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Sample sketch: Arrays (second grade)

Like the other sample sketches, this one is more complete than you will need to produce for your work. One striking feature of this sample is how much the teacher says, and how little the students say.

Discussion Sketch

Discussion Sketch, Side 1

On this side, quickly mark a few key characteristics of the discussion. On the other side, compactly record a discussion by noting the teacher’s questions, students' comments, and teacher’s responses to those comments.

Curriculum expectation or benchmark: Math Strand V. Numerical and algebraic operations and analytical thinking. Content Standard 1: Students understand and use various types of operations to solve problems. Benchmark 1. Use manipulative to model operations with numbers, develop their own methods of recording operations and relate their models and recordings to standard symbolic expressions and algorithms.

Objective (what students should know, understand or be able to do): 1. Students will use mathematical language to describe grouping situations. 2. Students will explore the relationship between repeated addition and multiplication.

Object to be discussed: (text, data, image, video, map, chart, representation of a phenomenon, students' personal experience, etc.):

Tile pattern blocks

Grouping pattern

____ Whole class: 5 students sit individually, 4 students in pairs

____ Small groups: 11 students divided into 3 groups of 3-4.

____ Pairs:

Uses of language

____ Students learn language

____ Students learn about language

____ Students learn through language:

Type of discussion

__ Recitation: Teacher asks what students know and builds upon that so students share a small body of knowledge.

__ Guided discussion: Teacher invites and helps students to comprehend, explore, analyze or evaluate a phenomenon, concept, problem or issue.

__ Open-ended discussion: Teacher invites students to synthesize and evaluate, and participates with restraint.

Map the classroom and enter student names


Discussion sketch, Side 2*


Teacher's question or statement

Student's statement or question

Teacher's response


Teacher begins by asking students if they remember what it means to be in a row. She asks students to put their thumbs up if they think a row goes up and down and sideways if they think a row goes across.

Class is about evenly divided between up and down and sideways.

I’m glad I brought this up. A column goes up and down and a row goes from side to side. I always remember because if you ever see a column on a house, they go up and down.


What’s a column?

A column on a house. Some large houses have columns, like on the porch.


A row goes from side to side. ... I’m going to show you a design. I’m going to make a design with my tiles. I want you to tell me about it using rows.

Teacher puts a 6X2 array made of pattern blocks on the overhead.

Various children calling out numbers.

I'm not done yet.


I want you to tell me how many rows there are in my design.

Various children call out “12.”


How many are in each row? I might be trying to trick you. How many tiles are in each row? Remember a row goes from side to side.

C: Oh, a row

C: six

C: two


9: 42

Put it up on your fingers. I’m hearing people call out. I’m seeing... How many tiles are there in a column? Put that up on your fingers. You have to use both hands.


Oh, I just gave you a hint. How many tiles are there in a column? A column goes up and down.


James doesn’t respond.


How might you tell me about this if I couldn’t see this? How could you tell me about this? Devin, if I couldn’t see this, what could you tell me about it?

D: What?


If I couldn’t see this ...

D: You wouldn't be doing anything.


But you could tell me what it looks like? What does it look like?

D: In the dark? In the dark?


If I couldn’t see this and you were telling me about it, what would you tell me?

D: I don’t know.


Look at it. What does it look like? How could you describe it to me?

(5 second pause)

If I asked you, what’s on the overhead right now, what would you say?

D: Tiles?


What kind of shape are they in?

D: Square?

They are square. They’re in a rectangle.


How many rows are there?

D: Two

There are two tiles in each row.


And there are 1, 2, 3, 4, 5, 6 rows. Going across we have two columns.


I’m wondering if it would change if I turned my paper. [Turns the transparency so the array is 2X6.]


Now how many are in each row? Ben? How many tiles are in each row now?

B: Six.


And how many tiles are in each column now?

B: Two


[Turns the array around again.]

So if I was going to write a number sentence about this, what might I say? Peter?

P: Well, you might count the first column and double it.

So I would count the column and then add that to it? So how many are in our column?


So how many are in our column?

P: Six.


So I would use six

P: Plus

Plus six.


P: Equals

Equals 12.


Is there any other way I can think about writing this in a number sentence? Teacher records 6+6=12 on overhead.


T: You could do 3+3+3+3.

I don’t have three of any groups though... But that would equal twelve. You’re right.


J: (calling out) I know six times four.


I’m wondering not how I could make 12, but how I could use this tile pattern to make a number sentence. Carole?

C: You could do 4+4+4

I could break this up into groups of four. I guess I could. I could do 4+4+4 equals 12.


And I guess by the same account, I could also take Tyrell’s idea and break this off. I could make equal groups of 3, couldn’t I? 1,2,3 ... 1,2,3 ... 1,2,3 ... yeah. I would write 3+3+3+3.


Megan, do you have another way?

M: 2+2+2+2+2+2

Oh, I could do 2 six times.


Are there still more ways I could do this?

Jason: I know, I know. I got a really good way.

Jason, what’s another way we could do this?


J: 6 times 4

We don’t have 6, four times.


Sasha: Oh, I know, I know.

Hmmm? Sasha?


Sasha: 6 times 2.

Thank you. Do we have 6 two times. We do. So we can do 6 times 2 equals 12.


Peter, do you have another idea?

P: We could do 2 times 6.

Two times six. Remember we talked about this. Two times six is the same as six times two. They’re going to be the same thing. And these all equal twelve. We have so many ways to make 12.


[Teacher asks students to take out math books. The page shows a variety of tile arrays. As a class, they write a number sentence for the first two. Then the children complete the third problem independently.]


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