This lesson helps my students understand that variable terms and constant terms can not be combined. We started by using the algebra tiles in previous lesson (Who's Right?, To change or not to change ), but it is difficult for them to transfer their knowledge from the physical model which makes it so obvious to the abstract algebraic model. Especially if students have learned that math is all about finding the final single answer, they think that whatever they end up with must be a single term. This lesson helps students see that different variables represent different values and that combining them before actual values are assigned does not work.
As they enter I read the warm up to them about my disorganized wallet. All the items that should go together are all in different sections and the students need to write a simplified list of the items.
Next I ask them to explain how they made their lists. As they go through their list of items (ID cards, coins, dollars, membership cards, photos, receipts, etc.) I write on the projector and ask them where they are in the original list so I can circle them. I continue to ask if there are any other like items they can combine.
At this point I put up the next screen which takes a closer look at the twelve coins. It tells students that I found in my wallet 2 dimes, 3 pennies, 2 nickels, 1 dime, 2 quarters, 1 nickel, and one penny. But then, at the bottom of my purse I found 3 more quarters, 6 dimes, and 4 pennies!
As they combine the like coins I start to write the list: 3 nickels and 8 pennies and …
I tell them this is going to take too long to write and it will take up too much room and I ask if it is okay if I use a letter for the coins. I don’t usually like to use the same letter from the item it represents, but in this case I do. I feel it is easier for students to see initially that different variables refer to different types of terms and I am less concerned with them not realizing that algebraic values stand for values and not labels. In this case the variable does end up representing a value anyhow. I also ask students what mathematical symbol I can use for “and” and I write the expression: 3n+8p+9d+5q
I ask my students to figure out how much money I have in coins and to keep track of the steps they took to figure it out. After they share their work I ask how the following expression might represent their thinking.
3(5) + 8(1) + 9(10) + 5(25)
As they make connections I ask what each number represents. The 5 represents the value of each nickel, the 3 represents the number of nickels, etc.
I bring out the three different pieces from the algebra tiles (the big square, long rectangle, and small square). Again I label the sides of each with dimensions and area: xbyx is x^2, xby1 is x and 1by1 is 1. Next students choose some values for x to show the resulting values of the three pieces. They have seen this before, but it helps to see it again now that they have made some mistakes and are looking for why. After choosing several values for x we should have a table of values algebra tile debrief notes and I ask a series of questions to help students see that even if the x^2 and x terms have the same variable they do not represent the same value and therefore cannot be combined and, even though they are both changing depending on the variable, they are not changing in the same way. Lastly, I want them to see that the value of the constant number does not depend on the value of x and that it doesn’t vary at all. The questions I may ask are:
I may also show them what a mistake might look like if they did combine some of these terms incorrectly. I may write the expression x^2+x and ask what someone might get if they did try to combine these? They may say 2x or 2x^2 or 2x^3. I would write each of these down and suggest that we try out some values of x and see if they are equivalent. I do the same for a variable term plus a constant. For example 2x+4. When I ask what someone might mistakenly get if they tried to combine the like terms. They may say 6x or 6, which I write down and we choose some values for x to test it.
When we do white boards I give them one problem at a time to do with their math family group and I have them all raise up their boards on a count of three so I can see everyone’s board at the same time. Once everyone’s board is up I can give individual corrective feedback. Some kids will hear feedback for another and correct their board before I get to them. Before I give them problems to work on I define what the variables represent:
As I write the first expression I tell them “In my purse I have 3 pumpkins, 5 candies, and 5 school busses in one section and 2 candies and 2 pumpkins in another section. Help me organize my really big purse!”
3x + 5 + 5c + 2 + 2x 4c + 6 + x + 2x + 6c + 2 3 + 2c^2 + 3c + 6 + 2c
I use the same coefficient for different variable terms so they don’t mistake “likeness” with the coefficient. I want them to notice that it depends on the variable. I also switch up the order so we I can show them the standard order of a polynomial as well. This also partly reinforces the focus on the variable. I want to bring out the details so students get the totally right answer and not a partially right answer. This is one way to focus on precision.