Here are the reformatted materials for Unit 3: teacher presentations, student task statements, and practice. Again, I have reformatted the student task statements as "student_summaries", with the summary moved to the top, as a resource for parents and students. When I needed the tables or graphs, I made "student_worksheets." I reformatted the teacher presentations to gather stray sentences that cross pages or make charts or tables and instructions are on the same page. This time I have also included materials from Grade 6 Unit 1 that covered the area of parallelograms and triangles, since my students did not come from a classroom that used the IM materials in grade 6.

Depending on when you view this, the materials may not be complete, but I will be adding to them as I move through the unit.

Unit 3 Teacher Presentation PDFs

Unit 3 Homework and Classwork Docs

Unit 3 Student Summary Docs

Unit 3 Student Summary and Classwork PDFs

Unit 3 Homework PDFs

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In trying to get them to value both the process of exploring and the connections that come from it, I realized that they might not actually see the difference between rote procedures and conceptual understanding the way I do. After all, from a student's point of view, if she knows how to do a problem, then she understands it. So this year I put a poster up in my room that says Little Steps / Big Ideas. When we started using fraction operations at the beginning of the year, and we re-examined the equivalence of multiplying by 1/3 and dividing by 3, or why we can multiply just the numerator in a problem like 3 · 4/5, we were able to talk about the Big Ideas behind the Little Steps and the difference between the two.

Area formulas are a perfect opportunity to highlight the difference between the memorized rules and the geometric principles behind them. Once students appreciate the difference, then they can see the value of assessment items that look for evidence of understanding the Big Ideas. Someone once told me the students know what teachers value by what they test and grade. Is that Big Idea going to be on the test? If you give tests, it should be.

I like to remind them of the long history of development in mathematics; that the procedures we have today weren't always obvious to even full-time mathematicians, and took a long time to evolve. Anytime your students' explorations culminate in a procedure or a "shortcut" that makes calculations quicker and easier, you can bring into relief the algorithm that makes the mathematics handy, and the big ideas that justify it and connect it to the rest of the field of mathematics.

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- how to write the general equation for a table and what it meant

- understanding the units attached to the constant of proportionality

- seeing the direction of the relationship in a table... the independent and dependent variables

So in the lessons and the practice, I made some changes.

Lesson 4:

- I asked more explicitly about the units of the constant of proportionality

- I included a multiplication sign in the equation so that they would understand m = 3 · h as "the number of miles is equal to three times the number of hours" instead of "1 mile equals 3 hours."

- I stressed the direction of the relation, using phrasing like "relating hours to miles" instead of "relating hours and miles."

Lesson 5:

- I broke this into two parts and reordered some of the problems. In part 1, I added more basic problems and scaffolded the problems to ask for two tables instead of one, emphasizing the two different ways to look at the same relation: relating miles to hours and hours to miles. We started with tables instead of equations.

- In part 1, I also used whole number/unit fraction constants of proportionality.

- In part 2, we moved into more challenging constants of proportionality, still beginning with tables. In the final activity, I asked the students to reason about equations.

- Also, as a warm up I made a note to ask the students to fill in tables using equations to again give them experience with the multiplication in the equation. They often want to interpret the equation as a ration.

- I also took a day before part 2 and after part 2 for a lab on tongue twisters as an application of proportional relationships.

Lesson 6:

- This year the students were not really using the equations to solve for the unknown quantities... they were really using tables. The math is the same, but they weren't introduced to the usefulness of equations. I rewrote the lesson to emphasize the use of equations, and I will need to rewrite the practice sheet next year.

Here are links to the teacher presentation PDFs (I use Openboard... let me know if you would like the Openboard files) and classwork and practice sheet docs and PDFs. I have reformatted the student sheets two ways. I usually just have students work off the screen from my teacher presentations. If there are tables or graphs for them to use, I make a student worksheet. I have also reformatted the students sheets so that parents can see what we are doing in class. I moved the summary section to the top, and tightened up the formatting to make them shorter page-wise. I call those Student Summaries.

Unit 2 Teacher Presentation PDFs

Unit 2 Homework and Classwork Docs

Unit 2 Student Summary Docs

Unit 2 Homework PDFs

Unit 2 Student Summary and Classwork PDFs

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As an example, my students are in the middle of Unit 2 in the Illustrative Mathematics/Open Resources Grade 7 curriculum. We have been doing a lot of problems where the students multiply a whole number times a fraction, and they have been doing pretty well with it. Some use mental math, and some want to write out the algorithm and cancel (which drives me nuts, but it works for them). To push all of the students to think a little more deeply, I thought about asking an OpenMiddle-style question:

The first round did not push the students much out of their comfort zone. I got repetitions of problems we had been doing already: 2/3 · 5 = 10/3 (I didn’t specify whole numbers in the boxes. But later on I was glad I didn’t). There were other students who answered with whole numbers, but they were pretty comfortable with those already.

I wonder what they will do with this:

With a little bit of constraint, the answers revealed some additional insights on their part. I got things like (I apologize for not taking the time to put all the fractions in a better format):

- 1/1 · 6 = 6 , and general rumbling there are an infinite number of those.
- And lots of ½ · 12 = 6, 1/3 · 18 = 6, ¼ · 24 = 6, and again more seeming consensus that we could keep going with those.
- One student offered 1/(1/2) · 3 = 6. That was pretty cool. I wish I could remember how he got there… I think it was continuing a pattern he saw, because he asked if that was possible. I don’t think he did the division, but I wonder…?

I know exactly what to do for the next round.

I wonder what they will do with this:

I got a lot of related problems: 2/4 · 12 = 6; 2/8 · 24 = 6; 2/16 · 48 = 6… which were cool, but somehow the thinking didn’t feel deep to me. I realized that I was hoping for a general insight that the two missing numbers had to form a quotient of three. That wasn’t what I had started out looking for, but as the conversation continued over three days, that was something I noticed was missing. The students naturally went to a doubling pattern and just ran that out infinitely, without looking for a broader, more general pattern.

**What did I notice? What do I wonder?**

First of all, I notice that the order of the steps was reversed. I started out wondering, running an experiment, and then noticing. Of course, it is not really reversed… notice/wonder/notice/wonder is really a cycle of inquiry; I guess I was just spending time on the other side of the circle. But it reminds me… as I have begun to I ask my students to notice and wonder more often in class, I need to continue the cycle around, and help them to formulate actions inspired by “I wonder” for another round of “I notice.”

While writing this, I noticed that my understanding of the math behind this problem and the levels of understanding that my students might achieve evolved and changed tremendously.

I noticed that I can’t tell you exactly who understood what, or who progressed. But I did see energetic, engaged conversation, and there were a variety of strategies shared. I don’t think I would spend more time assessing individual students, but I think it would be worth pushing their understanding further with more of these questions, and also worth the time to shift to a different modality working the same concepts, like a number talk looking at:

2/3 · 9 =

2/5 · 15 =

2/7 · 21 =

2/7 · 5 =

2/14 · 10 =

I noticed that I had just been taking answers from the students, and I never stopped to ask how any of the solutions were related. In using resources like Visual Patterns or Which One Doesn’t Belong? or OpenMiddle questions, you develop a patter that can help highlight the deeper connections. And it takes many, many uses of those formats to develop those ideas. I will definitely be adding “How are the solutions related?” or “Are there any patterns you see in the different solutions?” or something like that to my OpenMiddle patter.

As I mentioned before, sometimes I wonder what concept I am actually working to develop and sometimes that changes. In this case, I wonder if the more general idea is important enough to spend more time on. I am not sure. But I don’t think that’s a problem. I am taking a skill or concept that the students are familiar with, and making some new connections, stretching it a little bit. I added time and day notations so that you realize this is just a warm up, and we do lots of other things in a class period. What gets learned will play out differently for different students. And I think that is perfect. What they all will get is more time practicing mathematical thinking. And that is my number one goal every day.

]]>First of all, I notice that the order of the steps was reversed. I started out wondering, running an experiment, and then noticing. Of course, it is not really reversed… notice/wonder/notice/wonder is really a cycle of inquiry; I guess I was just spending time on the other side of the circle. But it reminds me… as I have begun to I ask my students to notice and wonder more often in class, I need to continue the cycle around, and help them to formulate actions inspired by “I wonder” for another round of “I notice.”

While writing this, I noticed that my understanding of the math behind this problem and the levels of understanding that my students might achieve evolved and changed tremendously.

I noticed that I can’t tell you exactly who understood what, or who progressed. But I did see energetic, engaged conversation, and there were a variety of strategies shared. I don’t think I would spend more time assessing individual students, but I think it would be worth pushing their understanding further with more of these questions, and also worth the time to shift to a different modality working the same concepts, like a number talk looking at:

2/3 · 9 =

2/5 · 15 =

2/7 · 21 =

2/7 · 5 =

2/14 · 10 =

I noticed that I had just been taking answers from the students, and I never stopped to ask how any of the solutions were related. In using resources like Visual Patterns or Which One Doesn’t Belong? or OpenMiddle questions, you develop a patter that can help highlight the deeper connections. And it takes many, many uses of those formats to develop those ideas. I will definitely be adding “How are the solutions related?” or “Are there any patterns you see in the different solutions?” or something like that to my OpenMiddle patter.

As I mentioned before, sometimes I wonder what concept I am actually working to develop and sometimes that changes. In this case, I wonder if the more general idea is important enough to spend more time on. I am not sure. But I don’t think that’s a problem. I am taking a skill or concept that the students are familiar with, and making some new connections, stretching it a little bit. I added time and day notations so that you realize this is just a warm up, and we do lots of other things in a class period. What gets learned will play out differently for different students. And I think that is perfect. What they all will get is more time practicing mathematical thinking. And that is my number one goal every day.

2.5.1 Lesson Plan

2.5.1 Lesson (Problems)

2.5.1 Practice (Word)

2.5.1 Practice (PDF)

2.5.2 Lesson Plan

2.5.2 Lesson (Problems)

2.5.2 Practice (Word)

2.5.2 Practice (PDF)

Looking ahead, I think I will need to adjust Lesson 6 as well. My students did not go through the Grade 6 materials, so some of the skills are new to them.

]]>I began by asking who knew a good tongue twister. Most of my students did, and we took a few minutes to share some. Then I explained:

“I am going to give each pair of you an unfamiliar tongue twister. One of you is going to repeat the tongue twister as many times as you can while the other one counts and watches the clock. You will time your partner for 10 seconds, 20 seconds, 30, 40, and 50 seconds, and record the number of times they say the tongue twister for each time in table. What I want to know is: do you think that table will show a proportional relationship?”

We had a good conversation about that. I was surprised that most of my students said no. They thought saying a tongue twister was just too unpredictable. You can challenge them to explain how they will know if it is or it isn’t, and see how much of the previous lessons they have understood and can apply on their own.

Then they got to work. We have just finished collecting the data. In conversations with pairs we looked at tables that seemed to not be proportional at first, but then seemed more proportional as we noticed other patterns. At one table we realized the number of times the student said the tongue twister was always 1 off from 1/5 of the seconds. At another it was always just above or just below. When we reconvene after our annual trip off-campus, I will be sharing all the tables with students, asking which tables have a constant of proportionality or something close, writing equations for those tables, and talking with them about the roles of approximation and modeling in applying math to science. I think this will be a great way to lock in some of the key concepts for this unit, and make them memorable.]]>

The first decision I made for myself was in fact a mistake: I was concerned about time for some of my own topics and activities that I wanted to keep, and I thought I could do without Unit 1. Scale drawings and Proportional Relationships seemed like they would be covering similar ground, and scale drawings weren’t as critical a topic as proportions. But the two units together develop the two kinds of relationships you find in proportional relations:

Scale factors are the relationships between pairs of ratios and the constant of proportionality is the relationship across each equivalent ratio … which stays constant for each row of the table. Unit 1 develops the idea of scale factors and Unit 2 applies scale factors to tables and adds the constant of proportionality. So they really build nicely on each other.

Fraction skills are also a concern of mine (more on that later) and Unit 1 is an important part of that sequence as well. In Unit 1, the students multiply by unit fractions and are reminded of the equivalence of dividing by 4 and multiplying by ¼. They also find 3/5 of a number or 2.5 of a number, and are reminded of reciprocals. In both Unit 1 and 2 IM ventures into fractions gently, introducing a skill with whole numbers, then using numbers that the students can make sense of mentally or with relatively easy calculations. I have tried to do that myself in the past, but it is nice to have the details thought through for me, and so far it has worked well for my students.

I have been able to make things work this year with a little extra time and attention to scale factors, but I will be using Unit 1 next year.

]]>Fraction skills are also a concern of mine (more on that later) and Unit 1 is an important part of that sequence as well. In Unit 1, the students multiply by unit fractions and are reminded of the equivalence of dividing by 4 and multiplying by ¼. They also find 3/5 of a number or 2.5 of a number, and are reminded of reciprocals. In both Unit 1 and 2 IM ventures into fractions gently, introducing a skill with whole numbers, then using numbers that the students can make sense of mentally or with relatively easy calculations. I have tried to do that myself in the past, but it is nice to have the details thought through for me, and so far it has worked well for my students.

I have been able to make things work this year with a little extra time and attention to scale factors, but I will be using Unit 1 next year.

Teachers sometimes ask students to think about symbols when they should be using the symbols to represent their thinking.

1) Pose problems that ask students to think mathematically about the big ideas we are interested in.

2) Introduce mathematical symbols and tools to represent their thinking.

3) When the symbols have meaning for the students, then they can use the symbols as tools.

Easy!

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Students can’t remember (or use flexibly) math procedures because they don’t understand them. If they explore and learn the concepts, they will remember the procedures that come from understanding.

Problem: they still forget the procedures

Memory and recall of facts and procedures happens in a different part of the brain than theory and conceptual thinking. We should be talking to the two parts in different ways. Once we arrive at the procedure through conceptual work, discussion and synthesis, then we follow that up with practice to engrain the procedure. The cognitive knowledge has been established and should remain, because it “makes sense.” Maintaining the fact/procedure part just requires repetition. Revisiting the theory (through reminders and mini-lecture, not re-engagement in thoughtful questions and problems) during practice uses up time and cognitive load, making practice harder.

Problem: That doesn’t seem to move knowledge forward overall. Eventually the procedural knowledge becomes disjoint from the conceptual knowledge. Also, I suspect that the way I revisited the theory, in the form of reminders from my knowledge, also tended to nurture “facts” (chunks of conceptual knowledge) that were disjoint from real understanding.

Maybe both procedural* and conceptual knowledge need active maintenance. And they should be integrated. My scales practice certainly benefits from thinking about the theory behind the scales – not at the same time, but consecutively… what is the major Ionian scale? If I don’t know that, that becomes my lesson for today... then I run a few scales based on that. A few minutes thinking, a few minutes practicing. If I do remember the scale, then I can go a little further… practice in C, now figure out how to practice in D… then practice in D.

*[I think we need to distinguish procedures (how to multiply fractions or get the area of a triangle) from facts. And maybe distinguish particular types of facts: multiplication facts, labels for mathematical objects (angles, denominators), labels for mathematical concepts or more abstract objects (integers, linear equations, commutative property)… worth thinking more about.]

- Short bursts – engage a concept, follow up with short practice session
- Students should only be practicing the procedures that come out of the concepts they currently understand

Appendix A - Ideas for maintaining procedural knowledge-

I like the idea of little chunks… pick an idea behind a procedure, revisit that in a thought-engaging way, with a clothesline or what do you notice or open middle/open beginning (create the problem)… then quick chunk of repetitive practice of that idea… if possible with reference to the idea… maybe the procedure is written on the board next to the work that we just did.

Represent a concept visually and have students write the symbols for the problem and the answer.

Practically, what happens when some students already have learned the procedure and understand the knowledge? Perhaps just tweaking my attitude a bit: rather than asking them to explain it or go further with a little bit of chip on my shoulder, saying out loud “You can always go deeper” while really thinking “Why can’t they all just be in the same place for once?!” or wanting to prove that the student doesn’t really know it all – I can get really excited about what they know, and maybe figure out a better type of extension than representing it a different way, or explaining their thinking…

What is a multiplication problem you can’t do? What is a multiplication problem you think I can’t do? What do you notice, what do you wonder? … Is there some category of activity (3 or 4) that they might look forward to doing after a quick session of practice that would extend the

BECAUSE… procedural practice should be concept driven… students should only be practicing the procedures that come out of the concepts they understand.

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