Today our lesson was about ratios, before this worksheet there was a PowerPoint presentation with sample questions very similar to this worksheet and we did problems 1-5 as a whole class. The learning goal was that students would be able to write ratios and unit prices. The students were shown 3 different was to write ratios, as a fraction (5/3) , as a sentence (five to three), or with a : (5:3). They could write the answer anyway for the problems but problems 10), 11), 21) it seemed it would be most beneficial for them to write in the fraction form. The sample problems similar to these ones were all shown as fractions and the students were told that the price always goes on top, as the numerator.
An anticipated response to this would be that they would feel comfortable writing these ratios in fraction form because that is what they have been working on for the past few weeks. It was surprising that a lot of the students decided to write them in the other forms, maybe because it was similar for them or they just liked something new.
A problem became aware to me with a lot of the students not seeing a fraction as a division problem, even though they have understood this before. They did not see the connection between the unit price problem and a division problem. The price is divided by the number of products for that price to get to the unit price. The students understood how to write it as a fraction but not how to convert that into a division problem. They also had difficulty with numbers that could not be divided easily, like $1.25 divided by 12 units(a warm up problem they had), the teacher briefly showed them they do not have to continue the problem and can just stop but they have never done that before.
I think a way to help the students with this would be to focus more on unit price problems and in general writing division problems as fractions. They have been working with multiplying, adding, and simplifying fractions but I think they really need to focus or reiterate that a fraction is a way to write a division problem.
I think that the teacher should make problems that are a little easier to divide since that is what the students are used to and then later after they have discussed harder problems more give them some of the harder ones. Giving them a problem where they can just round, or just stop doing the math is not something they are comfortable with yet because they have not really experienced it yet.
You do a very nice job of thinking about the different representations for this task. What do you think the "big idea" for this task is? Once you answer that, you can think about the different representations you might see (you start to do this). Think about a task that might give students opportunities to represent the big idea in different ways, and think of different ways in which you might "sequence" these representations to lead students towards thinking about the big idea in different ways (ratios can be represented as fractions, division, etc.)
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