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 What we had in mindA room full of sixth-gradersOne class period (40-60 minutes)Students who are new to ratio tables Goals of this activityStudents will be able to...Use a ratio table to scale values in a proportional relationship.Find missing and incorrect values in a ratio table.Use the population of a sample to calculate the total population. Supported Devices This lesson helps students count large numbers of things by using the mathematical structures of area and proportionality. Students use a ratio table to keep track of their work as they count the number of tiles required to cover a floor, and the time required to put those tiles in place.

How the activity works: 1. Tile
Each student tiles a square and learns the number of tiles that fit. 2. Estimate
Students estimate the total number of tiles and the total time required to tile a large area. 3. Compute
Using ratio tables, students compute these values. 4. Analyze
Students are given ratio tables to analyze and correct.
The Student Experience Set the stage for this lesson by talking with them about the context. Ask, If you were going to tile a floor, what are some things you would want to know before getting started?

Students begin by tiling a 4-square-foot section of floor. They are free to use the given tiles in any combination that will fill the section. Students estimate, then they calculate, the number of tiles required to for a 72-square-foot floor.

The tool students use for these calculations is a ratio table. There are many ways of working with this ratio table—all of them exploit the proportional relationship between area and number of tiles. For example, if a student knows that there are 16 tiles in 4 square feet, and that there are 48 tiles in 12 square feet, she may find the number in 20 square feet in any of these ways:

• Multiply 16 by 5, since there are 5 sets of 4 square feet in 20 square feet.
• Add 48 to two 16s.
• Double 48 and subtract 16 because 20 square feet is four square feet less than 24 square feet.

Look for different strategies that your students use—by examining their tables, talking with them, and by reading their written responses.

Keep an eye out forstudents who struggle to find the number of tiles in 8 or 12 square feet. These students may need help stepping back and thinking about this relationship: If there is twice as much area, I’ll need twice as many tiles.
The Teacher Experience

Observe student work both from the dashboard and by students’ sides as they work. Use the dashboard to keep track of student progress and to identify students to give individual attention. How do students find the number of tiles in 20 squares? Their responses tell you about their proportional reasoning.
Keep an eye on student progress. Click on a lagging student to see whether they need help or just more time to think.
Note the additional rows students put in their ratio tables. Discuss these values with the class at the end of the lesson to examine useful proportional reasoning strategies. Students use a ratio table to find the total number of tiles required to tile the floor. Students (and you) can type arithmetic expressions in the cells in order to keep the focus on the relationships involved.

Wrap up the lesson by drawing student attention back to the relationships in the ratio table. There are two sets of relationships that are important:

1. Relationships between columns; e.g. that the number of tiles is 4 times the number of squares.
2. Relationships between rows; e.g. that the number of tiles in 6 squares is twice the number of tiles in 3 squares.

The constant of proportionality expresses this first relationship. Use the tables students produce at the end of the lesson to draw out conversation about the constant of proportionality.