I introduce equations with a pan balance, cups and pennies (basically Timon's lesson, I've adapted slightly but apparently I've never blogged it). Most kids see the visual, draw a few models, internalize the concept and work with the variables. Some kids see the visual, draw a lot of models and I coach them on the transition to variables. One student I have this year needed to touch the model. So we spent a few classes going back and forth between drawing and counting out pennies. Then we got to equations with negatives and I stalled and gave him different problems until his model was really solid. Then the class was working on equations with no solution and infinite solutions and we had a great debate with a kid who thought the model was silly but then needed it to really understand no solution equations. We even modeled the distributive property with cups and pennies!

Finally, I was talking to a colleague about this pedagogical challenge and she mentioned hot/cold cubes. Today I grabbed a red and a black marker and we got started working through some equations with negative numbers. I was ready to pull out the red (hot) and blue (cold) cubes if we needed them but he got it! It was also interesting to see him get tired of drawing pennies and start writing the number instead. I didn't snap a photo of his work (plus marker on graph paper doesn't work well if you use both sides - it bleeds through!). It wasn't as neat as mine (plenty neat for his needs though), but I used the notation he used today. He also didn't rewrite things on every line, but this way makes it clearer for you what is happening and in what order.

Bonus 1: Integer problems are easier like this. At one point he had a pink 30 and two more pink pennies. It was obvious to him that made a pink 32 in a way that -30 - 2 = -32 is not obvious.

Bonus 2: the rest of the class was working on inequalities, which I have them solve by solving the equation and then testing values on the number line (both to check their work and so they don't forget to switch the direction of the symbol when dividing by a negative, not to mention we use this strategy in Calculus!). If you use the method above of always getting positive cups/variable by adding (instead of dividing by the negative number of cups) your inequality symbol never changes on you! I was psyched when he did that automatically and realized I wouldn't have to tell him any random rule about the symbol changing. So today this student successfully solved inequalities along with the class!

Teaching the contained math class is tough, but it's so exciting for me and my students when we find a strategy that works for them. I'm looking forward to a lot more color in this kid's strategy kit!

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