Posted onJanuary 28, 2013byBill McCallum
Once every few months or so I receive a message about the following standard:

6.G.2. Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V=lwh and V=bh to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

See if you can guess what people think the problem is before reading on. The most recent message said quite sternly that V=bh was NOT correct, and that it MUST BE V=Bh. This point of view is one of the starkest examples I know of the obstacles we must overcome in restoring the culture of mathematics in schools. The notion that certain letters MUST always stand for the same thing across different formulas is itself a mathematical error, a profound misunderstanding of how symbols are used. It’s like thinking the word blue must always be written in blue. And the misconception is not harmless. Students who come to college with it do not fare well amid the profusion of symbols in their science classes, unable to see that the function f(x)=sin(ax) in their calculus class is the same as the function A(t)=sin(ωt) in their physics class. Part of the power of algebra is that you can choose any letter you like to represent a quantity, as long as you specify what the letter stands for, and you can re-use that letter with a different meaning in a different problem. That said, the standard does not dictate the use of any specific letters; indeed, the core meaning of the standard is not about formulas at all, but rather about finding the volume of a rectangular prism by multiplying its length, width, and height, or by multiplying the area of its base by its height. So teachers and curriculum materials can use whatever letters they like, including V=Bh, or no letters at all. Indeed, formulas should always have words associated with them. Naked formulas like V=bh mean nothing by themselves without surrounding words, such as in the sentence

The volume V in cubic inches is given by V=bh, where b is the area of the base in square inches and h is the height in inches.

Changing the two occurrences of b to B changes the meaning of this sentence not one whit.

The Common Core 6th Grade Math standards do not address prime factorization; however, I realize it is a strategy for finding GCF and LCM which are standards. In the GA Performance Standards, students were required to express numbers in prime factorization. Is this not the same for Common Core? I want to make sure that Common Core is not suggesting that students only list factors and multiples in order to find GCFs and LCMs. Thank you.

Greatest common factors and least common multiplies are treated with a very light touch in the standards. They are not a major topic, and limited to numbers less than or equal to 100 (6.NS.4). For such numbers, listing the factors or multiplies is probably the most efficient method, and has the added benefit of reinforcing number facts. It also supports the meaning of the terms: you can see directly that you are finding the greatest common factor or the least common multiple. The prime factorization method can be a bit mysterious in this regard. And, as you point out, prime factorization is not a topic in the Common Core, although prime numbers are mentioned in 4.OA.4. So, the standards do indeed remove this topic from the curriculum. Achieving the focus of the standards means giving some things up, and this is one of those things. (Of course personally, as a number theorist, I love the topic!)

Dr. McCallum, in 7-G.4 and 8-G.9 the phrase “Know the formula” is used in realtionship to different area/volume formulas. Is the intent of the standard that the students are able to recall from memory these formulas on a summative assessment or be able to know how to use it in the solving of a real-world or mathematical problem? We have also been having a discussion over the use of b or B when talking about volumes of rectangular prisms, is there a preference in realtionship to the standards?

“Knowing the formulas” includes both of the things you mention, although personally I wouldn’t assess that by just asking them to repeat them, but by expecting them to be able to use them in solving problems (without having to be told them). There is no preferred choice of letters for the various quantities that come up in these formulas. Knowing a formula means more than just knowing “V = bh”. It means knowing what quantities all the letters stand for. So knowing the formula for the volume of a rectangular prism means knowing “if the height is h units and the area of the base is b square units then the volume is V = bh cubic units.” You could replace b by B in that sentence (both places where it occurs) and it will still be the same formula. The names we give to quantities are not essential components of a formula. Of course, it is useful to have conventions about the choice of letters. You wouldn’t want to say “the area r of a circle of radius A is given by r=πA2.

I feel it is clearer for students if the same letter always refers to the same quantity. Lower case b is most usually used for a linear dimension, such as length of a parallelogram’s base. Capital B is usually used for the area of the base of a prism or cylinder, or an irregular prismatic solid. Similarly, h is used for height in a plane figure, and H is used for a solid. Thus, when both occur in the same expression, students can easily tell the difference.

Michelle says: July 25, 2012 at 8:08 am Message from concerned teacher…I am in need of clarification about the extent of teaching 6th graders about inequalities. II have seen in some task and common core workbooks where the student is being asked to solve 1-step inequalities,but it is not explicitly stated in the standards (see below). In the standards, I do not see where the 6th grade student is asked to specifically solve an inequality as they are asked to solve an equation.The standard is specific about the type of equation the students should solve but does not indicate that the student has to solve an inequality.My understanding of the standards is that students are being asked to: * Identify from a given set whether a value is a solution of aninequality. They are using substitution to find solutions ofinequality.* Solve one-step equations* Write inequalities given a specific situation* Recognize that inequalities can have an infinitely number of solutions.* Graph solution sets of inequalities on a number line. 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. MCC6.EE.7. Solve 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. MCC6.EE.8. Write an inequality of the form x > c or x c or x r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid 50 per week plus 3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions. Bill McCallum says: July 27, 2012 at 7:10 pm Michelle, your understanding is correct, this is a good summary. Students are expected to solve one-step linear equations with positive coefficients and solutions; for inequalities, they are expected to work with and understand solutions in various ways, but not to actually solve them. I was confused by your MCC6.EE.8. It seems to say that students should solve inequalities of the form px + q < r, which is not in the Grade 6 standards. Maybe it is a cut and paste error or maybe that is a place where your state added something to the standards (but did they really add these sorts of inequalities without also increasing the demand on equations?). Elizabeth Oliver says: June 6, 2012 at 10:54 am Dr. McCallum, I have a few questions concerning 8th grade Transformations, Congruence, and Similarity Standards. 1) Will students need to rotate around a point other than the origin? 2) Will students need to reflect across any line other than the x-axis, y-axis, y=x, and y=-x? 3) Will students need to dilate using a center point other than (0,0)? We are in the process of writing units and wanted to make sure we were covering the rigor required by the standards.Thank you for assistance.Elizabeth Reply

June 14, 2012 at 10:42 am First a general comment on the Grade 8 geometry standards: it is not necessary that all the work on transformations in Grade 8 take place in a plane with coordinates. Students could get hands on experience with transformations using geometry software or a pair of transparences, in which they perform and analyze transformations in a blank plane without coordinates. I think this would be preferable for much of the work. However, there is one standard that refers to coordinates, and I assume this is the one you are talking about: 8.G.4. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. Here I think it is reasonable to suppose that the problem is limited to ones students can do with the algebraic tools at their disposal. That would include your (2) and (3), but it wouldn’t even include all rotations about the origin, since you need trigonometry for that. It could include rotations through 90 degrees, and I can imagine a few more transformations that could be used in instruction as challenge problems of an exploratory nature. For example, you could have students figure out that a dilation from a center other than the origin can be achieved by first translating the center to the origin, dilating there, and then translating back again. But I would hate to see this turned into some formulas to be memorized on a test.

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Posted on January 28, 2013 by Bill McCallumOnce every few months or so I receive a message about the following standard:

- 6.G.2. Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V=lwh and V=bh to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

See if you can guess what people think the problem is before reading on. The most recent message said quite sternly that V=bh was NOT correct, and that it MUST BE V=Bh. This point of view is one of the starkest examples I know of the obstacles we must overcome in restoring the culture of mathematics in schools. The notion that certain letters MUST always stand for the same thing across different formulas is itself a mathematical error, a profound misunderstanding of how symbols are used. It’s like thinking the word blue must always be written in blue. And the misconception is not harmless. Students who come to college with it do not fare well amid the profusion of symbols in their science classes, unable to see that the function f(x)=sin(ax) in their calculus class is the same as the function A(t)=sin(ωt) in their physics class. Part of the power of algebra is that you can choose any letter you like to represent a quantity, as long as you specify what the letter stands for, and you can re-use that letter with a different meaning in a different problem. That said, the standard does not dictate the use of any specific letters; indeed, the core meaning of the standard is not about formulas at all, but rather about finding the volume of a rectangular prism by multiplying its length, width, and height, or by multiplying the area of its base by its height. So teachers and curriculum materials can use whatever letters they like, including V=Bh, or no letters at all. Indeed, formulas should always have words associated with them. Naked formulas like V=bh mean nothing by themselves without surrounding words, such as in the sentence- The volume V in cubic inches is given by V=bh, where b is the area of the base in square inches and h is the height in inches.

Changing the two occurrences of b to B changes the meaning of this sentence not one whit.borBwhen talking about volumes of rectangular prisms, is there a preference in realtionship to the standards?Michelle says:

July 25, 2012 at 8:08 am

Message from concerned teacher…I am in need of clarification about the extent of teaching 6th graders about inequalities. II have seen in some task and common core workbooks where the student is being asked to solve 1-step inequalities,but it is not explicitly stated in the standards (see below). In the standards, I do not see where the 6th grade student is asked to specifically solve an inequality as they are asked to solve an equation.The standard is specific about the type of equation the students should solve but does not indicate that the student has to solve an inequality.My understanding of the standards is that students are being asked to: * Identify from a given set whether a value is a solution of aninequality. They are using substitution to find solutions ofinequality.* Solve one-step equations* Write inequalities given a specific situation* Recognize that inequalities can have an infinitely number of solutions.* Graph solution sets of inequalities on a number line. 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. MCC6.EE.7. Solve 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. MCC6.EE.8. Write an inequality of the form x > c or x c or x r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid 50 per week plus 3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.

Bill McCallum says: July 27, 2012 at 7:10 pm Michelle, your understanding is correct, this is a good summary. Students are expected to solve one-step linear equations with positive coefficients and solutions; for inequalities, they are expected to work with and understand solutions in various ways, but not to actually solve them. I was confused by your MCC6.EE.8. It seems to say that students should solve inequalities of the form px + q < r, which is not in the Grade 6 standards. Maybe it is a cut and paste error or maybe that is a place where your state added something to the standards (but did they really add these sorts of inequalities without also increasing the demand on equations?).

Elizabeth Oliver says:

June 6, 2012 at 10:54 am

Dr. McCallum, I have a few questions concerning 8th grade Transformations, Congruence, and Similarity Standards. 1) Will students need to rotate around a point other than the origin? 2) Will students need to reflect across any line other than the x-axis, y-axis, y=x, and y=-x? 3) Will students need to dilate using a center point other than (0,0)? We are in the process of writing units and wanted to make sure we were covering the rigor required by the standards.Thank you for assistance.Elizabeth Reply

- //Bill McCallum// says:

June 14, 2012 at 10:42 amFirst a general comment on the Grade 8 geometry standards: it is not necessary that all the work on transformations in Grade 8 take place in a plane with coordinates. Students could get hands on experience with transformations using geometry software or a pair of transparences, in which they perform and analyze transformations in a blank plane without coordinates. I think this would be preferable for much of the work. However, there is one standard that refers to coordinates, and I assume this is the one you are talking about: 8.G.4. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. Here I think it is reasonable to suppose that the problem is limited to ones students can do with the algebraic tools at their disposal. That would include your (2) and (3), but it wouldn’t even include all rotations about the origin, since you need trigonometry for that. It could include rotations through 90 degrees, and I can imagine a few more transformations that could be used in instruction as challenge problems of an exploratory nature. For example, you could have students figure out that a dilation from a center other than the origin can be achieved by first translating the center to the origin, dilating there, and then translating back again. But I would hate to see this turned into some formulas to be memorized on a test.

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