Some students failed the last test on simplifying expressions with distributive property and combining like terms. Those students will work on an intervention worksheet which provides more detailed directions and examples. These were students who still needed intensive individual support. Other students need support in our work with integers, because they are having trouble interpreting the numeric problem within the context of the hot and cold cubes (Cooking with Mathmaster Chef series). Some students are mistaking -5 for taking out cold cubes and need clarification that if there is only one negative symbol that it can refer to "taking away" OR "cold cubes", but not both. The other part students may struggle with is that they want to rush and finish quickly by not asking themselves what adding cold cubes or taking out cold cubes does to temperature, which is what tells them which direction to go on the number line. Students will be doing their "homework" in class so I can circle any problems that are wrong for them to take a closer look at. Those that finish early will do the asignment on the back which is to choose one of my "curiosity quotes" and write what it means to them. I post their quotes on my wall. When students become so accustomed to following the teachers lead they stop paying attention to their own thoughts and wonderings. I have noticed that this often prevents them from asking questions that will explore for deeper meaning and from looking for and recognizing patterns which lead to generalizations. I think it is important to encourage wondering and "what if..." questions by students in the classroom.
The warmup simplify intervention consists of three intervention questions about distributive property and combining like terms to help prepare those that need to retake the test as well as to give everyone additional practice. As students finish the warmup I direct them to try to answer the problem on the board which asks them what - 5 might be asking them to do with hot and cold cubes (Cooking with Mathmaster Chef series) and gives choices:
Take out 5 hot cubes
Put in 5 hot cubes
Take out 5 cold cubes
Put in 5 cold cubes
The first problem from the warm up, 3(4x+1), reviews multiplying throughout the parentheses. Some students forget to multiply 3 and 1, others may add them because of the addition sign. I would remind these students that the 3 outside is multiplied by all terms inside the parentheses and that the addition does not apply to the 3.
The second problem, 2n + 3c + 2 + 4c + n + 1, reviews combining like terms. The most common mistakes at this point are not recognizing the coefficient on the "n" and putting the final answer in the wrong order. I circle the 2n and ask how many "n's" is says we have, and then circle the "n" and ask how many they see here. If this is not enough I would ask "if we have 2 of them here and we are adding one more, how many will we have after we combine them?"
The last problem reviews factoring with the distributive property. The 3 most common mistakes at this point are to factor correctly but not the greatest common factor, to factor out the greatest common factor but leave the rest off, and to leave out the variable. In each case I make sure I start out with what the students are doing correctly. I remind them that if there is still a common factor that the factor they "pulled out" is not the greatest. I may suggest they use a factor tree at this point. If they do factor out the greatest common factor, but forget the rest, I circle the equal sign and ask if 24x+36 is equal to 12. (no) Then I tell them that I am asking them to undo the multiplication that was distributed and point out that they are doing the same type of problem as number 1, but backwards. I may go step by step and ask "12 times what equals 24x?", etc. This will also solve the problem of leaving out the variable..."12 times 2 equals 24, 12 times what will equal 24 x?"
The problem on the board is to help students notice that there is only one "negative" sign on the 5, so it can represent "taking out" OR "cold cubes", but not both, so it could be taking out hot or putting in cold.
Students who do not need intervention to retake the simplifying test will work on the homework integer adding subtracting on number line for the remainder of class. It consists of integer addition and subtraction with number lines and also a written response to famous quotes about homework curiosity. The sentence frames from the last lesson (which way do we go?) have been excluded, but I point out that this process still needs to take place in their heads because it is what tells them which direction to go on the number line. While I circulate I check for incorrect answers that indicate to me that they are skipping this step. I circle the incorrect answer and tell them that they are getting it wrong because they are not making sense of what the hot and cold cubes are doing to the temperature. I included the quotes on curiosity because I see a lack of it in the classroom. I think students have become too accustomed to waiting for directions and information from the teacher and have learned to suppress their questions and curiosity. I think this prevents students from asking deeper questions that could deepen their understanding and I think that it prevents students from looking for and recognizing patterns that lead to generalizations. I put insightful student responses up on sentence strips around the room.
Students who need intervention work instead on the simplify intervention that reviews multiplying with the distributive property, combining like terms, and factoring with the distributive property. They will take a Simplifying retest later in the week. I circulate and give individual instruction. Mistakes that I expect to work on are not multiplying to all terms inside the parentheses or adding instead of multiplying, combining x^2 terms into x^4, ordering terms in the polynomial, and factoring a lesser common factor or not writing an equivalent expression and just writing the factor. These are often the students who have become so accustomed to direct teaching and are having trouble taking a more active role in their learning.
I use recycled manila folders cut in half as flash cards. I show a card to the class for them to figure out. Students stand behind their desks and they don't raise their hand or call out. I call on a student and expect the answer in 1 second or less. This forces everyone to get their answer ready in their heads and ensures that all students (in theory) are doing the problem and not just the one I call on. If a student gets it wrong, doesn't answer in time, or says they don't know I call on another student. If a students gets it right they sit down and are done. If several students get it wrong or don't know I will briefly go over it after a student has gotten it right.
Today's cards have integer addition and subtraction on them and they have to tell what the problem might be telling us to do with hot and cold cubes (Cooking with Mathmaster Chef series). For example:
5 - (-3) means we put in 5 hot cubes and take away 3 cold cubes
-2 + 4 either means we take away 2 hot cubes and put in 4 hot cubes or put in 2 cold cubes and put in 4 hot cubes.
This is a way of making sure they can interpret the symbols. The tricky part is that their is no operation symbol on the first number. I could have taught them that this is just the cubes we start with, but I really wanted them to become familiar with the equivalence of adding positives (hot cubes) and taking away negatives (cold cubes), etc. Taking a little extra time to clarify this part is worth it because it gives them more exposure to this idea.