Category Archives: Investigations

The other day, I began writing up my lesson plan for a second grade class I was teaching today. I drafted the lesson, got feedback, revised and ended with this plan, around the 5 Practices, going into the classroom today.

I started the lesson, as I planned, with the students on the carpet like they typically are for a Number Talk. I wrote the sentence “There are 12 people in the park.” on the board and asked them to give me a thumbs up if they could give me a math question I could ask and solve from that statement. A couple students shared after a bit of wait time and I was getting a lot of even/odd talk or questions that involved adding more information to my original sentence. I asked them to turn and talk and one little girl next to me said they could find the number of legs. When I called the group back together I asked her to share her conversation with her partner and after that, hands shot up like crazy. It ended with a board that looked like this…

I asked them if we could think about any of these in the same way? I tried to underline the “same thoughts” in the same color, but they started making connections that is got a bit mixed. A lot of there conversation turned to numbers and so I started a new slide and asked what numbers they thought of when they read those problems and why. I recorded what they were thinking…

I really liked this opening talk (15ish minutes) and really didn’t want to let them go when it was time for their recess break in the middle of math class. So, they lined up and left for 30 minutes.

When they got back, we recapped the numbers and then I gave two groups question #1 and the other two groups question #2. They had individual time to get started and then they worked as a group to share their thinking. Knowing that I was going to be trading seats at groups for them to share their problem with another table, I was walking around looking for varying strategies so I didn’t trade seats and have a whole table who solved it all the same way.

They did a beautiful job working in their original group. I saw students who had different answers for the same problem talking out their strategies and arriving at a common answer. I saw students practicing how they were going to explain it to the new table they visited. I saw students who were stuck working through the problem with their tablemates. I can tell there is such a safe culture established by Lauren, the homeroom teacher. They trade seats, shared their problem and then I had to readjust my plans.

At this point, I wanted the tables talking about what was the same and/or different about the two problems but I was running out of time. In order to pick up with that conversation tomorrow, I decided to have them come to the carpet and I chose two papers (of the same problem) that had the same answer but different strategies. I asked the students privately if they would want to share and they were both excited so I put them both under the document camera and had them explain their work. I thought they was similar enough for students to easily see they both drew the figures out but as I walked around I heard the 1st student counting each one by ones and the 2nd student counting by twos after he wrote the equation. I had them explain their work and asked the class to think about what was the same and what was different and we discussed it. Here are the two I chose:

They pointed out all of the similar things such as feet, people, two’s (but were counted differently), and the same answer. The difference was the equation which was an important thing to come up. I saw quite a few students with the correct answer but incorrect equation. A lot arrived at 22 by counting by wrote 7+2 as their equation so that was an important thing that a student pointed out.

I only had 5 minutes left, so I decided to collect their papers and pick up with the sequencing and connections tomorrow. Which I kind of love because it gives me time to be more thoughtful about how they should share them and also time to talk to their teacher about what I saw today.

So, from my previous plan, I am picking up here:

Practice 4: Sequencing

In the share, after each group has presented to the other groups, we will come to the carpet for a share. The sharing will be sequenced in the way I discussed in the Selecting part, asking students during each student work sample how it is similar and different than the ones we previously shared.

Practice 5: Connecting

The connecting I see happening through my questioning as we share strategies. I am still working on writing this part out and looking for the connections that can be made, aside from the picture to number representation connections.

The connections I would love to see students making throughout the work and sharing, is how we can combine equal groups. For example I would like the student who is drawing ones and counting them all to move to seeing those ones grouped as a 2 or a 5 depending on the context. I would love the student who is seeing the five 1’s as one group of 5 to now see that if they have 2 of them it will make a 10 and if we have 4 of them we would have 20 and really start looking at different ways to combine those groups.

The problem I am seeing in this plan is the differences in the two problems. As I sit here with the papers all over the table, I am struggling to make a sequence involving both problems. So, do I sequence a set for each problem and give each 1/2 of the class time to talk about the similarities and differences? or just choose one problem and go with that?

For problem 1, I like this sequence in moving from counting by 1’s to grouping them and then to the finding half of 34.

For question 2, I see this sequence from pictures to grouping them by people and dogs, the third shows the 8 composed but broken apart on the number line and the paper before it, and the last one starting at 14.

I collected their papers and asked them, in their journals, think about how many people and dogs there could be in the park if I just told them there were 28 legs. I thought that after their share tomorrow of this problem it would lead them into a nice problem from which some great patterns could arise. Here were a few I grabbed before I left:

and this last one was getting at some really great stuff as she got stuck at 9 people and couldn’t figure out the number of dogs. I asked her to write what she was telling me!

Looking back, I would have probably chosen just one problem to work with to make it more manageable in sequencing and making connections during the share. Having two problems was nice as far as having them explain it to others, so I like that, but I am wondering if we did #1 through this process and then split for questions for #2 and #3.

I look forward to hearing how it goes tomorrow!

~kristin

Fractions as Division…Say What?

Last year I learned to appreciate the Investigations lesson in which students explore fractions as division in a Division Table: https://mathmindsblog.wordpress.com/2014/03/18/fraction-to-decimal-division-table-noticings/. However, as I was planning this year, I thought I really missed the mark in making it an explicit understanding that fractions represent division before exploring this table. I think I completely just assumed that students understood this from previous years and investigations with sharing situations involving fractional answers. I decided to check it out this year to see what they students knew/understood before beginning the division table work. I thought it could make some really nice connections evident.

I started by putting a few sharing problem on the board: 6 subs shared by 4 people, 9 subs shared by 4 people, 3 subs shared by 5 people, and 6 subs shared by 9 people. I asked how much each person would get if they shared the subs equally.  I gave the students some individual time to work through the problems and, after that, an opportunity to share their answers and strategies with their group.  In the majority of the class, I saw the work I had anticipated based on their third grade brownie sharing work in Investigations. A lot of drawing of subs, people, and “passing out” of the pieces.

One student thought about the whole being the number of subs, divided it into the number of people sharing and arrived at decimals, however struggled when he got to the 6 shared by 9. (The side written piece is after I asked them to write what they noticed and then he proved it worked with fractional subs to start).

I had a few students that provided the perfect transition between the visual drawings and the fraction being division. They intuitively wrote the problem as a division problem and solved it using what they know about multiplication. After sharing some of the visual representations, I had these students share their equations. They explained to the class that is felt like division because they were dividing it up among people.

After our sharing, I revisited the original problems, with the students proven answers, and ask them to write what they noticed about the problems. After a few moments, I heard so many “Oh My Gosh”s and “It was really that easy”s echoing about the room. One student exclaimed, “Why did I do all of that work?” pointing to his beautiful sub and people drawings.

Here are some of their noticings (I love that they automatically start proving it to see if will always work without me even asking anymore).

This one just absolutely cracked my up and proved once again that I cannot make assumptions about student understandings….

From this point, we tested out a bunch, talked about why it will always work and then starting looking at representing our “benchmark” answers as decimals. Tomorrow, I feel great knowing we will start looking into the division table with a deeper understanding of fractions as division. The word “explicit” sometimes makes me cringe in the way of “telling” students things, however I feel in this case the understanding of fractions as division was made explicit to the students through their own work group sharing and noticing today. I think that may be the piece I have missed before… I assumed they knew and could arrive at an answer, however never made the idea explicit as a whole group.

Today was a great day in math…Say What?

-Kristin

Fractions of Fractions

This is day 1 of multiplication of fraction by a fraction and I can already see this will dramatically increase my blogging! So much to write about (for reflection, excitement and possibly confusion). With the implementation of CCSS this year, this is new in the Investigations curriculum and I am finding some things I love about it already and some things I am struggling with just a bit.

Before this lesson, students have worked in the context of a bike race of “x” number of miles and found a fraction of the race various bikers have completed.  Looked like this:

This lesson went very smoothly and I found it was more of a struggle to have them model what was happening on the fraction bar since finding the fraction of the whole number was an action they could do mentally.  To some, it seemed like an unnecessary step and to be honest, I wavered between unnecessary and yet completely necessary to make their thinking visual. I knew how important it would be in fraction x fraction, so I made them construct the model of what was happening in the story.

Today we started fraction of a fraction. It incorporates the same visual image of the fraction bar, so I love that continuation from previous lessons. It did lack a context, which at first bothered me but as we continued working, and heard the discussions, I moved past that.  Tomorrow, I am actually going to have them come up with a story to go along with a few problems to see if they can contextualize the math they are doing.  We started with a fraction of a half and then a fraction of a third, writing the expressions (some equations) as we went:

Of course, you always have the students who fly through the work and finish early as I am walking around and having discussions with the students who need some extra help, so I asked those who finished early to think about the denominator each time. Why is the product’s denominator changing from the denominators of the factors? Did you have an idea what the denominator would be before you used the fraction bar?  There thoughts were so interesting:

Absolutely LOVE all of this scratching out, changing her reasoning!

This one brings up the issue of vocabulary….fours instead of fourths, eights instead of eighths. Something I have to bring out in our discussions.

This one I struggle with because of the words double and triple. I know the number itself is doubling and tripling, but I would like to have them expand that is it happening because there is another half to split or two other thirds to split.

I love that this makes the fractions factors and products are just like whole number factors and product.

Again with the “double” word. Is it just me that struggles with this one??

I am thinking this will be one of MANY multiplication and division of fraction posts! I am just amazed at the ease the students work with the fraction bars and I like what Investigations has done thus far with these lessons. One tweak I would like made would be the directions…students are asked to “stripe 1/2 of the shaded portion” and it is becoming a tongue-twister for me 🙂 I keep saying shaded when I mean striped, minor detail but they keep correcting me!

These conversations are so rich and valuable for this understanding that it blows my mind that a teacher could just say “multiply the numerators. multiply the denominators. That is multiplication of fractions.” If I had learned fractions this way, it would have all made SO much more sense!

To be continued…

A Fraction of our Time in Math Class…

I absolutely love fraction work with my students because there is always something interesting that leaves me pondering the whys and hows of my practice….

Being a K-5 Math Specialist for a couple years offered me the opportunity to really see the trajectory of our fraction work. Now being back in the classroom, I feel I have a much better grasp as to the work the students have previously done within our math program.  In third grade, they work tremendously with halves, thirds, and sixths using polygons to represent fractions of a hexagon whole for comparison and addition/subtraction. In fourth grade, students use arrays and known equivalencies to compare and add/subtract fractions with unlike denominators by choosing an appropriate array that works for both fractions (common denominator). In addition, at each grade level, students in need of RTI enrichment, work in Marilyn Burns’ Do The Math Program which utilizes fraction strips to compare and add/subtract fractions. All of this work focuses heavily on the students’ understandings of equivalencies.

Knowing all of this still never prepares you for the power of a new model….time! I have to admit, I am a huge fan of fraction strips and array work, however today I felt the power of clocks in developing equivalencies.  I have taught this lesson in previous years and to be completely honest, never really liked it. It felt contrived, like a pizza divided into slices in another form. This year I have realized it was not the context that was lending itself to the “pizza feel,” it was me.

The class began with a discussion of a blank clock face. I asked the class if the minute hand stayed at 12 and the hour hand moved to the 1, what fraction of the clock did it turn? They said 1/12 and we chatted about how we can prove that, divided it up and went from there. Next I asked if the hands were reversed, would that give us a different fraction? Some said no, some said yes and we talked about the equivalency of 5/60.

The student questions that followed took my appreciation of the clock to another level:

“Is this the same as degrees since it is a circle?”

“Could we do the fraction for a whole day (24 hours)?”

“Can we split the minutes in half to do eighths?”

“What fraction does the clock go at the time we go to lunch?”

Holy cow, how many directions could I take this lesson??  I moved forward with having the students work with partners to find all of the fractions they could represent on the clock.  Then I asked them to use that model to add 1/3 and 1/4 on the clock. It was interesting to see the students who know how to “find common denominators” by multiplying the numerator and denominator by the same number were challenged to make a proof of their equivalencies on the clock face, while the students who needed the clock as a tool had it as their disposal to see that 1/4 is 3/12 and 1/3 is 4/12.  That clock face immediately went from something I saw as just one more pizza, to both a tool and model at the same time in my classroom.

The follow up activity is called Roll Around The Clock (http://tinyurl.com/p8sm7wa). It has fantastic variations to the game and I have student work on the positive/negative scoring system that I will post soon, it was the perfect extension for the students who needed it!

So today, in just a fraction of time, I found a new appreciation for the analog clock and hopefully improved my practice by a fraction!

-Kristin