Just a quick one with regards to this - teaching primary mathematics, i'm curious as to how you'd appraoch teaching the conceptual understanding of multiplying fractions. My children are happy enough following procedures and conceptually understand fractions including adding and subtracting, but teaching the conceptual understanding of multiplying AND dividing fractions is proving problematic. I'm currently approaching this by using diagrams - grids especially at the moment are proving successful for some, but not all... Any guidance or help with this would be appreciated!
Multiplication relates to how many (repeated copies) "of", or how much (what portion) "of", a given original amount you can get. When you multiply by a fraction, you are finding that fraction, or portion, of the original whole. Assuming that you're dealing with "proper" fractions (which are smaller than 1), then you must end up with a smaller value, because you're taking only part of the original value. For instance, "(1/3)*21" might represent "how many people you can invite to your party, given that I've just told you to cut your proposed guest list to one-third its current length". Division relates to how many sets of a given size can be taken out of the original whole. For instance, "12/3" might represent "how many sets of three cookies I can make from a bag of twelve cookies"; in this case, four sets. On the other hand, "12/(2/3)" might represent "how many people can get two-thirds of a pizza, if I've ordered twelve pizzas"; in this case, eighteen people can eat. Hope that helps!
Similar to Stapel's response. I like to think of proper fractions as percentages (if you taught that yet) that are less that 100%. When you compute 70% of something than you are taking part of the something and will end up with less. I commend you trying to explain the reasons to your students! Everyday when my daughter comes home from elementary school I have to show her why what she was taught is correct.
What you are asking about
Suppose: Any fraction x 1 = The same fraction. All of the proper fractions are lower than one.
In just 12.5 minutes, I share knowledge that has never been revealed before:
6 = 2 x 3 1/4 = 1/2 x 1/2 Watch my video to see why both the above make sense because multiplication has never been generally defined by anyone before I came along, not unless you are talking about over 2000 years ago.
A
This is one of my favorite topics. It may be hard for some here to believe but I give he following to my advanced calculus( intro. to real analysis) classes. Given that \(0<\sqrt{a}<\dfrac{1}{\sqrt{b}}<1\) then arrange the following the following into ascending order: \(1,~\sqrt{a},~\sqrt{b},~\dfrac{1}{a},~\dfrac{1}{b},~a^2,~b^2\). |