# Patterns and Perseverance

Today in math was a test in perseverance. The students were working on the growth pattern of an animal called the Fastwalker. It was fairly easy for them to complete up to the 10th year, graph it and answer the questions regarding the line they graphed. The book did not require them to do any generalizing of a rule, however they had other plans! Here is a completed table of one of my students:

We worked on this up until lunch, discussed the line and how it was different than the constant rate of change graphs we had seen earlier. They kept asking if there was a rule for this one, so I asked them to play around with it and see what they thought. One of the students noticed that if you added all of the terms before with the term number you were trying to find, it gave her the height, so she started adding to see if it worked for the 100th term (nothing like starting small:).

While she was working on adding, another student, who had done a consecutive sums task earlier in the year in RTI w/me, realized there was an easier way to add those numbers, and you can see on the top of the page where he started playing around with pairing up the numbers:

It was so interesting to see the groups working with them and asking questions as they tried different ideas. The two of them finally saw that pairing up the numbers was working and began to generalize based on what they had done with the numbers. It was awesome because they began generalizing based on an even or odd number term because of the pairings and needing to divide the term number by 2. At the bottom of the first paper earlier in this post, you can see she wrote an even and odd rule for the pattern, while this student realized that if should work with even and odd because the decimal didn’t make a difference.

Shew…..they were sooo proud of themselves (and I was so proud of them) at the end of all of this hard work! The student who did the paper above said, “Wow, that one problem took us almost two hours!” And it was SO worth it to see the accomplishment on their faces. THIS is the reason we must make time for students to investigate their own mathematical curiosities and give them the time they need to persevere through these problems!

-Kristin

# Growth Patter Number Talk….3rd Times a Charm

Over the past couple of days, with my homeroom, I have tried a few strings of numbers to bring out the different ideas that are important when thinking about growth patterns and finding any term in a sequence, Here and here. Both days brought out many great ideas, conversations, and disagreements, however I couldn’t help but feeling the ideas we talked about in two days, could have been achieved in one and felt a little more connected. I knew it was completely the way I posed the problems, so when my second class came in yesterday, after missing a couple days of math due to testing, I was excited to try and adjust my previous work.

Apologize for the messy board, but I still cannot seem to get a handle on that recording thing…

After that, I asked them to to count by 3’s starting with 6 and stopped them at 15… Asked for the 10th term and got, as expected, 30 and 33. Then the conversation took off with proofs and some really important ideas that was hoping would emerge. I love it when the class is practically divided in half on an answer, we had the 30’s and the 33’s. I asked a 30 to explain how he got the answer and he quickly said 3 x 10=30. I saw a lot of agreement, so I asked for a 33 to share their reasoning. A student said that we “need the beginning number, three, to find out where the tenth one is. 3 x 10 is 30 but then you started three ahead of that so you add 3 to 30.” I wrote that down on the board.

A student then said something that made me have a realization, “It shouldn’t change because you are still doing 10 jumps of 3, so it HAS to be 30. 33 is 3 x 11.” In my last class I had a student who kept insisting that the 10th term remain the same no matter where we started and I could not figure out what they were trying to articulate. NOW, I understand. 30 will always be the distance between wherever we start in the sequence and the 10th term, but not the tenth from the true beginning. AH HA!

So, the beginning number was suddenly becoming very important and articulating “10th term from where” was having students agreeing that the 10th term starting from the 6 was going to be 33 but when thinking about a rule for the pattern we needed the true beginning. We were just about to head back to our desks to continue our work when a student (different than the one who had originally said it) said that we could write this one “3 x n + 3 = A” because you have to “add the three you are missing from the beginning to get the answer.” I had them turn, talk and try a few terms out and see what they thought. It was all wrapping up nicely (I was excited about it) when another student said, “You could also write 6 + (3 x n) since you are starting at 6” ….oh goodness, they just don’t ever let it end and I love it:) A disagreement arose that it would have to be “6 + (3 x n -3) because of that extra jump of 3 to start at 6.”

I always hate to say that time got the best of me, but I had missed this group for 2 days of math and I saw this conversation going lonnnnng so I had them write those ideas down in their journal to kick off our class on Monday!

I love when I have the chance to refine ideas that don’t go exactly as I had hoped they would, especially when I know it was completely how I posed the problem or asked the question. After a couple days of talks not connecting as I hoped they would, third time was a charm!

-Kristin

# Growth Patterns – Number Talk Attempt…

After Tuesday’s talk, I wanted to continue having the students look for patterns within sets of numbers. They found it really easy to find any future term in our last talk because our starting term was the change value. For example, they knew the 10th term when counting by 3’s and starting at 3 was as easy as multiplying 3 x 10. I was curious how they would think about predicting future terms if the series did not start with the same number they were by which they were counting. I brainstormed a few possible strings students could begin to think about this and, if time went well, they could look for connections between:

I decided on the following three series:

12, 18, 24, 30….

12, 15, 18, 21…

6, 11, 16, 21….

In each one I was going to have them figure the 10th term and discuss ways they were thinking about it. The idea that I wanted to emerge is the importance of accounting for the number at which they were starting and I also wanted to see how they made their predictions. I was going to end the talk by asking what the graphing story would look like if the first term was a starting height and it continued growing at this rate to connect back to our graphing stories from the previous day. As it sometimes happens, I did not make it as far as I had hoped because some amazing conversations were happening in the very first set of numbers.

I had a student count by 6’s while I recorded, starting at 12 and stopped him after I wrote 30 on the board. I purposefully stopped there because I was curious to see if students would think about the next (5th term) and double to find the 10th as they did the day before. There was an overwhelming agreement for 72 for that exact reason, but since I got a few different answers for the 10th term, I wrote them all on the board and the proving, agreeing and disagreeing began. There was one, lone 66.

There were two proofs for 72:

– Found the 5th term as 36 and doubled it to get the tenth term.

– Did 6 x 10 to get tenth term but then added 12 because he started at 12. I was excited to see he was acknowledging where the series started and the idea of the start being important.

The lone 66, then did a simple continuous count to the 10th term proving that it would be 66. Heads tilted and eyes squinted. I realized at that moment how much I appreciated that the students looked for ways to think about the 10th term without having to count up to it, but also realized that we needed to do a little more work in thinking about what was happening in the sequence.

Since I knew I was not making it past this series of numbers, I decided to connect this set of numbers to a set in which the start was 6. I wrote them on top of each other:

12, 18, 24, 30……

6, 12, 18, 24…..

It then became clear to most that the first set’s 10th term had to be 6 ahead of the bottom one because of the start. The idea of term number and the increase from term to term started to emerge. One student said the bottom series “started one term earlier so it has to end 6 earlier than the top one.” Another student bounced off of that with “A term is 6, right?”

The debate continued and great ideas were coming out about what being the 1st term meant and then one student (the 66) said well it has to be right because (the term number +1) x 6 works for every one of them. That threw most kids for a loop and definitely not a place that I thought most of the class would be ready to engage in conversation around. I let a few students question what he meant, and I let him answer them. The biggest disconnect for students was how the term number factors into finding the number in future terms. To them the term number was just labeling and not really relevant in the values.

It was time to move into our lesson for the day and I was happy with the ideas that were emerging so I had them go back to their journals and do a quick 3 minute writing of either: what they noticed between the two, what someone else said that cleared up something for them, or something they were confused about still. It was interesting to see the word “group” popping up when that really didn’t come up in the talk…

The “R” was my writing on the board….the SMARTBoard jumped and my 2 looked a bit like an R…they would not let it go of course:)

and of course there is always one that I want to find more about because it seems nothing like what the others thought about..

After this talk, we went into some pattern building with rectangular arrays and finding the nth term. So much to write about that too, but will have to save that work for another night!

Tomorrow, I want to go back to second number talk set I had intended to do today and see how the conversation builds on our thoughts from today. Do they think about the starting number now? Do they talk about the numbers as “terms”? I think I will have them journal about what they find is most important when predicting what future terms would be in the series.

-Kristin

# Growth Patterns…the Beginning

It is finally here, our last unit of the year and I have to say I am so excited to make it to Growth Patterns! Before we did some reorganization of units due to CCSS shifts, I could never get to this unit, however now having fewer units, I finally make it to the end!

If you have never looked at this Investigations unit before, here is a brief description:

This unit is really a beautiful culmination of all of the pattern noticings my student do all year long.

Being the end of the year, I know I am going to miss constantly having a class to try out my ideas on during the course of the summer, so I am making the most of the time I have left to test some different number routines.  After reading through this unit, two things jumped out at me that could potentially be incorporated into our number routine work before the lesson: predicting nth terms and relationships between two sets of numbers. So, I thought it may be interesting to do some counting and then think about relationships between our sets we make.

I first had a student count by 3’s four times: 3,6,9,12…. I asked the 10th term and everyone looked at me like I was crazy because this seemed too simple and said “30.” I asked how they knew and immediately thumbs went up and they all agreed that 3 x 10 = 30.I asked for another way you could get there? Next I got, “If you add one more 3 and get 15, you can just double it because it is doing the same thing all the way.” I wrote (12 + 3) x 2 = 30. Another student said, “You can double/half and do 5 x 6 = 30.” Me, “How does double/halving look here?” Silence for a minute, so I asked them to chat with a neighbor and then they came to thinking about jumps on a number line and explained that you were doing jumps twice as big so you need half as many.

Next I did the same process with 4’s: 4, 8, 12, 16 and I got much of the same as above. I asked if we could make any connections to the first set. I had an idea this one may be tough (but I had a plan for it in the next set), however I did have two students who noticed it was one more every time you counted by 4’s, so by the time you counted 10 times, you would have to add 10 to your total, 30+10=40.

The third problem, I wanted to give them the chance to make some connections to the 3 and 4 counts, so I did 12’s: 12, 24, 36, 48… Again, I got many of the above strategies, but then they made some connections to doing “3 times as many jumps of 4 in 12” and “4 times as many jumps of 3 in 12” and adjusting their 10th terms.

Because some students were really comfortable moving between the sets of numbers, others were still staying within one set of numbers in describing the 10th term, I gave them two sets to go back to their journal to reflect on: 2’s: 2, 4, 6, 8…. and 2/3’s:  2/3, 4/3, 6/3, 10/3….

I asked them to either talk about how they could arrive at the 10th term or make a connection to one of the previous sets of numbers…

It was interesting to see their connections between the sets, but I think for next time I need to think more about either the predictions of future terms or comparing two sets, this was a bit too much at once. Next time I may have them look at sets with constant change but not starting at a multiple of that number….and then as I was typing that, I think it would be cool to come up with a set with a missing number at the end that could be varying numbers depending on how they see the pattern, then give a term after the missing number and narrow it down to which pattern it actually is….hmmmm…have to think more about this one…

-Kristin

# Investigating Patterns

Due to ELA testing, I luck out with an extra 45 minutes of math time twice this week, and today was one!! I wanted my students to revisit the choral count we did on Monday and look deeper into the patterns they noticed. To extend that thinking, I wanted them to make some predictions about decimals that may or may not show up if we continued counting by 0.3 (Thanks so much Elham for the suggestion:)!

We revisited the count and the noticings…

I then wrote some decimals on the board, shown inside the rectangles (kinda) in the first picture above. I asked them to try and use the patterns they discovered to decide if the decimals would show up if we kept counting by 0.3. I was sure to choose a range of options so everyone had an entry into the investigation and focused on the patterns we had discussed. I loved the way they explored their patterns and it completely intrigued me the manner in which they do so.

Some explored by multiples of 3 by looking at wholes and then tenths…

Some used the patterns that involved just one place value but did not look at the decimal as a number…

This group looked at the decimal as a number and chose one pattern they know would work for any number. They broke each decimal into partial quotients to see if each part was divisible by 3…

Other groups used a variety of patterns, noticing that some would work nicely for certain decimals and not others…

The next two especially caught my attention because I had not anticipated the connections being made (I ADORE the way they think:)..

Let’s look at the first one…He saw the “switching the digits around and the other decimal always shows up” pattern working every time and decided to examine the why. His approach was so interesting. He decided to look at the missing addend between the number and its “switch” each time.  He noticed the missing addend was always a multiple of 0.9. He then started to look at the relationship between the original numbers and their missing addend. For example (and I so wish you could hear his thinking on this) the missing addend from 1.2 and its switch was 0.9 and the missing addend from 5.7 and its switch was 1.8, so what is the relationship between 1.2 and 5.7 that explains why the missing addend doubles? My curiosity is..what makes that be the next step for some students while others just notice it the missing addend is a multiple of 0.9 and are content. Loved this moment today because I got such insight into how students look at different pieces of a “puzzle” and choose to explore different relationships.

This one was so funn…

She noticed that any two numbers in her list (table), added together, had a sum that also appeared in the table or would appear, if extended. I asked her how she knew that and she showed me a few examples. “Ok, but why?” She thought for a while and then said, “Okay, it is kind of like the even plus and odd number will always give you an even number.” I could tell she was starting to make sense of the structure of numbers but having such a struggle in explaining it. To her, it seemed to just make sense and I think (hard not to make assumptions) that she was thinking about that 0.3 being a factor of both so duh, it just is.

She came back up, an hour later (she kept working on it when she left me:), and said she had it…”it is like DNA.” Ok, now I am intrigued. She explained it to me and I asked her if she could write that down for me because I thought it was so cool…

It seems like a stretch and I am still thinking about the connections, but I am stuck on the piece in which she says, ” …may look different but act similar…or act different but look similar….”

How many connections to factors and products, addends and sum and such ring true in this statement?? I love when they leave me with something to think about!!!

Another great day in math!

-Kristin