Forum Replies Created
-
Posts
-
I was a bit confused by your last paragraph, because I think of the phrase “multiplying 2 by 4” as meaning $4 \times 2$, not $2 \times 4$. But I take your overall point.
Multiplying fractions by whole numbers occurs in the previous grade, Grade 4. See 4.NF.4. And yes, 4.NF.4c is about story problems, and has an example of one. Your story problem seems to be more about $(a \times q)/b$ than $a \times (q/b)$. That is, we seem to double Ron’s beans before sharing, not after. Note that neither 5.NF.4a nor 4.NF.4c requires students to construct story problems, although that’s a good idea.
This a very interesting question. To summarize, you are saying the to you the diagram says
$$
\frac23 = \frac{2 \times 4}{3 \times 4}
$$
rather than
$$
\frac23 = \frac{4 \times 2}{4 \times 3}.
$$
First, your way of seeing it is fine and ends up with the correct understanding. The difference between your way of seeing the diagram and the way expressed in the progression is the difference between the two ways of viewing a $3 \times 4$ array: as 3 rows of 4, or 4 columns of 3. You are looking at the rows: there are two green rows of 4 squares each in an array consisting of 3 rows of 4 squares each, so the fraction is $(2\times4)/(3\times4)$. The other way to look at this is that there 4 columns of 2 green squares each in an array consisting of 4 columns of 3 squares each, so the fraction is $(4 \times 2)/(4 \times 3)$.Either way is fine, and it’s probably useful to go through both ways.
It might help to make a distinction between the plane and the coordinate grid imposed on it. The plane in which geometric figures live does not come automatically equipped with a coordinate grid; indeed, there was no such thing as a coordinate grid in Euclid’s day. You could think of the plane as a featureless plane in which geometric figures exist, and the coordinate grid as an overlay which can be used to measure positions in the plane. So when I perform a transformation, I move points in the plane to other points, but the grid stays where it is so I can use it to measure the new position of a point. This might be where the in/of confusion comes from: I can think of this as a transformation of the plane, but I can also think of points moving in the coordinate grid.
By the way, you don’t really need the coordinate grid at all to talk about transformations. You can talk about a rotation about a certain point through a certain angle without necessary giving coordinates to the point. It could just be some point in a geometric figure (say, the vertex of a triangle). It seems to me that the coordinates are almost getting in the way of things with your teachers.
As for dilations, I think the phrase “dilation of a point” is awkward. A dilation with center $O$ takes all the points in the plane and moves them along rays from $O$, scaling the distance from $O$ by a certain scale factor. Again, there is no need for coordinates, and no need for vectors.
There are two big steps in students’ understanding of number in Grade 6. The first is the unification of whole numbers, fractions, decimals, and negative numbers into a single number system as represented by the number line. But the number line doesn’t do everything you need. The other big step is a systematic use of properties of operations to understand how operations can be extended to include negative numbers.
For example, the relation between addition and subtraction helps in understanding subtraction with negative numbers. In earlier grades, students understand $6-4$ as the number you add to $4$ in order to get $6$, that is, the missing addend in the equation $4 + ? = 6$. In Grade 6 they understand $(-6) – (-4)$ as the number you need to add to $-4$ in order to get $-6$, the missing addend in $(-4) + ? = -6$. Since $-6$ is two units to the left of $-4$ on the number line, the missing addend is $-2$. So $(-6) – (-4) = -2$.
By the same token, the relation between multiplication and division helps with division of negative numbers. So $8\div (-4)$ is the missing factor in the equation $? \times (-4) = 8$.
In general in Grade 6 there is a consolidation of operational understanding of rational numbers, and a move away from concrete models, although concrete models like the ones suggested by molleyk are still useful.
No, I would say this does not included formalized significance tests, but is more a matter of understanding how sampling works through simulations of sampling distributions, for example. There are some examples in the high school statistics progression, for example the relation between caffeine and finger-tapping (an experiment I find particularly relevant as I type this).
The formal probability rules are in the high school standards, and in that case it is the general rule that does not assume the events are independent: P(A and B) = P(A)*P(B|A) = P(B)*P(A|B). The Grade 8 standards are really focused on understanding the underlying concepts of sample space, event, and probability model in concrete terms, without getting into the formal calculus of probability. So I would say that it goes beyond the standards to introduce the multiplication rule at this stage. Of course, if it arises naturally in an example there is no harm in pointing out that the probability of the compound event is the product of the individual probabilities, but that is not quite the same as introducing a formal rule. And it would be important to point out that this simplified multiplication rule does not always apply.
One point I would make is that the order of the standards does not necessarily correspond to the order of topics in the curriculum (see p. 5 of the standards). I could imagine teaching about rational numbers on the number line first, and then giving applications of negative numbers as in 6.NS.5. Or you could do it the other way around. In that case the temperature model lends itself particularly well to introducing the negative extension of the number line, since a thermometer is really just a vertical number line.
These are both curricular decisions, not dictated by the standards. There are limits on the number of digits in some of the Grade 4 operations standards, but not on the size of numbers that students see without operations. the term “multi-digit” does not come with an inherent limit, so it is up to the curriculum developer to decide the limit there. As for base-thousand units, I think it was an ad hoc term to refer to thousands, millions, etc., and not intended as a new piece of terminology to be taught to children.
I’m not sure there is a mathematical difference here, but rather a difference in point of view. A transformation takes points in the plane as inputs, and outputs other points in the plane. We conceptualize this as motion: the transformation moves the input point to the output point. This conceptualization is useful but not strictly part of the mathematical definition. Whether you think of this as moving points in the plane (a transformation in the plane) or whether you think of it as moving the entire plane all together (a transformation of the plane) strikes me as a matter of point of view, or taste as Dr. M. says.
Lane and Cathy have pretty well covered things, but just a couple of extra points. First, notice the footnote on page 15 that says students need not use the formal terms for the properties of operations. They should understand that you can add numbers in any order and use this fact, but they don’t have to know the name for it. Second, when you read a standard like 5.NBT.6, it is important not to interpret it is requiring every method listed for every division problem (that’s why the “and/or” is there).
Again, please tell me which unpacked document you are talking about. I looked around on the web a bit and didn’t find anything where “skip counting” meant starting from a number other than a multiple of 5, so I would say that starting from any number is certainly not required by this standard. That doesn’t mean it is forbidden, of course. And it does seem reasonable to start from any multiple of 5.
