Today's Number Talk
For a detailed description of the Number Talk procedure, please refer to the Number Talk Explanation. For this Number Talk, I am encouraging students to represent their thinking using an array model.
Task 1: 4 x 6
For the first task, 4 x 6, some students decomposed the 4 and found that 4x6 = 2(2x6). They modeled their thinking by drawing a 2x6 array and doubling it. Other students decomposed the six or even decomposed both numbers!
Task 2: 4 x 36
During the next task, we discussed 4 x 36. Some students decomposed the 36 into 30 + 6 and multiplied each by 4. These students created a 4 x 30 array and added on a 4 x 6 array. Other students decomposed both multiplicands: 4x36 = 2(30+6) + 2(30+6) or 4x36 = 1(30+6) + 3(30+6).
Task 3: 4 x 136
First, I told students the goal for today's lesson: I can represent (or show) the value (the worth) of a number in multiple ways.
I began by reviewing yesterday's lesson, Bank Teller & Representing Numbers Part 1. I reviewed key concepts by referring to the following posters: Digit Vocabulary Poster, Number Vocabulary Poster, and Place Value Vocabulary Poster. Who can explain what a digit is again? One student explained, "A digit is a symbol. For example, 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are the ten digits we use in numbers." And what does place value mean again? Students acted the meaning of this word out altogether. "Place Value.... the value of a digit's place in a number."
Place Value Chart:
I wanted call attention to the idea that "a digit in one place represents ten times what it represents in the place to its right," so prior to the lesson, I added the following to the bottom of the Place Value Chart:
____ ten = ____ ones
____ hundred = ____ tens = ____ ones
____ thousand = ____ hundred = ____ tens = ____ ones
____ ten thousand = ____ thousands = ____ tens = ____ ones
We began with the first equation. I filled in "1 ten" and asked: How many ones is equal to one ten? Students responded, "Ten!" Also, most students seemed to understand that 1 hundred = 10 tens = 100 ones. Then we got to the third line down and students were a bit confused with the number of hundreds in a thousand. To help break this task down, we counted the number of hundreds in 100... then in 200... then in 300... 400... and 500... Once students discovered that 50 x 10 = 500, they were then able to see that 100 x 10 = 1000. Here's the drawing on the board that resulted from this conversation: Scaffolding to find the # of 10s in 1000..
Bank Teller Time!
Next, I put my tie and glasses on, just like I had done the day before, and acted the part of a bank teller. I showed students the large Teacher Demonstration Withdrawl Slip and asked, Do you remember yesterday when we talked about withdrawal slips? When a customer uses a withdrawal slip, he is taking money out of the bank, right? As a bank teller, I'll ask the customer, "How would you like you money? Do you want your cash in $100 bills? $10 bills? $1 bills?" Also as a bank teller, it's important to know how to represent numbers in multiple ways. For example, if a customer wants to withdraw $50, I could give her five tens or I might give her fifty ones. This is because the value of numbers can be represented in multiple ways.
Using the Process Grid Labels, I created a Process Grid on the board. Next, I wrote "$4" on the withdrawal slip and asked, What if a customer asked to withdraw $4? I modeled how to represent a one-digit number using the process grid. If I had $4 (I wrote 4 in the Number column on the process grid), how could I could I represent this number using the money model? One student said, "Four ones!" Since this was review from the day before, the students knew where I was headed and were able to provide all the answers successfully.
Prior to moving on to the Expression column of the process grid, I reviewed the difference between anExpression and an Equation. We then discussed how we could represent the money model for the number 4 using an expression (4 x 1). At this point a student said, "I know another way we could represent 4 using the ones. We could give the customer 2 groups of two $1.00 bills." I was so happy to see students thinking outside of the box! I modeled this on the board, and then asked for the expression for this money model. Students said, "(2 x 1) + (2 x 1)." We then moved on to the Prove Equivalency column: (4 x 1) = (2 x 1) + (2 x 1) and simplified this equation to 4 = 2 + 2 and then 4 = 4. I knew that this simplification process would help students begin developing the foundational skills necessary for simplifying algebraic equations in middle school!
I continued in the same fashion: If I had $14 (I wrote 14 in the Number column on the process grid), how could I could I represent this number using the money model? After we represented 14 in two ways using the money model, provided the appropriate expressions, and proved equivalency between the two expressions, we moved on to $64 and $164.
Students were anxious to begin practicing on their own!
Prior to the lesson, printed copies of the Process Grid Labels for each pair of students. Then I placed each sheet in a page protector so that partners could create their own process grids across two desks: Student Process Grid.
I continued: I want you to all pretend you are bank tellers today! You'll come back to the counter to get your first withdrawal slip. Your job is to represent the withdrawal amount in at least two ways and to prove that these two representations are indeed equivalent, just in case the customer questions you!
I explained to students: Today, I'm looking for two things in particular! I'm looking for students to use "Sophisticated Math Talk" (I pointed to the Sophisticated Math Talk Poster from yesterday.) and I'm looking for you to prove that the value of a number can be represented in multiple ways using your own process grids! This means you'll need to provide clear, understandable, and convincing evidence.
Leveled Withdrawal Slips:
I explained the six levels of Withdrawal Slips at the Back Counter. Each level was marked with Labels. To add a little suspense, I turned the withdrawal slips over! I asked students to begin by choosing a withdrawal slip from Level A. By providing a gradual development of complexity, I was also providing a differentiated activity for all students.
Finally, I explained how students could move on to the next level of withdrawal slips: by checking their work with another group of peers. This will help develop Math Practice 3: Construct viable arguments and critique the reasoning of others.
Getting to Work!
I asked one student in each group to raise his or her hand. I asked this student to get a withdrawal slip from Level A at the back counter. Then I asked the student who was not raising his/her hand to obtain a process grid from the back table. Talk about some excited students!
During this time, I rotated the room to conference with students as they modeled multiple representations for each number.
Examples of Student Work:
Here's an example of a process grid for the number 87. This group did a beautiful job simplifying the equation to prove equivalency!
Then, students moved on to more challenging withdrawal slips:
Many students were successful at making it to the highest level of withdrawal slips! Here's an example:
I loved watching peers check one another's work. There were a couple amazing moments where students actually caught each other's mistakes, and instead of responding, "I don't think this is correct," I heard them referring to the Sophisticated Math Talk Poster and saying, "Can you tell me more about this?" Through explaining their reasoning, the students then realized their mistakes on their own! So great!
To bring closure to the lesson, I asked students to return their withdrawal slips, bags of money, and process grids.
Next, I passed out an Exit Slip, closely aligned with today's activity. I provided a bag of play money for a student who struggles with math.
I loved seeing the majority of student successfully represent the number in two ways and then prove equivalency! Here, a student turns over her paper and completes the problem again to make her work more precise: Front Side and Back Side.
Here are examples of proficiency levels:
Most of the students were proficient. Some were nearing proficient and only one student was at the novice level.