I review student homework prior to teaching a lesson if it contains concepts necessary for the current day's lessons. Upon review of student work, I saw few mistakes so I abbreviated the amount of time I might have usually dedicated to the review. I expected it would take more than 8 minutes. The students had to identify which part of an hour was shaded on a sheet on which clocks were divided into 1, 5, 10, 15, 20 and 30 minute increments. This was based on their work in yesterday's lesson "Fractional Parts of an Hour: An Investigation".
This was their homework page: HW Time as Fractions
We open with an Entrance Ticket prompt:
Write down 3 reasons that it's important to tell time, based on your own life and discussions we have had about telling time over the past week.
After students have had time to think, and write, I show the students the fractional pieces that they will be working today and they identify the number of minutes represented by each piece.
I have pieces cut out to represent 1 minute (1/60), 5 minutes (1/12), 10 minutes (1/6), 15 minutes (1/4), 20 minutes (1/3) and 30 minutes (/2).
I model how different increments of time/fractions can be added together to make one whole hour (MP4 - Model with mathematics).
I emphasize that for today, they are to look for pieces that fit together to make an hour (by building on the hour template) and record the equation (MP7 - Look for and make use of structure). I point out that we are not actually adding the numerators and denominators, we are just representing all the pieces that when combined make an hour. To add the pieces we would, of course, need a common denominator.
For today, the purpose is that they see that there are many combinations that can make 1 whole hour. They understand time increments because of our prior work so they don't reject the idea that 1/2 + 1/6 + 1/6 + 1/6 can equal one. Given that equation on paper, my experience has been that students do not think that can make one whole because, w/out knowledge of common denominators, they come up with all sorts of predictable errors, such as thinking the answer to the problem above would be 4/20.
I model several problems and have students write the equations because this is complex enough that I want to briefly take some of the cognitive load off by removing the demand of immediate verbal processing and allowing for think time, which is especially important for my ELL students who have a 3rd tier of complexity - learning new words in English that they probably do not have in their home language.
I pass out ziploc bags with pre-cut paper pieces in different colors and representative sizes for 1 min. (1/60), 5 min (1/20), 10 min. (1/6), 15 min (1/4), 20 min (1/3) and 30 min (1/2) .
Students work independently or collaboratively to come up with as many different ways to design an hour as they can using the Time Fraction Pieces (B&W) I copied them on colored paper. (If you have a color copier, here they are in color: Time Fraction Pieces (Color)). The students record the equation for each model they build.
Circulate and ask students to explain their thinking. Some examples of questions for students:
How many minutes are there in (1/60, 1/30, 1/20, 1/15, 1/10, 1/5) of an hour? What made you decide to use _______?
What fraction of an hour is representative of (1 min., 5 min., 10 min., 20 min., 30 min., 60 min.)?
How is this model different from __________? (a second model they have made, a neighbor's model)
Explain one of your equations for the parts that equal an hour.
What are two/three different ways to represent (1 hour, 30 minutes, 20 minutes, 15 minutes, 10 minutes, 10 minutes?)
Which fractional part of an hour is most commonly used? (Opinion) Why? Which one do you use the most? Do we use most at school? Are there any other ways in which it might be helpful to partition an hour? In what circumstances?
Here is an example of student work: Fractional Parts of Hour Student Work
I ask students to share a favorite example from their work. They read the hour off as fractions or as pieces of time. Expressing out loud the minutes that add up to an hour, or the fraction pieces that add up to one whole hour, give the students additional practice and provide support for auditory learners.
I am constantly asking them, "Why," and the boy who answers in this video explains his reasoning for selecting the example of 10 minute increments without prompting.