Draft of progression on Number and Operations in Base Ten

/ Bill McCallum

Here is the first public draft of the progressions project, on Number and Operations in Base Ten [corrected version uploaded 26 May 2012]. We welcome any comments or suggested changes, which will be considered for the final draft. Please post comments to this thread. We will be releasing other draft progressions for elementary and middle school over the coming weeks.

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

58 thoughts on “Draft of progression on Number and Operations in Base Ten”

  1. I downloaded the pdf, but was unable to print it. Some pages are corrupt. With some effort by our tech people, we were able to print it. You may want to check the file, as other people may not have the wherewithal to fix it. I am sure that many people will be interested in seeing and using the ideas in this document. Thanks to you and all who worked on this project.

  3. Could you explain what is meant by some of the recurring language that uses the word “level”? Example: Page 7, first paragraph, “…extention of the Level 3 make-a-ten…” Not sure what Level 3 refers to….
    Thank you.

  4. The levels are described in and appendix to the Operations and Algebraic Thinking Progression, which will be coming out soon. Here is a sneak preview:

    Strategies and representations used for single-digit addition and subtraction problems

    Level 1. Direct Modeling by Counting All or Taking Away. Represent situation or numerical problem with groups of objects, a drawing, or fingers. Model the situation by composing two addend groups or decomposing a total group. Count the resulting total or addend.

    Level 2. Counting On. Embed an addend within the total (the addend is perceived simultaneously as an addend and as part of the total). Abbreviate the counting by omitting the count of this addend. Instead, begin with the number name of this addend. Numbers are represented by the counting words in this counted total, and some method of keeping track (fingers, objects, mentally imaged objects, body motions, other count words) is used to monitor the count. For addition, the count is stopped when the amount of the remaining addend has been counted. The last number name counted is the total. For subtraction, the count is stopped when the total occurs in the count. The tracking method indicates the difference (seen as an unknown addend).

    Level 3. Change to an Easier Problem. Decompose an addend and compose a part with another addend.

  5. I have only read through Grade 2. I was debating whether or not to wait till I finish reading the whole document before posting my comments/questions, but I decided to just go ahead.

    First, a very global/general question. I was wondering who the intended audience of this document is. I think the authors were very careful about their language, but sometimes I wonder if the readers will catch the subtle differences. For example, in Kindergarten, we expect students to understand numbers 11 through 19 as “ten ones and some more,” as opposed to “one ten and some more,” which is in Grade 1. I wonder if it would be helpful to make it explicit that the phrase “ten ones” is used intentionally and different from “one ten.”

    As for our base-10 numeration system, I wish there is something about children’s understanding of 0. They need to understand 0 as a number – a number that represents the cardinality of an empty set. However, they also have to understand the role of 0 as a place holder in our number system. I believe those two are distinct understanding, involving the same symbol. Speaking of the idea of “place holder,” one important idea of the base-10 numeration system that is often left implicit is the rule that we write one and only one numeral in each place (except the leading 0’s before the first non-0 numeral for whole numbers and the trailing 0’s after the last non-0 unit to the right of the decimal point). Every first and second grade teachers have seen children who would write “212” as the answer for 15 + 17 because children write “12” in the ones place. A part of teaching goal of addition (and subtraction) of 2-digit number is to help students understand this rule. I wish there was something more explicit about it.

    Speaking of computation, I also wondered about horizontal versus vertical notation. In Japanese textbooks, they consider the vertical notation as a specialized notation for written calculation. So, students don’t see addition/subtraction written in a vertical format until they encounter a unit on addition/subtraction algorithms in Grade 2. Does the CCSS take any stance on that issue?

    Still in the Overview section, the document says (p. 3) “Standard algorithms for base-ten computations with the four operations rely on decomposing numbers written in base-ten notation into base-ten units.” I wonder if “decomposing” is the appropriate term here. When we look at 40+30, we want children to understand this as 4 tens + 3 tens, and when they also understand that (only) number that refer to the same unit can be added, they can understand 40+30 as 4 tens + 3 tens, or (4+3) tens. Perhaps “re-unitizing” is a better word for such an understanding.

    In Kindergarten section, I was wondering about the role of symbolic notation when the document discusses decomposition of numbers, for example 10 as 1+9, 2+8, etc. There are two questions with this: one is whether or not we are expecting students to use symbols to represent their decompositions, and the other is do they have to understand it in terms of addition operation, not just “10 is 1 and 9, 2 and 8, etc.”

    Another question in Kindergarten is the statement, “Difficulties with number words beyond nineteen are discussed in the Grade 1 section.” K.CC.1 says “Count to 100 by ones and by tens.” This document seems to suggest that “counting by tens” isn’t really a part of Kindergarten goal – twenty is just one more than nineteen, not really two tens. That seems to be reasonable, but I was just wondering.

    Well, maybe it was good thing I didn’t wait till I finished the whole document.

    Thanks.

  6. Another quick question.

    Both the progression draft and the CCSS itself uses the phrase “the standard algorithms,” but are they defined anywhere?

    • Good question, Tad. I do not know of any definition of this term that has consensus support (we tried to find one, believe me). So this is really up to the community.

      • Bill,

        I also am mystified by the semantics of “the standard algorithm”. Why not “A” standard algorithm? To me “the” implies that there is only one.

    • Whether you use the definite or indefinite article is a religious issue for some people. Personally I’m agnostic, but we had to make a choice. Perhaps this text from page 13 of the progression will help:

      “In mathematics, an algorithm is defined by its steps and not by the way those steps are recorded in writing. With this in mind, minor variations in methods of recording standard algorithms are acceptable.”

      • That quote, “In mathematics, an algorithm… are acceptable” was great to see. I think it gives valuable guidance to the field. I have a question regarding it:

        Would the first algorithm on page 9, “Recording combined hundreds… on separate lines” be considered a “minor variation”? As noted in the caption, this also works from left to right, which is more consistent with “the standard algorithm.” Would that make it a “minor variation?”

        It would be helpful to see an example of the “limits” of “minor variation” to help clarify it for the field. If this sort of partial-sums strategy is acceptable, note it. If the limit is closer to the second algorithm on page 9 (with the newly composed units placed below the initial addends), then please note that.

        … for what it’s worth,
        Brian

      • Bill,

        I appreciate the comment. However, I feel that the language is a little deceptive (or at least intentionally vague). For many in the elementary math education community and particularly in the non-education, parent, general public community “THE standard algorithm” for addition, subtraction multiplication, and division is the US algorithm for each operation. I know PD can inform the educators. How do we clearly articulate to non-professional stakeholders and the general public that are not receiving professional development that there is more than one “the standard algorithm”?

        Maybe I’m belaboring the semantic point here, but I fight this battle with parents and teachers on a daily basis already. I was hoping that the CCSSI would put this baby to bed, but it looks like additional burping is necessary.

  7. A couple of comments on Grade 3.

    RE: rounding
    I think teaching children to round numbers to the nearest tens is useless. The whole point of rounding (approximating) is to work with “easier” numbers, I think. In Grade 3, when children are working with only 3-digit numbers, I don’t know if 570 is any easier than 568. Moreover, when you start the instruction of rounding with “rounding to the nearest tens,” we must make the rule of “5 or above you round up” really an arbitrary decision, since 5 in the ones place (for whole numbers) is indeed right in the middle – equal distance from both decade numbers below and above. But if you start with something other than rounding to the nearest tens, all but one number with “5” in the place to the right are indeed closer to the number above. Thus, it makes sense to have the rule, “5 or above, round up.” But, in general, I think we should think we should delay teaching rounding as a formal technique for approximation till students start dealing with much larger numbers.

    The other comment may be more about Grade 2, but it is on the following statement: “For example, the product 3 50 can be represented as 3 groups of 5 tens, which is 15 tens, which is 150” (p. 11) I think it may be important for the document to discuss this idea of looking at 3-digit number using base-10 units more flexibly in Grade 2 when they study numbers up to 1000. So, not just thinking of 302 as 3 hundreds, 0 tens, and 2 ones, students should be able to consider it as 30 tens and 2 ones. With that knowledge, understanding of multiplying multiples of 10 by a single-digit number becomes easier. As much as possible, we should let students focus on one new idea at a time – in this instance, “how can I think about multiplying multiples of ten by single-digit numbers?” This way of thinking is also useful in simplifying the written notations of addition/subtraction strategies – for example, the left-to-right strategy shown on p. 9. This might be a good opportunity for a class of students to discuss if they can just write “11” starting in the tens place, instead of “110.”

    I think one of the “merits” of the base-10 numeration system is the ease with which we can look at any number and think of it in terms of any base-10 unit, like 1134 as 11 hundreds, 113 tens, etc. I think we want students to understand why our written number system is useful.

  8. 4.NBT.4 Fluently add and subtract multi-digit whole numbers using
    the standard algorithm….has multiple ways of deriving a product.

    I see the connections or ties between NBT/Place Value and OA/Properties of multiplication. I guess my understanding, or lack there of the standard algorithm needs clarification.

    I thought we were moving away from the partial products method?

    • Tony, are you referring to page 13? The addition and subtraction standard 4.NBT.4, requiring fluency with the standard algorithm, is a culminating standard that caps work on addition and subtraction begun in Grade 1. Following a similar development for multiplication, the Grade 4 multiplication standard, 4.NBT.5, says “Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.” This is building to the Grade 5 standard requiring fluency with the standard algorithm for multiplication (5.NBT.5). The standard do not explicitly require or forbid the partial products method, but it fits the description of a “strategy based on place value and the properties of operations”.

      • Bill,
        First, thanks for your time and efforts.

        In responding to the NBT progressions, I should have referenced;
        • 4.NBT.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm.
        • 5.NBT.5 for multiplication and division – Fluently multiply multi-digit whole numbers using the standard algorithm.

        I am familiar with alternative methods of finding sum, difference, product and quotient and have found them to be essential to a child’s pathway to fluency in arithmetic.

        Fluency, to me, is being efficient, flexible and accurate in the method that you use.

        It seems my confusion rests in what is a method, strategy and a standard algorithm?

        Thanks again,

        Tony

  9. Dear Bill,

    Thank you for posting the first progressions document. I have reviewed it relative to the work that I am doing with colleagues on the treatment of measurement, principally spatial measurement, in the CCSSM. With them, I will soon have a longish “teacher’s companion” to measurement in the CCSSM. I read the Number & Operations draft LP with that effort in mind.

    Two broad comments:

    1. Connection to the practices. I was concerned about the discussion on page 4. The connections between content and practices did not seem clear and strong (relative to the one paragraph on each practice in the CCSSM itself). I worry that this approach (of “reading in connections” to practices), if continued, will tend to make the practices anything the speaker wants them to be. Do you see strong connections between page 4 and the original one paragraph descriptions of the practices?

    2. Issues with unmarked shifts between discrete and continuous quantity as a basis for number and operation. The LP essentially stands on a discrete model of quantity, e.g., representation by base-10 blocks. But the discussion of multiplication leans a good deal on “area models” without acknowledging the sleight of hand that goes on in multiplying lengths to get area. The numbers attached to those area model diagrams are not clearly connected to their spatial referents. Presenting them without some attempt to explain how multiplying lengths turn into a count of squares perpetuates the confusion of what multiplication is doing. Those problems may not be there with arrays (which can be interpreted in a discrete perspective) but they certainly are for area. Referencing “area models” without acknowledging these issues hides the problem, when the clear empirical evidence is that neither teachers nor students understand what goes on in L x W = area. Though discrete/continuous quantity is a challenge for curriculum authors and teachers, not addressing it at all is really not an option. I think that the LP should at least call attention to the issue. Number & Operation cannot “draw on” understandings from Measurement to develop multiplication when the understandings are not there (for measurement).

    • Jack, thanks for these comments. Here are some thoughts in reply.

      1. The descriptions of the practices in the standards are a little thin when it comes to K–5 examples. The paragraph on page 4 of the progressions document is intended to be guide to help the reader look for such examples in K–5. There is also one place in the document itself where MP8 is referenced (page 9 near the top). Maybe there are other places where we could do this … any suggestions are welcome.

      2. You raise an interesting point here. Multiplication of whole numbers is linked to area in the Grade 3 Measurement and Data standards 3.MD.5–7, and multiplication of fractions is linked to area in the Grade 5 fraction standard 5.NF.4b (Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.) In each case the area is “unitized”, consistent in the latter case with the way the standards build fractions from unit fractions. So the issue of area (and, for that matter, length) as continuous quantities is skirted. How and where do you think this issue should be broached? Perhaps some comments in the MD and NF progressions could illuminate this matter.

      • Regarding the Mathematical Practices part of this discussion…

        I think the whole “Overview” section (pp. 2 – 4) of this document is very well-done. My questions are with regards to the Mathematical practices sub-section (p. 4):

        1. In the example given for MP5, I read that “special strategies” should be viewed as “tool” that could be used strategically. Is this correct?

        If so, this MP speaks strongly to computational fluency (among other things), correct?

        2. Accompanying the narrative of the MPs with a few brief illustrations in the margins would go a long way toward making these “real” to K-6 teachers – particularly the examples for MP2 and MP1. The example given in the narrative with MP 6 is perfect.

  10. Just a couple of comments on Grades 4 and 5.

    I hope there is a bit more emphasis on “0.1,” “0.01,” etc. written as decimal numbers as “decimal units.” Both 0.1 and 1/10 are representation of the same number, and it is that number, not how it is written, that is a decimal place value. I would like students to understand, for example, 2.4 as “2 ones and 4 1/10s,” “2 ones and 4 0.1’s,” “24 0.1’s,” “240 0.01’s” etc. The last example is just another example of how easy it is in our base-10 number system to see a given (written) number in terms of any decimal place value as a unit.

    RE: Properties
    In discussing the development of written algorithm (strategy?) for division, there is the following statement: “Division strategies in Grade 5 involve breaking the dividend apart into like base-ten units and applying the distributive property to find the quotient place by place.” I know exactly what this means, but I was also wondering if it is accurate to say we are applying the distributive property here. I also wonder if it is helpful to actually discuss some “patterns” of subtraction and division that are not quite the formal “properties” – or consequences of properties of addition/multiplication. For example, I think explicit treatment of the following patterns are useful:

    N – N = 0
    N – 0 = N
    N / N = 1
    N / 1 = N
    a – b = (a +/- K) – (b +/- K)
    a / b = (a * K) / (b * K)
    a / b = (a /K) / (b / K)

  11. Regarding placement of particular illustrations in the margins…

    I like the variety of examples for each of the operations (though I would have loved to see a number line model used as a strategy for adding or subtracting by place value).

    The color-coded multiplication illustrations (pp. 13 – 15) are terrific. Very well done!

    Placement of some of the addition and subtraction illustrations in the grade 2 section (p. 9) is very troubling to me. In an otherwise fantastic document, this really murkys the water for a couple reasons:

    In grade 2, the standards require fluency with addition and subtraction within 100 and addition of several 2-digit numbers. All given examples fall outside of these limits. They do fall within addition and subtraction within 1000 (2.NBT.7), but the second example in particular does not seem in the spirit of using “concrete models or drawings and strategies based on place value…”

    It is clear that the first example (partial sums) explicitly shows place value.

    The third example clearly uses a drawing, but also strongly implies that the standard algorithm should be taught in grade 2.

    While the second example (and the 3rd) are based on place value, the place value is obscured by the way these algorithms are recorded. To a second grade student (and teacher) these are not strategies based on place value, they are the standard algorithms. These examples really convince a second grade teacher that s/he is supposed to teach the standard algorithms.

    I know this was a big point of discussion during the draft feedback. To maintain clarity of the expectations in second grade, here are 2 suggestions for ways to clarify:
    1. Move these examples to grade 4 (p. 13) to illustrate 4.NBT.4, or
    2. Leave them on page 9, but include a VERY prominent qualifier with these illustrations that the grade 2 strategies should focus on using concrete models or drawings and strategies based on place value…; the standard algorithms are taught in grade 4.

    Without some sort of strong clarification here, the illustrations on page 9 will cause a lot of confusion and re-spark impossible conversations that I’m sure you would rather not have again.

    THANK YOU for all of the time and effort that clearly went into this. It is great to see this sort support documents FROM THE AUTHORS. I hope the curriculum/textbook publishing companies & authorship teams are paying attention to them!

  12. Brian, I see your point about the figures on p. 9. Partly this is a production error: there should be some artwork showing the drawings students might make to explain them. Using three digits rather than two allows one to illustrate the iterative nature of the algorithms, and emphasize the fact that the base ten system uses the same factor, 10, for each rebundling of units into higher units. But there is an inadvertent message here which we will try to fix in the next draft.

    • Bill,

      Thanks for the reply! I understand your point about the “repeated reasoning” of 3 iterations making the ‘base-10-ness’ more apparent. I am very appreciative that you intend to do something to clarify that:
      – the +/- standard algorithms are 4th grade expectations (not 2nd), and
      – second graders should bethe magnitude of numbers 2nd graders should be using.

      This will greatly help the field.

      • Bill,

        Thanks for the reply! I understand your point about the “repeated reasoning” of 3 iterations making the ‘base-10-ness’ more apparent. I am very appreciative that you intend to do something to clarify that:
        – the +/- standard algorithms are 4th grade expectations (not 2nd),
        – second grade +/- work should focus on fluency within 100, and
        – second grade work within 1000 should use concrete materials, drawings, and “strategies” (not algorithms) based on place value…

        This will greatly help the field. Thank you!

  13. Love the visual of the place value cards for Kindergarten but we can’t figure out how to actually construct them. If they are double sided with the tens frames on the back (of say 10 and 7) when you place the 7 over the ten (to hide the 0) and then flip it over to look at the tens frames, you can’t see the 7 dots. We could do it if the ones card is a “tent” so it layers over the 0 in 10 front and back so you see the dots. But teachers can’t make a class set of those. Do you have a prototype? or some other way of constructing these?
    Thanks for ALL the work you are doing. It is so helpful to those of us in the field.
    Selina and Ginger

  14. The place value cards have to be rearranged when they are flipped over because the ones are now on the left. Children can see this. They take apart the cards and decide where to put the ones circles (in the blank space on the right of the ten ones). Because left-rightness is not super strong at this age, children need models in the classroom of the tens quantities on the left and ones quantities on the right to support their correct visual building of the place value system. Working with the cards givens them experience in building this same knowledge with their motor system as well. The cards can be made on one page: numbers on one side and quantities on the back. Just be sure that the tens quantity is on the left side of the back of the card.

  15. Hello Bill,

    Thank you so much for these progressions- really adds a lot of context to the CCSS.

    My question relates to the place value layer cards and decimals. I LOVE the layer cards, and would love to incorporate them, but am reluctant to do so unless I can mirror this tool in decimals (4th and 5th grade). Larger place values are underneath smaller place values to the left of the decimal- the only way I can think of to do the decimals is to have smaller place values underneath to the right of the decimal; and clearly this could introduce some misconceptions.

    Any thoughts?
    Thanks-
    Jen

  16. When using the cards, emphasize only the place and the value of that place. Which cards you are putting on top does not matter. The cards do emphasize the symmetry of the place value system (around the ones place). Cards going away from the ones place in either direction go under cards closer to the ones place. So decimal cards work well, though it can take a couple of tries for students to get this symmetry if they have only been using the cards for whole numbers.

    • Thank you Karen! Helpful. Always looking for another way (besides arrays and “jumps”) to show place value understanding. Appreciate the clarification.
      Jen

  17. Bill,

    I have a question regarding 2.NBT.8- “Mentally add 10 or 100 to a given number 100-900, and mentally subtract 10 or 100 from a given number 100-900”:

    Assuming a subtraction example, is this standard intended to limit the minuend to multiples of 100 within the given range? Multiples of 10 within the given range? Or any number in the given range?

    1.NBT.4, 5, and 6 all deal with the same idea, but clearly specify the intent in the language… and they go both ways. Could the intent of 2.NBT.8 be clarified in the paragraph addressing this standard in the Progression document (page 10)?

    As always, thank you for your guidance and support,
    Brian

    • The minuend can be any number in the given range. I agree that there is a problem with the wording of 2.NBT.8: the flow should be from 1.NBT.5 to 2.NBT.8, and 2.NBT.8 could just say “Mentally add 10 or 100 to a three-digit number, and mentally subtract 10 or 100 from a three-digit number,” in parallel wording with 1.NBT.5. Meanwhile, 1.NBT.4 and 1.NBT.6 just flow into the catchall fluency standards 2.NBT.5. It seems that 2.NBT.8 picked up the range language from 1.NBT.4 and 1.NBT.6. This is a glitch and should be clarified in the progression, thanks for pointing it out.

  18. A reference to 4.NBT.5 is missing on p.13 – the paragraph that starts with
    “In fourth grade, students compute products of one-digit numbers”

  19. Bill,

    2.NBT.4- “Compare two three-digit numbers based on meanings of the hundreds, tens…” and 4.NBT.2- “Read and write…Compare two multi-digit numbers based on meanings of the digits in each place…” explicitly address comparing numbers using relational signs. Are they also intended to include ordering numbers (ex., Write these four numbers in order from least to greatest.)?

    If not, is ordering numbers addressed anywhere?

    Thanks,
    Brian

      • Thank you for clarifying that. Because this standard (and parallel standards related to comparing fractions and decimals) specifically talks about comparing using , I would not have assumed it is intended to address ordering numbers. This should definitely be explicitly stated in the Progressions documents so that it is interpreted consistantly by all classroom teachers (and curriculum writers).

        I understand that you are a very busy person and work a full-time job… THANK YOU for taking the time to respond to all of the questions we post here. This blog is the field’s most valuable resource!

        Brian

  20. Hi Bill,
    Could you tell me where the standard algorithm for addition and subtraction should be introduced? Traditionally, this is seen in math workbooks at the end of first grade or in second grade? However, it appears from the common core documents that it is more appropriate in 4th grade. So should 2nd & 3rd grade be focusing on addition & subtraction strategies based on place value understanding using concrete models, drawings, hundred charts, open numberlines etc…
    Also, can you comment on the use of timed test for fact mastery. Specifically in relationship to 2.OA.2, Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.
    Thanks,
    Lori

    • I think the standard algorithm would have to be introduced earlier than Grade 4, although fluency is not required until then. There’s a thread of standards

      2.NBT.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
      3.NBT.2 Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.
      4.NBT.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm.

      Notice that its “strategies” in Grade 2, “strategies and algorithms” in Grade 3, and “the standard algorithm” in Grade 4. Since you are aiming for the standard algorithm in Grade 4, the work in Grades 2 and 3 has to be building towards that. E.g., students might start by adding the tens and ones separately and combining, and then learn to compress that into the standard algorithm. Different curricula will take different approaches, but it’s hard to imagine reaching the necessary fluency in Grade 4 without some earlier exposure.

  21. Hi Bill,
    Thanks for getting back to me so quickly… Any thoughts on the use of timed tests for fact mastery. Specifically in relationship to 2.OA.2, Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers. For many, this translates into the use of timed tests. Is this your intent?

    • It was not our intent to dictate the pedagogical method. It is no accident that the standard says “know from memory” rather than “memorize”. The first describes an outcome, whereas the second might be seen as describing a method of achieving that outcome. So no, the standards are not dictating timed tests. Some curricula might go that route; others might encourage students to build up their knowledge of number facts another way.

      Sent from my iPad

  22. Hi, I noticed the progression document seems to use “like base-ten units” in some cases when they just mean “base-ten units.” When talking about addition and subtraction, like base-ten units makes sense because you add ones to ones and tens to tens, but when talking about multiplication and division, this is no longer true. For example, when multiplying 35 x 42, you would multiply tens with tens and ones with ones, but also tens with ones and ones with tens (unlike base-ten units).

  23. Bill,

    I was using your Progressions documents at a big “Unpacking the Standards for Mathematical Content” workshop and the Grade 5 group had a question regarding 5.NBT.6 (not mentioned at all in the Progression for NBT). They were unclear what it looked like to use ‘properties of operations’ to solve division problems. We thought it might just be inverse operations… then we thought maybe it might look like some of the examples given on the 4th grade pages. Please let us know what you are picturing here… and that might be something worth clarifying in the Progression.

    As always, your guidance is truly appreciated.
    Brian

  24. Brian, “properties of operations” here refers not to the relationship between division and multiplication, but to properties such as the distributive law, commutative law, and so on. There’s a complete list in the appendix. For example, any time you compute a quotient by breaking the dividend up into components and sharing out each one, you are using the distributive law: 135 ÷ 5 = (100 + 35)÷5 = 20 + 7 because 5 x (20 + 7) = 5 x 20 + 5 x 7. Not that you would necessarily say it that way to students, but you might very well use an area model, as in the Grade 4 examples, to illustrate the division, and an area model is really a physical embodiment of the distributive law. Likewise, problems involving breaking some 100s into 10s and grouping them with the other 10s are really instances of the associative law. So yes, something like the Grade 4 examples would be appropriate, although in Grade 5 one would be moving more to the notated algorithm.

  25. Hello All,
    A recent question was posed to me about estimation/approximation/rounding. In our state (NY) guidance has offered the concept that estimation happens at the beginning of a problem (making the numbers easier to use) and rounding is something that happens at the end after calculations are completed (this is a summation of a more thorough description). A teacher recently asked about where “approximation” fits into all of that. Can someone give me an explanation/description/useful response to this?
    Thank you!
    Dianne

  26. Hello, I was wondering if there will be or is a Progression Document on the Number System for grades 6-8. We are hopeful that the new CCSS-M is a game changer. The Progressions and the Blog discussions really help us change the language we use to communicate change in content focus to teachers. Thank you for your anticipated reply and all the work you do everyday. Hopefully the beauty of Mathematics will be more appreciated as a result of this important work.
    ~Nancy

    • I am also interested in Nancy’s inquiry. I’d like to know why, when students find the greatest common divisor of pairs of numbers up to 100, they only find the least common multiple of pairs of numbers up to 12. Two concerns there. First, since lcm(a,b) = ab/gcd(a,b), I don’t see why students shouldn’t find gcd and lcm of pairs of numbers up to [the same maximum]. Second, for finding common factors there aren’t many interesting examples of pairs of integers up to 12. There are only seven pairs that have a common proper factor, (4,6), (4,10), (6,8), (6,10), (8,10), (8,12), and (10,12), and only one, (8,12), has a common proper composite factor. There don’t seem to be enough examples for students to practice.

    • Dear Nancy, sorry to be slow catching up on this. Yes, there’ll be a Number System progression by the end of the summer. In answer to Brad’s question, yes, lcm’s are treated with an extremely light touch in Grade 6. Number theory is not a primary focus of the standards. It’s a beautiful subject (and my own research area), but there is much that is beautiful but not in the standards.

  27. Typo at the bottom of p. 19.

    7x + 33 should be 7x + 3.

    I appreciate your having included this side-by-side example.

  28. Would you please provide an explanation of the various types of tasks: Practice, scaffolding, constructing, and performance. Is there a site that goes in depth with an explanation of each?

  29. Pingback: Fly on Math Teacher's Wall - Put the Value in Place Value - The Recovering Traditionalist

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