Reflection on travels in using imagination in Math

My fifth grade class math instruction and our grade level instruction has benefitted from many of the journeys, sight seeing and experiences fostered through the Math Rocks Cohort. This was my sixth year as a fifth grade classroom teacher and I have enjoyed the teaching of math this year more than I ever have. The relationships and norms that our class set on the first days of the year in math were more exciting and initiated a community of dialogue, action and math talk than prior math travels.

Our grade level initiated the year with an activity that we did in our summer training –students created and shared 4 images or words that described how they felt about math. I took these words — very positive for the most part — the class created a mission statement and participated in a Blanket problem solving challenge.  The positive experience of Day 1 math has continued and has grown over the year.

Our team has two new grade level teachers, two new special education teachers as well as the first group of students that have attended our school since it opened when they were in Kindergarten.  The number sense they have come to us has become more visible in the Number Talk and various math talk practices that I learned about and shared with my team. My team talks about math at lunch, on the playground, before and after school. Sharing successful problem solving challenges, number talks and open middle talks.

Coming together to share Number Talks and implementing the signals and talk moves shared by Sherry Parrish and Kristin Gray is a favorite time in our classroom community. It has helped some of my more reluctant math students to take risks. Turning over more of the math talk to students carries over into other parts of the math and other content area instructional day. They quickly share their ideas and challenge one another (and me) when they disagree with math thinking. Using some of the Estimation 180 with high and low estimates prompted stronger estimation skills. I have learned better how to create Number Talks that are at the just right level for most of the class and to support the new instruction of the day. This week one of the lessons focused on using a double and half strategy in multiplying decimals. With earlier units, some of the students had shared this strategy so I chose some problems that encouraged doubling and halving with whole numbers before we started our lesson.

We have started using some Open Middle problems as introductory, closing and extension activities. It has become easier to create these and using open ended problems that have numerous solutions and strategies has fostered curiosity in individual and groups of students working together and with the teachers. One of the biggest obstacles we currently face is just how to find enough time to incorporate all of the great resources we have discovered!

A good journey is one where there just isn’t enough time to do all that you want to do. Math Rocks is coming to an end but it has made our schools adventure in math more interesting and exciting!

Tackling Open Middle Problems

As a teacher/mathematician, I hope that Open Middle Problems may improve problem solving strategies and bring excitement and challenge to all the students in my classroom. Our current topic of study was multiplying fractions by whole numbers.  I used boxes to create a whole number and a fraction. I wanted to see what students could do with the open ended problem and what it would show about their understanding of the concept without a word problem context.  I displayed it as an Engage activity at the beginning of our lesson:

Multiplying a whole number by a fraction Use digits 1 through 9 to complete the boxes below & create the greatest product possible. You may only use each digit one time.

Multiplying whole numbers and fractions

All of the students were able to attempt the problem at some level. Many of the students used friendly numbers such as 9 x 1/2.  As I walked around the room, I noticed that a few students had thought about using a denominator of 1 to create a larger product. I asked several students to share their answers. One of the students shared 3/1 x 9.  The lightbulbs went on with other students as he shared how a numerator over 1 is equivalent to a whole number. When I asked him if he could find a larger product by using different digits, he was able to do so.  One student who had discovered 1/8 x 9 shared his answer with the class.  I asked the class what the product would be if the denominator and whole number were switched and it led to a short class discussion of the commutative property. The problem allowed different solutions and sharing of math understanding in a fun and trusting way.

The next week, I used a few Open Middle problems as challenge/extension problems at the end of a review of prior concepts.  I am trying to find additional ways to differentiate so that the fast finishers and above level mathematicians in my class are challenged. Using it as an extension provided both benefits and a downside. The open middle problems provided enough challenge for the students who needed it but using it at that particular time did not give them time to find out if they had the best solution or not. One student wanted desperately to know if he had the “right” answer.  I discovered that open middle problems can provide the challenge and excitement that some of my students need but adequate time needs to be built in to discuss and share out their solutions.

This week we are concentrating on number patterns with multiplicative/additive patterns and graphing these realtionships on coordinate grids.  I am planning to use these problems as either an Engage or Enrichment/Extentsion with my class:

  1. Triathalon Training

Ms. Smith is training for a Triathalon.  Each day that she trains, she plan to bike twice as long as she runs. Her goal is to stay under 30 miles each training day. Which ordered pairs will fit this goal?

Running Distance Biking Distance

2.  Write an equation that would fit this graph:

Open Middle Graph

3. Equations:

Write four equations whose solutions is y = 5.  If you graph each of these equations, would the graps be similar or different?

4. Create a possible Input and Output table for y = ⅔ of x. What is the smallest digit you can use for x?



Math Talks – Fractions

My class has been enjoying and very participative in Math Talks over the past few weeks.  They have been sharing their thinking and even the reluctant math students have started to contribute.

We were starting a new unit — the first one on fractions this year (although we have compared decimal fractions with fractions in previous weeks). I decided to also let them bring whiteboards to the group meeting area so they could record their mental answers to free them up to listen to other students’ thinking and the recording.

Here is what I planned.   Math Talk 1 Plan

I decided to start the introduction to our unit on fractions with a string from Math Talks Matters.  The curriculum for the grade level begins with students adding fractions with different denominators, building on what students were taught in fourth grade.

I started with 2 1/4+ 7/8.  I was planning to then go to 3 1/4 + 1/4, and 7 1/2 + 7/8, 3 5/6 + 1/3, 1/3 + 8/9 next.

After explaining that the whiteboards were just to be used to record their mental answers, not to work out the problems, some of the students were drawing on their whiteboards and I needed to remind them again how to use them.  One of my most reluctant students showed his signal and eagerly raised his hand. I recorded his response: 3 8/12.  Several other students also said 3 8/12 so I put a check mark by this answer.  I realized that the students were adding across the numerators and denominators but held my tongue. Another student stated 3 1/8, one said 3 1/2 of 1/4 and another stated 4.  This math talk had more student answers than we have had recently – good chance for them to talk and for me to see their thinking. I quickly decided we would stick with the one problem for today and I would add a drawn model to the open numberline for recording.

I asked the student who offered 3 1/8 to explain his thinking and recorded it with a numberline, labelling his explanation with the equation. I then drew a model underneath it.  I decided to insert more scaffolding into the talk than usual by showing models of the most commonly chosen answer. Then I drew a model for 2 1/4 + 7/8 = with 2 3/12 below it and asked for thumbs up and down for agreement. Most students now disagreed with this answer but the original sponsor of the answer was sticking with it. I then asked the other two volunteers to explain their answer and the person with 4 stated he changed his mind.

I found this Math Talk to be the most difficult one I have done so far as it revealed so many misconceptions about fractions.  I decided to plan one for the next day with more friendly fractions.  It went much more successfully and showed me that the students had both learned and remembered more about fractions after one day.

Math Plan 2

Math Talks

In posting number talks with subtraction problems for my class, I have been pleased to see that many of my students are working flexibly with numbers.  Several of them have been using the Same Difference strategy but I didn’t realize that it had a targeted name until I read the chapter.  I have not had many volunteers share the Add instead strategy so next week I want to post some of the suggestions that invite students to Add Instead and use an open number line to record their thinking.  I am very excited about the different ways that were shown for teachers recording these talks as some of them are different than the ways I have been recording them.  I can’t wait to share these with my team.  I think they make it all make more sense to me and will help my students visualize better what other students are doing.

This week we were doing graphs in class and one of the graphs we made together recorded students’ favorite subject by gender.  I was pleasantly surprised to find that the majority of my students’ favorite subject is math – both among the boys and the girls and among those who are strong in math and those who are Inclusion students.  I credit the Number Talks and a few other problem solving strategies from the Round Rock Cohort for this attitude among my students.

Using Noticings & Wonderings

Last week I used noticings & wonderings as part of our review on volume.  Students had difficulty with the irregular shapes the previous day, so I wanted to help them build their confidence about what they could do to tackle these problems and to gain more information about where some of the students were having difficulty.

Here are some of the slides:

Students were engaged and the contributions were from a wide range of the students.  Although I was disappointed that the noticings did not go as deeply as I had hoped, I was pleased that some of the discussion and problem solving done later that day built on the beginning noticings and wonderings of the class.

Number Talks Reflection

For Thursday’s computational fluency activities, I chose to use two activities that I have used with the class two or three times before and that they have some practice with the routines.  We are working on expanded notation of decimals in our core curriculum.  The previous day I had used a Today’s Number activity to guage whether they had transferred over some of the decimal to fraction names and was pleased that some of the students were making the connections.

I wanted to preassess the connections students were making between models, fractions, decimals and expanded notation.  I wrote different number values to the thousandths place on a piece of paper for each table and gave them 5 minutes to record as many different ways they could represent the number.  Then they did a quick gallery walk.  They were able to participate in this activity smoothly and stay engaged in the short timetable they had since it was familiar to them.  Today’s examples included more expanded notation than I observed yesterday.  One group posted 1.036/1000 and we discussed whether there was a more accurate way to write this number.  The target number was 1.036.  One student interestingly came up with 1.000 + (.018 x2).  Others used expanded notation, a quick accurate model, mixed numbers and fractions.

We then convened at the carpet for a short Ways to Make the Number activity starteing with 28 + 24. Students are familiar with using the signals of 1 finger, 2 fingers, 3 fingers when they have 1, 2 or 3 strategies ready and holding it close to their chest to give others wait time.  I loved seeing how Kristin Gray used these routines in the video I have seen of her number talks.

I chose this activity because the student’s computational abilities and strategies vary widely in my classroom.  I recorded several students’ stategies and then asked them to add 48 + 26.  I used parenthesis to indicate the order they said them in and to reinforce the order of operations from our first math unit.  Finally, I asked them how they would solve 48 + 30 if they knew what 48 + 26 was.  I recorded two students’ thinking.

The class is really enjoying these activities and I am constantly amazed at some of their number thinking.  I have the Inclusion class and even some of my more struggling mathematicians are participating.

I hope that these activities will carry over into more success in computational problem solving.

One thing that I most wonder about is which numbers to use and how to tie into other concepts and topics in the curriculum.Tab1-p2 Tab1-p1

Classroom Setting

The first week of school flew by quickly.  Students loved doing an adaptation of the Blanket Flip challenge – using beach towels.  After drawing images that came to their minds when they thought of math and doing a gallery walk, the class brainstormed what they want math to look like in fifth grade.  (See the photo of the chart).


I set up the room in 5 table groups of 4 to 5 so students can work together in groups, partners and have their best view of the interactive screen in our room.  Students loved using the “magic pens” to demonstrate arrays.  We also have an ovalesque shaped open space in the room that we can use for math counting circles, number talks and other minilessons.


It was exciting using some of the great ideas I discovered in the first Math Rocks sessions and from the Problem Solving training I took this summer.

Mission #5

Would you rather has some great examples. I think it would be a great way to start the morning.

Womac Rocks

I love Open Middle for fun problem solving questions! I also really enjoyed the site Would you Rather. I think it would be fun to have one question on the board every morning and have them solve and explain in their math journals! The students could even discuss with their table groups before coming to a conclusion about what/why they would choose one or the other. Go check this website out! Math Mistakes is a cool resource for teachers. I read a post about kids recognizing keywords instead of actually understanding what is going on in the problem. I see this all the time in my classroom. We used to push key words so much in math, but we have stopped and push more discussion about WHAT is happening and HOW do you know that? Math Mistakes website could help with ways to promote discussion and ways to talk through  mistakes…

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Math Rocks Mission 5

I introduced our team to Estimation 180 and we are going to use one Explore some websites to use with our students as an introductory way of modeling and practicing ways to discuss math in our classrooms this year.

I explored and set up an account on 101 Questions.  I thought some of the questions/images were interesting and creative and like the way that viewers can see other questions that have been asked.  I am planning on checking out if I can link this site to Google Classroom for students to explore to discuss questioning.

I am impressed with the math thinking and analysis of corrections organized by grade level at Math Mistakes.  I find it helpful to see other professionals analysis of what students could be doing – I think that analyzing mistakes in student work can be difficult and it is great to know there is a place to practice this needed skill.

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