In this lesson, students explicitly connect and contrast double number lines and tables. They also encounter a problem involving relatively small fractions, so the flexibility of a table makes it preferable to a double number line. Students have used tables in earlier grades to identify arithmetic patterns and record measurement equivalents. In grade 6, a new feature of working with tables is considering the relationship between values in different rows. Two features of tables make them more flexible than double number lines:
- On a double number line, differences between numbers are represented by lengths on each number line. While this feature can help support reasoning about relative sizes, it can be a limitation when large or small numbers are involved, which may consequently hinder problem solving. A table removes this limitation because differences between numbers are no longer represented by the geometry of a number line.
- A double number line dictates the ordering of the values on the line, but in a table, pairs of values can be written in any order. 5 pounds of coffee cost \$40. How much does 8.5 pounds cost? You can see in the table below how being able to skip around makes for more nimble problem solving:
|weight of coffee (pounds)||cost (dollars)|
At this point in the unit, students should have a strong sense of what it means for two ratios to be equivalent, so they can fill in a table of equivalent ratios with understanding instead of just by following a procedure. Students can also always fall back to other representations if needed.
- Compare and contrast (orally) double number line diagrams and tables representing the same situation.
- Draw and label a table of equivalent ratios from scratch to solve problems about constant speed.
Let’s contrast double number lines and tables.
Make 1 copy of the The International Space Station blackline master for every 4 students, and cut them up ahead of time.
- I can create a table that represents a set of equivalent ratios.
- I can explain why sometimes a table is easier to use than a double number line to solve problems involving equivalent ratios.
- I include column labels when I create a table, so that the meaning of the numbers is clear.
A table organizes information into horizontal rows and vertical columns. The first row or column usually tells what the numbers represent.
For example, here is a table showing the tail lengths of three different pets. This table has four rows and two columns.
pet tail length (inches) dog 22 cat 12 mouse 2