Scaling up ratios

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Objective

SWBAT understand the concept of scaling up proportional relationships and begin to use a ratio table.

Big Idea

In proportional relationships a constant ratio is maintained.

Intro & Rationale

In this lesson students will be expressing their understanding of scaling up ratios. They have not been introduced to any terminology at this point, but they have had quite a bit of experience with the act of scaling up and down in the context of creating tile floor designs in earlier lessons (Designing the floor plan, Breaking down the design, Comparing ratios, Ratio soup assessment day). In last night's homework they described a given growth pattern and used the concept of scaling to continue the growth. They will be looking at a ratio table using the data from that assignment to see how that pattern shows up in the table and focusing in on the math involved.

My students do better with algorithms when they have a physical model or context to connect it to. Working with visual growth patterns helps give students a stronger contextual foundation for making sense of why we multiply by a scale factor because they can actually see it and reproduce it. The role of the teacher in this lesson is to listen, listen, listen! Listen for student explanations that reflect the idea of scaling and scale factor and highlight these for the class.

Warm up

10 minutes

Ask students to find the ratio of black tiles in each of the figures from last night's homework How does the pattern grow? When I hear someone suggest that the ratio is the same for all the figures I will bring that up with the class and ask who agrees and disagrees. Then I ask them to explain their reasoning and make an argument for or against. If all students are not finally convinced they are the same I will bring back the sentence frame "for every ___black tile(s) there are/is ___ white tile(s)" so they can see that the simplified ratio will be true for all.

 Also ask them to share with their math family groups how they answered the first question and described how the pattern was growing. As they do I listen for student descriptions that might help others understand the concept of scaling ratios: 

  • "the pattern is growing by just repeating the same pattern piece over and over"
  • "the same ratio of black to white tiles is being repeated over and over"
  • "two black and one white tile are being added each time"
  • "You just keep adding them in a ratio of 1 to 2"
  • "a new row of 3 tiles are being added each time, one white and 2 black"

As I hear good descriptions I ask these students to be sure and share this idea with the class when we go over it. 

 

Exploration

30 minutes

This lesson is a close examination of last night's homework. It is more of a direct teaching format to show them how to use a ratio table, but I want use as much of the student's insight as possible.

I ask the students to explain to each other how they answered each of the questions (one at a time) and listen for different methods that will help them understand how to use a table. For the last three questions I tell them to figure out how many of each color tile will there be and again listen for student insight that will help them understand how the ratio table works. homework How does the pattern grow notes.docx

When I introduce using a table (as seen in the videos) I want to refer to as many of their methods as possible. In the video I only mention one method (scaling up the ratio by the scale factor), but if they used other methods like doubling the number of white tiles I might say "if there are 30 whites there will be twice as many blacks" as well. Learning multiple methods makes students more mathematically flexible. The more patterns they recognize in the table the more sense it will make to them.

Homework

4 minutes

homework How does the pattern grow 2.docx Tonight's homework is very similar to last night's, but I ask them to try using a table to help them. I suggest they start the homework early so they can get help setting up the table. I keep telling them that if they get stuck to stop and look for patterns in the table that might help them. I also remind them that they already have strategies for figuring out the answers without the table, so they can fill in the numbers that way too.