Ratio – fractional notation

This topic contains 5 replies, has 4 voices, and was last updated by  Bill McCallum 1 month, 1 week ago.

Viewing 6 posts - 1 through 6 (of 6 total)
  • Author
  • #3162


    My understanding is that the CCSS express ratio with colon (3:2), or with words (3 to 2), but not as a fraction (3/2).

    Can you explain the reasoning behind it?

    The progressions document refers to the quotient 3/2 as the value of the ratio 3:2 (“3/2 is sometimes called the value of the ratio 3 : 2.”). And also “In everyday language the word “ratio” sometimes refers to the value of a ratio”.

    Can you elaborate on the difference between ratio and the value of the ratio? There are numerous websites that express ratio as a fraction, and we found assessment items that require identification of ratio as a fraction. Why is this considered “everyday language”? Why is it wrong?


    Bill McCallum

    A ratio is comparison of two numbers, whereas a fraction is a single number. All sorts of problems can arise by confusing the two. For example, suppose I have juice recipe which requires 2 cups of orange juice to 3 cups of peach juice. So the ratio of orange juice to peach juice is 2 to 3. Now I double the recipe, to get an equivalent ratio of 4 to 6. If I identify these ratios with the corresponding fractions, 2/3 and 4/6, then it sounds like I am saying that two times 2/3 is 4/6. A similar problem arises with addition of fractions. Suppose I have one class that has 9 girls and 10 boys, and another class that has 11 girls and 8 boys. So the ratio of girls to boys is 9:10 in the first class and 11:8 in the second class. Now suppose that I combine the two classes: what is the ratio of girls to boys in the combined class? I just add the 9 and the 11 to get 20 girls and the 10 and the 8 to get 18 boys, so the ratio is 20:18. But if I now confuse the ratios with the fractions, it seems like I am saying that 9/10 + 11/8 = 20/18. The misconception that you add fractions by adding the numerators and adding the denominators is a fairly common one, and I suspect it comes from the confusion of fractions with ratios. You are right that this confusion is rife in the field. I’ve even seen materials that promote the confusion and come up with this different way of adding fractions and say it is just as good as the “traditional” way. What a mess!



    Ok, so a ratio is a comparison of two numbers. It is not a number itself. If I’m a student, I think I can handle that. So what happens when I read the following on a Grade 6 PARCC practice test item….
    ….The ratio of the sales tax to the amount of a purchase is a fixed number in Town Q. The table shows the sales tax for a purchase of $1,200….
    I suppose what they are actually referring to is the value of that ratio. This will confuse students if they have been writing every ratio as a:b or a to b. I understand this is not a problem with Common Core Standards. Rather, whoever wrote this at Pearson did a poor job.
    Something similar will happen with the words rate and unit rate that the progressions define, yet it is claimed that the concepts can be presented to students however one wants (I don’t see how this is true when 6.RP.2 specifically refers to the unit rate a/b associated with a ratio a:b). I agree that you can get at these concepts using various language, but students will get confused come testing time if the language used in a question from PARCC or Smarter Balanced differs from the language in their book. I’ve seen a few items that use unit rate in a way that is more along the lines of a/b units to 1 unit, rather than as the value a/b.
    For now, this won’t cause an issue for most students as they won’t have textbooks using the language from the progression docs. However for students using Eureka and students that will eventually use the Illustrative Mathematics curriculum (I assume it will define rate and unit rate as in the progressions), how will they be able to handle the change in language on assessments?
    It seems to me as though testing consortia need to avoid the words rate and unit rate. But is it really that simple? Any thoughts?


    Bill McCallum

    The usage of ratio in that PARCC question is incorrect, I agree. I can’t see the table, but why didn’t they just say the sales tax is a fixed percentage of the purchase (assuming that they give percentages in the table)?

    As for the confusion about rate and unit rate, it’s not a problem for multiple choice items (as long as they get it right!). For student produced response items, assessment writers will have to make some decisions. I think it would be reasonable to accept an answer with the units even though only the unit rate was asked for. I don’t think it would be reasonable to accept an answer without units when the rate was asked for.



    Every time I read the progression document there’s something new that catches my attention. After reading this post I want to make sure I’m connecting all the dots correctly.

    It sounds like it’s inaccurate to write and refer to a ratio itself as A/B because A/B is the associated rate/quantity of the ratio A:B.

    It seems like proportions rely on equivalent ratios having the same unit rate. So in a proportion, A/B=C/D, are we saying A/B is the unit rate associated with A:B and C/D is the unit rate associated with C:D and because equivalent ratios have the same unit rate we are able to say A/B=C/D. Would this be an accurate interpretation or would you describe it differently?


    Bill McCallum

    That’s an accurate reading of the standards and the progression, yes!

Viewing 6 posts - 1 through 6 (of 6 total)

You must be logged in to reply to this topic.