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. For each task today, students shared their strategies with peers (sometimes within their group, sometimes with someone across the room). It was great to see students inspiring others to try new methods and it was equally as great to see students examining each other work for possible mistakes!
Task 1: 4024/8
For the first task, students decomposed the quotient into multiples of 8, such as 4000+24 and then divided: 4024:8. I asked students to try using the array before checking their work using the algorithm. Some students chose to show multiple strategies (array, partial quotients, verify with multiplication, standard algorithm): 4024:8 Multiple Strategies.. I then modeled one student's thinking on the board: Modeled on Board.
Task 2: 2012/4
Then, students solved 2012/4. To construct this task, I halved both the dividend and divisor from the last task and asked: What do you think will happen to the quotient? Some students thought that the quotient would halve as well. Other thought it would double. They were surprised to find it stayed the same: 2012:4. Some students found multiple ways to decompose 2012: 2012:4 Multiple Strategies.
Task 3: 2012/8
For the final task, students used the array method to solve and represented the amount left over by labeling: 2012:8. I loved listening to this student explain how he found that 8 x 250 = 2000: Modeled Student Thinking. I modeled his thinking on the board as he explained, "I knew that 25 cents x 4 = 100 cents o I doubled the 4 and then doubled the 100. Next I multiplied the 25 x 10 and then the 200 by 10 to get 250 x 8 = 2000."
Throughout every number talk, I continually model student thinking on the board to inspire other students. This also requires students to use math words to explain their thinking instead of relying on a model to represent the math. As students solved each task, I wrote the answers on the board to encourage students to use prior tasks to solve the more complex tasks.
Goal & Introduction
To begin, I shared today's goal: I can divide to solve word problems involving multiplicative comparison. I explained: Now that we are nearing the end of our division unit, it's important to take the time to apply our newly learned division skills to solve word problems. However, each of today's problems will be a special kind of problem called multiplicative comparison problems. We have solved problems like these in the past using multiplication. Today you'll be using division!
Pointing to George's Problem prewritten on the board, I continued: Let's look at a problem together! Let's say that George is shopping at a clothing store. (The students giggled and looked at George, a student in our class.) George buys 2 pairs of pants for $60. A pair of pants costs 3 times as much as a shirt. How many shirts could George buy with $48? After reading the problem, what do we know? Students responded:
To provide students with a workspace for George's problem and for problems they will be solving later on, I asked each student to get out 4 sheets of lined paper and staple the top edge.
When student were ready, I Modeled George's Problem and asked students to also complete George's problem using the first page of their lined paper packet. I wanted to model Math Practice 1 (Make Sense of Problems and Persevere) by showing students how to break down multi-step problems into steps. The following conversation took place as we solved this problem altogether:
T (Teacher): Please divide your paper into four boxes by folding and drawing lines as we have done in the past. We will use each box to represent each step we take to solve the problem. What do we need to find out first?
S (Students): We need to find the cost of one pair of pants.
T: What do we know about the pants?
S: Two pairs of pants cost $60.
T: Can we draw a bar diagram to show this? (I drew a bar diagram, which is also called a tape diagram, and split the whole into two equal parts. With the students' help, I labeled each part with one pair of "pants" and I labeled the total cost of two pairs of pants as $60.) What should we do to find the cost of one pair of pants?
S: Divide 60 by 2! (We then divided 60 by 2 using the standard algorithm and wrote, "one pair of pants = $30.)
T: Now what should we do?
S: Find the cost of one shirt!
T: How should we do that?
S: Draw another bar diagram! Put 3 shirts inside.
T: What are 3 shirts equal to?
S: The cost of one pair of pants... $30!
T: I modeled how to draw another bar diagram. With the students' help, I divided the whole into three equal parts and labeled each part, "shirt." What should we do next?
S: Divide $30 by 3 to find the cost of one shirt. Altogether, we used the standard algorithm to divide 30 by 3. We then wrote, "one shirt = $10."
T: Turn & Talk. What should we do next?
S: We need to see how many shirts George can buy with $48 so we need to see how many times $10 will go into $48.
T: This time, we solved this step using the standard algorithm before drawing the bar diagram. After arriving at the answer "4 r 8," I asked: How could we model the number of shirts that could be purchased with $48 using a bar diagram?
S: Divide $48 into 4 parts and put $10 in each part. Then show a remainder of 8. (On my paper (Modeled George's Problem) I should have made sure the $48 included the remainder of $8.)
T: Let's use this last box to explain the answer to the problem! (In order to avoid students spending a good portion of their math time on writing lengthy explanations, I modeled (with the students' help) how to answer the question using one sentence: Geroge can buy 4 shirts with $48.
We then moved on to student practice problems.
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 always love being able to develop a "game plan" with their partners!
I then passed out a set of Word Problems to each student and a copy of School Supplies to each pair of students. I asked students to cut out each problem and paste the problems throughout their stapled packet of lined papers (one problem at the top of each page).
As students were ready, I modeled one more problem, Modeled the Glue Bottle Problem, just as I had modeled George's problem, only with less teacher guidance.
Monitoring Student Understanding
At this point, most students were ready to solve problems independently. Once students began working, I conferenced with every group. My goal was to support students by providing them with the opportunity to explain their thinking and by asking guiding questions. I also wanted to encourage students to construct viable arguments by using evidence to support their thinking (Math Practice 3).
During this video, Lunchbox Problem, I encourage students to think about a reasonable answer before dividing. I meant to ask: What's 10 x 9? ...not 10 x 8!
Here, two students explain the cost of Mechanical Pencils & Crayons. These students did a great job representing their thinking using the bar diagram method and variables.
During this conference, Backpacks & Scissors, students understood the first two steps of the problem, but then became confused on the third step. Trying to decide which number to divide by (10 or 3) was challenging!
Most students were able to solve 3-4 problems during this time. Here are a couple examples of completed problems: