Today's Number Talk
For a detailed description of the Number Talk procedure, please refer to the Number Talk Explanation. For today's Number Talk, I let students take more time than usual.
Task 1: 2 x 12
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, 2 x 12, students took two jumps of twelve as well as twelve jumps of two. This student is gaining a true understanding of the Commutative Property: 12 x 2 = 2 x 12. Other students decomposed the twelve to solve: 2x12 = (2x10)+(2x2). I encouraged students to show me how to solve 2 x 12, Using Mulitple Strategies.
Task 2: 12 x 4
When we moved on to 12 x 4, students eagerly shared a variety of strategies. One student shows me that four jumps of 12 is the same as eight jumps of six: 12 x 4 = 6 x 8. Another student modeled how to multiply 3 x 2 four times, add the products, and then multiply the sum by two: 12x4 = 2[4(3x2)]. This student modeled how to Take jumps of 6, 12, or 24 to solve 12 x 4. I'm reminded of the importance of giving students time to "go deeper" in order to truly develop a sense of numbers and operations (Math Practice 2: Reason abstractly and quantitatively).
You can see that each of the number talks involve multiples of 12. 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, 12 x 4 is 12 x 2 + 12 x 2. 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, who sometimes struggles with focusing/learning as well as pushing himself to come up with multiple strategies, came up to the board. He first showed me a doubling and halving strategy and then showed me that he knew how to decompose: 24x2 = 2(12x2). I then took this time to reinforce the process of verifying: Verifying.
I also LOVED watching the level of Student Engagement! In this video, you'll see the number of students that were excited to share one of their strategies on the board!
I asked students to bring their math journals and gather up front, close the board. I wanted students to be as close as possible and ready to participate. I began by explaining the Goal & Problem to students as they wrote the goal on a new page in their math journals: After today's lesson, I want you to be able to say, "I can solve word problems involving ounces and pounds!" I problem solve every time I go to the grocery store! Sometimes I need to figure out how many packages of a product, such a sugar or chocolate chips, I need to buy to bake a batch of cookies. Other times, I'll want to make several batches of cookies and I 'll need to recalculate the number of each product I'll need to purchase.
I continued: Here's the problem format that we will begin with: I went the store to purchase one pound of ____ (a product). How many packages will I need to buy? In a row on the board, I set a variety of products in order from the lightest to heaviest: Au Jus Gravy (1 oz.), Green Tea (2 oz.), Walnuts (4 oz.), Croutons (5 oz.), Mustard (12 oz.). I set them in this order to purposefully build a gradual progression of learning, beginning with a simple problem and building up to a more complex problem. I also wanted to encourage students to look for and make use of patterns (Math Practice 7). For example, if walnuts weigh more than the green tea, we will have to buy less packages of walnuts to reach one pound. The bigger the package, the less packages we need to buy to get to a pound.
Let's start with the packets of Au Jus Gravy. How much does one package of Au Jus Gravy weigh? I went back to the problem: So, if I wanted to purchase one pound of Au Jus Gravy, how many packages would I need to buy? Several students said, "16!" I responded: But how can we show our work? Could we use an in and out table? This is a powerful problem solving strategy that we've been focusing on a class. I drew an in and out box and modeled how to represent our thinking using the chart. Students also created an in and out box for Au Jus Gravy in their journals. Turn & Talk: What should the label be for the first column? And what should the label be for the second column? After giving students some time to discuss, they came up with "Number of packages" as the label for the first column and "Number of ounces" for the second column. Okay, so what if we have one package of Au Jus Gravy? I placed a one in the left column of the chart: In & Out for Au Jus Gravy. How many ounces is one package of Au Jus Gravy? "One!" What about two packages? How many ounces are in two packages? "Two!" What about three packages? How many ounces are in three packages? "Three!" Three what? "Three ounces!" Should we keep just adding one more or is there something else we can do with an in and out box? One student responded, "We can double. Double the three and you'll get 6 packages. Double the other three and you'll get 6 ounces." We continued on, by doubling the 6 packages to get to 12 packages. At this point, I asked: Should we double the 12? Students responded, "No!" Why not? "Because that's too far." "We only need 16 ounces." Why? "Because 16 ounces is a pound. Once students established that we would need to buy 16 packages of Au Jus to buy one pound of Au Jus, I wanted the students to try solving the next problem on their own.
We moved on to the product, Green Tea. I asked students to create an in and out box for the green tea in their journals. I wonder how many packages of green tea we would need to buy to purchase a pound of green tea. After giving students time to work on their own, I asked students to turn and talk about their thinking (supporting Math Practice 3: Construct Viable Arguments). I brought students back together by saying: Okay, ready in 3... 2... and 1... Who can tell me how to solve this problem? I called on a student and as she explained her thinking, I modeled her thinking by constructing an In & Out for Green Tea. We moved on to Walnuts and followed the same process, ending with another In & Out for Walnuts. I purposefully chose this order of products to build a staircase of complexity. The first three products, au jus, green tea, and walnuts, weighed 1 oz, 2 oz, and 4 oz, which were all factors of 16 oz (one pound). The next few products will challenge the students' thought processes as the weight of these products are not factors of 16 oz.
At this point in the lesson, all students were catching on quickly and became excited to move on to the next product, Croutons! Once students had time to work independently and to Turn & Talk, I asked a student to come up to the board to model the in and out box. This happened to be a student who often struggles with math, so I was especially pleased to see him successfully solve the problem, with the help of his classmates. At first, this student found the number of ounces for one package, then two, and then three: How Many Bags of Croutons?. Once we got to three packages, which is equal to 15 ounces, I heard a student say, "We can round!" At this point, I said: Well, what if I worked for a restaurant and my boss sent me to the store to get a pound of croutons? What if I only come back with 15 ounces? Do I have a pound? I even joked around a bit with the students and said: I might get fired! Heres' what happened next: Do I Have a Pound?. This was a very important turning point in our lesson. It was the point in which students began to realize that you sometimes have to buy too much in order to have enough. This conversation supported the development of Math Practice 1: Make sense of problems and persevere in solving them, as students were discussing whether or not it made sense to round or to buy one more package.
Finally, we moved on to the last product, Mustard. A student came up to the board to solve for this product as well: How Many Bottles of Mustard?. Once again, we revisited the same conversation. How many should we buy to have one pound. This reinforced newly developed ideas about reaching one pound.
To bring closure to our math lesson today, I simply asked students to Turn & Talk:
Can you always buy an exact pound of a product? Give an example to support your thinking.
During this time, I observed student comments and gestures. Many students pointed to the board and referred to the croutons, which weighed 5 ounces, or the mustard, which weighed 12 ounces, "You can't buy one mustard and part of another bottle." I loved hearing students personalize mathematics by saying, "Yeah, you can just stop and get a hotdog on the way home if you don't want the extra ounces of mustard."