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Which equation is correct?

On Friday, I had the opportunity to hang out in some classrooms with some awesome teachers. They were having their 7th graders work in math stations. The students were working in self-selected groups of twos or threes. They worked on one question per station. During the last round of stations, a group got stuck on this question:

Set up and solve an equation to find the value of x. Find the measurement of ∠AOB and of ∠BOC.

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Each student had worked on the question individually, and then the three kids compared answers. They each had come up with a different equation.

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My immediate thought was FABULOUS!!  The second thought was what are the teacher moves so that the outcome is the students will figure out which of the equations are correct.

The teacher joined the group, and asked the student who had written m∠AOC = 180° to explain the thinking she used to write her equation. The group all agreed that the equation was correct and it was helpful to know, but it would not help them find the value of x.

The teacher then asked the student who had written m∠BOC + 11= 180 to explain the thinking she used to write her equation. Then, the teacher asked the first student to describe how her equation connected to the equation, m∠BOC + 11= 180.

The teacher asked the student who had written 13x – 15 to explain how her equation connected to the previous equation, m∠BOC + 11= 180.

At this point, the teacher left the group to finish their work on the question.

We stepped off to the side of the room to chat about what had just happened and what the next steps might be. I pulled up Michael Fenton’s recent blog post Visual Patterns + Error Analysis on my phone, and our immediate thought was we could use his format and these 3 equations (and maybe find one more) for Monday’s class.

We decided that class will start with a list of three or four equations (per Michael Fenton) and ask the students to determine which equations are correct. As Kristen moves through the room while the students are working, she’ll take some notes to help her decide which 3 kids she wants to call on and in what order she wants them to share their thinking.

The teacher moves are super important at this point because it is about organizing classroom discussions so the kids are able to see how their ideas connect to other students’ work. We want this to happen for the kids as a result of the class conversations. We want the connections to be made as a result of student-to-student conversations not as a result of the teacher TELLING kids how ideas connect and relate. We don’t believe in teachers taking that part of the learning away from kids; we do believe in kids building the connections between ideas for themselves.

So, a huge thanks to Kristen, Brittany, and Michael for putting together such great ideas that give this week an awesome start, and for letting me join in the fun.

Resources:

1.  The teachers were using questions from Lesson 2 in module 6 of the grade 7 materials published by Eureka Math. The materials can be found on the EngageNy website.

2.  Reason and Wonder:  an awesome blog written by Michael Fenton

3.  5 Practices for Orchestrating Productive Mathematics Discussions by Margaret S. Smith and Mary Kay Stein

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91

Occasionally, my job requires that I do some travel.

As I was getting ready for my most recent trip, my husband said to me, “You do realize that you are 91 miles away from 900,000.”

In airline speak, that means that I have flown almost 900,000 miles with the airline since I signed up for their frequent flier program.

So, my first question is how many miles have I flown?

My goal this year is to hit the one million mile mark.

And that makes my second question, how many miles do I need to fly to reach the one million mile mark?

How do you think your kiddos would answer my two questions, and what strategies and reasoning do you think they might use?

Multiplication and the SPRITE Christmas Tree

The last week before school let out for Winter break, I shared a photo of a “tree” made from a bunch of 12-packs of SPRITE topped with a 12-pack of FANTA. The top row had one 12-pack of FANTA, the second row had two 12-packs of SPRITE, the third row had three 12-packs of SPRITE, and the bottom row was made of thirteen 12-packs of SPRITE.

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Every time I walked past the SPRITE tree, on my way into the store over the 2 glorious weeks of winter break, I was reminded that I never went back and finished the second post about the tree. And then today, when I stopped to pick up a few things at the store, the spot by the front doors where the tree had been, was empty.

When I got home, I found the box of one-inch tiles and created a picture of the SPRITE tree.

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Then, I started moving the rows of “12-packs” so that they formed a rectangle.

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Now, I had 7 rows of thirteen “12-packs of soda” sitting on my table.IMG_6067

As I looked at the rectangle, I thought about how powerful it would be for kids if we could help them see that if they split the one large rectangle into two smaller rectangles; one rectangle made up of 7 rows of ten and a second rectangle made up of 7 rows of three, finding the amount of cans in the SPRITE tree becomes a much more manageable task.  At this point in the whole process, the tiles were getting to be annoying and not very helpful. It was time to use a different representation of the problem–a drawing.

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It was much easier to continue to work on the task using the drawing, Labels, ideas, computation notes and solutions can be recorded, and used for further reference and in conversations. Annotated illustrations are a favorite tool of mine.

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Now, all that was left was to figure out how many cans of soda would be inside all of the 12-packs that were used to create the tree.

One method that students might use to determine the number of cans of soda would be to multiply 91 by 12. I am really a fan of kids using properties of place value to figure out the count in a multiplication situation. They can do quite a lot of the work using mental math and computation strategies, and these are very helpful skills to have.

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Another method that kiddos might consider using is to determine the number of cans of soda in the seventy 12-packs and add that to the number of sodas in the twenty-one 12-packs, as seen in the array (or the area model) in the drawing below. This method, often referred to as partial product, is a really helpful strategy for students to understand and be able to use with confidence. The total number of cans is 1092, when 840 is added to 252.

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A third method might be to find the total number of cans in the 7 rows of 156 cans. When students apply the properties of place value to 156 x 7, they find that there are 1092 cans in the tree.

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At this point in this project, my son came in from after school practice and asked what I was doing. (I did have quite a bit of stuff out on the table.) I explained that I was thinking about the different methods that elementary kids might use to figure out how many cans of soda were in the SPRITE tree. He looked at the tiles and all of the notes and drawings I had on the large gridded chart paper, and said, “Mom, why don’t you just multiply the 91 x 12? You can borrow the calculator on my phone if you don’t feel like figuring it out in your head or writing it out on your paper.”

Just a some thoughts on multiplying multi-digit numbers and helping kids make some sense of how to find the count of objects in a multiplication situation.

Sprite Christmas Tree

How many cans of soda were used to make this Christmas tree?

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Our neighborhood Albertson’s grocery store has this soda can display sitting right outside their front entrance. It is built from 12-packs of SPRITE, topped with one 12-pack of FANTA. I’ve been walking past the display for a few weeks now. Yesterday, the conditions were such that I could finally snap a picture. No other shoppers out front. My Assistant Superintendent of Educational Services was not doing her weekly shopping. Most importantly, neither of my kids were with me, so they were spared the public embarrassment of their mom taking pictures of store displays.

Guess how many cans of soda are in the SPRITE Christmas Tree.

Use the too high, too low, just right system of guessing from Estimation 180.

It may help them to know that the tree is only one 12-pack deep.

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There isn’t a primary source available to verify the number of cans (12-packs) that the salesman used to create the display like there was for the Candy Cane Tree. (@modernhonolulu on Intagram)

So, here is how I figured it out:

1.  There are 13 rows. The first row has one 12-pack, the second row has two 12-packs, the third row has three 12-packs…. You get the idea. So, in the 13 rows of 12-packs there are seven sets of thirteen 12-packs or ninety-one 12 packs.

2.  Each of the ninety-one 12-packs has 12 cans, so multiply 12 x 91.

3.  There are 1092 cans in the SPRITE tree.

Candy Cane Christmas Tree

How many candy canes did the design team of Curate Decor + Design use to make this fabulous tree in the lobby of The Modern Honolulu?

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First, make a guess that is too high. Then, make a guess that is too low. Finally, make your guess. Be sure to write down each guess. (This nice system of having kids work with number and estimating is courtesy of Andrew Stadel’s work found at Estimation 180.)

If you need a little more information before you give a final estimate, here is the very top part of the tree.

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Record any revisions. How confident are you that your estimate is really, really, really close?

I am sure that by now you JUST can’t stand it and desperately need to know how many candy canes in the Christmas tree.

That was where I was at as soon as I saw the tree–just had to know how many candy canes were used. The folks at the front desk at The Modern Honolulu were super helpful and they just told me. They didn’t even make me guess.

Are you ready? Are you sure you don’t want to revise your guess one more time?

Drum roll please…

You are looking at 4510 candy canes.

Beatriz’s Number Talk

Every once and awhile a conversation happens to you that is beyond amazing. I was really lucky and that happened Monday of last week. One of the teachers who was attending the workshop came back from lunch early. We started chatting, and the amazing conversation just appeared.

I have been trying some of the Number Talks you shared with us last year. My kids are amazing. I can’t believe how much they all contribute when we do a number talk, and not just some kids, but all of my kids. This is what I have been doing and what do you think I should do next.

We talked about her work, the kids’ work, what she was learning about her kids, and how much confidence they were building. We talked about how she might go about stretching her kids’ thinking a bit more. This is the very best kind of conversation, when the magic happens, because we were talking about how to encourage kids.

The very next day, this photo appeared in my inbox.

Number Talks Practice

Beatriz’s awesome kids’ thinking all right there.

Number Talks by Sherry Parrish is book about “helping children build mental math and computation strategies”. I first ran across the book about 4 and a half years ago, and have shared it and recommended it everywhere I work. It describes an incredible process and routine to use with kids that cultivates awesome results: flexible thinking, confidence in their own ideas and strategies, and a classroom community of mathematicians.

It is a beautifully simple set of steps that produces amazing results: select a strategy and a set of expressions from the book, write down one of the expressions, ask kids to think about what the answer is and what strategy/strategies they used, ask them to share their answers, and, then, ask them to explain how they found the answer (you record their strategies). You want to collect and record ideas from 3 or 4 kids. Now comes the best part, the class talks about their ideas.

Number Talks is a great resource that includes the how-to, descriptions of strategies for each of the operations, examples of how to model kids’ thinking, pages and pages of a day’s worth of expressions, commentary from teachers and the author, and a dvd that offers glimpses into classrooms of kids and teachers using number talks as part of their daily work.

Back to Beatriz. We worked together earlier in the year. One of the pieces we worked on is how to get kids to talk in math class. We used the Number Talks book in the conversation with her grade level at her school. She took it from there. And her kids are doing amazing work–all because their teacher believed that they could.

We all need to work with people like Beatriz who create magic for kids every day.

Summertime Math Conversations

A friend and I had a wonderful opportunity to spend a day with a wonderful group of teachers who work with kids, kindergarten through grade 2. Our day was organized around three main points: developing understanding of the Standards of Mathematical Practice and what that might look like in our daily work with kids; getting organized as we begin to use a new math program; and building confidence in adjusting tasks and lessons to meet our kids’ at their current levels of mathematical thinking, reasoning, and understanding.

The mid-morning grade level conversations were organized around a game from Jamee Petersen’s book Math Games for Independent Practice. This created an opportunity to explore the impact and power of the instructional practice of creating a variety of games/tasks/activities around a specific mathematical concept, idea, or strategy.

We asked everyone to play the game Build Ten. The goal of the game is to provide an opportunity to explore the part-part-whole relationships inherent to the number 10. The game is played in groups of 2. Each pair needs a regular die-with the numbers 1 through 6, two base-ten rods, 20 base-ten cubes, and 2 work mats. Each child takes a tens rod and sets it down on his/her work mat. Player 1 rolls the die and sets out that many cubes, against the tens rod. Player 2 does the same thing. They keep doing this until one player builds ten. The two players clear their mats and play again.

The teachers all played the game a few times to gain a sense of how the game works, to have an idea of where kids might get stuck or where they might experience some confusion, and to decide what questions to ask and at what point(s) in the game. It also provided an opportunity for the teachers to experiment with how they might introduce the game to their students, what learning and understanding they want their kids gain from playing the game, and how/when/where they might use the game.

The conversation then moved on to when using the game in class what questions might you ask to help you understand how or if your students are developing specific strategies, such as, counting all, counting on, doubles, or make 10.

The final component of the morning was organized around the specific skill of adjusting a task to meet kids’ at their current levels of mathematical thinking, reasoning, and understanding. We set out a bunch of math tools: ten frames, Unifix™ cubes, centimeter cubes, rekenreks, two-color counters, and base-ten blocks. Our directions to the grade level teams were to create a task, game, or activity that offers students another opportunity to develop and use strategies for adding or subtracting whole numbers.

So, what would you choose as your focus if you were to create a new game? What tools and materials would you use? What questions would you focus on as you chatted with your students as they played the new game?

The next post will have some samples of the games and tasks that were created.

Hitting the Target

Earlier this summer, a group of us met for a few days to build a series of assessments (for kindergarten through grade 2) for our district. We spent a quite a bit of time discussing, debating, and deciding on what mathematical concepts the grade level assessments should focus on: what information do we need to know about kids’ levels of understanding, what information do we want available for our PLC conversations, what information is important to share with parents, and what information might the district need for compliance for funding? We also want to connect the assessments to the district grade level days, where we can spend some time on the ideas and the role of formative assessment in our daily work.

One of the tasks we included in the assessment plan for grade 2 is entitled Hitting the Target and comes from Illustrative Mathematics. It is designed to “help students develop flexible strategies for adding and subtracting within 20” (from the Commentary section of the task).

We selected this task for a number of reasons:

  1. The task is in a game format. We like having a game format available to support kids’ learning of really important ideas.
  2. Hitting the Target offers so many instructional uses: a game that can be shared with the entire class, an activity that pairs of kids can play throughout the year, a formative assessment task that can be used across the school year to monitor students’ development of number sense (and fluency).
  3. The task supports the development of flexible strategies—as noted above. That is a really important aspect of Hitting the Target. It also connects to other routines and opportunities that are being woven into classrooms in our district: Number Talks and Counting Circles.
  4. Formative assessment is an inherent component and attribute of the task. We can learn so much about students’ ideas and mindsets about math as they play Hitting the Target.

SUMMARY of Hitting the Target

Play the game as a whole class. This allows all kids to understand the technical aspects of the game, as well as, provides opportunity for them to see the range of ideas and strategies that can be used.

  • You need a set of number cards from 1 to 10. (pdf provided by Illustrative Mathematics)
  • Ask a student to select 5 number cards from the set of 1 to 10.
  • Have another student select a target number between 10 and 20.
  • Students must add and/or subtract 2 or more of the 5 number cards to get the target number.
  • Ask students to share the number combinations and explain the strategies they used to hit the target number. Record the expressions.
  • As the strategies are shared, provide labels/names for them: making 5s, making 10s, using known facts.
  • Once students are comfortable with the game and understand how it works, they can play in pairs.

VARIATIONS and Suggestions

  1. Use a regular deck of cards: aces through 10. Aces have a value of 1. We like this modification because the possibility for doubles is available.
  2. Use dice. You can use dice that range 0 – 5, or 5 – 10, or make your own dice using blank cubes or blank dice.
  3. You can use a deck of ten frame cards. (www.52pickup.me)
  4. Give kids a record sheet so that they can keep track of their work.
  5. Kids can keep track of their work on big white boards (http://www.whiteboardsusa.com). They can also use chart paper/construction paper and markers.

Regardless of the materials and the target number, the task is designed to support learning opportunities for kids to build computational fluency (and confidence). “Computational fluency refers to having efficient, accurate, generalizable methods (algorithms) for computing numbers that are based on well-understood properties and number relationships.” (From NCTM—Principles and Standards for school mathematics, as quoted in the Commentary section of the task.)  The idea of computational fluency is one that is the focal point of lots of conversations; teachers are talking about the depth and range of fluency in mathematics, what fluency looks like in classrooms, and how this is the same and/or different from the previous set of state standards.  Lots of opportunities for professional dialogue.

We really liked all of the possibilities this task offers kids and their teachers. It provides kids with multiple opportunities, over time, to build number sense. Using Hitting the Target in many settings and situations within their classrooms provides teachers with a range of data and information about their kids and how their students’ thinking and strategies are being developed. The options to vary and modify Hitting the Target means that it is accessible to all kids – everyone has an entry point into the mathematics conversation – and that is so important.

Planning for Grade Level Work Sessions

Tomorrow is a day filled with grade level work sessions focused around starting number talks in classrooms.  We will be using Sherry Parrish’s book, Number Talks, to help frame each of the conversations as we discuss the purpose of number talks–to help students build mental math and computation strategies.

We will select an operation, a focus strategy, and the set of number talk problems that will fit the group of students in each of the classrooms where we’ll be working.  Each of the grade level conversations will then focus on making some predictions about the strategies the kiddos will use; which students will participate; how the level of participation might change during the number talk; and how kids use of strategy might change over the course of the number talk.

And, then we’ll try out each of the number talks we planned–fishbowl style.  I’ll facilitate the number talk with the kids and my colleagues will collect data, make notes about how kids’ thinking and ideas change over the course of the number talk, and record the various strategies kids use.  Afterwards, we’ll debrief the learning that occurred during the number talk:  what did we see and hear, why do we think those things happened, what did we learn, and what’s next. (Debriefing protocol based on the work of Barry Tambara.)

Tonight, I’m organizing the notes I made yesterday from the book Intentional Talk:  How to Structure and Lead Productive Mathematical Discussions by Elham Kazemi and Allison Hintz to include in tomorrow’s conversations, as well as, selecting some number talk problems that we might want to consider using.  The note taking/record sheet is all printed and ready to go.

Here’s one of the number talk problems that I predict might generate a variety of strategies as kids determine how many dots there are in total in the three ten-frame cards, by using the question structure from Sherry Parrish’s Number Talks:

How many dots do you see and how do you see them?

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What strategies do you think the students will use to determine the number of dots?

 

 

 

 

 

Addition Strategies

Last month at NCTM 2014 in New Orleans, I picked up a copy of Intentional Talk:  How to Structure and Lead Productive Mathematical Discussions by Elham Kazemi and Allison Hintz.  Chapter 2 is focused on the idea of open strategy sharing.  The authors define an open strategy sharing talk as one that is designed around a mathematical problem that “lends itself to multiple solutions.  Students listen for and contribute different ways to solve the same problem….The goal of open strategy sharing is to bring out a range of possible ways to solve the same problem and build students’ repertoire of strategies.”(page 18)

The authors’ idea of an open strategy sharing talk connects very nicely to the work of Sherry Parrish in her book, Number Talks.  Cathy Fosnot discusses the idea of number strings in the Young Mathematicians at Work Series.  Their collective ideas offer suggestions to and encouragement for teachers as they design learning opportunities for students that are focused on building strategies, developing flexibility in thinking, and analyzing how to approach mathematical situations.

If I were to set out these three ten-frame cards for a number talk with some first graders, what strategies do you think the kids would use to answer:

“How many dots do you see and how do you see them?”

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(question structure is from Number Talks by Sherry Parrish)

Here’s my list.

1.  Counting All:  The child (typically) starts at the top left and counts each and every dot.  She may start at a different spot or with a different card, but she count by ones.

2.  Counting On:  The child knows that the top card has 9 dots.  She moves to the second card and continues counting, “10, 11, 12, 13, 14, 15” and then goes on to the third card and counts each of the five dots, stopping at 20.  As with the counting all strategy, the child may not necessarily start with 9.  She may begin with the card that she just knows, that she can subitize, and move into counting on from there.

3.  Making Tens:  The child looks for dot combinations that make 10 or ways to make a ten.  In this case, the student visually moves the dot from the second row of the 6 card into the empty space on the 9 card, decomposing the 6 into a 5 + 1.  Now, he has created a 10 and a 5.  Then, he can combine the 5 (from the 6 card-with one dot moved to the 9 card) + 5 (the third card); 5 + 5 = 10.  So, 10 + 10 is 20.

The child might start with the 5 card, and decompose the 6 into a 5 + 1, and say that 5 + 5 is 10.  Then, one dot from the 6 card is put with the 9 card, creating ten.  10 + 10 is 20.

4.  Doubles:  In this example, the doubles strategy and make 10 strategy overlap in the fact that double 5s is 10.  The change in language is subtle.  The child might start out decomposing the 6 into a 5 + 1, and move the 1 with the 9.  Double 5s in the top card is 10.  That leaves two 5s in the last two cards.  The child says that I know that double 5s is 10.  So double 10s is 20.

5.  Other:  Kiddos will come up with interesting ways because they want to share and they want to have a contribution when the group is asked by their teacher, “Does anyone have a different way?”  Kids can get really creative with their strategies and provide you with a great opening to talk about efficiency and that is a good problem to have.

FINAL THOUGHTS:  If you have been using number talks regularly in your class, at this time of the year most of your students will be able to move the dots around, visually, and hang onto and organize their moves into logical and easily-followed explanations.

You may also have some students who will benefit from working with you on this question (or a similar question) in a small group setting.  They will be more comfortable using counters and blank ten-frames to help them describe their thinking about the number string.  The counters allows kids (and you) to physically move the dots from one card to another; to directly model the thinking being shared in each student’s explanation.  If you use two-color counters as your dots or different color counters (such as UNIFIX™ cubes or base ten ones’  blocks) for each number in the number string, kids can actually ‘see’ the decomposing and composing of the numbers.

Another option you might want to consider is to use white board markers and write and draw arrows and other notations right on the cards.  The marker easily comes off the cards with an eraser or a wipe.

So go ahead, organize a number talk for your kids and see what happens.  This is a great time of year to try some new things–you know your kids and they know you–everyone is comfortable.