Reasoning for Teaching Multiple Strategies
During this Addition and Subtraction Unit, I truly wanted to focus on Math Practice 2: Reason abstractly and quantitatively. I knew that if students learned multiple strategies of adding and subtracting numbers, I wouldn’t only be providing them with multiple pathways to learning, but I would also be encouraging students to engage in “quantitative reasoning” by “making sense of quantities and their relationships in problem situations.” By teaching students how to use a variety of strategies, such as using number lines, bar diagrams, decomposing, compensating, transformation, and subtracting from nines, I hoped students would begin to see numbers as units and quantities that can be computed with flexibility.
For today's math lesson, I wanted student to work together in groups of three during guided practice. Collaborative learning almost always supports Math Practice 3 (Construct viable arguments and critique the reasoning of others) as students are in continual conversation and defending their thinking. Assigning partners is always quick and easy as I already have students strategically placed in groups of 4-5 students throughout the room (based on abilities, behavior, communication skills, etc.). I simply divided these larger groups into smaller groups of 3 students.
I passed out 3 colors of paper (orange, blue, and green) and three page protectors per group. Next, I asked each group to place the papers inside the page protectors and label each page with the following: Standard Algorithm, Decomposing, and Bar Diagram. I included the bar diagram in today's lesson to connect the lesson with yesterday's lesson and to provide extra practice with the bar diagram method.
Real-World Application & PowerPoint Presentation
Keeping in mind Math Practice 4 (Model with mathematics), I wanted to provide students with another real-world situation in which they would add and subtract multi-digit numbers. This way, they would also have the opportunity to use another strategy, decomposing, to check their answers. So, prior to the lesson, I created a Powerpoint Presentation called Comparing Salaries. To begin, I showed students the first slide, which was the Goal: I can check addition and subtraction algorithms by decomposing.
After discussing the goal, we moved on to the next slide, What is a Salary?. I explained: Today, we will be discussing the average salaries of different careers. A salary is a fixed compensation paid periodically to a person for regular work or services. For example, if you work for a company, many times, you'll get paid the same amount of money every month. If you are an independent contractor that builds homes, you get paid each time you build a home.
We then went to the next slide, Zola's Real World Problem. I included this slide to provide students with a real-world problem: Everyone, I want you to meet Zola. Zola is researching different career options. She needs your help comparing the average salaries of different professions. This will help her make a decision about the career that she wants to pursue. Do you think we could help her today? (At this point, I couldn't help but entertain a conversation about other ways people are compensated for their work besides money!)
I continued this lesson by reviewing the word, decomposing, with my students: Today, we are going to be using the standard algorithm, bar diagram, and decomposing to make sense of addition and subtraction problems. Let's talk about the word decomposing. What does decomposing mean again? After a brief conversation as a class, I introduced the Decomposing Poster and continued: Decomposing is when you break numbers a part. For example, 36 could be decomposed into expanded form: 30 + 6 or it could be decomposed however you'd like, such as 10 + 20 + 6!
Building a Staircase of Complexity and Gradually Releasing Responsibility
To create a staircase of complexity within the PowerPoint, Comparing Salaries, I created three levels of comparison tasks. I wanted students to first compare Daily Salaries. (presentation slides 5-8) for today's lesson. For tomorrow's lesson, the students will then compare Monthly Salaries (presentation slides 10-13) and Annual Salaries (presentation slides 16-19). This way students would begin by computing 2-3 digit numbers and work their way up to computing 5-6 digit numbers.
Explanation of Comparing Method
In order to compare salaries today, students used subtraction to find the difference between salaries on each page of the Powerpoint Presentation.
During the guided practice time, we started by modeling, solving, and discussing the Daily Salaries slides. I modeled each strategy (algorithm, decomposing, and bar diagram) on the board, while students modeled the strategies on their strategy pages. Also, students rotated their strategy pages with each new task so that all group members were given the opportunity to practice each strategy. With time, I released more and more responsibility to students.
Modeling the Bar Diagram
We began with Daily Salaries Example 1. I reviewed the process of making a Bar Diagram using the average daily salaries of a grocery store bagger and a bank teller. In this example, I drew two bars to compare two parts (two careers). I labeled the larger bar "$115" and the smaller bar "$82." We then represented the difference between the two bars with a question mark.
Modeling the Algorithm
Next, we moved on to reviewing the standard algorithm. This strategy was quite simple to review. I reminded students to line up digits and reviewed the borrowing process.
For the last strategy, I modeled how to use Decomposing to Subtract 115 - 82. I first wrote the problem, 115 - 82, in algorithm form. Then, I decomposed the minuend and the subtrahend into expanded form: 100 + 10 + 5 and 80 + 2. I made sure to line up the ones, tens, and hundreds. At this point, we subtracted the expanded form of the 115 to the expanded form of 82. This did require some borrowing: First, we crossed off the 10 and changed the 5 ones into 15 ones. Later, we had to borrow 100 in order to subtract: 100 - 80. The process of subtracting using decomposing definitely supports Math Practice 2: Reason abstractly and quantitatively. Instead of simply following a set of procedures to solve the standard algorithm, students had to truly think... Is this 8 or 8 tens?
More Guided Practice
After modeling, solving, and discussing the first two occupations, we moved onto the next three slides: Daily Salaries Example 2.png, Daily Salaries Example 3, and Daily Salaries Example 4. With each slide, student understanding and success increased: Group Working Together to Solve 237-196. This was also my signal to provide students with less and less teacher support.
Conferencing with Groups
During this time, I also would walk about the room and conference with groups of students. Here are a couple examples of conferences with groups: Group Conference 374-365 and Group Conference 578-529.
For independent practice time, I created 2 practice pages by copying & pasting portions of worksheets found at Math-Aids.com. I wanted to provide students with the space necessary to check the addition and subtraction algorithms using decomposing: Decomposing Practice Page 1. As students finished, they compared their answers with others at the back table.
Rationale for Practice Page
I chose to use this above practice page to check that students were able to practice and apply concepts learned during the lesson.