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 explore the associative property of multiplication. I began by explaining the Associative Property of Multiplication Poster: When three or more numbers are multiplied, the product is the same regardless of the order of the multiplicands. If we have (a x b) x c, we could rearrange the multiplicands: a x (b x c) and get the same product.
Modeling the Associative Property: (6 x 3) x 2
I wrote this task on the board: Teacher Model, (6 x 3) x 2. I reminded students: We always solve whatever is inside the parenthesis first. So what's 6 x 3? Students responded, "18!"What's 18 x 2? Students responded, "36!" What would happen if we rearranged the multiplicands? I then wrote: 6 x (3 x 2) and (2 x 6) x 3 on the board. Students did the same on their own white boards: Student Whiteboard (6 x 3) x 2. Students began solving equations on their own. In no time, I heard expressions of astonishment, "It worked!" "Wow! We go the same answer!" I then responded: I wonder if this happens every time!
Students Testing the Associative Property
Next, I asked students to test the associative property using their own numbers. Here are a few examples:
Some students even tried four multiplicands!
As students completed this task, I asked them to share their findings with others. We also shared strategies as a class. I modeled students' thinking on the board while the students used math words to explain how they used the associative property. I then asked: Is the associative property always true? Does it always work? Through investigating this property on their own, most students were convinced that they could trust the associative property.
Goal & Introduction
To begin, I invited students to bring their whiteboards and sit on the front carpet, closer to the board. I projected a Powerpoint Presentation: Checking for Reasonableness. I introduced today's Goal: I can check for reasonableness when multiplying. I explained: Sometimes I'll go to the grocery store with a $20 bill. Perhaps I want to buy as many cartons of eggs as possible with twenty dollars. If I see that a carton of eggs is on sale for $1.99, I might think, "I can buy 10 cartons." But, just to check for reasonableness, I'll round $1.99 to $2.00 and multiply $2 x 10 cartons to get $20.
Modeling Checking for Reasonableness
I continued on to the Next Slide in the lesson. I modeled Task 1, 5 x 11 on the board, step-by-step as students also completed the task on their own white boards: Students Modeled on White Boards 5 x 11.
I began with the algorithm, 11 x 5, asking students: What is 5 x 1? (5). And what is 5 x 1 again? (5). So what is the product for 11 x 5? (55),
Next, we moved on to checking for reasonableness. I rewrote the problem 11 x 5. I explained: When checking for reasonableness, I normally only round one of the multiplicands. I know that 5 is already a friendly number. What could we round the 11 to in order to make it friendly as well?Students responded, "10!" I rewrote the problem: 10 x 5 and asked students: What's 10 x 5? (50).
How do we know our answer is reasonable? Students remembered when we checked for reasonableness with addition and subtraction and responded, "Because 50 is close to 55."
At this point, we moved on to the next slides and followed the same process. With each new slide, I released more and more responsibility to students by slowly modeling steps (which gave students the opportunity to work ahead of me). Here's the modeling for each task:
As students became more comfortable with the strategy, they began experimenting with different ways of rounding (such as rounding 714 to 720 instead of 700 or rounding 714 x 9 to 715 x 10). This gave students the opportunity to attend to precision (Math Practice 6) as students experimented with rounding in order to get a rounded product that was closest to the exact product.
Encouraging Mathematical Discourse
As students finished each task, I asked them to turn and talk about the strategy that they used. This provided students with the opportunity to engage in Math Practice 3: Construct viable arguments and critique the reasoning of others. Many times, conversations helped solidify student understanding of the check for understanding method. Other times, students were able to find mistakes by talking through it.
I knew students were ready to continue practicing with their partners!
Picking math partners is always easy as I already have students placed in desk groups based upon behavior, abilities, and communication skills. Before students began working, I asked them to discuss how they would like to support each other today. I gave them many examples: Do you want to take turns talking out loud? Do you want to solve quietly and then check with each other? Or do you want to turn and talk anytime you get stuck? Students loved being able to develop a "game plan" with their partners!
I passed out a Game Board (found at this site) to each student. I wanted students to have a fun way of practicing the checking for reasonableness method. I projected a Teacher Gameboard and modeled how to place 2-digit, 3-digit, and 4-digit numbers on the board. I wanted students to begin with 2-digit numbers (first 1/3 of the game board), move on to 3-digit numbers (second 1/3 of the game board), and then be exposed to 4-digit numbers (final 1/3 of the game board).
As students finished writing in their numbers, I asked them to gather the remaining game materials: Game Materials.
I then asked two students to help model the game as I explained the following directions:
1. Both partners roll one die to determine who goes first. The partner with the highest roll will go first.
2. When it's your turn, roll the die and move your marker forward that many spaces.
3. Pull a card. Multiply the card by the number on the space. Solve the problem using the algorithm and checking for reasonableness method.
4. The winner is the player who gets to the end first.
Students caught on quickly and couldn't wait to begin playing: Example of Students Playing Game.
Monitoring Student Understanding
Once students began working, I conferenced with every group. My goal was to support students by asking guiding questions (listed below). I also focused on not giving away answering or correcting student mistakes.
During this conference, I found the perfect opportunity to encourage a student to focus on Being More Precise. Providing immediate feedback is one of the best ways to impact student learning and I anytime I can encourage students to engage in Math Practice 6 (Attend to Precision), I do!
I just loved listening to this student as he explained how he was Seeing Connections! Instead of just saying that 24,003 is reasonable because it is close to his estimate: 24,000, he explained how far off he was from the estimate and why!
Most students were able to finish the game during this time. Consequently, most students were given the opportunity to solve 2-digit, 3-digit, and 4-digit problems using the check for reasonableness method.
In the middle of this lesson, I discovered two ways that this game could be improved. I simply shared out the solutions to problems mid-lesson. If I had composed this lesson using the backwards planning model, I think I would have found these problems prior to the lesson!
Problem #1: Some students included 2-digit, 3-digit, and 4-digit numbers ending in 0, such as 7000. Then, when they multiplied 7000 x 3, checking for reasonableness became pointless!
Solution #1: In order to make sure the game would help students meet today's math goal, I asked students to replace all numbers ending with 0.
Problem #2: The game seemed to become a bit long. After playing for 15 minutes, students didn't get very far on the board. Students became less motivated because the end was so far off!
Solution #2: I asked students to grab another die so that they could roll a pair of dice in order to advance on the game board.