This warm up Warm up scaling up a ratio.docx gives the loss to win ratio (3:5) for the Dixon Rams and asks if this means that the Rams lost exactly 3 games and won exactly 5. They are asked how many games the Rams might have played. The athletes in the room will hopefully take the lead on this and point out that this is just the simplified ratio, the Rams could have played a total of 8 games, but that they could have lost 6 games, won 10 and played 16. In the unlikely case that no one comes up with this idea I will use our sentence frame again ("for every ___ games lost there were ___ games won") and ask if there are any other number for which this would still be true? Did they really only play 8 games?
Once they realize the possibilities I suggest they use a table to help them figure out the possible numbers of games. Warm up scaling up a ratio with a table.docx This gives them additional framework for understanding the reading they are about to do about "scaling up ratios".
Scaling up a ratio reading.docx This reading is about "scaling up" ratios. They have already had some exposure, but I want them to have something in as kid friendly language as possible that they can refer to without having to ask me. When they work together on the table in the reading their group is much more self reliant with the text to refer back to. Instead of asking me to explain something to them I hear them showing each other. "No, look here where it says...", "If you read this part here...", "You can see in the diagram right here..." I will circulate and point out to the class when someone is sharing where they found something in the text.
When a group has agreed on an answer to one of the questions in the reading (table) I may ask them to come up and show the class how they figured it out. If they don't explain how the table helped I will ask someone to come up and show us how this shows up in the table. If they say "The pattern is repeated 2 times, so I just doubled the black and white tiles" I might say "Jessica says the black and white tiles are doubling, can someone show me what that looks like in the table?" I may ask students to explain in words ("Samantha has a good explanation") or show us just the math ("Jose did a great job of showing what's happening with the math")
Whenever I have created something in writing or a powerpoint I like to ask the kids what part of it was the most helpful, what was confusing about it, and what might have made it more helpful.
During the wrap up I want to go over the questions from the text first and I display the table from the text under the document camera so we can do this together.
I make sure to model the questions using the table and the arrows to show that the black tiles are being multiplied by a factor of five. Then draw the same arrow for the white and total tiles as I ask the two parts of the question. It may also be helpful to refer back to the sentence frame from the "which is blackest" lesson (for every 1 black tile there are 3 white tiles). Refering to and adding on to the repeating tile design in the text and then counting the number of times the pattern has been repeated is also helpful. Visuals are always helpful for ELL students, but also for everyone at this introductory level.
After going through the questions in the reading I want to ask them to discuss what is staying constant and what is changing as the pattern grows. I purposely use the new vocabulary term so they start to become familiar with its use. They may say that the ratio is staying constant since for every 1 black tile there are 3 white tile. They may suggest that we are adding the same number of black tiles and the same number of white tiles each time. This is evidence of additive thinking and I want to model this pattern in the table or in the design. They may say the scale factor is staying constant and that if black tiles are being repeated a certain number of times then all the rest will be repeated the same number of times. This is an example of multiplicative thinking. Eventually I want all students to develop multiplicative thinking, but until then both types need to be modeled so that those who are still developing additive thinking skills can see the connection between the two.