This is the first lesson in the unit so we will begin by talking about proportional relationships. I will ask some questions to assess how much the students know and remember. I will show a simple pattern on the SmartBoard and ask students to describe what they see. This will be presented as a turn-and-talk. I will accept any reasonable and serious answer, but will lead students towards answers that help describe the ratios.
I may choose to show 12 green color tiles to 8 blue color tiles. I may arrange them visually in rows to show the value of the ratio in simplest form: GGGBB GGGBB GGGBB GGGBB. This will allow for students to describe the ratio in a variety of ways. Some may describe part to whole or whole to part ratios. Some may given the ratio in simplest form. I will record all of these ideas.
Next I will ask them to discuss where do we use ratios in the real world. I will try to get students to see that ratios(rates) are used regularly in daily life - hourly wages, sharing, describing speed, grocery shopping, etc. Many of these ideas will probably come out without much prompting.
Following the discussion, I will introduce a ratio problem. I will guide students through solving the problem using a double number line. In the previous grade, my students learned how to use bar/tape models but the double number line will be new. It will be important for students to know that the value of each interval should be the same for the respective number line.
As I am filling in values on the number lines, I will emphasize the idea of "for each" or "for every". Part iv is designed to help students answer the essential question using a scale factor from ratio to ratio. The double number line should help students see this shortcut (MP8). I will draw arrows from the value of the ratio to various equivalent values showing the multiple being used.
The guided problem solving serves as a check for understanding to see if students understand how to solve and model problems using a double number line.
The first problem has the same structure as the example problem from the introduction. Parts iii and iv are designed to get students to move beyond the values of the number line and see how a scale factor can be used. Part iv asks students to explain their method or reasoning for part iii (MP3).
The second problem brings in an extra layer of difficulty. This time the ratio is not given in simplest form. Students will need to first find this value before numbering the number line. I expect some students will want to place the values 9 and 12 directly to the right of 0 on the number line. If so I will ask them about a common factor of 9 and 12. Once they identify 3, I will show them how we can place the values 9 and 12 three intervals to the right of 0 on the number lines.
Students will now work on 4 problems. The level of difficulty increases for each problem. The first two problems are similar to the first two guided practice problems. The third problem has about the same difficulty as the 1st problem, but this time I ask that students construct the double number line themselves. They may need to be reminded to draw the lines parallel and to make sure the intervals of each number line are aligned.
The last problem gives a whole to part ratio not in simplest form. This will be the most challenging. Not only will students have to simplify the ratio, they will have to ask questions to determine the other parts of the ratios.
Before beginning the exit ticket I will help students to summarize what we have learned: 1) Proportional relationships can be modeled using a double number line; 2) A scale factor can be used to find other equivalent values on and beyond what is on the number line.
Students will then take the exit ticket. I will likely make this exit ticket worth 5 points. These points are not for grades; they are just a way for the students and me to see how well we are doing with the lesson. Parts A-C will be worth 1 point and Part D will be worth 2 points - 1 for a correct answer and 1 for a valid explanation.
The criteria for success will be to earn at least 4 out of 5 points.