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Bill, would you be able to elaborate on why estimation and rounding is mentioned in 3.OA.8 rather than 3.NBT.2 ? I appreciate that you’ve said estimation doesn’t need to be limited to 3.OA.8, and also that it shouldn’t be a big focus, but why is it there under that standard and not also/instead in 3.NBT.2, or even 3.OA.3 or 3.OA.4?
I think for the purposes of this standard, the word “complicated” is relative; that is, an expression is complicated if it has parts that can be regarded as a single entity (or not). For example (x+y)^2 could be viewed as complicated because it can be viewed as something squared; the key ability here is the ability to clump the x+y into a single something in your mind and see the expression that way. It might have been clearer to use the word “compound” rather than “complicated”.
The emphasis in 4.MD.1 is on “times as much” or “times as many”. Since a foot is 12 times as long as an inch, there are 12 times as many inches in a measurement as there are feet. The corresponding statement the other way around involves multiplication by the fraction 1/12, and so waits until Grade 5 where such multiplications are introduced. Thus you get conversion both ways in 5.MD.1. This is not to say students couldn’t use the table to convert both ways, but it is not required.
The set of fractions contains the set of whole numbers. So, a standard about fractions includes the possibility that those fractions might be whole numbers. But no, it’s not true the other way around; a standard about whole numbers is limited to whole numbers. In general fractions includes mixed numbers, yes, but see the discussion here.
As for your question about interpreting graphs, I thought Cathy answered it. Interpretation might well be part of a curriculum designed to help students meet the standard. The same goes for using picture graphs to solve problems. That’s a preparatory step for using bar graphs, which involve a little more sophistication with regard to understanding the meaning of the unit. As for other sorts of word problems arising from bar graphs, the over-riding criterion for whether to include something or not is whether it leads to students being able to meet the standards, so if you can make a case for including something based on that, then it’s fine to include it. In this case, the relevant standards would be 3.MD.3 and 3.OA.8.
Standards are not curriculum. Standards set goals; curriculum is designed to achieve them. From page 5 of the standards:
These Standards do not dictate curriculum or teaching methods. For example, just because topic A appears before topic B in the standards for a given grade, it does not necessarily mean that topic A must be taught before topic B. A teacher might prefer to teach topic B before topic A, or might choose to highlight connections by teaching topic A and topic B at the same time. Or, a teacher might prefer to teach a topic of his or her own choosing that leads, as a byproduct, to students reaching the standards for topics A and B.
Thanks Bill. I think the final sentence of your first reply sums it up: some exposure would be good but fluency can wait for later.
In essence, and I know we’ve discussed this in other posts, the wording of the Standards regarding fractions is difficult to interpret. Sometimes “fraction” is used inclusive of common fractions, mixed numbers, and decimal fractions – sometimes not. In a recent post you’ve also stated that whole numbers are fractions (http://commoncoretools.me/forums/topic/bar-graphsline-plots/#post-1581). Overall, this elastic approach to the topic makes it a real struggle to identify what is required for any given standard.
I understand that the aim is to shake teachers out of their beliefs about different forms of fractions being different numbers, but at some point teachers, curriculum developers, and test-makers need to make marks on a page. They need to know what forms are expected for a given task at a given grade, even if they understand conceptually that all forms are equivalent. The “complexity of numerical expressions” you mentioned is exactly what needs to be known – leaving it open for potential conflict between curriculum developers and test-makers is not ideal.
The terminology of “common fraction”, “improper fractions”, or “mixed numbers” may be reduced or absent from the Standards to avoid creating misconceptions but these terms are very clear, concise, and convenient labels. Although the Standards cannot be altered to increase clarity, perhaps the NF Progressions could be – explaining what such terms are, how they relate to each other, and how they apply to each relevant standard would be a great help for others not reading this blog.
I just want to add that whole numbers are fractions, so a line plot with whole number labels is not excluded by 4.MD.4. Of course, students should also see line plots with labels that are not whole numbers.
A further thought, which should perhaps have been my first answer: mixed numbers are basically not a very important subject in the standards. Students should be able to interpret expressions in mixed number form, and should be able to deal with them, but they should not be a separate subject, with long lists of drill exercises associated with them. They will arise naturally in word problems, as in 5.NF.6. That is enough. There shouldn’t be a separate module called “multiplying mixed numbers”. A student who avoids them entirely by always converting them to a/b form is doing just fine.
Duane, sorry, this is going to be a long answer, and in the end it will be the same sort of “figure it out for yourself” answer that I’ve been giving lately. But I will try to throw some light on the matter.
You are using the terms “mixed number” and “common fraction” in ways they are not used in the standards. Indeed, the first term only occurs twice and the second not at all. This is not unusual. One often hears people talk about fractions and mixed numbers as if they were different types of numbers, with their own ways of being added and multiplied.
By contrast, the standards use the word “fraction” to refer to a particular sort of number (one that you get by dividing the interval from 0 to 1 into b equal parts and putting a of them together), not to a particular sort of expression. That number can be expresssed in different ways. It can be written in the form numerator/denominator (“fraction” in conventional terminology) or in the form whole number + fraction less than 1 (“mixed number”).
From that point of view, a flippant answer to your question is: if students are multiplying fractions they are multiplying mixed numbers. However, there is more to it than that. Although any requirement about fractions potentially includes fractions expressed in mixed number form, one has to think about how the complexity of numerical expressions grows across grade levels. A fraction expressed as a mixed number is really being expressed as a sum (with the plus sign missing), so a product of two mixed numbers is really a product of sums. You can deal with this a couple of different ways: compute the sums first then multiply (i.e. “convert the mixed number to a fraction first” in conventional terminology) or multiply out the sum using the distributive law. Exactly where you expect which level of complexity in expressions and their manipulation is not a matter for the NF progression per se, but rather part of the general question about the progression in numerical expressions. The standards leave some room for different decisions by curriculum writers there.
So, to the specifics of your question, 5.NF.4 is not really about the form of the number, but about the number itself. Students should be able to multiply fractions, and that includes multiplying mixed numbers, if only by the first method above. But, although the standards do not impose a limit here, considerations of expression complexity do. These are largely up to curriculum developers. The grand culminating standard 7.EE.3 expects fluency with number in all form, but how you get there could vary from curriculum to curriculum. I would certainly expect students to see some products involving mixed numbers in Grade 5, but not fluency with the most complicated ones (e.g. mixed number by mixed number).
I agree with the consensus here: 3 x 1/5 in Grade 4, 1/5 x 3 in Grade 5. As Brian says, the two are conceptually quite different: the first is what you get by putting 3 segments of length 1/5 together, the second is what you get by dividing a segment of length 3 into 5 equal parts. The fact that these are the same can be shown by some reasoning on the number line; it’s not obvious. As Cathy says, this doesn’t prevent curriculum writers from developing a pathway that starts talking about simple instances of fraction x whole number before Grade 5, but it does constrain assessment developers.
Here’s what we know from the standards: students should be able to read and write numerical expressions without parentheses in Grade 3, and they should be able to read and write numerical expressions with parentheses in Grade 5.
The question Brian and Duane are discussing is: what is the pathway from Grade 3 to 5 with regard to numerical expressions? It’s a good question and a good discussion, but the standards are largely silent on the answer. Remember, standards are not curriculum. One curriculum might decide to introduce parentheses in Grade 3, another in Grade 5. Yet another might follow Brian’s suggestion about 4.OA.3 and introduce them there. That’s certainly a point at which their usefulness becomes evident. But not every multistep word problem leads to an equation with parentheses, and anyway I don’t read the standard as requiring that every single word problem be accompanied by an equation, merely that students should have experience writing equations for word problems.
The point is, all this discussion takes place within the space of flexibility allowed by the standards. The standards require that there should be a pathway, but don’t say what it is.
Brian, your second interpretation is closer to the truth. “Time intervals in minutes” doesn’t have to mean “less than an hour.” For example, there’s no reason why Grade 3 students can’t say how many minutes it is from 3:30 to 5:00. But multiplication, fractions and decimals, and unit conversion open up the scope on Grade 4. The one of these you didn’t mention was the inclusion of fractions and decimals in Grade 4. Students in Grade 4 might be expected to be able to answer a question like “If Don can peel 3 potatoes in a 5 minutes, how many can he peel in 3/4 of an hour?” (Not saying this is a great problem, but you get the idea.)
Brian, I’m keen on knowing the answer too. My best guess is that only the repeated addition model is expected in Grade 4.
One related thing I’ve been considering is how to interpret 4 x 1/5 when no context is provided. One way is to think of it as a set problem (what is 4 multiplied by 1/5) and the other is as repeated addition (what are 4 groups of 1/5). Do you think that if there is no context that students should be able to still get the correct answer by approaching it more or less abstractly?
