Are 6 X 5 And 5 X 6 The Same? - Math Coach's Corner

This post from 2014 was originally titled 6 x 5 and 5 x 6 are NOT the Same, and it has generated the most amazing conversations over the years! Based on that dialogue, I changed the name of the post a while back, but I left the post as written.

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Greetings from the long lost Math Coach! I am now officially a math interventionist and lovin’ life! My 4th graders have been representing multiplication with square tiles and writing equations to match pictures of arrays (like the ones shown above). These investigations have uncovered a misconception in their thinking that I wanted to share with you tonight. While 6 x 5 and 5 x 6 have the same product, they are not the same thing. The multiplication sign really means “groups of“, so I encourage students to read 6 x 5 as 6 groups of 5. This practice helps students make sense of multiplication.

I have two soapboxes to stand on tonight, so please humor me. First up is my CRA soapbox. Making the switch from additive thinking (6 + 5) to multiplicative thinking (6 x 5) is a huge transition for students, and it must begin with LOTS of concrete (hands-on) experiences. Over time, you need to overlap the concrete, representational, and abstract stages of learning to help students smoothly bridge the distance between concrete and abstract. Look, for example, at the cards below. After selecting these two cards, students can use counters to make 3 groups of 5 (concrete), draw 3 circles with 5 stars in each circle (representational), and write the equation 3 x 5 = 15 (abstract). All three stages of learning in one activity.

Next soapbox. As teachers, let’s be sure we are using precise language. If you are describing a multiplication equation, be sure that you are calling the numbers being multiplied factors and the answer to the multiplication problem the product. I’ve also found that students are easily confused about rows and columns. A simple anchor chart showing this vocabulary is a great reminder for both you and the students.

The images shown above are from my There’s Nothing Alien About Multiplication resource. It includes 4 different games and a themed multiplication division chart. Click here to download one of the games for free!