Landscape Design

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Objective

SWBAT multiply a 2-digit number by a 2-digit number using the array method.

Big Idea

Students will be able to create a visual representation of multi-digit multiplication.

Opening

20 minutes

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 "someone new 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: 18 x 2

For the first task, many students decomposed, 2x10+2x8=36, while others used compensation: 18 x 2, Compensating. I was proud to see so many students showing multiple strategies: 4 x 43, Multiple Strategies.

 

Task 2: 18 x 32

During the next task, I began by asking: How many more 18s will we have than the last task? With time, students responded, "30 more because 30 + 2 equals 32!" Some students chose to solve this task using a simpler approach: 18 x 32, Simpler Method. Other students challenged themselves to use a more complex method: 18 x 32, Complicated Method! Of course, the more complicated the approach, the more likely students will make errors. At the same time, I love watching students showcase their number sense. 

 

Task 3: 43 x 4

Then, students decomposed 43 x 4 in a variety of ways. Here's two: 43 x 4 = (20+20+3) x (2+2) and 43 x 4 = (40+3) x (2+2). As students shared strategies with one another, I loved watching them respectfully disagree and then work together to discover discrepancies between their work. 

 

Task 4: 43 x 54

For the final task, I asked: How many more 43s will we have than the last task? Again, with time, students responded, "50 more because 50 + 4 equals 54!" Many students then decomposed by place value (tens & ones): 43 x 54 = (40+3) x (50+4).

 

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.

 

Teacher Demonstration

30 minutes

Goal & Introduction

To begin, I introduced today's goal: I can multiply a 2-digit number by a 2-digit number using the array method. I then explained: Today, you are going to be solving two problems that have to do with landscape design. With each problem, you'll be asked to solve the problem using the array method. 

Lesson Design

For today's lesson, I wanted students to not only solve 2-digit x 2-digit multiplication problems using the array method, but I also wanted them to see how arrays can be used in a real world setting to solve a multiplication problem (Math Practice 4: Model with Mathematics).

So I created an activity where students would help design a rectangular backyard by dividing the yard into four equal sections. Throughout this activity, I was reminded of the fact that math is so much more meaningful when taught within a given context (such as backyard designs)! 

Student Handouts

I passed out a sheet of 18 x 23 Grid Paper to each student (modified from this site). I also asked students to get the laptops out (one per student). I then shared a Google Document: Multiplication ArraysStudents copied the presentation and saved it in their math folders under their Google Drive. This document could also be emailed to students as a Microsoft Word document (Multiplication Arrays) or printed as handouts: Multiplication Arrays.

Designing Mr. Green's Backyard

Altogether, we looked at the first page, Problem 1, Mr. Green's Backyard Design. As we read through this problem, we discussed important information, such as "rectangular backyard" and "18 feet by 23 feet." I then asked: What do we need to do next? Students said, "We need to make an 18ft x 23ft yard on the grid paper. Without any prompting, students immediately began counting the number of squares on the grid paper. Excitedly, students said, "There's the perfect number of squares! There's 18 squares here (pointing to the horizontal side) and 23 squares here (pointing to the vertical side)!"

Okay, so what you're saying is that this whole grid represents Mr. Green's backyard? I then modeled how to draw two lines on the grid paper, one vertical and one horizontal (from one side to the other) in order to divide Mr. Green's backyard into four sections. Here, a student explains how she divided her grid paper: Four Sections

Next, I showed students how to use the chart on the second page of the document to record their thinking: Modeled Chart. I wanted students to organize the backyard sections and the total square feet in each section. 

Encouraging Precision

As students continued working on this project, I encouraged them to attend to precision (Math Practice 6) by setting the following expectations: 

  • Label each section of the backyard (Pool)
  • Label the dimensions of each section (12 feet)
  • Label the area of each section (108 feet squared)
  • Use "feet squared" when representing area
  • Double check your calculations with a partner (Total size of Mr. Green's Backyard) when finished

 

Here are a couple examples of student conferences: Encouraging Precision and Encouraging Precision 2.

Final Product

Many students had time to color their backyard designs while others finished the task: 

Student Practice

50 minutes

Partners

For student practice time, students continued working with their partners. 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! 

Getting Started

We continued on to the third page of the Multiplication Arrays document: Problem 2, Laying Sod in Mr. Green's Backyard. I explained: Let's say that Mr. Green decided to just lay sod (grass) to cover his entire backyard. He now wants your help calculating how may square feet of sod he will need. Let's look at the first scenario. We all went to the next page: 15 x 12 Array.

I continued: What if one side of Mr. Green's backyard measures 15 feet and the other side measures 12 feet? Let's use the array method to find how many square feet altogether! Next, I modeled how to complete the graphic organizer to represent the array method for 15 x 12: Modeled 15 x 12. I used colors to help students color-code and organize their steps. 

Partial Products

Since we have worked with partial products so frequently in the past, I simply modeled the process of completing these problems using this method. Otherwise, I would have provided explicit instruction and examples of the partial product method before students practiced this method further on their own. 

Monitoring Student Understanding

Students then moved on to the next task on the next page: 30 x 12 Array

While students were working, I conferenced with every group. At times, I would provide students further instruction. Other times, my conferencing goal was to support students by asking guiding questions (listed below). 

  1. How can you decompose the first multiplicand?
  2. What is easier to multiply by?
  3. Does this product make sense?
  4. What are you going to multiply by next?
  5. Can you explain this part to me?
  6. How did you know ____?

 

Here's an example of Supporting a Student. Through conferencing and peer support, this student became successful: Successful Student

During another student conference, Solving 90 x 12, I asked guiding questions to encourage the student to explain his thinking. The same student then showed me how he used the calculator as a tool to check his thinking: Using a Calculator to Check