This is a "zoom in on the math" lesson. Students have a pretty good grasp of the concept of scaling ratios up and down. Here we just look at how to find the scale factor from information given in a ratio and in the table. Some students are still thinking additively and it is hard for them to make the leap to multiplicative thinking. I expect them to understand that the relationships remain, for example that for every 1 black tile there are 3 white tiles. However, some students may still be more comfortable scaling up by adding one more black tile and 3 more white tiles each time rather than using a scale factor and multiplying both black and white tiles by the same factor. This context is really nice for helping them make this transition, because we can talk about how many times the original ratio or design is being repeated. Understanding that if the design is repeated a certain number of times the black, white, and total numbers of tiles will be repeated that same number of times helps them develop the multiplicative thinking needed to make sense of scale factors.
Students revisit one version of the table warm up interpreting table.docx they made in yesterday's lesson (Let's get organized), which shows the numbers of white, black, and total tiles needed to cover 1, 2, and 3 square feet using a repeating pattern/design. Students are asked questions that can be answered using the information in the table just to remind them how to navigate the data in the table. warm up interpreting table handout.docx
The first two questions ask students to write ratios. I expect several to write three ratios rather than just the one in simplest form. Although they are able to simplify ratios and identify equivalent ratios this is the first time they are writing ratios from a data table. Some students will just write three ratios, some will write just one, and some will ask how many to write? I turn the conversation back over to the groups. If I notice that some members of the group have written one and other three I may just point it out and ask them to discuss. If a student asks me I may ask the group or pose the question to the whole class. I expect several to bring up simplifying as a way to prove that only one is needed. If students share all three ratios with the class I will ask the class if they are different and then have them discuss and explain why they are the same before asking which one we should use.
The three videos show students asking how many ratios they need to write, explaining why they only need one, and why they changed their mind.
This exploration helps students figure out the scale factor they need to multiply by in order to calculate the missing information in the table. finding scale factor in a table.docx In addition to focusing on the mathematics we also continue to make sense of how information is given in a table and how it can be used as a tool (mp5). The lesson leads students through a series of questions that help them relate the scale factor to the number of times the pattern is being repeated which helps them move from additive to multiplicative thinking. The video (multiple methods of scaling) shows the difference between using additive and multiplicative reasoning.
If there is time left I want students to have a little practice using the table to find the scale factor. Students work together on individual white boards and raise their white boards on the count of three. That way I can quickly scan who needs additional feedback and no one can opt out.
I will show them the table from their warm up and write a 100 in the "black tiles" row" and ask how many white and total tiles will be needed for 100 black tiles. The students who figure it out quickly know where to look in the table for the information they need and are using multiplicative thinking. The students who are taking longer are probably trying to "add up" and the students who are completely lost probably don't know where to look for the information they need. I remind them that if they figured it out before their math family that they should show their group members where in the table they can find the information they need to figure out the scale factor.
When I go over each practice problem (similar to the first) I first ask what the scale factor is, then how they know? This prompts them to locate the information they needed "well, I know that we started with 1 black tile and if we use 100 black tiles the pattern is repeated 100 times, so I multiplied the white and total tiles by 100". As the student explains I model on the board.