when do i ever need that? the students protest.
even if we assume we will never use it, later, still this provides future options and all students must have options so it is in the curriculum. equal opportunity for all students you cannot be lazy.
so firstly the students DO NEED IT NOW for school. it is in the curriculum because it prepares for advanced math but also in recipe word problems and doing recipes-for example:
a recipe which is for 6 people and uses 6 potatoes and half cup of flour. what if i only cook for myself and my wife? altho easy to divide the six potatoes.... but i do not want to use too much flour!
so can we use a calculator for dividing 1/3? nope. i tried it. (the calculator answers show both division and multiplication have the same answer but we know that they are not the same. i tried it 1/3 / 1/3 the issue is the calculator is computing whole numbers)
so- how do we solve it?
Lesson one:
1.1
we begin with the obvious simple idea first. we know any number divided by itself the answer is one.
for example 2/2=_? 3/3=_? we can use the calculator for the whole numbers and see the number divided by itself the answer called "quotient- the same as in "IQ" the q means quotient- is one.
therefore obviously a fraction divided by itself for example "one third divided by one third" [using the symbol : below, as in other books] must also be one and still what is the meaning? by knowing the meaning we can compute 1/3:1/2 and 1/2:1/3 and identify the difference.
PART 1.2
we will learn a new meaning for the : symbol meaning divide [commonly written as "vertical pair" two dots and a line between them or simply : because keyboards only have : and yes the symbol for ratio which is : has the same meaning but that is another topic so i will not demonstrate that here]
a smart good way to teach division is by writing the LARGER number first on the left of the smaller number and only later the smaller number [will be first on the left].
after you know 8:2=_? means a group of 8 if divided into 2 GROUPS each group must be? the answer is 4, because equal groups [and sometimes a remainder]
we will learn an additional meaning the same numbers IN THE SAME ORDER means to ask "how much or many" of the RIGHT SIDE NUMBER can FIT INTO the other number.
for example 8:2 same as above, has an additional meaning asking "the group of 2 [written second and appearing on the right side] fits inside 8 must be_?
the answer is the same number 4 times. the same numbers with two meanings.
we need to use the second meaning for fractions so we must practice it.
8:2 means a group of 2 marbles or a twin-pack of pudding FITS INSIDE 8 must be_? 4 because four times like filling four twin-packs of cupcakes as we are familiar totals 8.
we can use this idea for:
*smaller number written first on the left side-
not 8:2 as above but instead 2:8 which means a different idea.
note: this differs from multiplication when we can swap the order and sometimes should swap the order, because only in multiplication the totals are the same 2*4=4*2 but changing the order in division truly changes the meaning and we cannot swap the order as i will explain.
the SMALLER NUMBER first on the left for example
2:12 [in division! not on a clock!] means how much/many of "the number on the right twelve" fits inside the other smaller number? must be less than one so how much? only one sixth fits inside.
practice USING A CALCULATOR.
a. 2:4=_?
b. 4:2=_?
not the same!
in question a, "how much of 4 fits into 2" we understand not even once so less than once [only half fits into it.] we see that 2:4 is not the same as 4:2 and even in multiplication only the 'total is the same" but the meaning is different see endnote, later.
practice [NOW NO HELP CALCULATOR]
c. 12:6=_? means to ask 6 [the number on our right side] FITS INSIDE 12_?
the solution is twice like two of the common and familiar six-packs.
however when we write the SMALLER NUMBER first on the left the meaning is DIFFERENT.
d. 6:12=_? this means how much from the right-side-number 12, fits inside it? it does not fit but a portion of it fits. so the answer is "half" 6:12=1/2
e. 2:12 meaning to ask "how much 12 fits into 2"?
less than one group of 12 the answer is one sixth. 6:12=1/6
after this introduction using "whole" numbers we can use this idea for FRACTIONS
PART 3
remember the two meanings:
20:10=2 can mean break the 20 into ten equal parts so each is 2 and an additional meaning the number TEN on the RIGHT fits inside TWENTY, how many times_? the same number with another meaning.
similarly, when the smaller number is written first:
f. 10:20 means the number on the right 20, how many of it fits into 10? so we understand that in division the order changes the meaning and we cannot switch the order the way we can in multiplication. 10:20 does not fit... not even once but how much of it fits?
only HALF fits into 10 so when we type the smaller number first in a calculator the answer is less than one because it does not fit even once. the answer is 0.5 half fits in.
now that we have demonstrated the idea in whole numbers and used the calculator we can proceed to fractions a subject which calculators are not built for.
PART 4
similarly for questions which the calculator is not built for 2/3:1/3 "two thirds divided by one third" using : as the divide symbol, means THE NUMBER ON THE RIGHT 1/3 that fraction fits inside the fraction 2/3 and we know it fits twice because there are two "thirds" so 2/3:1/3=2 but what is the meaning?
a portion which is one-third can fit 2 times in 2/3 because there are 2. it is like 2:1 without the denominators BECAUSE THIS TIME the denominator IS THE SAME IN EACH FRACTION. [for that we can use the calculator 2/1 and see it fits twice but for the fractions however the calculator will not show. i checked for you to save time and it showed something else and the issue is as above].
until now we have covered 1/3:1/3 same and the larger number left 2/3:1/3 so next is:
PART 5
now we write the smaller number FIRST on the left
1/3:2/3=_? the meaning changes. the number on the right 2/3 how many of them fit into 1/3?
less than once but how much? only half fits so the solution [called quotient like "IQ" means quotient] is half so 1/3:2/3=1/2 meaning "half" fits INSIDE as above only half.
since this time the denominator is the same it is like 1:2 the answer is less than one. in other words
since the denominator is the same it is similar idea to 1:2
note: in some calculators using / as soon as you type 1/2 you see appears 1/2 which is already the answer too. 1/2 is a half and 1:2 is a half therefore 1/3:2/3 =1/2
we have already used the calculator above but for this it is not built. end lesson one
lesson 2 is for different deniominators but the ide is "how much of the fraction fits in the other fraction" as above. you will use the idea of common denominator which you alreadyt learned for adding fractions and then once they are the same denominator use the method above so actualy no need for lesson 2!
end-note about multiplication 2*3=3*2 but not the same meaning
we can demonstrate the two multiplication we can say a group of six like the familiar SIX-PACK times 2 groups totals 12 written 6*2=12 and spoken in the reverse commonly "2 six-packs" is the idea of 6*2=12. in contrast to 2*6=_? the idea is a group of 2 like a twin-pack of cupcakes times "six of them" and only becausethe totals are the same can we swap the order but not in division as above.
so we see a minor difference between 6*2=2*6 altho the total is the same number still the idea is different
6*2 is a six-pack but two of them so the total is twelve in contrast to 2*6 a twin-pack times 6 of them is 12 total 12.
a dozen donuts 12*1=12 but two dozen donuts truly is 12*2 a dozen but two of them and can be written the reverse order only because they are equal total.
the parallel of "break the 12 into 2 equal groups" is precisely 6*2=12 because 12:2=6 in the simple meaning.
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