Difference of Squares through Pictures

Our unit on Quadratics is probably my favorite unit to teach. There is so much growth! One of the topics we cover in our Quadratics unit is differences of squares. I love looking at concepts through pictures (Have you seen the post Pythagorean Theorem Proof Without Words from Mrs. E Teaches Math? It's awesome.) The format of a factored difference of squares seems so simple, but I find that giving it context through pictures helps the concept sink in. 

Here we have a square with side length x.

A Math Word Wall PEMDAS Mobile

PEMDAS mobile tor a math word wall

Some ideas seem so simple but for some reason have details that take forever to work out. This PEMDAS mobile was one of those ideas. For a long time I had wanted to find a way to show that division and multiplication and subtraction and addition can switch order depending on which of the pairs comes first in an expression. I love mobiles because they are dynamic. So a PEMDAS mobile just seemed like a good idea!

I know that PEMDAS is a little controversial and that a lot of teachers would rather use a different acronym, or maybe forgo acronyms altogether in favor of Order of Operations. In my experience, kids understand that brackets are used to line up grouping pairs, similar to computer coding. Still, if you like GEMDAS or BEMDAS, I included a G and a B can be subbed in for the P.

PEMDAS mobile tor a math word wall

The first thing I did was color then cut out the 8 pieces. I like rainbow theme, but you can color it to match the theme of your room. I printed on thick paper, almost like that oak tag we all used to pine over in the 1980s.

PEMDAS mobile tor a math word wall

I then flipped one of each P, E, MD, AS piece face down, put glue in the center and ran a ribbon down the middle on the glue.

PEMDAS mobile tor a math word wall

Next came the matching pieces glued on top and then a big book to flatten for a bit. 

PEMDAS mobile tor a math word wall

I only flattened for about 5 minutes in case some pieces stuck together, which they did. The glue was still wet to they were easy to peel apart. 

PEMDAS mobile tor a math word wall

And here's the final product! If I were to make it again, I might use a thinner ribbon so the pieces could move more independently of each other. I also made the mobile match my other math word walls so that it can be used as a supplement. 

math word walls

You can download the PEMDAS mobile for free here. You can also check out the math word walls that are in my classroom and the ones that I have made for 5th grade, 6th grade, 7th grade, 8th grade, Algebra, Geometry and Algebra 2 through this link.

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Zero on the Rounding Roller Coaster and my Growth Mindset Moment

Rounding roller coaster in a 5th grade math word wall

This past week I had to do a fair amount of pride-swallowing to hear some critical feedback. This often seems to happen when I post on Facebook!

This time it was about this seemingly innocuous rounding roller coaster in my 5th grade math word wall. I left out the zero on purpose. After all, 1.70 is the same as 1.7, right? So we're not really rounding it. Right? 

Effective Frequency: Do we teach Slope too much?

One of the hardest high school math topics I teach in Algebra 2 is slope. I know what you're thinking-- slope is not an Algebra 2 topic. This is exactly what makes it so hard.

I have this theory, and you can disagree with me in the comments. My theory is that slope is so hard to teach because the word is so familiar. My students may see it first in 8th grade, then definitely in 9th grade, then again as a review in 10th grade before the state test. By the time they get to me in 11th grade, eyes are glaze over at the mere mention of the word. But very few have really mastered the topic.

So why is retention so low? My belief is that slope is either introduced too early before students are ready, and/or too little time is spent on the topic the first time it is introduced. 

This video is about an advertising strategy called "Effective Frequency". When I saw it, I immediately thought about my students. "Sales" is buy-in. "Potential prospects" are our students. 'Advertising" is teaching. "Conversion" is learning. Sometimes I feel like the idea of working hard, completing work, and that school is important is a hard sell. In this way, this video applies directly to teaching. Is slope overexposed? After the 3rd time students see it, is it too late?

Years ago when doing research on negative numbers for my thesis, I came across an eye-opening article from William Schmidt, Richard Houang, and Leland Cogan titled A Coherent Curriculum, The Case of Mathematics. In this article, the number of topics covered per year in the United States is compared to the number of topics covered per year in those A+ countries that the US is always pointing to with a, "Our math scores need to be more like theirs." SIDE NOTE: I wish we would celebrate all the good things about our students rather than focusing on what makes them less than kids in other countries. But that's for another day.

William Schmidt, Richard Houang, and Leland Cogan, A Coherent Curriculum, The Case of Mathematics

Above is a screenshot of the graphic included in the article that shows the progression of topics in A+ countries in grades 1 through 8. Only a few topics are covered each year, allowing teachers and students to get deep into these topics.

William Schmidt, Richard Houang, and Leland Cogan, A Coherent Curriculum, The Case of Mathematics

And here is the graphic from the United States. To be blunt, we're all over the place. Many topics are taught every single year, at least in the grades shown. 

How may times have you said to yourself, "I wish I had more time to spend on this topic because my kids ALMOST get it"? I know I have said it a lot. Do you think student understanding and retention would improve year-to-year if we were allowed this time when first introducing topics?  

Teaching Students the Pitfalls of Simple Interest Loans

Simple interest loans sound so, well, simple. But teens need to know what to look out for when deciding on the terms of their car loan. Here is an extreme video I found where a woman gets completely ripped off by a simple interest car loan: