over time (at the moment). The other 10 lessons do.

There are a few reasons for time-releasing new questions: (1) This ratchets up the challenge level for a lesson. The answers for questions revealed on and after Day 4 are not included in the Instructor’s Notes. And, more questions in a lesson means that it becomes slightly more difficult to raise one’s practice meter up to any given level (though the difficulty increase is very minor in most cases). (2) It helps break the repetition a little. (3) Transfer is facilitated when students revisit a previously learned topic in a slightly new context.

Contact and Future Work

I’ve written here about work on Guzinta Math that I’ll be getting to in the near and farther future. If you have any questions or technical issues, please email me at qanda[at]guzintamath.com.

Some Polite Suggestions

Here are some recommendations for how to think and behave around this material.

  • Ideally, provide guidance in one form or another on EVERY module the first time students go through it. The Instructor Notes provide the answers and some guiding questions for adults for the original practice questions (not the time-released ones). Use those notes to help you guide students. Guidance doesn’t mean you are perched on top of their shoulder, making sure they don’t get anything wrong.
  • Take. Your. Time. Students should live with this content—revisiting it regularly to maintain their Practice Meter levels—throughout the entire year. They don’t need to “get through” the content fast.
  • It’s possible to revisit the same module of a lesson every time to raise one’s Practice Meter level. That’s okay! But encourage students to complete other modules over time as well.
  • Forget whatever label you’ve assigned to your student. “Quick learner”? They still need to work through and revisit this material throughout the year, just like everyone else. Revisiting deepens their understanding and often reveals patches of misunderstanding—where they learned to play the game well but don’t really get it. “Slow learner”? They should be challenged and given high expectations like other students. Again, revisiting over the entire year is key.
  • If your student has memorized the answers to questions, that’s fine, at least once. That’s a good cue to let that module sit and allow forgetting to set in for a while and do other things. Also, if you suspect that a student is moving through stuff without thinking, that’s a clue to SIT DOWN WITH THEM AGAIN and work together. Watch the videos again and discuss. Have them explain things to you. Read the Notes. Have them generalize. Work through the exploratory module even when it doesn’t have questions to answer. In short, make them think and elaborate when otherwise they wouldn’t.

Guzinta Math 6 (Windows & Mac)

guzinta math windows

This morning, I wrapped up some final work on compiling all 15 Grade 6 Guzinta Math lesson apps into one Windows application (and a few days later into one Mac application). To try it out, visit the download page.

At the moment, since there have been few downloads of the application, you may see a warning from your SmartScreen or other defender, which you can ignore. Once it has been downloaded enough (I don’t know the number) the defender warning will go away.

Modules Zero

guzinta math windows

My main project for the next year is to build a Module 0 for each lesson. At the moment, there is some useful but mostly placeholder material there. Here is a look at the lesson homepage for Fraction by Fraction Division.

What will go in Module 0? Instruction inspired by the principles of variation theory. This Module 0 will not be monitored by the Practice Meter, and will serve to provide students a scaffold into the learning in the other modules as well as provide a differentiation tool if educators should need it.

Grades 7–8 Package and Other Platforms

Before writing the zero modules, however, I’d like to finish updating the lesson apps to Version 5.0. This version includes the zero and fourth modules, and has a bit slicker look. As of this writing, I’ve got 14 lesson apps in Grades 7–8 to update. These will all be updated as individual Chrome apps first and placed on the Chrome Web Store for download to Chromebooks.

Once these are updated to Version 5.0, I can package them together (all 16 lesson apps for Grades 7–8) into one application, just as is done for Grade 6. Done.

Other Enhancements

I’ve got a lot of different enhancements in mind as well. A version for classroom use—where there is a single computer or just a few—with student login would be pretty simple to knock out. That way, individual practice meters could still be tracked while students use just one version of the application. Done. Also, timestamping and individualizing end-of-module certificates is on my mind for the future Done. —as is a repository of practice problems.

What else? A version completely online has been mentioned. I would also like to put together some how-tos and “better” practices documentation for different use cases (home school, school and home, classroom only, etc.). New content for statistics and probability standards, correlations to popular curricula. So many things that will have to wait for another day!

Notes

Another task I have on my list to do is to build up the information around the lessons. For example, as mentioned above, each of the lesson apps in this package (and in future packages) has a Module 0 that is not tracked via the Practice Meter, although it does contain questions with correct answers. This will not change—students should have at least one fairly robust module in each lesson that isn’t about being assessed in some way. Even if these modules zero will be used less frequently, it’s important to have them, for reference, for a way to test yourself before diving into the practice meter work, etc.

In some cases, too, the fourth module of an app is not linked to the Practice Meter. These are usually exploratory modules. If a Module 4 has questions whose answers can be checked (by submitting an answer and pressing the check button), then it is linked to the Practice Meter. Otherwise, it is not.

Building Systems

It’s always fun to build things that allow for (a) generative responses and (b) flexibility in responding. This “in action” video from our upcoming lesson app on systems of equations does those two things.

Students are asked to build a linear system with a given solution (generative), and there are an infinite number of ways of doing this (flexibility)!

We’re always looking for ways to incorporate generativity and flexibility in students’ work, along with the more typical stuff. It helps make the learning a little more interesting without dispensing with the rigor.