Reasoning

Relevant Decimals Lesson

This is a lesson that I tried with a 5th grade class to give a context to decimal addition and subtraction. Most of the math problems I’ve found involving decimal computation seem “artificial.” They have a “real world” connection, but the connections are irrelevant to most 5th graders. In order to make the connections more relevant (as Dan Meyer posted in a recent blog: students want to solve it) I came up with a context for a problem that had the math content embedded, but also involved the students in the problem itself. Credit for this lesson needs to go to a 3-5 EBD class at my school. The students in this class about 3 yrs ago, loved to make tops out of connecting cubes. They did this because they were told that they couldn’t bring in any toys to class (Bey Blade was the hot toy at the time). Since they couldn’t bring in these spinning, battle tops, they created their own with connecting cubes.

The first time I witnessed these students spinning their tops, the big question they wanted to know, was whose top spun the longest. I filed the idea away until about a week ago when some 5th grade teachers at my school asked for some help with decimals. The following is the lesson I used – thanks to this class of students. It’s written as it was done. I know what I’d change when I do it again. Please take a look. Use it if you like. I’d love to hear about your results and how you change it to make it better!

Standards:

5.NBT.1 Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.

5.NBT.3 Read, write, and compare decimals to thousandths.

a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000).

b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

5.NBT.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Materials:

Connecting cubes

Decimats, or Base-ten manipulatives for modeling

Stopwatches (we used an online stopwatch that measured to thousandths of a second)

Opening:

Give students a copy of the decimat and ask what decimals might be represented. Follow up with these possible questions: What might hundredths or thousandths look like? How could you use this to model 0.013? 0.13? Share your thoughts with your partner/team?

Work Session:

The task is to design a spinning top, using connecting cubes, that will spin for as long as possible. Your group may want to design 2 or 3 tops, then choose the best from those designs. Once a design is chosen, students will spin their top and time how long it spins using a stopwatch. Each group will do this 4 times. Students should cross out the lowest time. Students will then use models and equations to show the total time for the top three spins. Students will show, on an empty number line, where the total time for their three spins lies. Students must justify their placement of this number on a number line.

Here is a sample top (thanks for asking for this Ivy!)

Top

Closing:

Students present their tops and their data, then compare their results.

Possible discussion questions:

Whose top spun the longest?

How do you know?

How much longer did the longest spinning top spin than the second longest spinning top?

Show your thinking using a model.

How many of you would change your design to make it spin longer?

How would you change it?

Decimat model 2

Students used models to explain their thinking to each other and construct viable arguments.

Students used models to explain their thinking to each other and construct viable arguments.

A Number Talks Reflection – A Look Back Over the Past 3 Years. . .

 Before reading this post, you may want to check out the GloSS and IKAN diagnostic math assessments from nzmaths at this website:   http://www.nzmaths.co.nz/mathematics-assessment.  These assessments are diagnostic interviews that teachers use with students.  When students respond with answers to the problems posed by the teacher, the teachers have to listen to students’ reasoning, not just the answer.  The information gathered is incredibly powerful and has driven our teachers to ask for resources and strategies that will help their students grow and progress through these developmental stages.

I introduced number talks to my school during pre-planning three years ago.  I read the book the previous summer and knew it would be a success if I could just get my teachers to try it.  The challenge for me was to find a reason for them to want to try this new thing called number talks in the midst of all of the other new initiatives.  I looked at what we had been doing over the past year and a half and tried to find where these number talks would fill a need.  When I discovered that need, several (more than I expected) teachers wanted me to introduce number talks with their classes immediately – during the first week of school!

 

The need I found was to improve strategies for computation to help students achieve higher strategy stages on the GloSS assessment.  Teachers had noticed that students were getting stuck on stage 4 (basically, the majority of students – even those in 5th grade – had one strategy for everything, counting on).  They were stuck because we continued to assess, but hadn’t looked at the data gathered from those assessments to come up with a course of action to help students.  The ideas were out there and we had discussed strategies before, but few teachers were implementing these ideas daily.  We wanted the pig to grow, but we were weighing it instead of feeding it!

 

When I started introducing the number talks, teachers were very interested and many were excited about out how this would work.  I worked with each teacher/class for an entire week.  For four days, I would model the number talks.  On the fifth day, the classroom teacher would take over and I would observe.  We would meet after to talk about the experience and we would discuss how the teacher would move forward from this point.  Sometimes these were difficult conversations.  What I learned from these discussions was that many teachers thought of this as a magic bullet, where teachers would talk about strategies first and then have students practice a few verbally.  This myth was dispelled as soon as I walked into the first classroom.

 

I introduced number talks to every class that year.  Some teachers wanted to see them, and then decide whether to use them.  Some knew they wanted to use them, and some just wanted 4 days with someone else teaching for 20 minutes.  And there was one skeptic, who did number talks with the expectation that they would not work.  And that was ok.  It wasn’t mandatory, just a strategy.  A tool to use to help kids help themselves.

Number Talks Assessment from 3rd grade with teacher commentary (September)

Number Talks Assessment from 3rd grade with teacher commentary (September)

I would check up on teachers every so often to see how teachers and students were doing with their number talks.  Some had stopped doing them after a while, some only did them 3-4 days a week, but there were some… Some who saw the value right away and did them religiously (I apologize for this blatant disregard of separation of church and state) every day.  These teachers took number talks and ran with them!  They not only used them to help students develop strategies, they used them to assess those strategies. They were asked to share. And they did.  During professional learning, faculty meetings, and through emails, other teachers began to notice that the number talks were beginning to show results.

 

Teachers were amazed, and so was I, when one month after introducing number talks to a third grade class, I walked in just to see what was happening and saw student after student mentally adding two three digit numbers using strategies based on place value, friendly numbers, and compensation.  These were a mixture of Special Ed., EIP, Title, and Gifted students.  They were all at different places in their understandings of the strategies they were hearing and using, but because they were developing the strategies, they were empowered to keep trying to use them and develop new strategies that were efficient (quick, easy to think about, and work every time).

 

The teachers who did the number talks consistently and with fidelity were the ones whose students reaped the rewards.  When the teachers assessed with the GloSS at the end of the year, those teachers were the ones tracking me down to tell me their stories.  I heard things like:

 

“All but two of my students went up 2 strategy stages.  The others went up 1.  It has to be the number talks.  That’s the only thing that really changed this year.”

 

“Number talks was a great way to really listen to my students and hear what they know.  The GloSS makes more sense now.”

 

“I can’t believe what my lower students said during the last GloSS assessment.  They really used what we did in those number talks.”

 

“Number talks really helped my kids with their strategies, and it shows in their other math work.  I love number talks!”

 

Number talks have been a huge success for all teachers at my school who have used them with fidelity.  We’ve hired some new teachers this year and they seem just as eager to learn about number talks as the teachers I worked with a few years ago.  Now, with all of this experience and several number talks experts, our school can offer more support than ever to these new teachers.  We’re all expecting the best.

 

Oh, and remember the skeptic. . . well, she’s one of the experts now!