Forum Replies Created
-
Posts
-
A ratio is comparison of two numbers, whereas a fraction is a single number. All sorts of problems can arise by confusing the two. For example, suppose I have juice recipe which requires 2 cups of orange juice to 3 cups of peach juice. So the ratio of orange juice to peach juice is 2 to 3. Now I double the recipe, to get an equivalent ratio of 4 to 6. If I identify these ratios with the corresponding fractions, 2/3 and 4/6, then it sounds like I am saying that two times 2/3 is 4/6. A similar problem arises with addition of fractions. Suppose I have one class that has 9 girls and 10 boys, and another class that has 11 girls and 8 boys. So the ratio of girls to boys is 9:10 in the first class and 11:8 in the second class. Now suppose that I combine the two classes: what is the ratio of girls to boys in the combined class? I just add the 9 and the 11 to get 20 girls and the 10 and the 8 to get 18 boys, so the ratio is 20:18. But if I now confuse the ratios with the fractions, it seems like I am saying that 9/10 + 11/8 = 20/18. The misconception that you add fractions by adding the numerators and adding the denominators is a fairly common one, and I suspect it comes from the confusion of fractions with ratios. You are right that this confusion is rife in the field. I’ve even seen materials that promote the confusion and come up with this different way of adding fractions and say it is just as good as the “traditional” way. What a mess!
Thanks for these comments. I agree that we don’t want to make too pedantic with children and parents; certainly it would be crazy to test them on this vocabulary point (but yes, I know, people do crazy things). And the progressions documents should not be read as dictating what we say or do with children and parents. Their main purpose is to clarify the underlying issues for curriculum writers and others who need to make a deep examination of the standards. From that point of view, there is an issue that needs to be clarified. Suppose I say that apples cost $1.15 per pound, and then write the equation y = 1.15x to represent this, where x is the number of apples and y is the cost in dollars. That number 1.15 appears two times in the sentence: once with units (dollars per pound) and once without (in the equation). The progressions document calls the second one the unit rate: you could also go the other way and call the first one the unit rate (your preference) and call the second one the constant of proportionality, or the numerical rate, as you suggest. Either way, it’s useful to have some conventions about this. Notice that the standards themselves sidestep this issue, by requiring the use of ratio and rate language, but not specifying what that language is. So this is something the field has to sort out.
Yes, using a tree diagram for this is entirely consistent with 7.SP.8. The main thing would be not to get hung up on teaching general rules for calculating the probability of compound events—that really is high school, as in S-CP.B—but rather to give students concrete experiences that prepare them for those abstract rules.
I really don’t think this is an important mathematical or pedagogical issue. Clearly there is some confusion about the meanings of the terms “histogram” and “bar chart.” While it would be a good idea for the field to come to some common understanding of the meanings of these terms, I do not think it’s an important concern for students of mathematics in these grades. In the end, we want them to be able to correctly read and produce these graphical representations of data. That’s the most important thing. What words they use for them is less important. Textbooks and teachers should be consistent in whatever terminology they use, of course.
I wouldn’t read to much into this in the sense of it being a format required by the standards. It was, as you say, mostly a matter of making the meaning clear. Stacked fractions are clearer, but not always typographically possible.
I’m not sure I completely understand the question, but here are some reactions. On the one hand, I think the activity on page 2 might strike students as a little weird: “we already know that 2/3 of 15 is 10, why are we doing this the hard way?” On the other hand, it is illuminating in showing a connection between previous knowledge and general rules that have now been developed for operations on fractions. My general feeling about reducing fractions is that there will be situations where it is clearly beneficial to replace a fraction with an equivalent simpler one, and this is one of them. So I’m not opposed to reducing fractions, just to the idea that it is always a necessary thing to do. We want students to know how to find equivalent fractions and choose useful ones in cases where there is one that is clearly useful.
Well, really, we were not trying to be word police here; I don’t see any reason why you couldn’t just use the word “simplify” on occasion. But when there is a more precise term available, use it. For example, when you multiply two polynomials you might expand the product, or you might leave it in factored form. When you substitute numbers into the quadratic formula, then I think it is appropriate to talk about simplifying the resulting numerical expression.
You have a sharp eye. Geometry in Grade 6 is, I think, a not-quite-critical area, and the grade level introduction reflects that ambiguity. I would also say that the statements of critical areas are really just attempts to summarize succinctly what is in that grade level. So they are not necessarily an indication of priorities about allocating time; there is no statement about which critical areas should receive more attention than others.
I think you meant “will not need to be able to read a z-score table” in the first paragraph, right? Your approach sounds reasonable, although I could also imagine an approach that does not mention the term “z-scores”. There are lots of possible approaches here.
“Vetted and approved” is a bit strong. I helped review the Arizona ones early on, but I don’t think there was time for a thorough review. So they are probably basically pretty good, but you should not attribute oracular authority to them. I don’t know about the Kansas ones.
I can imagine what excesses people might be going to with this standard. The intent of the standard is to prepare for the discussion of similarity and congruence in Grade 8, with some hands on problems that give students a concrete feel for similarity. All of the things you mention in your second paragraph are possibilities, but some judgments have to be made about how much time this takes up in the curriculum and how it supports other more central topics, such as ratios, proportional relationships, and work with rational numbers and their representation as decimals and percents.
I do not think it is necessary to introduce congruence and similarity as formal concepts here; as I said, the point is to prepare for those concepts, not try to do everything at once.
And, any geometric figures that kids have seen are fair game, but you could go a long way with figures made up out of triangles and rectangles.
Slope is a geometric concept; it is a property of a line in a coordinate plane. So, a line can have slope. But a function is not a geometric object, so it can’t really have a slope. When people talk about the slope of a linear function, they really mean the slope of its graph, a geometric object which is attached to the function. And if the linear function relates quantities with units, for example distance versus time, then makes more sense to talk about the rate of change rather than the slope.
In a document like the Progressions you want to use the same word throughout to emphasize the unity of mathematical concepts across grade levels. Ellenberg’s article served a different purpose; it explained why the term “number sentence” made sense mathematically, and might help students understand what an equation is. Personally I don’t have a strong opinion here; my main interest is in discussions about meanings. So I’m happy either way as long as the meaning is made clear.
