Artifacts

We started Multiplication this week. I needed a quick way to determine if students knew their multiplication tables or not that segued quickly into me working with the students who had no concept of multiplication. I decided to give 3-minute Multiplication Fact quizzes all week – the students who knew their facts completed the quizzes quickly and bought in from the challenge of improving their scores and the competition of comparing their numbers with their peers. The students who needed me to teach them multiplication were clueless on the quiz, but since it only lasted 3 minutes, I could quickly see how they were doing and start remediating immediately. By the end of the week, here’s what their work looked like:

Some notes:

• I defaulted to showing the area model because it’s the most visual of the models. Most kids gravitated to that. By the end of the week, some students got tired of drawing all the circles and wanted a faster method, which let me show them how keeping track of each column let you build a multiplication chart so you didn’t have to draw circles for every problem.
• Some students used the ‘count the tallies to create a chart’ method (similar to skip counting), but since language is an issue with my students, counting was also an issue and these students tended to make smaller counting mistakes (pointed out about with red arrows). The students who used the visual models tended to fair better than the students who tried to rely purely on the symbolic/procedural models.
• In one of the pictures above is the Lattice Method, but its crossed out. I showed this to several students, but no one latched onto it, which I thought was curious because it’s also very visual. In thinking about next week (division), I’m actually kinda glad no one latched onto it – it’s pretty straightforward to show how to divide using the same area model and the tally model that you use for multiplication, but the lattice method is a little less straight-forward to reverse-engineer to get division.
• Check these out:

This is from my student who had absolutely no concept of multiplication – she thought the ‘x’ symbol still meant ‘add’. I showed her the area model, but she has trouble counting past 15, so I also told her to cross off circles as she counted them – every 10th circle, write ’10’ on the side, then start over. Using this strategy, she was basically grouping circles by 10s, then adding all the groups together to get her answer. This was the basis for my comment on twitter: I’ve never appreciated place value and grouping-by-10 than when my students can’t reliably count past 15.

• I thought this was curious: several students used this strategy

In other words: students doubled one multiplier while halving the other multiplier, then added the former number as many times as the latter number. So 7 x 8 is the same as 14 x 4 which is 14 + 14 + 14 + 14. I was surprised several students did this by default without needing to be shown this ‘trick’ – very clever.

• I give a quiz every Friday over whatever weekly skill I decide to cover. This week was a 20 question quiz that was all multiplication problems. Every student passed, even the ones who spent the whole period creating their models and only learned multiplication on the Tuesday (we had Monday off from Labor Day). I feel like this is the only way this class can work – every skill needs a model that students can create on their own and use to solve problems, even if it means it takes much longer than might be considered ‘reasonable’.