See the Do Now video in my Strategy Folder for more details on how I begin class. Often, I create do nows that have problems that connect to the task that students will be working on that day. For this do now, I picked a multiple choice problem from my released state test that is important: graphing polygons on the coordinate grid and finding the area of them. To check the do now I plot the points on the grid and connect them to form a quadrilateral. I ask a student o explain how they would find the area. I am looking for a student to explain that you can split the quadrilateral into two triangles and use the formula (Area of a triangle = ½ bh) to find the area of each triangle and then add them together.
After the Do Now, I have a student read the objectives for the day. I ask students why it is important to be able to compare ratios and rates. Comparison shopping and racing are just two of many ways these skills can be found in the real world.
On page 2 we fill in the notes together to activate students’ prior knowledge. For the example, “She made $49 for every two lawns she landscaped” I ask students to change it into a unit rate.
On page 3 we work through example A and B together. For example A a common mistake is that students see a pattern occurring for the first 4 values in the table and then they extend that pattern to the next value in the table, even though the number of students is 18, not 15. I emphasize that the information in the table is proportional, therefore each new number of students and pizzas must be equivalent to the ratio of 3 students: 2 pizzas. I am looking for students to use the ratio 3 students: 2 pizzas to figure out #2 as well.
For Example B we talk about why you weigh more on Jupiter. Students are interested to talk about the size of Jupiter and how gravity on Earth compares to gravity on Jupiter. I have students work on #1 independently for a couple minutes. Some students may find a unit rate (1 pound on earth/ 2.6 pounds on Jupiter) to solve the problem. Other students may strategically use the data in the table to find the answer (80 pounds on Earth/208 pounds on Jupiter so then 40 pounds on Earth/104 pounds on Jupiter so 208 + 104 = 312 pounds on Jupiter). I ask students to share their ideas. I share the strategies if they don’t mention them.
I have students work in partners for pages 4-8. I circulate checking students work and observing the strategies that students use to solve problems. Students are engaging with MP1: Make sense of problems and persevere in solving them, MP2: Reason abstractly and quantitatively, and MP5: Use appropriate tools strategically. Students must break down the problems to understand what the information and units mean. Students choose what tools to compare the rates and ratios. Some students use tables, some students find unit rates, and other students find equivalent ratios/rates. Once they have an answer, they must put that answer back into the context of the problem in order to understand what it actually means.
For problems #1-3 students most students will create rates with the same number of laps or same amount of time. Other students will calculate and compare unit rates. A common mistake for #1-3 is that students think that the student who has the higher time for a particular distance is faster. If many partner groups are making this mistake, I stop the class and have a student volunteer to help me act out an example. I go 6 feet in one second, while the student goes 4 feet in one second – who’s faster? I am faster because I go farther in the same amount of time. Next example: it takes me 4 seconds to go 2 meters, the student takes 2 seconds to go 2 meters – who’s faster? The student is faster because it took him/her less time to go the same distance.
I ask students to share strategies for #5. Some students may find and compare unit prices. Other students may find the price for a common number of apps (like 60 apps) to see which store is cheaper. I put the student’s work under the document camera and have him/her explain their mathematical thinking. See the Closure video in my Strategy folder for more details.
Rather than give a ticket to go, I collect student work to analyze how they are doing with comparing rates and ratios.