Association of Teachers of Mathematics in Maine
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Prompt 2: Option A

09 Nov 2014 4:00 PM | Anonymous member (Administrator)

Use and connect mathematical representations.
Analyze samples of student work from a lesson that you have taught this year. Find examples in which students have used different representations to solve the same problem. Make a plan to connect those representations explicitly in future lessons. Find relationships between and among the representations and think about how you could use the students' work to develop their understanding of a concept.

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  • 23 Nov 2014 9:07 PM | Anonymous member (Administrator)
    Students in the seventh grade have been learning about negative numbers. They've worked with red and black chips, used number lines, discussed scenarios involving integers and played a number game using integers.
    In working with students in small groups, I've noticed that they try to use the rule that they've been developing in class but they are often applying the multiplication rule to the addition of negative numbers. I've had students use + and - cards to represent the situations and to talk through a scenario that represents the problem. When I present a scenario to a student or have them develop the scenario, they generally get the problem correct.
    My plan has been for the students to remember to fall back on the scenarios or number lines as they work through problems on their own. The number line relationship between subtracting a positive and subtracting a negative is one that I'd like students to use. I'm going to try using their work on a recent assessment to see if they can use this relationship (subtracting a positive and subtracting a negative) as a way to help them understand the concept of subtracting integers.
    Part of the battle is to have students recognize that they can do the work - it might be hard but that doesn't mean they can't learn it. Their frustration with math and it not usually making sense is exactly what I want them to confront. I want them to learn how to make it makes sense.
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    • 30 Nov 2014 1:00 PM | Anonymous member (Administrator)
      How do you show subtracting a negative using a number line? Is it about direction? In which case this is a connection to vectors, possibly. When my students struggle with subtracting a negative number, I go back to the idea that to "subtract" means to "add the opposite." I'm not sure where I got this from, but it helps to clarify the concept for them, at least when working with algebra ideas.
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  • 30 Nov 2014 9:39 AM | Anonymous member
    In my 8th grade Pre-Algebra classes I use a specific problem either at the beginning of the unit or after students have reviewed solving equations by combining like terms. However, I encourage students to use any strategy that they'd like in solving the problem. The most popular strategies are usually guess and check, looking for patterns in organized tables, and model drawing. When I use this problem in class tomorrow, I will provide students ample time to discuss and critique their strategies and I will pre-plan some specific questions which are more in tune with the 'focusing pattern' of questioning. While my ultimate purpose is for students to add equation writing and solving as a strategy, I want students to see the connections between the equation and the other strategies and to analyze the strengths of the strategies.
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  • 10 Dec 2014 2:47 PM | Deleted user
    We were working with slope intercept and graphing data. I had a wide rage of student understanding and ways of showing the slope intercept formula for the data. Some students were able to take the data and convert it into x, y values subtract ordered pairs to find the slope and then solved for the y intercept. Other students were able to find the differences in x and y in a less formal way by looking for the pattern of change in the data and making that into the slope, then plot the ordered pair closest to the y axis and step the slope backwards until it crossed the axis to find the y intercept. I had others that had to graph the data and pulled the data from the line, to come up with the formula. We talked about the various methods as a class and we discussed pros and cons of each method including accuracy.
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  • 06 Jan 2015 10:25 PM | Deleted user
    One strategy used for making a ten while completing addition problems, was to draw circles. This strategy was difficult for me to understand at first because I had never seen it before. I will try to explain how it is meant to work. It is not just an illustration of the two addends, and counting the total number of circles. No, this is different. Given the addition equation 9 + 6 = ?, the student would draw one circle to make a ten, make a vertical line, and then draw circles counting to a total of six circles ( the second addend...). The visual then is one circle, a separating vertical line, and then five circles. i.e. o I ooooo
    Only two students could use the drawing to see and understand what the answer would be quickly with out counting up from nine. You can see ten and then add five more for 15, or one group of ten and five ones. This strategy draws your attention to the tens and ones parts of a number. Once I figured it out, I have found that it really makes the teen numbers and addition facts make sense and seem easier to break apart and put together.
    I worked in another lesson to show the two student drawings and examples. I really tried to focus on the making ten and then adding more ones. I encouraged the students to look at their answers and say how many tens, how many ones in the answer. Then we checked the drawings to see if they matched what was said. I do think that a few more students gained a bit more flexibility with this strategy.
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