We decided to combine the geometric measurement part of the MD Progression with the Geometry Progression (coming soon!). Here is a draft of the data part of the MD Progression (covering both categorical and measurement data): ccss_progression_md_k5_2011_06_20.

We will be releasing a draft of the Fractions progression before the end of June, and the Ratios and Proportional Relationships progression in early July.

[29 July 2012] This thread is now closed. You can ask questions here.

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## About Bill McCallum

I was born in Australia and came to the United States to pursue a Ph. D. in mathematics at Harvard University, met my wife, and never went back. I am a professor at the University of Arizona, working in number theory and mathematics education.

Any chance that the Fractions progression is ready? I am definitely looking forward to it!

Thanks,

Jen

Also wondering when the draft of the geometry progressions will be available. We have found the counting and cardinality progressions very useful while aligning and writing our kindergarten units. We are ready to start working on the Geometry unit and are eagerly awaiting the release of the progressions document. Thanks!

I’m wondering if, even though the geometry progression isn’t ready, you might be able to answer a question I had from an 8th grade teacher yesterday. She was wondering about 8.G.5. In the past, students were expected to find angle solutions algebraically when given the “two parallel lines cut by a transversal scenario”. There is no example or mention of this, but should this still be expected of students in grade 8?

Thanks for any feedback!

I agree that doing this in Grade 8 is a very natural extension of the Grade 7 standard

especially since students are doing lots of work with solving linear equations in Grade 8. This is a case where a connection between two different domains leads to certain type of task, and it would be a reasonable choice for curricula to include such tasks. That which is not mentioned in the standards is not thereby forbidden.

However, you ask if such questions “should still be expected of students in Grade 8.” Well, I wouldn’t want to see this added to a long list of “must-do” activities in Grade 8 that are not mentioned explicitly in the standards. The standards try to provide focus, to describe what is the most important job in each grade level. In this case, I wouldn’t want such an activity to move from “not forbidden” to “required.”

Dr. McCallum,

We in the field are so grateful for your insights and your clearly articulated guidance on matters such as the one posed by Dianne. Our teachers are fully vested in the new CCSSM. They are questioning about how to sequence the grade level content standards. Our curriculum programs have “crosswalks” and teachers would like to know if they should follow the sequence of the “crosswalks” which is in order of the standards themselves or if there is some other more coherent sequence. Since the various curriculum programs are not yet pure in their alignment with each domain, teachers are asking for guidance on sequence. Thanks in advance for your reply.

~Nancy K-8 Director of Math (Massachusetts)

Dear Nancy,

The sequence of the standards is not intended as an indication of the sequence in which topics are taught in a grade level. Indeed, it says in the standards, on page 5:

The standards have to be translated into curriculum (by textbook writers, district leaders, etc.). They cannot be ported raw to the classroom without any processing; in particular, I hope we can get past the practice of asking teachers to put the “standard of the day” up on the blackboard. The standards don’t work that way.

Bill,

This question is with regards to 4.MD.2- “Use the four operations to solve word problems involving distances…including problems involving simple fractions or decimals…”

I have no problems solving problems involving simple fractions – foundations for this are solid in the NF standards at this grade. However, operations with decimals are absent from grade 4. (4.NF.5 through 7 deal with decimals, but not operations with them. Operations with decimals are introduced in 5.NBT.7 and the progression culminates at 6.NS.3.)

Are students supposed to learn to operate with decimals in grade 4 or not?

I guess one could argue that students can convert the decimals to fractions out of 10 or 100 (4.NF.6 going backwards), then perform the calculation on the fractions (4.NF.5), then convert the fractional answer back to a decimal (4.NF.6). But operations with decimals are clearly and explicitly addressed in grades 5 and 6, not grade 4.

Please advise.

As always, thanks for all of your guidance,

Brian

Bill,

I realize this question is related to the geometric measurement portion that is not yet posted, but I am looking for your guidance…

3.MD.2 clearly seems to apply only to the metric system. Is this correct?

4.MD.1 seems to apply to measures of time and money, as well as metric and customary measures of length, mass/weight, and capacity. Is this correct? Is it intended to address any other systems?

5.MD.1 the example given is a metric one… is this standard intended to apply only to metric measures, or U.S Customary, too? If U.S. Customary measures are included, which units (ex., tablespoons, fluid ounces, cups, pints, quarts, and gallons)?

Thanks,

Brian

The wording of all these standards is a bit different, with slightly different intentions. The overall intention is to provide some milestones in measurement (hah hah) without letting measurement get out of hand and crowd out other subjects. So yes, all that’s required by 3.MD.2 is metric units. That doesn’t mean teachers are forbidden to mention anything else, but it does indicate the target for assessment. 4.MD.1 one prefaces the list of units by the word “including”; so you have to cover at least those, plus systems that have already been covered in previous grades, such as feet and inches. The word “including” indicates other systems are possible if there is time, without requiring them. As for 5.MD.1, I would say any system covered so far is fair game; the examples are meant to be illustrative, they don’t qualify the meaning of the standard.

A follow-up regarding 4.MD.1 –

Are students supposed to memorize the conversions for the stated units, or just apply understanding about the relative sizes and conversions to solve problems?

While the latter provides an awesome opportunity to connect this standard to much of the OA domain at this grade (especially 4.OA.5) and the NF domain, the former would require a bit of time to develop and it doesn’t connect to any of the Critical Areas in grade 4.

Thank you for your support,

Brian

Well, the standards says they should “know relative sizes”. This does not dictate how they should come to know them—through memorization, or through experience with problem solving. But it is an expectation that they will know them eventually, although I wouldn’t make this a sole target for assessment.

Bill,

Last Geometry question for now… I was working with a group of teachers at a conference for the Assoc. of Math Teachers of New York State and questions were raised regarding the limits of 2.G.1 (quick, easy question) and 3.G.1 (much messier).

2.G.1 – Many people thought this would include irregular shapes, but some thought students would only have to recognize or draw regular pentagons, for example. Please clarify.

3.G.1 – This one had us all stumped… “Understand that shapes in different categories may share attributes, and that the shared attributes can define a larger category.” Is this intended only to include the specific example given? If not, this could be limitless…

It could include only quadrilaterals. If this is the case, do students need to understand every relationship among sub-families of quadrilaterals (i.e., rectangles and rhombuses share the attributes that define parallelograms… squares share the attributes that define rectangles, rhombuses, parallelograms, quadrilaterals, and polygons)?

It could include all polygons (i.e., By some definitions, equilateral triangles are special types of isosceles triangles. Do third graders need to understand this?)?

It could include three-dimensional shapes. If this is the case, students might have to know that cubes are a special type of square prism, which is a special type of rectangular prism, which is a special type of prism, which is a special type of polyhedra.

Please advise of limits!!!

Thank you,

Brian

2.G.1. No, there’s no suggestion the shapes have to be regular.

3.G.1. Agreed there are no explicitly stated limits, but common sense must prevail. Judgement is called for here, and a respect of the focus of this grade level on multiplication, so you can’t spend forever on this. Some examples are more difficult than others (e.g., your equilateral triangle example is more difficult than understanding that rectangle is a special case of a quadrilateral, I think). My reading of the standard is that you have to (a) cover at least the explicitly mentioned categories and (b) see enough examples to get the general idea of a hierarchy of shapes (that’s the “understand” part of this standard). Again, there’s a difference between what you might do in the classroom and what assessment might target; the assessment consortia frameworks will give more guidance on now much time should be spent on this.

I have a question about the relationship between 2.MD.5 and 2.OA.1. The MD standard looks like a subset of the OA one restricted to the context of length. Is this intentional or the underlying intention for 2.MD.5 was different but ended up sounding similarly?

Thank you,

Alexei

I think it helps to look at the cluster headings here:

Although the standards are both about word problems, the first has an orientation towards the using operations to represent contexts, building towards modeling, whereas the second has an orientation towards understanding addition and subtraction as putting together or taking away lengths, building towards an understanding of the number line.

Bill,

I have a question regarding some of the Measurement standards…

2.MD.9 and 3.MD.4 both begin “generate measurement data by measuring lengths…” (These are the standards that include measuring lengths to the whole unit and 1/4 unit, respectively.)

The comparable standards 4.MD.4 and 5.MD.2 never use the words “generate” or “measuring lengths.” They have students “display a data set of measurements” and solve word problems from the data (to 1/8 inch for both standards), but they aren’t asked to generate that data themselves by measuring. Is it the intention of these standards that students “generate” and “measure” to the 1/8 inch even though this is not explicitly stated?

This might be something worth addressing in the final draft of the Progression.

Thanks for your guidance,

Brian

Bill,

I re-discovered this question while doing work with my fourth grade teachers last week, then, yesterday, I received an email from a Math Coach from another state asking me if I had any answers on the same topic. Can you offer any clarification regarding the intention of 4.MD.4 and 5.MD.2? Are students supposed to measure to the 1/8 inch in grade 4, or is 1/4 inch (in grade 3) the most precise measuring the standards require?

Thanks,

Brian

That’s right, they are not required to go beyond 1/4 inch in Grade 3. That doesn’t mean they are not allowed to, of course!

HI Bill,

Is cups, pints, quarts, and gallons included in 4MD1 and 5MD1?

Shannon

HI Bill, Just decided to repost because I haven’t received a response yet. Thanks

Is cups, pints, quarts, and gallons included in 4MD1 and 5MD1?

Shannon

The standard doesn’t require these units, no, although of course kids might already be familiar with them. The units of liquid measure explicitly mentioned are liters and milliliters. Note however 6.RP.3d: “Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.”

Bill:

I have been curious about the decision to stick with only measurement data when using line plots. There are many classic activities used to introduce line plots like “how many pockets” or “how many siblings” does each kid have, but these seem to be counts more than measures since you can’t have 3 ½ siblings. Where would these activities fall under? Do you recommend continuing to do some line plots using counts or should it always be measurement data?

Thank you,

Ivan

Ivan, a few thoughts here. First, these examples could equally be interpreted as being about categorical data, although the categories in this case are numbers. Second, line plots are leading to the concept of a continuous distribution later (in high school), so it seems natural to associate them with continuous data. Also, the data standards in K–5 play a supporting role, so in this case they are supporting measurement.

We have a 5th grade question: “The following fractions represent the measure in cups of nine beakers, filled with water.” (Fractions given: some 1/2’s, some 1/4’s, some 1/8’s). This question is taken from the AZ unpacked standards.

a. Use the information to plot the measurements on a line plot on your Student Answer Sheet.

b. Record the title and label the axis.

c. Make a statement comparing the number of beakers filled with cup to the number of beakers filled with cup.

HERE is our question. Since the line on the plot is an axis, do students need to include 3/8 on that axis to have equally scaled intervals?

I guess the student has to show in some way or another how they chose the placement of data points, and this is one way to do it. But I wouldn’t make that a hard and fast rule; you could also have the line marked in fourths, with one smaller tick mark halfway between 0 and 1/4.

Is the draft of the Data Part of the Measurement and Data Progression available as a WORD file?

I’m afraid not.