To begin today's lesson, I wanted to set very high standards for the mathematical discourse expected during math time. Even though we set some expectations at the beginning of the year and listed them on a poster, Turn & Talk Guidelines, I wanted to review these guidelines and challenge my students to meet even more sophisticated goals!
I explained the goal of our opening activity: I know how to participate in sophisticated mathematical talk. Today, I'd like to make sure we are all participating in high-level math talk with each other. This is because research shows that talking and explaining our thinking during math helps all of us learn more! Let's think about what sophisticated math talk looks like and sounds like. Together, we created an anchor chart on Sophisticated Math Talk.
Before moving on, I reread each expectation aloud and asked students to choose the most important word in each expectation. For example, when I read, "look and lean in to listen," students thought that listen was the most important word in the expectation. I wanted to give students the opportunity to negotiate the meaning of each expectation and to truly think about the importance of each talking guideline. At times, students defended why one word was more important than another. I loved listening to them defend these high-level expectations!
For this lesson, I assigned math partners (based on skill level, behavior, personalities, etc.) and asked partners to gather on the carpet at the front of the classroom with their whiteboards and Bag of Money. Each bag of Money contained: about twenty of each of the following fake bills: $1, $10, $100, $1,000, $10,000, and $100,000, each printed on different colors to help establish the fact that each bill represents a different value. I purposefully included bills that would allow students to model base ten numerals.
I began by reviewing key place value concepts from yesterday's lesson, Day 1: Place Value Introduction. First, I reviewed the difference between a digit and a number and the meaning of place value using vocabulary posters: Digit Vocabulary Poster, Number Vocabulary Poster., and Place Value Vocabulary Poster.
Next, 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 wanted students to understand the purpose of the lesson so I explained: Being able to represent large numbers will help us understand numbers as well as the processes of adding, subtracting, multiplying, and dividing large numbers.
Today, we will be using the same Bag of Money as yesterday to represent the value of numbers. Can anyone tell me why we won’t be using fives? (Because we have a base 10 system. This means that our number system is based on 10. We use the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. It takes 10 ones to make a ten, 10 tens to make a hundred, 10 hundreds to make a thousand, and so on). This was a great point to introduce the Base 10 System Vocabulary Poster.
Bank Teller Time!
At this point, I put on a pair of glasses and a tie and pretended to be a bank teller. I showed students the large Teacher Withdrawal Slip and explained the difference between a withdrawal and deposit slip. Thereafter, I explained: Often times, customers will come into the bank and hand me a withdrawal slip. This means that he customer would like to withdraw a sum of money. I'll ask the customer, "How would you like you money? Do you want your cash in $100 bills? $10 bills? $1 bills?" As a bank teller, it's important to know how to represent numbers in multiple ways. For example, if a customer wants to withdrawal $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. Here's an example of a Modeled Process Grid. Next, I wrote "$6" on the withdrawal slip and asked, What if a customer asked to withdraw $6? I modeled how to represent a one-digit number using the process grid. If I had $6 (I wrote 6 in the Number column on the process grid), how could I could I represent this number using the money model? Please take out your money and represent the number 6. Students were excited to use play money during math time! Using the money from their bags, students found the only way to represent 6, using six ones. I then placed six one dollar bills in the Money Model column of the process grid.
Prior to moving on to the Expression column of the process grid, I reviewed the difference between an Expression and an Equation We then discussed how we could represent the money model for the number 6 using an expression (6 x 1). I skipped over the Prove Equivalency column for the number 6, knowing that I would address this column with the next number, 12.
I continued in the same fashion: If I had $12 (I wrote 12 in the Number column on the process grid), how could I could I represent this number using the money model? Please use your money and represent the number 12. After giving students some time, I asked: Who can show me one way to represent 12? Many students said, "One $10 bill and two $1 bills." I then placed one $10 bill and two $1 bills in the Money Model column of the process grid. How would I express this money model using an expression? Students caught on quickly, "10 x 1 + 2 x 1." Is there another way to represent 12 using your money? Students said, "Twelve $1 bills." I then placed twelve $1 bills in the Money Model column of the process grid. How would I express this money model using an expression? Students caught on quickly, "12 x 1."
Now it was time to move on to the Prove Equivalency column. I explained: What if I give a customer one $10 and two $1 when he was expecting twelve $1? Is there a way to prove equivalency and show that both representations of the same number equal the same value? Placing both expressions on either side of an equal sign, I simplified the equation:
(10 x 1) + (1 x 2) = (12 x 1)
10 + 2 = 12
12 = 12
I knew that this simplification process would help students begin developing the foundational skills necessary for simplifying algebraic equations in middle school!
Students were ready and excited to try this on their own! I asked students to return to their desks (next to their partners).
Prior to the lesson, I laminated a sheet of bulletin board paper for each group and taped Process Grid Labels across the top of each sheet of paper.
At this point, I passed these sheets out to each partner along with Vis-A-Vis markers, a small cup of water, and paper towels. (I had originally thought that dry erase markers would easily erase on laminated paper, but I was wrong and decided to go with plan B!)
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 our the poster.) and I'm looking for you to continue representing the value of a number in multiple ways using your own process grids! Students love the idea of having such a large workspace and recreating a "teacher-like" poster!
First, I asked students to model $26. Students followed the same steps as I had using the model process grid. Students represented 26 in several ways using the money model and written expressions: (2 x 10) + (6 x 1), (26 x 1), or (16 x 1) + (10 x 1). They proved equivalency. This was definitely the most complex part of the process grid.
After checking with me, students went on to represent $56 and then $216.
Here, students are Counting Money.. It's clear that representing the value of each digit in each number with money truly helped solidify Math Practice 2: Reason abstractly and quantitatively.
Here, students show how they have represented $216 in two different ways using the money model: Representing $216 with the Money Model. Then, the students move on to proving equivalency: Proving Equivalency.
I encouraged early finishers to find two more ways to represent $216. Before cleaning up, I found that one group had proved the following during this time:
(100 x 3) + (10 x 2) + (1 x 6) = (1 x 216) = (21 x 10) + (1 x 6) = (2 x 100) + (1 x 16)
To bring closure to this lesson, we first cleaned up and returned materials to the back table. Wiping off the large laminated papers became easier with time, however, it wasn't the best case scenario. Once our classroom was put back to order, I provided some students with the opportunity to celebrate partners for specially using "Sophisticated Math Talk." This is such a important component to math class. I want students to feel comfortable and willing to take risks!