**1. Equivalent fractions-**-The multiplication table is filled with row upon row of equivalent fractions.

Here you can see a simple multiplication table that I created on Google Sheets. The thing I love about my Google Sheets multiplication table is that I can customize it in whatever way is useful. So if I want to talk about equivalent fractions for 3/8, I can highlight those rows. But then I can easily change to something else. What a great way for my students that may struggle with equivalent fractions to have a quick reference to find them, but also a visual way to see that the reason by 12/32 is equivalent to 3/8 is because both 3 and 8 were multiplied by 4! These realizations that may at times seem to be no big deal for teachers can absolutely blow the minds of our students.

**2. Equivalent ratios/ratio table--**Just as we can use the multiplication table for equivalent fractions, it can also be used for scaling ratios up and down to find equivalent ratios. So now our students are trying to answer some proportional reasoning question: "It takes James 4 minutes to solve 7 problems. How many problems can James solve in 12 minutes?" Again, this idea of using the multiplication chart flexibly, even attaching a label or meaning to some quantity, can be a stretch for kids at first.

**3. Proportional relationships versus non-proportional (but linear) relationships**--Let's say you were focusing on the proportional relationship of 6x = y. What about showing the multiplication chart as a place to see this?

The multiplication chart can show the table of values for a proportional relationship by simply looking at the column with the correct constant of proportionality. This gives another way to think of a proportional relationship....it is a relationship that if you had an infinitely large multiplication table, it would have a row on there. You could also build the connection between 6x = y and 6x + 1 = y by having students add one to all of the values in the 6 column.

Ok, and honestly, some students will use the multiplication chart because they don't know their multiplication facts. As much as I wish this weren't true, it just it. So rather than fighting against it, I've decided to help all of my students see that the multiplication table can be a great tool to help us learn about a lot of middle school concepts far beyond simply multiplying. After writing all of this, I think this year, we just may create a digital multiplication table in the first few weeks of school to establish right away what a great tool it can be.

How do you think multiplication tables can be helpful in middle school? What other "elementary" tools do you rely on to make your classroom a better place?