[Bill McCallum]

Experiences in Grade 8 should be hands-on and experimental, showing transformations in motion, so to speak, with physical tools such as transparencies or with dynamical algebra programs. Once students have a good visual understanding of what the different transformations do, they can start developing the notions of congruence and similarity using them, as in this task. We’ll try to get more Grade 8 geometry tasks up there, but in the meantime it might make sense to look at some of the high school tasks and imagine toning them down a little, such as this one or the various ones attached G-CO.8. I can imagine working with the same basic diagrams here but not getting into the congruence criteria.

[Bill McCallum]

I hope this doesn’t turn out to be the case and I agree it’s not a good situation. I’ve heard similar things from other states. It seems to me that at the very least one could remove items from the test bank that are no longer in the Core. However, even if that doesn’t happen it’s conceivable that sticking to the focus in the Common Core will be beneficial because it could improve performance on most of the test. For example, if 20% of your current consists of material that is no longer in the core, it might still be a good strategy to focus strongly on the other 80% and let the 20% go, in the sense that it could improve performance on the test overall. The promise of focus and coherence is that greater proficiency with fewer topics will serve a student better than a weak grasp of many topics. Of course, I understand how difficult it must be to make that leap in practice.

[Bill McCallum]

Yes, that’s correct, written expressions involving decimals are certainly included.

[Bill McCallum]

Sorry, I don’t know what documents you are talking about here.

[Bill McCallum]

“Knowing the formulas” includes both of the things you mention, although personally I wouldn’t assess that by just asking them to repeat them, but by expecting them to be able to use them in solving problems (without having to be told them).

There is no preferred choice of letters for the various quantities that come up in these formulas. Knowing a formula means more than just knowing “V = bh”. It means knowing what quantities all the letters stand for. So knowing the formula for the volume of a rectangular prism means knowing “if the height is h units and the area of the base is b square units then the volume is V = bh cubic units.” You could replace b by B in that sentence (both places where it occurs) and it will still be the same formula. The names we give to quantities are not essential components of a formula.

Of course, it is useful to have conventions about the choice of letters. You wouldn’t want to say “the area r of a circle of radius A is given by $r = \pi A^2$.

[Bill McCallum]

The thread that Duane is referring to is here. As Duane says, the Common Core views 0.2 and 2/10 as different names for the same number. So adding 0.3 and 0.04 is the same thing as adding 3/10 and 4/100. In order to add them you have to convert 3/10 to 30/100. If you write this down as
$\frac 3{10} + \frac{4}{100} = \frac{30}{100} + \frac{4}{100} = \frac{34}{100}$
then people might call it fraction addition, and if you write it down as $0.3 + 0.04 = 0.30 + 0.04 = 0.34$ then people might call it addition of decimals. But if you ask a child to explain either one of these, the explanation is exactly the same: “I have 3 tenths and 4 hundredths, and to add them I have to express them in the same units, and I know that 3 tenths is 30 hundredths so the sum is 34 hundredths.” Thus, when you are teaching for understanding, the distinction between fraction addition and decimal addition melts away. They are only different operations requiring “conversion of decimals to fractions” if they are each taught as blind procedures.

Of course, once students have a secure understanding of the underlying meaning, we are not going to expect them to keep repeating the explanations. At some point, when they acquire fluency, they can just “see” the answer. This is similar to adding 30 and 4, where students initially think explicitly in terms of 10s and 1s, but at some point have a sufficiently robust understanding of place value that they just “see” that 30 + 4 = 34.

[Bill McCallum]

I hadn’t written a reply to the other post yet. Your point about similar variability is well taken. Basically the idea of this standard is to get across the idea that if two distributions have different means, you have to look at the variability of each before you try to infer something from the difference in means. If they have very large variability, and so a lot of overlap, the difference in means doesn’t mean much; if they have narrow variability then maybe it does. The simplest case of this would be if they both have similar variability, so you can get a sense of how big the difference in mean is compared to the variability.

[Bill McCallum]

Yes, we are working on it.

[Bill McCallum]

Golly, no, I don’t think I do have an opinion! But I will say that in Illustrative Mathematics we have adopted the convention that number lines have arrows at both ends (at least once you get beyond elementary school, where number lines are really number rays).

[Bill McCallum]

Admittedly it doesn’t say so explicitly. But the main point of 8.EE.7 is to expand into the entire universe of linear equations with rational coefficients, relaxing the restrictions of 7.EE.4 on the type of equations. The application of linear equations to solving problems doesn’t go away, and you could call on the practice standard MP4 (Model with Mathematics) to serve as a reminder of that.

[Bill McCallum]

On the issue of withholding diplomas, that’s a matter of state and local policy. It’s certainly not mandated by the standards; it’s not even mandated by NCLB. I don’t have a preconceived position on it; it depends so much local conditions, how many students are effected, the school culture, and so on.

As for how the standards were chosen, that’s a long story. The list of sources in the bibliography gives some idea of the answer. Certainly recent reports such as Curriculum Focal Points, the NRC early childhood learning report, and the report of the National Mathematics Advisory Panel were influential. Also the standards of high-achieving countries and states. And, perhaps most importantly, the progressions documents written by the work team, which we are no slowing bringing into final form.

[Bill McCallum]

Fred, sorry I wasn’t more helpful, but apart from my prejudices expressed earlier I don’t have much to say without knowing more of the specifics. If were at a PTA meeting at my kids’ school where this was being discussed, I would base my decision on all sorts of local considerations, including how the school handles the different tracks (does it take them all seriously or treat one of them as a dumping ground, for example).

[Bill McCallum]

Mary, I began to have second thoughts on doing this as an official effort, because I try to maintain a distinction between the standards and my interpretation of them. In principle my interpretation is no better than anybody else’s, although of course I can provide insight into the thinking behind them. That isn’t to say this wouldn’t be useful as an independent effort.

[Bill McCallum]

I agree it can be confusing, and that’s partly because we are trying to accommodate old usage at the same time as promoting new usage. In the Common Core, a fraction is a certain type of number on the number line. The name of the number doesn’t change what it is, so in that sense mixed numbers and finite decimals are fractions (also some infinite decimals of course, but that’s a story for later). The terms “mixed number” and “decimal” really refer to certain ways of expressing fractions. So really every time we we talk about them we should say something like “fraction expressed in mixed number form” or “fraction expressed in decimal form.” But that would get old very quickly, so we use the shorter terms.

And, the terms “proper fraction” and “improper fraction” are deprecated entirely in the Common Core.

[Bill McCallum]

The equation $5 \div 3 = \frac13 \times 5$ is certainly an instance of invert-and-multiply. But that rule does not arise until Grade 6, and in Grade 5 the equation arises in a slightly different way, as you suggest. In Grade 4 students were multiplying fractions by whole numbers, because can be explained in terms of the repeated addition interpretation of multiplication by a whole number. But that does work for multiplying a fraction by a fraction, which students start doing in Grade 5, so in Grade 5 students have to get a meaning for expressions like $\frac13 \times 5$ that did not have a meaning before. They know from 5.NF.3, discussed on page 11, that $5 \div 3 = \frac53$. That is, $latex \frac53$ is the quantity that when you multiply it by 3 you get 5. On the other hand, it makes sense to give $\frac13 \times 5$ the same meaning, since it fits with the idea of taking one third of something. So $5 \div 3$ and $\frac13 \times 5$ are the same thing.

The situation is a little different for $\frac13 \div 5$, because that can be related directly to a multiplication they already know, namely $5 \times \frac{1}{15}$, as you explain.

So, in answer to your question: the ultimate goal is to understand division as related to multiplication. But you have to have the concept of multiplication defined first. To give a meaning to $\frac13 \times 5$, we think of it as “one third of 5”, or “one part of a division of 5 into 3 equal parts”, and then we use the reasoning behind 5.NF.3 to see that this is the same as $\frac53$.

I think all this could be said more clearly in the progression, thanks for pointing it out.

• This reply was modified 12 years, 4 months ago by Bill McCallum.
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