We have all seen ratios and fractions over the last couple of years. In this unit, we will dive deep into the concept of what really is a ratio. Where do we use them? Where can we NOT use them? What is the difference between ratio and fraction? Understanding the concept of "ratio" is key to truly understanding fractions. We start here:
Introduction to Proportions
A proportion is a simple yet confusing term. It is something that all middle schoolers have a background knowledge of, but we have never truly looked at in much depth. In 7th grade, proportions is the single largest concept that we learn.
Watch the video below for an introduction on proportions:
Solving Simple Proportions
Using our knowledge of how we know two fractions are equivalent (in the video above), we can begin to solve proportions for missing values. For now, we will only be focusing on comparing the ratios between both fractions, or comparing the ratios within each fraction. We will NOT be using cross products until later in the year.
Solving Difficult Proportions
The proportions above were "simple" because the values that you are multiplying by are nice easy whole numbers. However, what if your proportions were using numbers that didn't produce "simple" answers? The concepts we will use are the same as we used previously, however, the methods may be a little more abstract.
Rates and Unit Rates
Rates, Ratios, and Unit Rates are all types of fractions. We generally only differentiate between them when we are dealing with real-world circumstances. That's because they all having something to do with the specific units of measurement being used:
It is important to note that a Unit Rate is still a type of Rate. However, a Unit Rate is a more specific type of Rate. It is the Rate per one unit of something. For example, if I told you that a 6-pack of soda costs $3, what is the cost per one can? In 7th grade, we will be using proportions to show our work here:
Proportion Word Problems
Once we are comfortable examining proportions and rates, we will talk briefly about word problems. This can also be taught as the first lesson of this unit, but is sometimes saved for last until we are more comfortable with the concept of proportions.
Determining Proportionality
The next concept, like word problems, could also be taught first in this unit. However, I have found that covering this concept now can be more helpful because it helps summarize proportions and leads us nicely into solving equations.
Percents, Decimals, and Fractions:
To start off this unit, we will refresh our brains with converting between fractions, decimals, and percents (something you have been doing for years now!!). I will introduce a new idea with 7th grade percents, however, and that is using proportions to understand the relationship between fractions, decimals, and percents. Take a look at the video below:
Using proportions to convert between F,D, &P:
Once we get acclimated to converting between fractions, decimals, and percents again. I like to introduce the idea of using proportions (equivalent fractions) to do this. This will help us as we transition to using proportions with percents later on.
Comparing and Ordering F, D, &P:
Once we are familiar with converting between fractions, decimals, and percents the next natural step is to compare them to each other. One way we can do this is to determine which value is greater (say for example you were given a decimal and a percent), while another is to simply order a list of fractions, decimals, or percents. Thism again, is something that you have been doing for years now, but we will look at the concept a little differently by introducing the use of proportions.
One thing to always remember is that in order to confidently compare one value to another, you MUST present the values in the same format. In other words, you need to convert all numbers to fractions, decimals, or percents depending on your preference (I am going to try to persuade you to like fractions more!!).
Using Proportions to find percents, parts, and wholes:
Now we recall how to write and order fractions, decimals, and percents. We also know how to express percents using proportions. The next step is to find missing numbers using proportions.
Percents by Chunking:
Chunking with percents is meant to be more of a "mental" way of calculating percents of numbers. It offers you a quick and easy way to find percents of numbers by 'chunking' those percents into more manageable pieces. For example: you can find 26% of 50 by finding 10% + 10% + 5% + 1%.
See the video below:
Word Problems with Percents:
Now that we are comfortable manipulating and calculating with percents we can move onto word problems. I like to save word problems for last because you can choose the best or most preferred method based on the needs and parameters of the problem.
Percent Increase and Decrease:
Lets say a store was having a shoe sale for ____% off, and your favorite pair of shoes was reduced from $20 to $15. Can you figure out the percent off? Conversely, what if the sale is over and the shoes rose in price from $15 to $25. Can you figure out the percent increase? That is what this lesson will entail.
Percent Error:
The Percent error is the ratio between the amount something is different from an original amount, compared to the original amount. In other words, if you guessed that there were 5 kittens in a box, but there actually 4 kittens, then you were off by 1 kitten total. This 1 kitten compared to the actual amount (4 kittens) has a ratio of 1 to 4, or 1/4. This means that you were off by 25%, or your percent error is 25%.