LAP 1 – Balancing Equations Using Multiple Representations
When balancing the equations, Student A chose to mostly use pictures to make sense of the math. However, on the last question, he wrote out numbers and multiplication to help him make sense of the three different variables, and then later drew pictures to represent the numbers. I had intentionally designed this activity so that my students were encouraged to draw connections between those different representations.
Student B chose to consistently use numbers, words, and pictures to solve each problem, showing an advanced understanding of the connection between those three representations.
Student C provided great written explanations of his mathematical reasoning, which was unusual for him. Reading how he thought through each problem made it clear that he understood the idea of both sides of the scale always needing to be equal, for he made sure to hold the values of each shape consistent throughout the two problems.
In the second activity of this lesson, students worked to interpret more real-world scenarios to balance equations. Student D chose to use variables to represent each object, while Student E chose to first use pictures and then eventually variables. I wanted students to start with whatever representation worked best for them, as evidenced in these samples of student work.
On the second day of the unit, a group of students worked on the coconuts problem extension while the rest of the class finished the Algebra Balance Scales packet. I was really impressed by their perseverance on this problem. They were using drawings, manipulatives (blocks), and equations to try and figure it out, and I gave them some hints from time to time. It took them a full class period to complete, but they were so proud of themselves when they finally got it. Here is one student’s work: Coconuts Problem
LAP 4 – Working Backwards
As students began working backwards to solve equations, the same freedom of multiple representations carried over from the previous lesson. Students learned that to work backwards, they needed to “reverse with the inverse,” or do the opposite operation. While Student F preferred to write out the numbers and inverse operations and solve as he went, Student G preferred to write out both the forwards and backwards sequences using a flow chart, and then do the actual math to solve. Each student figured out which way worked best for them, which is an important part of mathematics and education in general.