In this lesson students are introduced to decimals and their equivalent percents and fractions. We are also exploring an idea that came from a student mistake in an earlier lesson (Fractions in a box). There is a high demand on prior knowledge and scaffolding and differentiation is a big part of this lesson. The most important part for the teacher is keeping calm and taking the knowledge gaps that arise in stride as if you fully expected them. To make the scaffolding appear to be an intended part of the lesson make sure you are ready with required materials like fraction circles. Working in groups is a really valuable and effective way to differentiate for students. Students feel a great deal of pride when they are able to help one another and because of this they do so with a positive attitude that is very supportive.
warm up box diagram and decimal equivalent. The warm up asks students to read 0.01 and 0.1 using their place value names and then write them as fractions and percents. This idea was introduced in the previous lesson (Percents in a box). The next two problems ask students to find 30% of 40 and 40% of 30. I am hoping students will notice they are the same value. If they don't I will ask them if they have noticed anything weird that makes them wonder and will circulate to ask groups what they are noticing. As students begin to notice similarities I share this with the class "Angel is noticing some similarities, I wonder..." This is actually something that came up in an earlier lesson (Fractions in a box) when a student made a mistake in another class. His classmates thought it was strange that he got the right answer, but did the problem backwards and some of them wondered if it was always true. Once students in this class notice the similarities I point out how it came up, because I want to emphasize that mistakes are good and can lead to discoveries!
I really like allowing student ideas and questions to guide instruction because it is really engaging for students and they are more likely to persevere when exploring their own ideas!
I expect there to be some need for scaffolding because of the common and wide spread gaps in prior knowledge. Its a good idea to be ready with mini intervention lessons and materials. warm up box diagram and decimal equivalent notes. Concrete materials like fraction circles are particularly helpful for ELL students.
After students notice that the two equations from the warm up are equal I ask how many of them think that's weird. I ask if anyone thinks it might not be weird and I expect a few to say that, because all the numbers are the same we are multiplying and dividing by all the same numbers maybe it will always be the same. I ask them to talk in their math family groups for just a minute and see what they think about that. Then I ask for a show of hands for those who think it might be a coincidence with just these numbers and who thinks it might not be a coincidence and it might work with all numbers. Then I ask how we could test it. "We could just try it!" is what I'm hoping to hear. I really want to encourage my students to "play" with their numbers. It gives them a sense of joy in math class and a sense of power over the symbols both of which are new to my students.
I give them two more to compare on white boards which they work out and find are also the same:
60% of 50 and 50% of 60
I expect them to predict that this might always be true because the numbers we are multiplying and dividing with are the same. This is the perfect lead in to explaining why cross products are equal for equivalent fractions. Most of my students have been taught cross products as an algorithm that they don't really understand and misuse in all the wrong situations.
For now I leave them with this last thought. I break down each of the percent equations as follows:
60% of 50: Can we represent 60 as 6x10 and % as 1/100 and 50 as 5x10? (yes)
Then we can rewrite the equation as 6x10x1/100x5x10
Doing the same for 50% of 60 we can write 5x10x1/100x6x10
Putting them side by side students can see they are the same numbers being multiplied. I ask them if they know which property says these are equal (commutative). This is a great way to get students to make connections between the maths.