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Re “to what extent should we focus on these points – their names?”: Note that
the few standards that prescribe terms do so explicitly, e.g., 1.G.3: “describe the shares using the words halves, fourths, and quarters.”I think the answer is the division analogue of this: http://commoncoretools.me/forums/topic/multiplying-mixed-numbers-grade-5/.
Do you mean that most containers for measuring (e.g., beakers with measurement scales) are only 1 or 2 liters? Students could manage to measure larger volumes if they used a third container for putting already measured amounts of liquid.
Are you asking about the “estimate” in “Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l)”? I interpret this as being able to look at a bottle or other uncomplicated container (e.g., a glass), and be able to make a reasonable estimate its capacity in liters. (“Reasonable,” of course, is in the eye of the assessor.)
I don’t see that one needs another unit of measurement, for instance, I estimated the width of my hand was 4 inches (it’s actually more like 3.5 inches) so I have some sense of how long an inch is, even though I haven’t been doing a lot of length measurement.
“e.g.” doesn’t mean one
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use it. In this particular example, drawing or using a number line seems like a lot of work for students who have learned strategies for finding sums of multiples of 10 in previous grades. (Use appropriate tools strategically.) As you note, there are decisions to be made about how to format a number line that goes up to 90. For finding sums of smaller numbers of minutes that are not multiples of 10 a number line might be more appropriate.
My guess is that what Bill was thinking was along the lines of “3:30 to 4:00 is 30 minutes (perhaps looking at analogue clock face), 4:00 to 5:00 is 60 minutes. 30 plus 60 is 90, so it’s 90 minutes from 330 to 5:00.” No unit conversion. No number line.
Using a number line to calculate in hours and minutes would be complicated if “1 hour” were also labeled “60 minutes.” That’s what I mean by “not in base 10”—there’s something labeled “1” but it’s not made of 10 subunits but 60 subunits, which would be 1, 2, 3, . . . if subunits were minutes or 1/60, 2/60, . . . if subunits were hours. In the first case, the situation might be better represented by a double number line (which is introduced in middle grades, see RP Progression). In the second case, the denominators are large (not in grade expectations) and students would need to convert minutes to hours anyway.
Students aren’t expected to do measurement conversions until grades 4 and 5.
From NF progression:
At Grades 4 and 5, expectations for conversion of measurements parallel expectations for multiplication by whole numbers and by fractions. In 4.MD.1, the emphasis is on
times as much” ortimes as many, conversions that involve viewing a larger unit as superordinate to a smaller unit and multiplying the number of larger units by a whole number to find the number of smaller units.4.MD.1, 5.MD.1For example, conversion from feet to inches involves viewing a foot as superordinate to an inch, e.g., viewing a foot as 12 inches or as 12 times as long as an inch, so a measurement in inches is 12 times what it is in feet. In 5.MD.1, conversions also involve viewing a smaller unit as subordinate to a larger one, e.g., an inch is 1/12 foot, so a measurement in feet is 1/12 times what it is in inches and conversions require multiplication by a fraction (5.NF.4).
Using a number line to represent hours and minutes seems complicated because it doesn’t arise from measurement experience with physical units (e.g., rulers) and wouldn’t be in base 10. Students aren’t expected to use number lines to represent feet and inches though they might use drawings of rulers.
Students are not expected to know parallel lines in grade 3. They need only be able to draw shapes and analyze them as it says in the paragraph below the one from which you quote. That analysis doesn’t necessarily involve saying that opposite sides determine parallel lines although students’ descriptions might involve properties that we’d recognize as properties of a pair of parallel lines, e.g., same distance apart when measured by dropping a perpendicular (language we would not expect until later grades).
Check out the grade 3 section of the NF progression (here) and you’ll see it as an example.
As noted in the Functions Progression, the main focus is on functions of one variable:
Undergraduate mathematics may involve functions of more than one
variable. The area of a rectangle, for example, can be viewed as a
function of two variables: its width and length. But in high school
mathematics the study of functions focuses primarily on real-valued
functions of a single real variable, which is to say that both the
input and output values are real numbers. One exception is in high
school geometry, where geometric transformations are considered to be
functions. For example, a translation T, which moves the plane 3 units to the
right and 2 units up might be represented by T: (x,y) –> (x+3,y+2).The Khan video seems to pull a rabbit out of a hat. To factor a quadratic, it begins (without explaining why) by asking students to find pairs of numbers A, B such that A + B is the coefficient of the linear term and A x B is equal to the product of the other two coefficients. This still involves guessing and checking, so I don’t understand Lane’s remark about eliminating it. (Maybe the frustration is meant to be eliminated by the choice of quadratics?)
It seems to me that working backward eliminates the rabbit (and the need to memorize the trick). If you have a quadratic 4x^2 + 25x – 21 (as in the Khan video), then if the quadratic could be factored, it could be written as (ax + b)(cx + d). Working backward, a x c = 4, so the possible values of a and c are (in some order) 4 and 1 or 2 and 2.
So, if it can be factored (using whole-number coefficients), 4x^2 + 25x – 21 = (4x + b)(x + d) or 4x^2 + 25x – 21 = (2x + b)(2x + d).
The second possibility can be ruled out because (2x + b)(2x + d) = 4x^2 + 2bx + 2dx + bd which would make 2(b + d) = 25.
Similarly, b x d = -21, so possible values are (in some order) -1 and 21, -7 and 3, 1 and -21, 7 and -3.
Trying them out in (4x + b)(x + d):
It seems unlikely that 4 x 21 or 4 x -21 would be involved, so trying other possibilities first.
(4x + 21)(x – 1) = . . . 21x – 4x . . . No.
(4x + 7)(x – 3) = . . . 7x – 12x . . . No. And switching the negative sign to the 7 won’t help.
(4x + 3)(x – 7) = . . . 3x – 28x . . . No, but switching the negative sign to the 3 would work.
Factorization is (4x – 3)(x + 7).
This example seems so long-winded that the idea of working backward might get lost in the mass of details. The same ideas about working backward to factor might be illustrated with 2x^2 – x – 1.
Such an example might serve as fodder for discussion of when one might want to use the quadratic formula instead (“Solve quadratic equations by inspection (e.g., for x^2 = 49), taking square roots . . . as appropriate to the original form of the equation” A-REI.4b). Or, it might point out how completing the square is a form of working backward that doesn’t involve checking several possibilities.
Also, note discussion of “complicated” here: http://mathematicalmusings.org/forums/topic/definitions-complicated-expression/
