Decomposing to Compare Monthly & Annual Salaries

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SWBAT check addition and subtraction algorithms by decomposing numbers.

Big Idea

Students will use the expanded form of numbers to compare salaries.


20 minutes

Today's lesson is a continuation of yesterday's lesson, Decomposing to Compare Daily Salaries

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. 

Choosing Partners

For today's math lesson, I wanted student to continue working together in groups of three during guided practice. Collaboratively 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. I asked students to continue working in the same groups of three as yesterday. 

Getting Ready 

For each group, I passed out 3 colors of paper (orange, blue, and green) inside page protectors. Next, I asked each group to label each page with the following: Standard Algorithm, Decomposing, and Bar Diagram. By including multiple strategies within a lesson, student are provided with more opportunities to practice these strategies and notice connections between them.

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 used the first slide, to remind students of our Goal: I can check addition and subtraction algorithms by decomposing.

Building Relevancy

After discussing the goal, we moved on to the next slide to review the meaning of a salary: What is a Salary?. Next, we reviewed Zola's Real World Problem


Guided Practice

45 minutes


I continued this lesson by reviewing the word, decomposing, with my students: What does decomposing mean again? Students responded, "It's when you break numbers apart." 

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. During yesterday's lesson, students compared Daily Salaries (presentation slides 5-8).  For today's lesson, students will 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. 

Guided Practice

During the guided practice time, we started by modeling, solving, discussing, and comparing the Monthly Salaries and then the Annual Salaries. 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

With each task, we first created a bar diagram. Here is an Example of Modeling the Bar Diagram, where we are comparing the monthly salaries of an aerospace engineer and a veterinarian. In this example, I drew two bars to compare two parts (two careers). I labeled the larger bar "$8,104" and the smaller bar "$7,808." We then represented the difference between the two bars with a question mark. After some discussion, we decided to replace the question mark with the variable, y. We also discussed how we could round each bar to estimate the difference. We rounded 8,104 to 8,100 and 7,808 to 7,800. We then found the estimated difference between the engineer and veterinarian's salaries would be about 300. We'll look back to our estimate later on to make sure our algorithm solution is reasonable. 

Modeling the Algorithm

Next, we moved on to reviewing the standard algorithm. Here's an Example of Modeling the Standard Algorithm. This strategy was quite simple to review. I reminded students to line up digits and reviewed the borrowing process. 

Modeling Decomposing

For the last strategy, I we used decomposing to solve: Example of Modeling Decomposing. In this example in particular, I first wrote the problem (8,104-7,808) in algorithm form. Then, I decomposed the minuend and the subtrahend into expanded form. At this point, we subtracted the expanded form of the subtrahend from the expanded form of the minuend. Borrowing was a bit tricky (such as crossing off the 8000 to borrow 1000....), but with time, students became more comfortable with this process. Here's an example of a student decomposing to solve 5,125-4,255: Decomposing Example.

Student Practice

30 minutes

For independent practice time, I created 2 practice pages by copying & pasting portions of worksheets found at I wanted to provide students with the space necessary to check the addition and subtraction algorithms using decomposing: Decomposing Practice Page 2. As students finished, they compared their answers with others at the back table.