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Perhaps the wording here was a little too compact. The point is that so-called terminating decimals are really just a special case of repeating decimals, in which the repeating pattern is 00000…. It was not intended to exclude terminating decimals.
It seems this is a question for your state assessment writers, not for me! Or are you talking about the PARCC or SBAC assessment? I don’t think we know this level of detail about them yet. Anyway, here are some thoughts.
I guess you are talking about the following standard
7.RP.3. Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
The thrust of the standard is problem solving using proportional relationships. It’s hard for me to imagine students working with lots of simple interest problems without ending up knowing that you calculate the annual interest by multiplying the principle by the annual interest rate and that you add that interest every year (or month or whatever time period you are using), leading to proportional relationship between the interest and the number of years. So, you might want students to be able to solve:
Write an equation expressing the relationship between the interest $I$ and the number of years $t$ if the principle is $\$500$ and the annual interest rate is 4%.
by first calculating the annual interest to be $0.04 \times \$500 = \$20$ and then using that information to come up with the equation $I = 20t$. You might do a similar problem where the number of years is given and the goal is to come up with a proportional relationship between $I$ and $P$. Or you might ask them to come up with a graph or a table.
Is this the same as knowing the formula $I=Prt$? Yes and no. One the one hand, a student who knows all this has all the ingredients to construct the formula. On the other hand, in Grade 7 you might want to stick to proportional relationships where only two of the quantities are represented by variables and the others are given constants, as in the problem above.
So, without knowing exactly what your teacher means by “memorize $I=Prt$,” it seems to me that it is starting out in the wrong place. If it means sticking the formula into your head without understanding what it means, then you might as well use a cheat sheet. And if you have worked lots of problems with simple interest then it’s hardly necessary to memorize it.
We want students to see formulas as useful ways of expressing relationships between quantities, not as black boxes that you plug numbers into.
I wouldn’t say “whenever possible”, but rather “whenever there is a natural mathematical connection.” (Maybe that is what you meant.) In other words, I don’t think it makes sense to blend domains just for the sake of blending. But there are places where it is very natural to do so. For example, throughout K–5 the domains Operations and Algebraic Thinking and Number and Operations in Base Ten are naturally entwined. And there are many places where the Measurement and Data standards and the Geometry standards support the work with number. The progressions documents point out some of these connections.
The first question is really about teaching, not about the standards. Obviously a student who writes 14 backwards does not yet meet the standard. But it seems equally obvious to me that if the same students says that he or she means 14, then this is a matter of development, not mathematical understanding. I leave it to the expert Kindergarten teacher to decide how to handle this matter.
The same applies to your second question; a Kindergarten teacher knows much better than I do what to expect and how to guide a student.
As for the third question, students are expected to read numbers up to 120 in Grade 1:
1.NBT.1. Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.
Students in Kindergarten will probably start identifying numbers 1–20 as they work to meet the standards K.CC.1–3, but you are correct that it is not a formal requirement in Kindergarten. The emphasis in Kindergarten is on knowing the count sequence and representing the number of objects in a group with words and written symbols. I’m not sure what assessments you are talking about, but putting this into a summative assessment goes beyond the standards. Of course, teachers might well want to know what students can do with writing numbers in order to inform their teaching.
I’m not sure what your fourth question is. The cluster heading for K.CC.1–3 says “Know number names and the count sequence.” So, it is in the math standards that students should know number words. But are you talking about reading or writing them? That could be part of a literacy assessment, I agree.
First, thanks for pointing out the error in the labeling of the graphs, it should say either $f$ or $y=f(x)$. The fact that the error crept in is an example of just how widespread this usage is, and maybe it is futile to fight it. But it can’t literally be true that $f(x)$ is a function, because it’s a number, and a number is not a function. The letter $x$ refers to a specific but unspecified number in the domain of $f$, and $f(x)$ refers to the corresponding output. That’s the way function notation works. I would worry that not being precise in this usage leads to confusion and misconceptions later on. I think your desire to use $f(x)$ to refer to the function comes from a sense that $x$ in some way represents all the input values at once. But this itself is dangerous, I think: a lot of the trouble students have with algebra comes from a feeling that $x$ (or whatever letter you are using) isn’t really a number but is some vague mystical thing they have to perform mysterious rites on. So the more we can keep students anchored in the idea that the letters in algebraic expressions and equations are just numbers, and that the things you do to expressions and equations are just the things you can do to numerical expressions, the better.
As for the $y = $ notation, when we say something like “the function $y = x^2$” we are using abbreviated language for “the function defined by the equation $y = x^2$, where $x$ is the independent variable and $y$ is the dependent variable.” You can’t say that every time, so we have a shortened form, which depends on certain conventions: the dependent variable occurs on the left and an expression in the independent variable occurs on the right. So, it would be problematical to say “the function $2p + 3q = 5$” because it doesn’t specify which variable is the independent variable and which the dependent variable, and because, although this particular equation is solvable for one or the other, that’s not always true with an equation in two variables.
With these conventions, the terminology “the function $y = f(x)$” is o.k. If $f$ is the function, then $f(x)$ is the expression in $x$ giving the value of the function at $x$.
Lots of questions here and I’m not sure I have answers for all of them. I’ll take the easiest one first: it seems to me that some software for dealing with large data sets is a must in the statistics standards. A lot of the statistical measures don’t make much sense for small data sets (why on earth would you want to know the median of the set {1, 2, 2, 3, 5, 9}?). Also, it is only for small data sets that the different definitions of quartile make any serious difference, and getting into those distinctions is most certainly a waste of time. I think for most of the measures you discuss above, students should be looking at large data sets using technology, in which case the software will be reporting the measures. Students should understand what they mean and how they are computed.
Some of your questions seem to be about vocabulary. There’s no harm in introducing vocabulary (skew left, skew right, five-number summary) as long as the vocabulary doesn’t become the topic. For example, I would want students to be able to look at two distributions, one skewed one way and one skewed the other way, and identify the one in which the median is greater than the mean. I don’t much care whether they know which one is skewed left and which one is skewed right, at least not on a test, although obviously it is useful to have consistent terminology that students can use as they talk to each other. The same goes for five-number summaries. Students should be able to identify the minimum, the maximum, the median, and the two quartiles, and they should be able to talk about the differences between two data sets using these measures. The term is good as long as it is understood as useful in those conversations, but not if it becomes a pre-occupation of instruction and assessment.
You are right about the divide. Whichever arrangement is chosen it’s important to have a clear organizational principle. One reason for putting exponential functions earlier is an emphasis on patterns of growth, with linear and exponential functions representing the simplest functions from that point of view. This would occur in a function-based approach to algebra. A reason for putting linear and quadratic equations together is an approach based on looking for algebraic structure and reasoning about equations. This is an approach to algebra based more on developing symbolic fluency. In the end, at the end of Algebra II, you want students to have both symbolic fluency and a good understanding of functions, so the endpoint should be the same. But it’s important to have a clear organizing principle as you arrange topics into courses, a story to tell that will be clear to teachers and students.
I’ve just created a new forum on arranging the high school standards into courses, you can go on over and ask there.
I agree, and also want to make clear that students should eventually be able to interpret and work with fractions in mixed number form.
This is an example where the cluster heading provides valuable information not obvious from the standard itself. The cluster heading here is “Know number names and the count sequence”, and so this standard just refers to the counting words, not to counting objects.
I’m not sure which you mean by the simplified version and which you mean by the unsimplified version. Both 2×(8+7) and 2×8 + 2×7 are equally simple to my eyes. The standards use the word “term” for summands in a sum, so the second of these expressions has two terms (each of which is a product). I suppose you could say the first expression has one term, but that seems a bit odd to me; I would say it is a product with two factors, the second of which is a sum of two terms.
This question has come up before, here. But I agree that it is odd to exclude equations like $x-7=11$. I don’t think division is excluded; an equation like $x \div 5 = 3$ could be written as $(1/5)x = 3$, and probably should be at this grade level.
Duane, the answer here parallels the one there. Division is defined in terms of multiplication; $a \div b$ is by definition the number that gives $a$ when multiplied by $b$. So every division problem can be viewed as a multiplication problem (and vice versa). If I calculate $300\div 25$ by saying “I know $100 = 4 \times 25$ and $300 = 3 \times 100$, so $300 \div 25$ must be $3 \times 4$,” I am implicitly using the associative property $3 \times (4 \times 25) = (3 \times 4) \times 25$. Once again I must emphasize that it is not intended that students necessarily be able to name the properties. A student who can explain the reasoning above or illustrate it using drawings or equations meets the standard.
