Rationale for teaching with a task:
After I have worked directly with the students on a skill, I like to use a task. A task gives the students more practice on the skill while working in groups. Allowing the students to work in groups gives the students different perspectives from their classmates. Students can learn from each other. As the students work on a task, I am the facilitator, walking around monitoring and questioning the students to lead them to the solution.
I let the students know that today we will do a task. I remind the students of the structure and routine of a task. First, you have private work time to think about and plan how to solve the task. Next, you work in groups to explore the concept of the lesson. Finally, we come back together as a whole class to share/analyze/and discuss the task.
To begin the lesson, each student is sitting at his/her desk. I say, "Today, you solve a task using skills that you have already learned." I give each student a copy of the task at their desk. I remind the students that they learned their multiplication facts in 3rd grade, and that they should be familiar with fact families. I go on to tell the students that we have discussed the meaning of multiplication and how we can use models to help solve multiplication problems.
I tell the students, "In today's lesson, you use your understanding of multiplication to solve this real-world problem. You have to find the answers by identifying the factors of a number." (4.OA.B4)
I give the students about 5 minutes of independent time to read and plan to solve Is that a Fact Task (MP1). The students have been given counters to help them with their models (MP5). They can use the counters during their private work time to begin solving the problem. After the 5 minutes of independent planning, the lesson goes to the next phase of group exploration.
There are 72 students waiting to be seated in the library. The principal wants the students to all sit in the same area. The teachers are busy setting out chairs for the students. How many ways can the teachers arrange the chairs?
With a partner, find the multiplication facts that equal to 72. Write the equations and draw a model for each equation that shows how the chairs should be arranged.
As the students work independently, I walk around to monitor. I notice that some students use the counters to lay out their arrays of 72. Other students, who know their multiplication facts, begin by writing down the facts that equal 72. During this phase of the lesson, I do not question students to help lead them to the answers. I want to give the students the opportunity to struggle and come up with a plan before they work in their groups. (I find that this allows the students to take ownership of their learning.)
During the group exploration/discovery phase, the students work in pairs. Each group has a copy of the task. The students must work together to complete all requirements of the task. The students are required to find all the possible ways the chairs can be arranged in the library by using multiplication facts (4.NBT.5). The students reason abstractly and quantitatively by decontextualizing the information from the task and representing it symbolically (MP2). During this phase, the students do not receive direct instruction. In this lesson, they apply skills previously learned. The students are guided to the conceptual understanding through questioning by their classmates, as well as by me. The students communicate with each other and agree upon the possible ways to arrange the chairs. This takes discussion, critiquing, and justifying of answers by both students (MP3). As the groups discuss this task, they must be precise in their communication within their groups using the appropriate math terminology for this skill (MP6). Each pair has counters to help them with their models, thus giving them a visual of how the chairs can be arranged (MP5). As I walk around, I am listening for the students to use "talk" that will lead to the answer. For example, in one group I hear a student ask his partner to show him the answer with the counters. I am holding the students accountable for their own learning.
During the phase, I monitor and assess the students' progression of understanding through questioning. Possible questions to help lead to the solution are as follows:
1. What is the task asking you to find?
2. How can the counters help you solve this problem?
3. All even numbers are multiple of what number?
4. How can division help you solve this problem? Can you divide 72 with your counters?
As soon as the group exploration phase began, the noise began. The students were very excited about the chance to"talk" with their classmates to learn. Because most students have learned their multiplication facts, the students easily found 9 x 8 and 8 x 9 as arrays for 72. The other arrays were not so easily found. I used questioning to help guide the students to use patterns that they have learned to help find the solution. Also, I encouraged the students to use the counters by laying out 72 counters in even rows.
Any groups that finish the assignment early, can go to the computer to practice the skill at the following site until we are ready for the whole group sharing: http://www.ezschool.com/Games/FactFamily2.html
During this phase of the lesson, student solution paths are shared. While the students were working in groups and I was walking around questioning, I identified solution paths to be shared as a whole class for this phase.
I call groups to the front to share their solutions. This is a teaching opportunity for the few students who may still not know the skill. This part of the lesson is lead by the teacher through asking assessing questions. The students may also have questions that they would like to ask.
During this phase, I like to organize the sharing of the solution paths in a strategic manner. First, I have a group share all of the possible multiplication equations for the chairs to be arranged in the library. These students share: 1 x 72=72, 6 x 12=72, 4 x 18=72, 2 x 36=72, 24 x 3=72, and 8 x 9=72. Next, using my document camera, I show models of different groups. I try to share as many students' work as possible. I feel that this gives the students a sense of pride and build the type of classroom environment that is pleasant for all. As the student work is displayed on the document camera, the students share their answers and how they found the answer. As the students share, I use this as a chance to discuss strategies and patterns that the students can use to find the different arrays. For instance, I discuss with the students patterns that they have already learned, such as, they can add the digits of 72 and get 9. I remind the students that because these two digits added together to equal 9, then 72 is a multiple of 9. Therefore, 9 is a factor of 72. Also, I remind the students that 72 is an equal number. All equal numbers are divisible by 2. Therefore, 2 is a factor of 72.
After the share/discuss/analyze phase of the lesson, I close the lesson out by having the students do an Exit Ticket Is that a Fact. This enables me to see how well the students understood how to use find the factors of a number.
After the students find the solution, I collect the exit tickets to evaluate the students' understanding. Those students who need remediation will work with me in small group the next day.
I found that the students who knew their multiplication facts finished the exit ticket quickly. Finishing quickly was not a criteria for doing well on this assessment. I wanted the students to use modeling to help them solve the problem. Overall, most of the students found all of the factors of 24. Because 24 is not a large number, I feel that this helped the students be successful on this assignment. To make it more difficult in the future, I can use a larger number to challenge the students.