Draft of progression on Expressions and Equations

Here is the draft of the progression on expressions and equations. Comments and suggestions welcome in the comment area of this post.

[2012/08/31. This thread is now closed. Please ask questions here.]

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.
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27 Responses to Draft of progression on Expressions and Equations

  1. Tad Watanabe says:

    Just a few comments, through Grade 6 section.

    First comment/question is a more general one that goes beyond the EE domain. Is there something in the CCSS That describes which mathematical terms are expectations for students at what grade? So, for example, at what grade, do we want middle school students to use the phrase, identity, to describe a particular type of equations?

    p. 4 The comment on the side with a red dot, “Because these two expressions refer to the same quantity in the problem situation, they are equal to each other,” seems to be very important but very difficult for students to understand. Students are much more likely to think of $2.30 as simply the result of calculation – or “x + 5:50 – 9:20” simply as indicating steps of calculation, not a quantity. That idea can’t be overemphasized.

    Also, on p.4, and this is somewhat a general question, as well. The standard, 6.EE.7 says, “7Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.” So, is the example equation on p. 4, “x + 5:50 – 9:20,” appropriate for Grade 6? How should we read these examples? Does “x + p” literally mean the equation of the sum of a variable and one number? Since equations of the form, px + q = r is a Grade 7 expectation, we know that we are not supposed to combine these two types in Grade 6. But, what about x + p + q = r?

    Still on p. 4,and still in the same paragraph, I am not sure if I like the “rather than” language in “students find greater benefit in representing the problem algebraically by choosing variables to represent quantities, rather than attempting a direct numerical solution, since the former approach provides general methods and relieves demands on working memory.” Aren’t they (algebraic representations and numerical strategies) obviously related? I’m not sure if I understand the claim about “relieves demands on working memory,” either. I don’t think students will find it any more difficult to think, “well, before he paid $9.20, he had 2.30+9.20, $11.50. And this was after he got $5.50 from grandmother…” It seems like a benefit of an algebraic equation in a complicated process is that it suggests a numerical strategy to find the solution.

    Now on to p.5, and this is the last comment/question. The draft says, “Students in Grade 5 began to move from viewing expressions as actions describing a calculation to viewing them as objects in their own right.” This statement made me wonder what 1.OA.7, “Understand the meaning of the equal sign” means. In general, I wish the document will more clearly articulate the contrast between what we are expecting from students in Grades K-5 from what we should expects in Grades 6-8. As the draft note, students start using letters for unknowns in Grade 3. So, what are some of the specific differences in their understanding we are expecting? I think the more clearly and specifically the document can articulate the contrast, the more useful it will be for teachers and curriculum developers.

  2. Justin Yoshimoto says:

    My overall thought is that this is what we expect of students at that level to know and understand and that is quite different from what our students truly know and understand about mathematics. I have 11th and 12th grade students who struggle with many of the concepts detailed in this draft. For example, something as simple as the idea that multiplying by 1/3 is the same as dividing by 3 are two different concepts for students. They do not see the connection or the relationship. I also have students who struggle with linear functions. I think that their understanding of lines is very robotic. They know y = mx + b, but they have no conceptual understanding of why that is the equation of a line or the relationship between the variables x & y. Mathematics for a lot of students are disjoint concepts and skills.
    With that being said, I agree with the philosophy of common core standards. It’s an ambitious expectation and a huge burden for everyone to carry. We all need to be accountable at every level in order for this to work and we cannot operate in isolation. I haven’t read the other drafts for K-5, but I think before we can start having a conversation about what we want students to know by grade 6 we need to be confident that students have a good number sense. My personal opinion is that students have a weak number sense and that contributes to their overall ability to think abstractly in algebra. Students can learn how to use the distributive property, but their lack of number sense is what holds them back from understanding why the distributive property works and why is it useful in mathematics. I really feel like we are moving in the right direction with mathematics education, but it’s just going to take time.

  3. Kimlan Trinh says:

    Just a few thoughts from page 11, section grade 8.
    Per 8.EE.1:”Know and apply the properties of integer exponents to generate equivalent numerical expressions.” The article says “We define 10^0 = 1 because we want 10^a10^0 = 10^(a+0) = 10^a, so 10^0 must equal 1.” This method of deriving a random number x to the power of zero equaling one seems to be too lengthy and ambiguous to a first-time learner. Instead of showing how 10^0 = 1 through implication and reasoning, it will be clearer to show how to get 10^0 = 1. We could introduce rule of division of powers having the same base (10^a : 10^b = 10^(a-b)) in order to explain why 10^0 = 1. For example, many students, if not all by the eighth grade level, will know a number X divided by itself, X, will equal 1. Hence, we can show them that 10^2 /10^2 = 10×10 : 10×10 = 1. Thus, 〖10〗^2/〖10〗^2 =〖10〗^(2-2)=〖10〗^0=1. Then students can “extend these rules to other bases, and learn other properties of exponents.”

    • This is certainly another good way of doing it, and fits the principle of extending the rules.

    • Ruth in PA says:

      I agree completely. I have tried a variety of means to accomplish “meaning” in the rules for exponents, this is the only one that seems to stay with students.

      Nicely done.

  4. Owen Matsunaga says:

    Thank you for the great work on illustrating and detailing the progression that students should make under the expressions and equations standards over their middle school years. It really helped me to understand how the CCSS builds on itself each year. I have only one minor suggestion, which involves the ticket price inequality example on page 10. While I understand the concept being discussed (i.e., negative numbers reversing the inequality), I was slightly confused by the use of the numbers 1000 and 50 in the problem without explanation. To better help the reader see the connection with a real-world problem perhaps it could be stated that the auditorium holds 1000 people and that past experience indicates that for every dollar increase in ticket price you lose 50 potential concert goers, or something to that effect. I also note that there appear to be two typographical errors. The first on page 7 – “of of” on the second line of the first full paragraph. The second is on page 9 – where I believe the number “64” in the equation 3s + 37.5 = 64 (11 lines from the bottom of the page) should be 63 instead.

  5. Brian Cohen says:

    Bill,

    Compliments:
    • The examples noted throughout the document that illustrate specific connections to the Mathematical Practices will be extremely helpful during professional development.
    • The “Some common student difficulties” note on p. 5. Current research/publications point to the power of teaching through misconceptions. More notes like this one will help teachers and students!
    • The discussion of PEMDAS – clear, concise, and with a good example to make it concrete.

    Thank you,
    Brian Cohen

  6. Brian Cohen says:

    Bill,

    Questions:
    • I think there is a typo in a side-note on p. 5. The note regarding “The ‘any order, any grouping’ property” notes that “the sequence of additions and subtractions may be calculated in any order…” This should read “additions and multiplications,” right?
    • On p. 6, the formula for the surface area of cubes is used to illustrate 6.EE.2 (Write, read, and evaluate expressions in which letters stand for numbers). Am I wrong to interpret that grade 6 work with surface area is limited to using nets (6.G.4) and grade 7 may take it to the level of algebraic generalization (7.G.6)?

    Thank you,
    Brian Cohen

  7. Evelyn Bovey says:

    A few comments on the Grade 6 section:

    The draft states that “as word problems get more complex, students find greater benefit in representing the problem algebraically by choosing variables to represent quantities, rather than attempting a direct numerical solution, since the former approach provides general methods and relieves demands on working memory”. I feel that this is easier said than done, particularly for my special education students. Currently, the majority of my 8th grade special education students continue to struggle in representing math problems algebraically. These students have previously experienced 3 years of middle school mathematics (grades 6-8). From what I’ve seen of their work in solving word problems, these students have difficulty coming up with solutions derived from the use of algebra. Rather, they work out their solutions arithmetically, which leads me to believe that the concept of algebra is not easy for these students to grasp. In fact, some of my students have even told me how much they dislike algebra.
    These students appear to understand the Order of Operations (PEMDAS) when solving problems. For example, when they see the expression, 8 x (5+1), my students will add 5+1=6, and then multiply 8×6 to arrive at the solution, 48. However, these students do not appear to be able to determine other ways to calculate this problem (i.e. using the distributive property) without teacher guidance.
    Many of my students with special needs struggle with abstract reasoning, critical thinking, and reading comprehension skills when it comes to solving word problems. I don’t believe that this draft takes into considerations that students in special education do not learn at the same pace as typical students. In general, the progressions stated in this draft, according to grade level, seem too high for students in special education to achieve.

  8. Evelyn, you have described well the fundamental contradiction of standards. In the real world, students arrive in Grade X with wide variation in their readiness for that grade. Some are not ready at all, as you describe; others might be ready for a much higher grade. The only solution I can see is to use standards wisely (what an idea!). We can’t view standards as saying that every single student must study such and such material in such and such a grade. Rather, we should think of standards as a scale against which to measure progress. Some students will be behind, some on schedule, some ahead. Standards are useful in providing a reference point, and in setting common expectations, but the system has to deal humanely with the range of abilities that students bring to the classroom.

  9. Dianne says:

    Hello,
    We have a question regarding 8.EE.7. Can you give us any guidance or clarification on whether specific methods of solving a system of linear equations will be expected/required? And also, whether there are specific types of systems that should be illustrated? We are wondering about whether either or both substitution and/or elimination (with adding/subtraction/multiplication, etc) should be experienced, along with graphing as possible methods.

  10. Lane says:

    I agree with Kimlan Trinh with regards to how a zero power is defined…”because we want…” seems to provoke a sense of argument (do I want?) rather than inspire a buy-in for a continuum of logic from “one less factor.” My students would follow the continuum more readily. After they buy in and absorb, then sometime later the sum of powers is a good, “Oh, and see how….”

    Another question I would have is are we allowing the students enough time to “see” the properties for themselves before we start having them memorize them? Personally I never teach the properties, rather provide enough practice writing out (a^2b^4)(a^3b^4) aabbbbaaabbbb in gradual complexity over time (also having the students apply the communitive property to rearrange). Next (a^2b^4)(a^3b^4)^2 as (a^2b^4)(a^3b^4) (a^2b^4)(a^3b^4) = aabbbbaaabbbb. Then when the negative exponents are introduced, I emphasize using reciprocals over subtracting until they can fluently break down nested expressions. It is much more tedius for the students, but mine (almost) never ask “Do I add or do I multiply?” because they “see” which is appropriate. In common assessments, my students do stand out for that particular skill. I use this practice for Algebra I and our Algebra 3 which is actually a remedial class following Algebra 2.

  11. Lane says:

    Searching the documents for “like terms,” the expression only came up when simplifying expressions following distribution. So it would appear we will not be extending “like terms” into expressions like 2xyz^2 + 3xyz^2, but I can’t find anything definitive about that. Thoughts?

  12. Pingback: I don’t think they mean that | Overthinking my teaching

    • Thanks Christopher, do you have a suggested fix? I’m guessing you are worried about the second part, “terms may be grouped together in any order,” which could be interpreted as saying something like 3 – 5 + 4 can be calculated either as (3 – 5) + 4 or 3 – (5 + 4). What I had in mind when I wrote this is that the terms in this case are 3, -5, and 4, but the internet does not support me in this usage. Which is weird, because nobody calls 4 a factor in 3÷4. But never mind. Maybe the easiest fix is to say “addends may be grouped together in any order”. And then perhaps add a note that at some stage students learn to see a sequences of additions and subtractions as a sequence of additions with negative terms.

  13. Typo Police says:

    At the bottom right side of page 7, in the table next to the first red dot, the value under 4 should be 1.76 instead of 1.75.

  14. Brian Cohen says:

    Bill,

    I’m resubmitting an old question that went unanswered, but came back up during curriculum work:

    On p. 6 of this progression, the formula for the surface area of cubes is used to illustrate 6.EE.2 (Write, read, and evaluate expressions in which letters stand for numbers). Am I wrong to interpret that grade 6 work with surface area is limited to using nets (6.G.4) and grade 7 may take it to the level of algebraic generalization (7.G.6)?

    Thanks,
    Brian

  15. Brian, your suggestion seems reasonable to me, although I can imagine using the nets to get the formula in Grade 6. But this is really a matter of curriculum design; it depends how much time is allowed, and how the curriculum materials want to handle the connections between EE and G in this grade level or the next. In the final analysis, I don’t think the standards specify the level of detail you are at here.

  16. Peggy Britton says:

    Hi,

    In Grade 6 of the EE progression document, under “Reason about and solve one-variable equations and inequalities”, very little is said about inequalities. The sentence “In Grade 6 they start the systematic study of equations and inequalities and methods of solving them” suggests that students might begin solving simple inequalities in Grade 6, such as inequalities of the form x + p > q or px > q for cases in which p, q, and x are all non-negative rational numbers. (This would parallel with the work done with equations (6.EE.7).) However, the only other mention of inequalities in Grade 6 of the EE progression document is in reference to inequalities of the form n > 0 used to represent a domain for a real-world situation. Is the expectation that solving inequalities will not begin until Grade 7 with two-step inequalities?

    Thanks,
    Peggy

    • You are right that there is not much emphasis on solving inequalities in Grade 6. The main emphasis is on understanding what it means to be a solution to an inequality, and testing whether a number is a solution, as in

      6.EE.5. Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

      I could imagine a curriculum pursuing this as far as solving single step inequalities as you suggest, but that’s not required by the standards.

  17. Lynda says:

    Is the expectation with 6.EE.7 that students only solve addition and multiplicaiton equations in Grade 6 and not solve equations such as x – 2 = 5 until Grade 7?

  18. It seems a bit silly to exclude $x-2=5$, doesn’t it? I think that might be a glitch in the standards; the restriction to positive $p$ and $q$ makes sense for $px=q$, since we don’t multiply or divide negative numbers until Grade 7. But it seems reasonable to include $x-p =q$.

  19. Lynda says:

    I agree, and hoped the restrictions on the variables were only intended to restrict equations to those that did not require students to operate with signed numbers. The progression document only has examples of addition equations, so that still left me wondering. Perhaps a different example or simply a note for clarification would be helpful in the progressions document.

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