Thought for the day: Inserting blank spaces into a multiplication table encourages students to predict products of fractions or decimals before they learn the rules.

Examples of noticing and wondering

I notice that this multiplication table includes 0.
I notice rows and columns inserted between each row and column in a regular multiplication table.
I notice that the entries that are shown in the table match the ones in a regular multiplication table.
I notice that the whole numbers in each row (column) increase by the amount of the first number in the row (column).
I notice that the blank squares will probably have to include fractions (or mixed numbers) or decimals.

I wonder how I can figure out the missing numbers that belong in the table.
I wonder if the numbers in each row and column will still increase at a steady rate when I fill in the blank squares.
I wonder if there are other patterns I can find that will help me fill in the missing squares.
I wonder if some of the new squares will contain numbers less than 1.
I wonder if I could create a table with even more rows and columns between each of the whole numbers.
 

Notes

This prompt guides students to think about the why behind multiplying fractions and/or decimals.

The idea—though it may take some time to get there—is for students to predict how to complete the table before you teach them rules for doing the calculations. For example, they may predict that the "1/2" row (or column) starts at 0 and increases by 1/2 every other square. When they fill in the intermediate squares, they discover that they need to count by 1/4s in order to make the numbers in the row increase steadily.

Even (or especially) if some students complete the table fairly quickly, be sure that they step back and pay attention to what is happening. For example, they may be able to explain why it makes sense that 1/2 • 1/2 = 1/4 (because 1/4 is half of one-half) or why 3 1/2 • 1  1/2 should equal 5 1/4 (because it is 3 1/2 plus another half of 3 1/2). 

More to explore:

  • Try to discover a shortcut for multiplying fractions.
  • Explore thirds by inserting two rows and columns between each pair of whole numbers!