This introduction is intended to provide students with more various models that could be used to solve proportional reasoning problems. The previous lessons have focused mostly on double number lines. This lesson will include double number lines, bar models, and even a concrete model using pattern blocks.
We will model the example problem in 4 ways moving from concrete to pictorial to abstract: 1) using pattern blocks; 2) using double number lines; 3) use bar models; 4) using multiplication.
I will distribute the following pattern blocks to pairs of students:
2 yellow (whole)
4 red (halves)
6 blue (thirds)
12 green (sixths)
We will read through the problem and then students will have to decide which pattern block could be used for which ingredient. The yellow one may confuse students because it does not represent an ingredient, it serves as a reference for 1 cup of the batch.
I will say that a small uses 1 cup of carrot juice. Model that with your blocks. How does this compare to the given ratio of ingredients? Answer: twice as much. So how much of the other ingredients should we use? Answer: twice as much. I will then ask students to model it and fill in the values. We will proceed in a similar manner for the other values.
Next we will use a double or perhaps triple number line.
Then we will use a bar model. The model can be fairly simple. For example we could represent beet juice by having one-third of a bar shaded. We see that a medium uses 1 cup of beets or 3/3 which means a medium uses 3 times the given amount. We can than apply this modeling to the other ingredients.
Finally, we can find each new amount simply by multiplying.
Students will now be mostly on their own to solve this next problem. They should feel free to use any of the methods given, though the concrete will be a bit tricky for this problem
As students work I will be looking for which models they choose to employ and how accurately they are using the models.
I have noticed that some students have difficulty interpreting the table. They may need the teacher to point out that each batch or set of ingredients reads from top to bottom in a column. In other words, they may find it helpful to focus on 1 column at a time.
Students may need some prompting so I may ask questions like:
How much salt is used in batch 1? How does this compare to what was given in the recipe ratio? Show me.
For students who find this task exceptionally easy, I will given them an additional problem. How much of each ingredient will be used for a 24 cup batch? What about for a one-half cup batch?
As we review, I will look to display the various models used to solve the problems.
This lesson assessment not only assesses how well students can scale a recipe but also apply what they have learned over the previous days. I think the assessment is a slightly easier problem that the one from the problem solving section.
There are 9 blanks for students to fill in. Therefore 7 out of 9 will be considered a successful assessment. Students are not required to simplify answers.