While my other posts addressed more mid to upper level math skills (within a K-8 range), this week's topic deals with more simplistic concepts. A nice, engaging, educational method for younger students is to introduce a concept and then give them lots and lots of opportunities to practice that concept. With this chart below, once you get the hang of what's happening, there's lots of room for organizing the information in different ways to offer lots and lots of practice with the idea of fractions and ratios.
Let's start by organizing our information into a chart:
A fraction is a number that tells you what part of a whole group you have. A fraction is written as one number over another with a bar between them (eg. 1/2). The top number, or "numerator", is the number that tells you how many pieces you have from the group; the bottom number, or "denominator", is the number that tells you how many pieces were in the group all together. So, the fraction 1/2 tells us that our group had 2 pieces all together, and we had 1 of those pieces.
A ratio is a comparison of numbers within a group. A ratio can be written as a fraction, with a colon separating the numbers that you're comparing, or writing "to" between the numbers. When written as a fraction, one number is written in the numerator spot and the other is written in the denominator spot; when written with a colon, the numbers are written next to each other with a colon separating them. So, if we have 15 marbles composed of 8 blue marbles and 7 red marbles, the ratio of blue to red is 8/7, 8:7, or 8 to 7. We can flip them around, too, to say that the ratio of red to blue is 7/8, 7:8, or 7 to 8.
Both fractions and ratios can be reduced to use smaller numbers to describe a situation. For example, if I have 8 cookies, and I ate 4 of them, I could say that I ate 4/8 of the cookies, or I could reduce that fraction to say that I ate 1/2 of the cookies. Another example- if I have 20 marbles composed of 5 purple marbles and 15 green marbles, I could say that the ratio of purple to green is 5:15, or I could reduce that to say that the ratio is 1:3. If both numbers in a fraction or ratio are divisible (can be divided) by the same number, then you can divide them to get to a reduced fraction or ratio.
Using our chart of the tribute gifts that Jacob set aside for Esav, we can use fractions and ratios to make different descriptions and comparisons of the types of animals. (Note: To avoid confusion, I'll use the colon for ratios and the bar for fractions.)
What fraction of the animals were female? 490/550, or reduced- 49/55. This means that if all the animals were divided into exactly the same groups, for every group of 55 animals, 49 of them would be female.
What was the ratio of female to male animals? 490:60, or reduced- 49:6. This means that if all the animals were divided into exactly the same groups, each group would have 49 female animals and 6 male animals. Notice that when you add the ratio numbers together, you get the total number of animals (490 females + 60 males= 550 animals) and this works for the reduced fractions and ratios, as well (49 females + 6 males= 55 animals in each group).
What do the comparisons look like for individual types of animals?
Goats:
--Fraction of females: 200/220 or 20/22 or 10/11. So, for every 11 goats, 10 were female.
--Fraction of males: 20/220 or 2/22 or 1/11. So, for every 11 goats, 1 was male.
--Ratio of females to males: 200:20 or 20:2 or 10:1. So, for every 10 female goats, there was 1 male goat.
Do you see how nicely the fractions and ratios fit together? How the numbers are consistent, even when they're reduced, so that you can make comparisons between males and females and also compare sections of the whole group to each other?
Let's try another animal. Ewes and Rams have the same ratio as Goats, and the Camels only had females and children- no males, so nothing to compare within the category.
Cows/Bulls:
--Fraction of females: 40/50 or 4/5. So, for every group of 5 cows & bulls, 4 were female.
--Fraction of males: 10/50 or 1/5. So, for every group of 5 cows & bulls, 1 was male.
--Ratio of females to males: 40:10 or 4:1. So, for every 4 cows, there was 1 bull.
Donkeys:
--Fraction of females: 20/30 or 2/3. So, for every 3 donkeys, 2 were female.
--Fraction of males: 10/30 or 1/3. So, for every 3 donkeys, 1 was male.
--Ratio of females to males: 20:10 or 2:1. So, for every 2 female donkeys, there was 1 male donkey.
There are lots of other fractions and ratios that we could look at here:
--What fraction of all animals were goats? What fraction were camels?...
--What was the ratio of goats to camels? goats to donkeys?...
--What fraction of all female animals were female goats? ewes? camels?...
--What was the ratio of female goats to female donkeys? male donkeys to bulls?...
If you're really getting into the comparisons, you can actually calculate how many different comparisons could be made between the animals, but that's getting into combinatorics- another topic for another time.
What interesting comparisons can you make? Do you see any interesting patterns in your fractions or ratios?
Lovely lesson. Thank you Leah. What about percentages in relation to ratios and fractions?
ReplyDeleteAny reason that Jacob did not include a male camel for Essau?
Two good questions!
ReplyDeletePercentages would be the next logical concept to be taught following along these lines. Looking at any fraction, if you can find an equivalent fraction that has 100 for the denominator, then the numerator is the percentage ("per cent"= out of 100; this fraction gives you the amount you would have when you have an equivalent group of 100 items). If your numbers aren't as easy to convert to 100, then you can divide numerator divided by denominator; this will give you a decimal number, where the first two decimal places are the whole numbers of the percent.
When you turn ratios into percents, it's a little more complicated, because you have to think about how much the whole group is while you're only looking at the individual pieces that make up the whole. You always want the equivalent ratios to add up to 100, because the two groups together create your whole amount. For example if you have 4 donkeys- 1 male and 3 female, the ratio of male to female is 1:3, this ratio would be 25:75; 25% are male and 75% are female.
The calculation skills are similar for changing fractions and ratios into percents, but the ratio calculation is just slightly more complex than the fractions. Adding percentages into the calculations brings the exercise from a lower/mid elementary level up to a mid/upper elementary level because of the depth of calculation and number manipulation.
As for the male camels- My original post was really based strictly on the text. With a little more exploration, if we look at the explanations of Rashi and Bereishit Rabbah (76;7) on these verses it seems that while the other animals were all a different ratio of females to males, the ratio of the camels was 1:1, and for this reason the text did not explicitly state the number of male camels that were given. Of course, following this explanation, my numbers above would be changed, and certain ratios and fractions would be different.