# VISIBLE LEARNING IN MATH-USING TOOLS TO PROVE THEIR THINKING

In one of the classrooms, the teacher wanted all of her students to access the task. She had a large number of English Language Learners and students with specific Math IEPs. The teacher had worked with students on using fraction models and drawing diagrams while naming the fractional pieces. Often times, students will use fraction circles and not remember what each fractional part represents. The teacher was keen on having students name the fractions as they worked through the problem. Many students drew a model with fraction circles and split these into the appropriate segments for each child. Some did not have to convert to equivalents to arrive at an answer, while others used equivalent fractions.

During the class discussion, the teacher made it a goal to give students opportunities to share their representations and especially to link the process of finding equivalent fractions. Student engagement was high partly because of the introduction with a discussion of pizza and their favorite types, and because little background information was needed as all students were familiar with pizza and different-sized slices. Students demonstrated their problem-solving skills through explaining their work with words, numbers, and with pictures. Through this task, students demonstrated their mathematical understanding of fractions, equivalent fractions, and comparing fractions with unlike denominators.

In the first example, Student A (see Figure 8.1) used fraction bars and labeled the type of pizza and displayed the correct portions and Student B used fraction circles and labeled the fraction names. They both were able to show that the amount was the same, but the process was more about finding the correct portion and comparing the two amounts than naming the amount of pizza each boy ate.

In the following example (see Figure 8.2), the student actually combined the fractions by renaming each one to a common denominator of eighth and adding them. She showed her work through her pictures, numbers, and words. This was the last student to share their work after the task.

**Figure 8.1 Student work using concrete manipulatives.**

Another student (see Figure 8.3) used fractions and converted each fraction to a decimal. Decimal units had been taught previously, but no other students chose to work with decimal equivalents.

One of the teachers who retaught this lesson in her own class stated, “The goals of our Stuffed Pizza lesson study task were for students to use mathematical reasoning & language, compose & decompose fraction pieces while renaming fractions, compare fractions, and translate their pictures to numbers and words. When I implemented the lesson with a fifth-grade class, I felt as if all lesson and teaching goals were met.

The task was set up to support students’ reasoning skills. Students had to determine which boy was correct in regards to who ate more pizza. Students worked with and

Figure 8.2 Student work using equivalent fractions to combine. *Source:* **Authors.**

Figure 8.3 Student work using decimals to combine amounts. *Source:* **Authors.**

manipulated fractions to figure out the total amount that each boy ate. They combined fractional parts of three different pizzas, to find a whole amount for each boy. In order to answer the final question, of who ate more, students had to compare and reason regarding their work.”