I will begin with the essential question. This is the same essential question as the previous lesson - How can we find the unit rate (constant of proportionality) for rates involving fractions . We saw that unit rates can sometimes be found using multiplication - especially when the 2nd term is a fraction. We saw this yesterday when working with unit fractions. This helped us to see that if the 2nd term of a rate is a unit fraction, when we scale the rate by the denominator of the 2nd term we find the unit rate (ie 1/4 * 4 = 4/4 or 1; 1/5 * 5 equals 5/5 or 1).
To make this point, and to tie the concept to a variety of fractions other than unit fractions, I will pull up the previous exit ticket. In both of these problems we end up scaling the rates up using the value of the denominator as these were unit fractions. I will then define the word reciprocal and have students determine if we used the reciprocal to solve the problems from the exit ticket. The answer should be clearly YES. The resource has a few examples of a reciprocal, but it may be helpful to provide a few more examples for students to quickly solve so they will see that the product of a number and its reciprocal equals 1. This is the key concept for today's lesson: we can scale a rate by the reciprocal of its 2nd term to find the unit rate. This makes sense as we know with rates we can divide and that dividing is equivalent to multiplying by the reciprocal.
The rest of the lesson moves in a direct instruction format. I will present example 1; students will copy what I copy. I will write the rates in the form of a complex fraction. Students will then do a similar example on their own. This is a brief check for understanding to see if students can apply what was learned in the example. Students will need to use the reciprocal of the denominator as the scale factor. Some students may just try to use the denominator without using it's reciprocal.
The second example follows the same format, yet students will have to convert the mixed fraction into an improper fraction first. On the check for understanding, be on the look out for students who ignore the whole number of the mixed number and only "flip" the fraction.
This section has four problems. They are of a very similar structure to the previous problems from the examples. If a class was very successful with the check for understanding questions, then I will let them work through these problems while I check in on students who I know were having difficulty. If a large portion of the class struggled with the check for understanding questions then we will go through these problems one at a time. I will set a time of about one minute per problem and walk the room while students are working. If I see that there are still a lot of errors, I will take the problem to the board. If a lot of students have the correct answer, I will have students move on to the next problem.
This may seem obvious, but I always find it helpful to have my solutions in hand as I check students work. While I can generally solve most of the problems quickly in my head, it just allows me to be more present to keeping an eye on the entire class instead of working out problems in my head.
Now students work on their own to solve about 6 problems. The first 3 problems shouldn't cause too many problems by this point of the lesson. On the 4th problem, I expect some students to find miles per minute instead of miles per hour. They will need to be reminded to determine what fraction of an hour 25 minutes represents. I will encourage them to simplify the fraction to make the computation easier.
Problem 5 and 6 ask students to apply the idea of a unit rate to solving a problem - #5 is a unit price problems, #6 is a miles per gallon problem. On problem 5, students may be tempted to use decimals for the money, even though the money is presented in fraction format (ie ten and one-half dollars). On #6, decimal values are used for gallons of gas, but these can be easily written as fractions in lowest terms. This is a simple example of students having to use MP1.
Before we begin the exit ticket I will ask: How did we find the unit rate when the rate involved fractions? Students will briefly discuss and should come up with an answer that states: we scale the ratio by the reciprocal of the 2nd term.
Students then take a 3 problem exit ticket. Each items mirrors the three basic types of problems that are repeated throughout the lesson.
Students should be able to answer all 3 problems to show they have mastered the lesson. The skill required is the same for all of the problems.