One answer I've settled on is that each form has its own strengths and benefits that are related directly to the flexibility with which they can represent different ratios. Percents are the most restricted, representing ratios with a single divisor (100). Decimals are in the middle (power of 10 divisors). Fractions are the most flexible (any divisor except 0).

The cost of this flexibility is increased challenge in comparing the numbers directly as well as the creation of redundant representations (2 = 2.0 is fairly trivial, 1/2 = 234/468 is much less so). So, in many (most?) cases, it is easier to compare the most-restricted forms directly. The question then is whether the effort to change forms is greater or less than the effort saved in the comparison.

When I was a student (or have a calculator handy), I find/found it usually saved me effort to convert to decimals. But, that cost-benefit analysis was the result of having * very* underdeveloped fraction number sense. Having taught grade 5 and grade 6 for a long time now, I can see how much easier it can be to compare fractions in their fractional form. The relative benefit of that conversion is significantly decreased.

As a teacher, I also like keeping fractions as fractions because I find it really helps my students develop number sense around these new numbers. So, to make a long story short, I agree that converting the fractions to decimals leads to an easier comparison, but I often find that conversion step to be more work than it's worth and a stealer of learning opportunities.

]]>Good luck using SBG. I think you will find that, once you try it, you won't go back!

]]>What is your process for creating/designing a new lesson on a curricular topic (e.g. completing the square, anti-derivatives, etc.). What do you do? What do you think about? You can think of a specific lesson if you want, if it makes this easier! (I know we don’t often have the opportunity to think about the things we have come to do naturally!)

Can you talk about a lesson that you designed and taught recently that you were super excited about? Did it go well? Did it flop? How do you know?

We like to collaborate in our department -- though it admittedly isn’t easy because of our busy schedules. Have you had the opportunity to collaborate in a meaningful way in your career? Can you talk about how that went? If not, what would be the ideal collaboration for you?

As a department, in the Upper School, we’re moving away from traditional textbooks and more into student discovery and investigation to uncover ideas/concepts. This means that there is more of an onus on teachers to develop curricula -- which can be a lot of work, but it is exciting work. Have you ever had the opportunity to develop parts of a curriculum (or an entire curriculum)? Tell me about that? If not, what are your thoughts about teachers designing their own curricula?

What are ways you’ve experimented in your classroom -- whether they worked out or not?

Many in our department have kids work in groups. Do you use groups in your teaching? If so, how do you use them? If not, how do kids interact with each other in your classroom?

What is something you’ve been consciously working (with regards to your teaching) in your own classes?

What are some things you think students would say about you? Why?

I’m personally trying to become more intentional about a lot of different things that go on in my classroom. Can you think about ways you’re intentional in your practice? Things you’re hyperconscious of either when planning or executing a lesson?

This is just out of curiosity -- there is no right answer! In the math department, different teachers feel differently about real world mathematics and applications (though we all recognize that students really appreciate seeing mathematics become “real” to them). What are your thoughts about pure mathematical work versus real world mathematical work?

]]>I had something a bit more general in mind that is not restricted to adding the same value to numerator and denominator. I'll tweet it out!

]]>It is essentially adding the same value to numerator and denominator of a fraction. a+n/b+n

One concern I have is that some users may not recognize the limitations of this strategy (ex: works differently for improper fractions). Of course, that doesn't mean that it can't be used; we just need to understand when it does and does not apply.

Interestingly, I recently used this same idea as a Sometimes/Always/Never prompt with students: "Adding the same number to the numerator and denominator of the fraction increases the value."

]]>For the 7/13 vs. 5/11 example, imagine if you have completed 11 questions on a test and got 5 of them right, so you are at 5/11. Then you did two more questions and got them both right, so now you are at 7/13.

Well, your overall grade should have INCREASED because you just got the last two questions right, so 7/13 should be larger than 5/11.

]]>Thanks for the comment. I haven't seen my students returning to the flowchart as a reference. However, it does seem as though the creation of the flowchart has helped them process the different strategies and needs of different numbers.

Thank you also for the suggestion of Popplet. I will have to check that out. I have had them work on whiteboards because I didn't like any of the online options. So, maybe this one will change my mind.

Tyler

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