In the second slide, I’m going to place a chart with ratios comparing inches to feet. The chart will be missing some quantities and I want the students to find those missing quantities and tell me how they could find any missing quantity. By asking this question, the students are making a generalization about this chart by saying that you can find any quantity by multiplying or dividing by 12. (SMP 1,2,7)
This chart will move us directly in to our first real world ratio discussion; conversions.
During this part of the instruction, it will be very important to keep in mind that students have not had much exposure to the metric system and may not be familiar with ratios that accompany both systems. I don't want students to give up just because they don't know the ratios. So I will be writing the conversions on the board for the students to use as a reference.
Using a conversion problem, I’m going to have the students set up a table to find the missing quantity. Since students are familiar with the ratio table, I’m going to let them work on their own. I will move around the room checking for understanding. It will be very important to have students verbalize what the ratio means when they find the missing quantity. This type of exposure will help students apply the concepts better. (SMP 6)
To reactivate prior knowledge, I’m going to have a discussion about the double number line. I want them to remember that it is important to create equal spacing and intervals and that not all numbers have to be on the number line, but they should be in the order as they would appear on the number line. Remind students that the double number line diagram compares two quantities that have a multiplicative ratio.
I’m going to use the same problems from the ratio table and represent them on the double line diagram. I want the students to see that the double line diagram and ratio table function in the same manner. The difference is that the line diagram needs to be in order, whereas, the ratio table can be used in different ways.
Tape diagrams (also known as fraction strips, bar models or length models) can be used when comparing similar part to part ratios. In this section, I’m going to demonstrate the type of problems that can be used to solve tape diagrams. As students work out each example, ask them what they notice about the types of problems we are solving? Are there any common words? Does anything sound the same? We want the students to realize that the tape diagram is a part:part comparison. For example, 3 parts blue and 2 parts red make purple. How many of each color can be used to make 20 pints of purple? Tape diagrams can also be used with conversions. For example, if 2 pints make 1 quart, you would draw 1 rectangle to represent the quart and 2 smaller rectangles (2 halves) to represent the pints. This pattern would continue to find as many quarts or pints as needed.
The students have tape diagrams in their tool box and you can refer them to their tool box if they need extra assistance. Today, we are looking at types of problems used with this diagram. Go through the tape diagram problems with the students. After a couple of examples, have them try some on their own.
To close this day’s lesson, I’m going to give them 3 different types of real world ratio problems. They will need to decide which visual representation they would use for each. In addition, they will need to say why they chose that particular visual.(SMP 4)
The tape diagram is the only one that would stand out. Tape diagrams are most useful when you are comparing part to part and each part makes up a certain quantity. If students choose a tape diagram for one of the other situations, have them justify their answer by showing you how it is done. Otherwise, accept any reasonable answer and justification.
Allow time for students to compare their strategies with other students. Be sure they know to justify their answer with mathematical language and reasoning (SMP 3)