Today's Number Talk
For a detailed description of the Number Talk procedure, please refer to the Number Talk Explanation.
Task 1: 4 x 3
For today's Number Talk, I asked team leaders to pass out the Number Line Model to help students show their thinking later on. For the first task, 4 x 3, students took three jumps of four and four jumps of three. Other students tried Decomposing 4x3 or Doubling & Halving 4x3.
Task 2: 5 x 4
When we moved on to 5 x 4, students eagerly shared the following strategies: five jumps of four, four jumps of five, and decompose: 5 x 4 = (2x4)+(3x4).
Task 3: 4 x 9
During the final task, we discussed 4 x 9. I loved hearing the various strategies students used. One student. Other students decomposed: 4 x 9 = 4 x 5 + 4 x 4. Another student solved 4 x 10 and then took away one four. One student really really surprised me when she used halving and doubling in this way: 4.5 x 4 = 16 + 2. You can see that she halved the 9 and got 4 and a half Then multiplied 4 x (4 + 1/2). You can see that she got 4 x 4 = 16 and 4 x 1/2 = 2. Then she added 16 + 2 = 18. Finally, she doubled the 18 and got 36. You can see that many student begin to naturally push themselves past what they already know when they become familiar with a strategy. In this video, you'll see me Walking a Student through 4x9. You can tell he is starting to grasp this concept, but isn't quite there yet.
You can see that each of the number talks involve multiples of four. By working with a common multiple, students will be able to connect and apply the learning from one task to the next task. In addition, I'm hoping students will discover patterns between the given tasks. For example, 5 x 4 is double 3 x 4 + 2 x 4. This will help students develop Math Practice 8: Look for and express regularity in repeated reasoning.
My favorite part of today's Number Talk was when a student began experimenting with the distributive property: (a x b) + (c x d) = (a x c) + (a x d) + (b x c) + (b x d)! She first came up to the board with this: 4 x 9 = (2 x 2) + (4 x 2) + (4 x 2) + (4 x 2) + (5 x 2) + (5 x 2) + 4 x 5). Then, after solving, she realized that she had too much so she eliminated a (4 x 2) and a (5 x 2). She was still coming up with an answer that didn't make sense to her. I loved where she was going with this! I decided to provide more directed support than usual to help her get on the right track. Here's the video: Distributive Property Discovery. (I should also mention that today was crazy hair day!)
I held up the bag holding one gram of gravel and the bag holding one kilogram of gravel. I placed both bags in students' hands and began reviewing what students already knew: Okay, so which is bigger a gram or a kilogram? "A kilogram" How many grams are in a kilogram again? "1000 grams!"
I moved on to explain the goal of the lesson: I can solve word problems involving grams and kilograms. Today, you are going to pretend you are all apple farmers. Prior to the lesson, I drew an Apple Farmer on the board by projecting clipart I found online. Why would it be important to understand how many grams apples weigh if you are a farmer? Students responded, "You need to know to sell them." "Yeah, if you are putting together a bag of apples." Well, I wonder how much an apple actually weighs. Turn & Talk: How much do you think an apple weighs? When I ask students to turn and talk, I'll sometimes ask for student responses whereas other times, I want to keep the lesson going and I simply keep going. At this point, I asked: Who wants to find the answer to this question using a platform scale? Every hand shot up! I asked team leaders to retrieve the platform scales and passed out an apple to each group. Without my asking, students began weighing the apples. They were naturally motivated to find the answer to the question posed. I walked around the room, making sure all students in each group were given the opportunity to try weighing the apple.
Prior to the lesson, I printed Apple Problem Solving on blue paper and cut each problem out into strips so that I could give students one problem at a time. You'll notice that each problem builds upon previous problems and gradually build with complexity. I included word problems that involved fractional units, even though I hadn't taught students the Fractions Unit yet. With implementing the common core standards and focusing on teaching math content in depth, I'm trying to integrate as many mathematical concepts as possible across all math units. We'll only get to the first few problems today! Students will work on solving the rest of the problems tomorrow.
I passed out problem number one to students and modeled how to paste the blue slip of paper down at the top of a new page in their journals: Modeled Problem Solving. I explained: Remember, today's goal is to solve word problems involving grams and kilograms. Here's your first problem. I'm looking for you to solve each problem using two strategies and to correctly label your answer. Let's complete the first problem together: About how many apples would it take to make a 1 kilogram bag of apples? I purposefully set a platform scale on the desk of a student who struggled with measurement yesterday. Well, what do we know about one apple? I set an apple on the scale. Students responded, "It weighs 200 grams!" I said: Well, let's begin by using an in and out box! This is a problem solving method that I've been working on with my students. Also, I knew that this strategy would help them with more difficult problems later on. I drew a 2 column chart on the board. If we place "# of apples" on one side, what do you think we should write on the other side? "# of Grams!" Does an apple weigh exactly 200 grams? "No... about 200 grams!" Well, what's another word we can use for about? Students struggled with this, even with some prompting: E...st...i...m...a.... "Estimate!" Yes! Let's write "estimated grams" on the other side of our in and out box. Now, if we have one apple, about how many grams do we have? "200!"
I continued: What should we do next? "Add another apple!" Why? "Because we want one kilogram." I added another apple. The struggling student was enthralled with the scale and the movement of the red arrow, bouncing up with each added apple. And if we have two apples, about how many grams do we have? I walked back over to the scale and place another apple on, making two apples. Students responded, "400!" We continued this until all five apples were on the scale: 5 Apples = 1 Kilogram. I also took the opportunity to discuss doubling within the in and out box. If you double 2 apples, you get 4 apples. Then you can double 400 grams to get 800 grams. Here, a student remembered how to do this from a previous lesson so she came up to the board to model it: Doubling. I also wanted students to be able to tell me how they know when to stop with the in and out box. So I added 6 apples and 1,200 grams. Then I said: Okay, so the answer is 6 apples, right? Students responded, "No! You've gone too far!" "You only need 1,000 grams." "Yeah, you only need one kilogram and a kilogram is 1000 grams!" Oh... maybe I should go back and read the problem! Does anyone else ever do that when they're problem solving? We continued on. I modeled how to take jumps of 200 grams as a second strategy while many students tried their own strategies (adding, multiplying, dividing). One student came up and share how she showed her work using division: Other Strategies. I then encouraged her to verify her work by multiplying.
I passed out the second problem and directed students to paste it down in the same fashion as the first problem. I asked students to work together or independently. During this time, I went from student to student, conferencing with each child. I wanted to monitor student understanding and ask probing questions to help guide students to discover connections between concepts, such as grams and kilograms, and between problems. Questions I asked included: What is the problem asking? What did we do on the last problem? Will that help us on this problem? What can we do to make this easier?
Most students chose to work with others. One student who had been out of the room began Completing Problem #1. This student at first told me that the answer is 500 apples. I think he was getting mixed up with Problem #2. Instead of telling him his mistake, I simply encouraged him to reread the problem. He then came up with the answer on his own.
This student explains how she solved the second problem: Solving Problem #2. I loved how she drew pictures of the apples along the number line to further demonstrate the answer to the problem. Also, she is attending to precision by labeling each whole apple "200 g" and the half apple "100 g."
This student truly struggled with the concept of half: Tough Concepts!. I walked him through the problem step by step. Instead of splitting five 200 gram apples to make a 1/2 kilogram bag, I asked him to split the apples between two people. I thought he would be able to relate to this more. You can see that his understanding of the concepts is developing, but he needs to be provided with a simpler problem or hands-on manipulatives.
Instead of asking students to complete an exit slip, I collected student journals to check for understanding. Here's an example of Problems #1 & #2. For the most part, students who incorrectly answered the problem made simple mistakes. For example, this student used an Incorrect Label (grams instead of apples). Tomorrow, I'll remind students to make sure to provide accurate answers. Here, Problem #2 So Close!, a student successfully demonstrates the answer to the problem when using a second strategy but didn't realize that she had gotten two answers. Again, I'll address this tomorrow!