The Math Blog 19-20
Rates and Unit RatesPosted by Mark Rinaldi on 2/3/2020
We are taking our work with ratios and applying it to rates. A rate is a ratio where the two quantities being compared are measured in different units. A simple example of a rate is speed. Speed is measures as distance over time. The most common rate of speed seen in everyday life is miles per hour. Notice the two values are measured in different units: miles and hours.
An important type of rate we are working with is the unit rate. A unit rate is a rate in which the second quantity in the comparison is always one unit. For example, speed is almost always given as a unit rate. We don't see signs on the highway saying the speed limit is 195 miles per 3 hours. It is always something per 1 hour.
Many of the examples we are working with in class are related to finding unit rates. For example, if you are told that one bottle of juice costs $3.84 for a 16 ounce bottle and another bottle costs $4.50 for 25 ounces, you would want to look at the unit price to see which is the better deal.
$3.84 / 16 ounces = ? / 1 ounce
We would divide 16 by 16 to get to 1 so we divide $3.84 by 16 to get to $0.24 so the first bottle is $0.24 per ounce.
$4.50 / 25 ounce = ? / 1 ounce
We would divide 25 by 25 to get to 1 so we divide #4.50 by 25 to get to $0.18 so the second bottle is $0.18 per ounce. The second bottle is a better value.
Khan Academy links:
The Rates section has some videos about unit rates and some practice problems.
Ratio Word ProblemsPosted by Mark Rinaldi on 1/28/2020
We have been working on solving word problems using ratios. I've been trying to get them to set up their problems the same way every time. Let's look at an example:
The ratio of boys to girls in the school is 8 to 11. If there are 336 boys in the building, how many girls are there?
The first thing we do is write the ratio using the words of what we are comparing. In this case, our starting ratio would be:
Then we write the starting ratio:
boys/girls = 8/11
Then we write the new information they give us to solve the problem:
boys/girls = 8/11 = 336/?
Now we have equivalent ratios and we can find the multiplicative relationship between the ratios. We need to figure out 8 times what equals 336. So we divide 336 by 8 and get 42. The multiplicatve relationship between the ratios is x42. So then we multiply 11 by 42 to find the number of girls.
11 x 42 = 462.
There are 462 girls in the building.
RatiosPosted by Mark Rinaldi on 1/27/2020
We have begun our unit on Ratios, Rates and Proportions. Our basic definition of a ratio is just a comparison of two quantities.
Let's look at this set of letters for example:
A D A B A A B C C A D A C
If I asked for the ratio of A's to B's, we would count the A's and count the B's. Since there are 6 A's and 2 B's, we would say the ratio is 6 to 2. The order is important as the numbers have to match the quantities we are looking for.
We also discussed that there are different ways to write ratios. The answer in the above example could be written as 6 to 2, 6:2, or 6/2.
Khan Academy links:
Introduction to Ratios with some sample problems.
Division of DecimalsPosted by Mark Rinaldi on 1/20/2020
In division, the students already know that a decimal in the dividend does not cause problems. We just divide like normal and bring the decimal point up. They just have to make sure they are keeping their columns lined up so they bring the decimal point up to the right place. The part that we have to watch out for is if there is a decimal in the divisor. Let's look at an example.
In this math problem, we see there is a decimal in the divisor (the number outside the box). To make this easier to solve, we are going to move the decimal point over as many places as needed to make it a whole number. We would have to move the decimal two places to turn 0.91 into 91. Since we moved that decimal two places, we have to keep it equivalent and move the other one two places so 50.96 becomes 5096.
We then solve it as a normal division problem and get our answer.
You can check out Khan Academy for review:
Multiplying DecimalsPosted by Mark Rinaldi on 1/13/2020
Students have seen that multiplying numbers with decimals points in them is the same as multiplying whole numbers except for the last step where they determine where the decimal point goes in their final answer. Let's look at a simple example.
4.57 x 9.3 =
We can ignore the decimals and look at this problem as 457 x 93. If we do out the problem we get an answer of 42501. Then we go back and look at the original problem. If we look at our two factors, the first factor has two (2) decimal places after the decimal point and our second factor has one (1) decimal place after the decimal point. If we add that (2+1 = 3) we get three. That means our answer needs to have three decimal places. Therefore, the answer would be 42.501 since that gives us three decimal places.
Khan Academy video: Mutiplying Decimals
Khan Academy practice: Multiplying Decimals practice problems
Unit Test ReviewPosted by Mark Rinaldi on 12/16/2019
Dividing Mixed NumbersPosted by Mark Rinaldi on 12/4/2019
We are now looking at the division of fractions. The trick to doing division of fractions is to not try and divide fractions. Instead, we turn it into something that we do know how to do, which is multiplication of fractions.
As we have been saying all along, the first step to any division or multiplication of fractions/mixed numbers problem is to turn everything into a fraction. In previous posts, we've gone over how to turn a mixed number into a fraction. You multiply the denominator by the whole number, add that product to the numerator, and keep the denominator the same. If it is just a whole number, put it over one (example: 9 = 9/1). Once everything is in fraction form, we can continue.
There is only one new step for division compared to multiplication. We change the division sign into its inverse operation which is multiplication. After that, we "flip" the second fraction into its reciprocal or multiplicative inverse.
multiplicative inverse examples:
4/7 --> 7/4
8/13 --> 13/8
4 2/3 = 14/3 --> 13/4
6 = 6/1 --> 1/6
One of the most important things to remember is that only the fraction after the division sign gets "flipped". Once you have done that, the problem has become a multiplication of fractions problem which is something we have been doing for a while. We also discussed why the process works.
9 1/3 ÷ 5 5/6 =
28/3 ÷ 35/6 =
28/3 * 6/35 =
Now we solve that as we would a normal multiplication of fractions problem. We can look for anything that can simplify before multiplying. Look at the previous posts to see the methods for simplifying before multiplying.
28/3 * 6/35
4/1 * 2/5 = (4*2)/(1*5) = 8/5 = 1 3/5
This video and the one that follows it are good refreshers
Multiplying Mixed NumbersPosted by Mark Rinaldi on 11/19/2019 9:00:00 AM
Since we have been working on multiplying fractions, today we decided to throw some mixed numbers into the problem to see what happens. We discussed the fact that the easiest way to multiply mixed numbers is to not multiply mixed numbers. Instead, we turn them into fraction form and just multiply the fractions as we have already been practicing.
Step 1. Turn any whole numbers or mixed numbers into fractions. To turn a whole number into a fraction, just make it over 1.
For mixed numbers, you multiply the whole number by the denominator and add it to the numerator. The denominator stays the same.
3 * 2 = 6 + 1 = 7
Step 2. Once they are in fraction form, we can see if there is anything that can simplify. If you look at the previous post, we discussed a couple ways to simplify before multiplying.
Example: becomes 27/4 and 32/15 when we turn them into fractions. As we already looked at, we can simplify before multiplying at this point. We can either look for common factors or we can break them down into prime factors. Look at the previous blog post to see an example of how we did that.
Step 3. Multiply across and it becomes 72/5
Step 4. If the answer is greater than a whole, you use division to make it back into a mixed number.
with a remainder of 2
So the final answer is
Multiplying FractionsPosted by Mark Rinaldi on 11/19/2019 7:00:00 AM
We've started looking at the multiplication of fractions. Students very often find it easier then addition or subtraction of fractions because there are fewer steps involved overall. In fact, when given a basic multiplication of fractions problem without having worked with it before, many students were able to make an educated guess and get the correct answer.
3/4 * 5/6 = ?
We multiply across (numerator * numerator and denominator * denominator).
3 * 5 = 15
4 * 6 = 24
3/4 * 5/6 = 15/24
Of course, we always have to simplify our final answer if possible. We have practiced simplifying in a couple different ways in previous posts. 15/24 would simplify to 5/8.
What we are working on in class is trying to simplify before multiplying so that we can break down smaller numbers. Let's look at a tougher example.
25/36 * 27/50 =
We could just multiply across and get 675/1800, but then we would have to break down those bigger numbers to get it in simplest form. Instead, we can use a couple different methods we have been practicing in class to try and simplify before multiplying.
One method is to break down both numerators and both denominators into prime factors and write them out as one big fraction.
25 = 5 * 5
27 = 3 * 3 * 3
36 = 2 * 2 * 3 * 3
50 = 2 * 5 * 5
So then it would look like this:
We can then cross out any number on top that we can pair off with a number on the bottom since any number over itself equals 1. In this example, we can see that we could cross off two 3's from the top and bottom and two 5's from the top and bottom and we would be left with:
or 3/8. Looking back at the original problem, if you had 675/1800 and simplified it, it would eventually get down to 3/8, but it is easier to break down the smaller numbers then do the multiplication.
Now for those students who have their facts memorized very well, they can often look at the fractions and simplify without the prime factors.
Looking at 25/36 * 27/50, the would see that 25 and 50 are both divisible by 25 and that 27 and 36 are both divisible by 9 and the problem would become:
1/4 * 3/2 = 3/8
This is one of the big areas where knowing your calculation facts makes you much more efficient by saving you a lot of time and helping you avoid the rising frustration level.
Multiplying Fractions Section
Simplifying FractionsPosted by Mark Rinaldi on 11/19/2019 6:00:00 AM
Simplifying a fraction means finding the equivalent fraction with the lowest possible whole number numerator and denominator.
There are a couple ways we have looked at simplifying.
#1 - If you are good with your math facts, you very often can tell when the numerator and denominator are both divisible by. We can divide the top and bottom by the same thing over and over again until we can't go any smaller.
36 ÷ 2 = 18 ÷ 3 = 6 ÷ 2 = 3
48 ÷ 2 = 24 ÷ 3 = 8 ÷ 2 = 4
36/48 = 3/4
#2 - We can use the prime factors of the numerator and denominator to find the simplest form. You can review prime factorization on Khan Academy. If we had the fraction 72/162, we could turn the numerator into prime factors and the denominator into prime factors.
72 = 2 * 2 * 2 * 3 * 3
162 = 2 * 3 * 3 * 3 * 3
We can then look at the top and bottom and simplify out any top numbers that can be paired up with a bottom number. We see that a 2, a 3, and another 3 from the top could be paired up with a 2, a 3, and another 3 from the bottom. We take the factors that are left and that gives us our simplified fraction.
72 = 2*2 = 4
162 = 3*3 = 9
You can review simplifying on Khan Academy.