Things Aren’t Always Equal

Enter a 5th grade class.
The lesson has already begun.
Students are quietly representing one-third in as many ways as they can in their math journals.
They begin to share their representations on the whiteboard.

Notice anything concerning?

The class begins to discuss the representations. They quickly focus on,

They discuss why these are not equal thirds.

One student sitting near the front quietly says, “Don’t worry if they aren’t equal. Things in life aren’t always equal.”

The initial warm-up activity was designed to review and solidify representations for one-third. How can we increase students’ thinking level using the drawing.

Question: Suppose this drawing was on the test in April. What questions might they ask?

Student 1: What is wrong?

Student 2: What didn’t they understand?

Student 3: Could you fix the drawing to show one-third?

Student 4: What fraction is represented?

Question: What tools would be helpful?


After a long pause a student says, “A ruler?”


Question: How would a ruler help?





“I don’t know”

AHA 1: The students have few opportunities to choose the appropriate tool. I think this was the only math “tool” the student could think of. It was the only possibility offered. He didn’t know how it would fit. I asked if grid paper would be helpful. The students agreed that it would be helpful.


AHA 2: Students don’t think of grid paper as a tool.

The student continued to work on dividing a triangle into thirds. His revised drawing,

Extension Questions: Will this technique work with all triangles? Other polygons?

Do you have a similar story?

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