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 2
Task 2: 4 x 32
During the next task, some students decomposed the 4 and some students decomposed the 32, while other students decomposed both multiplicands. I loved hearing the various strategies students used. I supported students as they experimented with Using a Model to show 4 x 32. Here, a student shows how 4x32=3(10x4)+2x4. Surprisingly, another student connected the equation with our recent capacity unit and showed us how to solve 4 x 32 with the Gallon Guy Model.
Task 3: 4 x 132
During the final task, students decomposed the multiplicands in a variety of ways, including: 4x132=4(100+30+2).
I began by asking students to get out their whiteboards, student journals, and glue sticks. Once students were ready to learn, I introduced the goal for the day: I can explain how the placement of a digit determines the value in a number. I asked students to turn and talk: What is a digit? After some time, we came back together and I explicitly taught students the meaning of "digit" using the Digit Vocabulary Poster. I then asked students to write "all the digits in the entire world on their white boards." Students went right to work, writing the digits 0-9. We then counted all the digits together 0... 1... 2... 3... 4... 5... 6... 7... 8... 9... To test the students' understanding, I excitedly continued counting and shouted out 10! The students immediately protested and explained why 10 is not a digit, "A digit is only the numerals 0-9. Ten is a two-digit number. It's made of two digits."
We then went over the meaning of a number. I asked: If 0-9 are digits, then what do you think a number is? After students had time to turn and talk, I explained the meaning of a number using the Number Vocabulary Poster. I asked students to write a number on their whiteboards. Each student held up their whiteboards as they were ready. I asked some students to share how they knew they had written a number. Students would explain, "A number is made up of digits." Then we came to one student who had the digit 9 on his board. Referring to the vocabulary posters for evidence to back up his thinking, he explained, "Nine is both a digit and a number because it is a numeral between 0-9 and it is a symbol used for counting." By allowing students opportunities to participate in student discourse and develop evidence-based explanations, I knew I was supporting Math Practice 3: Construct viable arguments and critique the reasoning of others.
Next, I moved on to creating a Place Value Model on the board using Large Money on colored paper. Students loved looking at the money! It was a high-interest topic to say the least.I could hardly stop them from blurting out all sorts of comments about who was on each bill or the fact that one of them has a two dollar bill at home. I knew immediately that this would lesson would successfully connect place value to a meaningful real-world application (Math Practice 4: Model with Mathematics).
I also wanted students to be able to create the same place value model in their student journals: Hundreds, Tens, & Ones so I passed out bags of color-coordinated Money to every two students (my students' desks are arranged in groups so they are always ready to collaborate with each other). Each Bag of Money contained the following fake bills: $1, $10, $100, $1,000, $10,000, $100,000, and $1,000,000, each printed on different colors to help establish the fact that each bill represents a different value. I purposefully included only bills that would allow students to model base ten numerals.
Step by step, I placed each bill on the board, from right to left, starting with the one dollar bill and ending with the million dollar bill. I held up the one dollar bill and explained: Whenever you see a one dollar bill, I want you to think of it as the ones place. I wrote "ones" under the bill on the board. Each student also pasted a one dollar bill in their journals. I continued this with the tens and hundreds place. I then drew a house around the ones, tens, and hundreds and said: This is the Ones Period. We continued the same procedure with the thousands and millions periods, by placing the thousand, ten thousand, hundred thousand, and million dollar bills in order both up on the board in in student journals: Millions, Hundred Thousands, Ten Thousands, & Thousands. Of course a million dollar bill has never been in print so I created a Meadowlark Million (named after our school, Meadowlark Elementary).
This was a perfect opportunity to discuss how the placement of a digit determines its value in a number. I asked students: Which would you rather have... five ones? or five hundreds? Why? Students responded, "Five hundreds, because the hundreds are worth more." I ran with this and responded: Okay, so what you're saying is that the place value of the five can vary depending on its location in a number? What is place value anyway? Turn & Talk. After giving students some time to discuss the meaning of place value, I introduced the Place Value Vocabulary Poster and altogether, we came up with a fun way to act out the meaning!
Let's practice determining the place and value of digits in a number. Today, we'll start by looking at the value of the digit 5. Look at the number 5. Take a moment and work with your partner to show me $5 dollars using your bags of money. Then, use your whiteboard to answer the following questions: 1. What place is the digit 5 in? (ones place) 2. What is the value of the digit 5? (5) I modeled how to Represent the Place & Value of a Digit on White Boards. After giving students a few minutes, I asked students to turn and talk about their answers. I took this time to check for understanding. Then, we followed the same procedures and found the value of the digit 5 in the following numbers.
I choose to use similar digits in each number to take the focus off the numbers themselves. I wanted students to focus on the increasing value of the digit, five. Here, a student use money to explain the Place Value of 5 in 516 while another student uses money to explain the Place Value of 5 in 50,016.
After providing students with guided practice, they were ready to practice on their own!
To provide students with independent practice, I passed out copies of Place Value Practice A and Place Value Practice B (front to back). I printed these practice pages from the following math worksheet website!
On the front side, I asked students to work together with their partners to model each number using the bag of money before responding to each question. Here's an example of students Representing the Value of Underlined Digits using the paper money.
After completing the front side with their partners, I asked students to independently complete the back side on their own with or without the money. By asking students to model their thinking, consider the units involved, and attend to the meaning of quantities, they will also be practicing Math Practice 2: Reason abstractly and quantitatively.
As students finished the second side, they corrected their work with each other at the back table. It is through student discourse that many students are provided with the opportunity to realize and correct mistakes.
To bring closure to this lesson, I first asked students to turn in their place value papers and to place their bags of money into each group's basket. This way, the bags of money will be ready-to-use and accessible for tomorrow's lesson. I also took the time to celebrate students who were high-level mathematicians, working hard, and successfully modeling the value of digits using the bags of money.