Objective: Students will be able to use the array method to represent and compute one-digit by two-digit and two-digit by two-digit multiplication problems.
Core Decisions of Lesson Design
What is the curricular content to be learned by the students? What are your learning goals for your students? What concepts, strategies, and/or skills do you want students learn? Your goals should specify both content and process goals.
The goal of this math lesson is for the students to be able to use the array method to represent and compute multiplication problems. [Note: in the original lesson, I had planned to cover both the array method and partial product multiplication. However, I decided, after talking with Professor Remillard, that this would be too much content to cover in one lesson. I also had planned to jump right to the open array method, but decided it would be best to start with arrays with marked boxes so that the method would be more transparent.] The pedagogical focus of the lesson is selecting and using representations to make mathematics meaningful and draw connections between mathematical concepts, so I am presenting the array method in the hopes it will make multiplication, specifically when dealing with place value, more meaningful. Time permitting, I would also like students to be able to connect what they have learned about the array method to explain why the standard algorithm works. Specifically, students will understand why the must line numbers up and carry the one when multiplying two-digit numbers. In other words, these steps are done because one is actually multiplying a group of tens.
Starting with two-digit by one-digit multiplication problems, students will learn the process of representing a multiplication problem as an array and practice using an array to keep track of the numbers that need to be multiplied. Once the process appears to make sense, the students will apply what they learned to two-digit by two-digit multiplication problems, which is currently a goal for the entire class.
How will you teach the content? What are your underlying teaching methods and strategies? What kinds of activities will you engage students in to support their learning?
The lesson will begin by modeling a one-digit by two-digit multiplication problem. First, I will ask the students how to divide the number into their place value groups and demonstrate how to set up an array with the information they provided. We will start with using graph paper so that the arrays have boxes, but then move towards using open arrays depending on student abilities. The array will act as a way to organize the numbers that need to be multiplied. Next, I model how to use the array method to solve a multiplication problem, explicitly writing out each step and ensuring that the students understand where the numbers are coming from.
Next, the students will then attempt a one-digit by two-digit multiplication problem on their own. If necessary, we will do more one-digit by two-digit practice until it seems like the majority of students understand the array method. Then, we will apply it to two-digit by two-digit multiplication.
After I model how to do the first problem, I am hoping that the students will be able to explain their thinking to each other for the rest of the lesson, thus, removing myself from being the only authority figure. [Note: In my original lesson, the pedagogical focus was facilitating mathematical discussion. Although I changed the focus in the final lesson, I still feel that having a discussion around mathematics is an important part of all lessons.] Furthermore, by reasoning through one’s thinking the student is more likely to have a better understanding of the material. To do this, I plan to use some of the strategies from the “Why? Let’s Justify” chapter of Intentional Talk.
In addition to practicing with several problems throughout the lesson, the lesson will end with an exit slip with problems the students are to do entirely on their own.
Why have you selected the topic and goals and the particular teaching methods? What factors have influenced your decisions? Here you should discuss how your core decisions about the what and the how were influenced by factors such as standards, curriculum, particular methods, theories about learning and teaching, your educational philosophy and beliefs, and what you know about your particular students, their experiences, and the curriculum in the class.
Currently, the class is working to develop fluency and accuracy with two-digit by two-digit multiplication problems. The main methods students use to solve these problems are the traditional method and the lattice method. While many students are able to use these methods effectively, some still struggle and not all of the students who are able to use the other methods seem to fully understand why they work. Therefore, the array method seems to be more transparent, and I hope this algorithm will make multiplication more transparent for the students who are either struggling with or do not fully understand the other algorithms. Furthermore, in learning the array method, I hope that the other algorithms also make more sense to the students.
By the common core, by the end of fourth grade, students should be able to “Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models” (CCSS.MATH.CONTENT.4.NBT.B.5). The array method touches on all of these objectives.
Lesson Plan Template
Goals / Objectives
In clear and specific terms, say what it is you hope your students to accomplish.
Students will be able to represent and compute two-digit by two-digit multiplication problems using the array method.
Standards (and Assessment Anchors, if applicable)
Note which the standards and assessment anchors this lesson will help students to accomplish.
Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
Materials and preparation
List the materials you will need to prepare for the lesson.
- White board or chart paper
- Graph paper (1/2 inch rule and 1/10 inch rule) [Note: this was added after deciding to use arrays with marked boxes rather than open arrays. Using graph paper makes the arrays easier to see.]
- Exit slip questions
Classroom arrangement and management issues
Briefly describe the physical arrangement of the class with respect to managing your plans in the particular space:
- Describe the classroom arrangement you will use; explain why you have chosen it. Consider where students will be for each part of the lesson and how they will get there.
- Describe how students will get the materials needed.
- Anticipate management concerns likely to arise and describe steps you will or have taken into address them.
For this lesson I will be working with a small group of students who fall in the middle of the class in terms of their multiplicative understanding. The students will be sitting at a table facing each other in order to facilitate discussion, and I will be sitting between them so that everyone can easily see and hear my instruction but so that I am also seen as a participant in the discussion. The students will talk with each other throughout the lesson, but they will not get up from their seats.
Students will only need a pencil and paper. I will ask them to bring pencils with them before they move from their desk and I will provide paper, which I will pass out at the beginning of the lesson. At the end of the lesson, I will pass out the exit slips.
In order to manage behavior, I will discuss the norms I expect at the beginning of the lesson. I will also emphasize that everyone’s voice is important to the sense-making process and that it is important to listen to and respond to each other in a respectful way. When done effectively, I will praise students on their ability to discuss material with each other. If needed, students will be reminded of these norms throughout the lesson.
One management concern that is likely to arise is that students will become restless and/or frustrated with the lesson. In order to prevent this, I will try to make sure students are not working on the same problem for too long but also that students understand the process before we move on.
Plan [Total Time: approx. 35 minutes]
Include the imagined sequence of events (with a time estimate for each part of the lesson)
- the “hook”
- the body of the lesson
- closure (if appropriate)
For each portion of the lesson, specify focus questions that you plan to ask or problems that you will pose that will help you structure the activity. This is particularly important for sharing/discussion times. It is not enough to indicate that you will bring the class together for a discussion. You need to specify how you will shape the discussion and what kinds of things you will be listening for and attending to during it.
Hook [15 minutes]
1. The lesson will begin by discussing a multiplicative situation and asking students what type of problem it is. Since it is close to Thanksgiving, the problem I plan to use is, “There were 15 people at my Thanksgiving dinner. We eat ate 5 slices of pumpkin pie. How many slices of pumpkin pie did we eat all together?” In linking multiplication to a real life situation, I am hoping it will be easier for students to see how these numbers can be divided into place value groups. [2 minutes]
2. I will then introduce the array representation of a one-digit by two-digit multiplication problem (the problem above), asking for student input as for how to divide the numbers into their place value components. Based on student feedback, I will draw an array on the board that represents the first multiplication problem we will work on. [2 minutes]
3. We will discuss how to compute the area of a rectangle and use this information to determine how to determine the area of the array. (If students seem to be struggling with how to compute the area of a rectangle, I will draw attention to what the students think is the relationship between the small blocks composing the rectangle and the entire rectangle.) In doing this, we will discuss why you can add all the smaller (place value) boxes for form the entire array. [3 minutes]
4. Students will then attempt to independently set up an array for another multiplication problem. [5 minutes] [Note: This step was added after Professor Remillard reminded me that watching my set up the array and actually setting up the array themselves are often two very different things.]
Body of the Lesson
5. Now that we have covered how to set up an array to represent a multiplication problem, we will examine how to use the array to solve the multiplication problem. We will first got over the example I have set up as a group. First we will estimate the answer so we can later determine if our solution makes sense. Then, using color coding to help students to see each of the parts, we will use the concepts of area and place value to simplify the multiplication problem. For example, in the first problem (15 x 5), we will discuss where the 10 x 5 appears on the array and there where we have represented 5 x 5. [3 minutes]
6. Students will then try a one-digit by two-digit on their own, continuing the problem they had previously set up. [Note: originally the release of responsibility was not very gradual. However, I adapted to the lesson to do one step at a time, first with me modeling and guiding the students and then having the student attempt the step individually. By breaking up the problem, I thought it would become more manageable for students.] Students will be instructed to try the problem individually, but if necessary after a few minutes, they may discuss it in groups of 2 or 3. Once the group has solved the problem, I will ask one student to share and explain his/her solution. (If it does not seem clear from the explanation, I will ask the student how he/she did certain steps or picked certain numbers.) I will then ask the other students if they agree or disagree or if they have any questions or comments. [7 minutes]
7. If students seem to understand one-digit by two-digit multiplication using the partial product method, we will to a two-digit by two-digit multiplication problem. (If they seem to need more practice, we will repeat step six.) After discussing how this type of problem is both similar to and different from the problems we just solved, I will once again encourage them to try the problem on their own first, again using the array method, hoping that they will be able to further understand the method by expanding and generalizing it to a more difficult problem. I expect that some of the students may have more difficulty, so I will encourage them to work through the problem in a group of two or three, emphasizing that all group members must play an active role and that even if you do not fully understand the problem, asking questions about your partner’s thinking is still an important contribution. Again, one student will explain his/her solution and receive feedback from other students. [10 minutes]
8. Students will receive an exit slip with two multiplication problems where they will be asked to use the array method. One problem will be one-digit by two-digit, one problem will be two-digit by two-digit. [5 minutes.]
Assessment of the goals/objectives listed above
Describe how you will assess students with respect to your goals. What evidence of student learning will you gather (written work, class discussion, observations) and how will you gather it? Explain how this evidence will help you assess progress toward your goals.
Throughout the lesson, students will be discussing their mathematical ideas and understanding, which will help me to get a sense of who understands the material and who could use extra help. Furthermore, students will be working to solve problems throughout the lesson. While I will encourage group work and discussion, I want all students the attempt each problem on their own first. This will allow me to see who seems to understand the partial product method, but it will also encourage students to try rather than copying their partner’s work. Moreover, by eavesdropping on students’ discussions, I can judge how much each student seems to understand.
Specifically, I will look for whether each student can: divide a number into it’s place value components, set up an open array using place value, set up a partial-product problem by using the information he/she gathered from the open array, correctly compute the multiplication problem. On the assessment checklist I will be looking for the students ability to: identify a multiplication problem (during the hook), solve a one-digit by two-digit multiplication problem through guided instruction and independently, solve a two-digit by two-digit multiplication problem through guided instruction and independently. Some more abstract concepts I will also be assessing include: accurately estimating a multiplication and using the estimation to determine if the final answer makes sense, accurately modeling both one-digit by two-digit and two-digit by two-digit multiplication problems with an array, and understanding how to use the array method and why it works. Lastly, I would like students to be able to connect the array method and the traditional method for solving multiplication.
The assessments throughout the lesson will help me determine if I should repeat a step or whether I can move on. However, the exit slip at the end of the lesson will help to confirm my judgments of each student’s understanding.
Anticipating students’ responses and your possible responses
- Management issues
- Response to content of the lesson
1. Hopefully management issues will be avoided because we will review the expectations at the beginning of the lesson. However, if necessary, we will review norms as needed throughout the lesson and if it is still an issue we can discuss why certain expectations were set and how that helps the entire group learn. Furthermore, I will praise students who are respectful when sharing and listening to their fellow group members
2. One anticipated response to the content of the lesson is that students will have difficulty learning another method and/or will have difficulty seeing the value in learning another method if they already know how to use the traditional or lattice method. In response to this, I will try to help students see how the array method is connected with the other methods, specifically in the way each method handles place value. I will also encourage students to think of this on their own as we go through the lesson, and I will check in with them at the end to see what they thought.
1. Accommodations for students who may find the material too challenging
2. Accommodations for students who may need greater challenge and/or finish early?
1. Although students are encouraged to try each problem on their own before working with other students, I will also encourage them to work together and ask questions about each others thinking. If necessary, students who are still finding the material challenging can have extra practice with the one-digit by two-digit multiplication problems and try to really understand why those work. Furthermore, we can talk about how the one-digit by two-digit problems are connected to the two-digit by two-digit ones.
2. For students who may need a greater challenge and/or finish early, I will encourage them to really think about why the method works and about how they will explain their thinking to their partner/group. If there are students who are way ahead of everyone else, I could give them a two-digit by three-digit problem to try and we could discuss how these problems are related to the other problems we had done.
Kazemi, E., & Hintz, A. (2014). Intentional talk: How to structure and lead productive mathematical discussions. Stenhouse.