This is about week 3 of flipping the classroom. We are just starting our last unit of the year, which is on equations, equivalent expressions, and inequalities. I decided to start this unit with a lesson on writing expressions and equations. The students had previously learned the distributive property, so I thought I could extend that into distributive property with negatives, and combining like terms.
Common Core State Standards:
Use properties of operations to generate equivalent expressions.
1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
2. Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”
In a flipped lesson, I have the students watch a video for homework that gives them background information or reviews old topics. I made this video. It’s not an amazing video, but I wanted them to review their understanding of profit, income, expenses, and how variables are used in equations.
In class, I checked to see if they had taken notes on the video. About half the class did. NOT GOOD! This is a classic struggle of flipped teaching, isn’t it? I had planned that if this happened, I would go ahead and do the opening problem anyway. This one was designed to be *less* accessible without the video, but not completely inaccessible.
The students worked the problem while they also logged onto their Netbooks. After a few minutes, I asked for answers. They called out…. “350.” “2100.” “negative 50.” “I also got negative 50.” “1500.” “Wait. I changed my answer. Negative 50.” “Negative 50.” “Negative 50.” I said we seemed to be reaching a consensus. I asked for a few students to share how they got negative $50 for the profit.
One student said: “First, I had $1000, and then I multiplied $100 by 5 and got 500 and added that on. Then, I took the $1200, and I multiplied $70 by 5 and added that to the $1200 and got $1550. Then I subtracted $1500 minus $1550 and got the answer.” I wrote: (1000 + 100×5) – (1200 + 70×5) and she said it captured her method. I said she must have watched the video! We discussed whether parentheses were necessary and the class decided they were.
I asked if anyone got the answer a different way. Another student volunteered: “I got the $1500 to start, but then I just subtracted the $1200 and got $300 left to spend on T-shirts and legos. But the T-shirts and legos for 5 students costs $350 and so that’s more than you have.” I wrote: 1000 + 100(5) – 1200 – 70(5) and asked if that basically explained it.
A third student offered that we basically start $200 “in the hole” and so we would need to make up the $200 with the student fees. But if you take the -200 and add 500 for the fees and then take away 350 for the other expenses, you still end up $50 behind where you started. This expression was: -200 + 100(5) – 70(5). I told them I thought it was a clever strategy to combine the fixed income and expenses into one number.
We took note of the different strategies and it was time for one more quick mini-lesson before diving into work time. The students went into Google Docs and opened a spreadsheet. I taught them directly how to create a table for the function y=30x+12. We created a column of x values, increasing by 5 each time. I showed them how to enter a formula for the y-values and drag down the table to complete it. Most students had not done this for a long time and needed the review.
For fun and spiraling, 🙂 I asked volunteers for a real-world situation that might match the equation y = 30x + 12. One student offered (I loved this answer): “You have a countertop.” I said, “Describe the countertop.” He suggested, “It, like, has a part sticking off the side you could set your baby on, and it’s the 12. Then the big part of the countertop is 30 wide and the height is x.” He drew:
Another student offered: “You have 12 dollars now. And you’re selling something, and the thing you’re selling is 30 dollars each.” I said, “So what’s x?” And she said, “However many of the thing you sell.”
It was time for the cookie problem. I love this problem because it doesn’t have an easy mental-math solution – the spreadsheet is a very handy tool for reasoning out how many cookies you need to sell. I presented the problem of owning a business making cheap, processed cookies and told students they should write down important information. Students asked if they could take pictures of the problem with their phones, and I said they could.
I showed the students just how to get started with “Cookies” and “Income” columns in their spreadsheet, and I gave them the instructions to use the spreadsheet to figure out how many cookies they should sell to break even. As students worked on their spreadsheets, one young lady waved me over and asked “Don’t you basically make 7 cents per cookie, but since you have $2000 and you pay $6000, you just need to see how many times 7 cents goes into $4000?” She was combining like terms in her head! I said that she should give it a try and see if her answer comes out reasonable. I also asked if she thought she could calculate profit for any number of cookies using what she said. She nodded and started in on her spreadsheet.
Many students had a spreadsheet that looked like this after a while, so we did a catch-and-release to see what everyone thought and where we were going next.
Their income was not getting close to the expenses very quickly. The students could tell they were not making a profit yet, because the expenses were so high. Some couldn’t tell if the cookies would ever make a profit. Others had used the formula “0.5 * A2 + 6000” for expenses, making their expenses grow faster than their income, which was really confusing. I asked the student who combined like terms to share how many cookies she thought we needed to sell. She explained that 7 cents goes into $4000 over 50,000 times. Students wondered if they had to add that many rows! I suggested they just increment x by a different number. Perhaps 1,000 or 2,000? Who could figure out the exact number of cookies needed to break even? Some kids did make a third column for “profit”, subtracting the expenses from the income. This was helpful to them as they looked for the break-even point.
I released them and they worked a bit longer. Eventually almost everyone determined the number of cookies would be between 50,000 and 60,000, and some students even figured out the answer precisely, although by different means. One student figured out if he made x increment by intervals of 584, he could get within $6.00 of the break-even point. Another student played with two rows of the spreadsheet until the profit got as close to zero as she could get. A few more students used calculators and divided $4,000 by 0.07 to find the exact number of cookies needed.
The lesson was overall a very good one and the students were really engaged. I can tell when a lesson is a good one by the number of kids that approach me for bathroom breaks. NONE!
I would like them to play with the ideas of combining like terms, distributing a negative, and solving equations more tomorrow – and then they can start learning how symbolic manipulation can help them solve equations more efficiently… and that is when it makes sense to use symbolic manipulation! I’ll have to look around for videos! I know there are a lot available already.