Two-Step Multiplication and Division Problems
It is expected that students entering the Integrated Math 1 class as freshmen would have prior experience evaluating two-step and multistep expressions, as well as solving two-step and possibly multistep equations. However, frequently these skills requiring inverse operations and “undoing” of problems are taught in a context-free manner.
From the nine types of common one-step multiplication and division problems, we can conceivably construct up to 81 different types of two step multiplication and division problems. It would be difficult, and inefficient, to provide students with examples and practice regarding each specific type of combination of problems. The goal of the next set of problems within this unit is to provide several two-step problems, of varying types, that will allow students to develop problem solving strategies.
Personalization of Two-Step Multiplication Problems
As students access the one-step equal groups and comparison multiplication problems, introducing additional considerations and constraints to the scenario will lead to two-step multiplication and division problems. In the one-step example problems discussed above, the only consideration was cost. However, in reality there are numerous factors that should be considered before a definitive decision is reached.
Once students are demonstrating success while working with one-step problems, adaptation of the problems to yield two-step problems could include the following scenario:
Quality Scenario
The uniforms being purchased can be reused for multiple years, depending on the quality of the material we select. Based on previous experiences, we know how long each brand of uniform should last. We can use this information to build new one-step problems and begin extending the scenario into two-step word problems.
For example: Consider the two options for uniform types. Option A: The Name Brand uniform costs $32 each. Each uniform can be used for 6 years. Option B: The Decent uniform costs $24 each. Each uniform can be used for 4 years. Option C: T-shirts with school logos for $8 each. Each uniform can only be used for 1 year. They must be repurchased annually.
One-Step Problem: If we assume that Name brand uniforms last 6 years, what is the cost per year of the $32 name brand uniforms?
In this example, the question allows students to reason about a group size - units unknown problem. Adaptation of this problem is possible by changing the number of years the uniforms are expected to last. By decreasing the number of years, students will see the cost per year increase. Increasing the number of years will bring the cost per year down. This problem can also be used to consider the other brands under consideration with their respective use periods, which all can be modified to form a larger set of problems.
Two-Step Problem: Suppose that the t-shirts last only 1 year and we need 450 uniforms. If we expect the same number of athletes for the next 4 years, how much money will we spend on t-shirts after 4 years?
This two-step problem is a combination of two group size – product unknown problems. The first computation could be either considering the cost of 450 uniforms for one year, which is the calculation 450 x $8 = $3600 and using the product, multiplying by the number of years 4, leading to the value $3600 x 4 = $14,400. Students are expected to have prior knowledge reasoning with inequalities. An intention of this problem is for students to reflect upon their solutions in the context of the scenario. Since our budget constraints are limited to $10,000, students would expect questions such as,
Question: Will we have enough money to buy enough t-shirts for 4 years at the current cost per t-shirt? Why or why not?
How much more money do we need for us to afford t-shirts for 4 years?
Two-Step Problem: If spend $10,000 on name brand uniforms and each uniform is used for 6 years, how many total students would have worn an updated uniform after 6 years?
This is a two-step multiplication problem where we examine how many uniforms we can afford for one year, but also consider how many students will benefit from the new uniforms. The first step is determining how many uniforms can be afforded, which is a group – size unknown problem, since we know the total amount we can spend, $10,000 and we know the cost of each uniform, $32 for name brand uniforms. This will yield the equation: $32 × number of uniforms = $10,000.
Taking the result of the first step, we then have a group – total unknown problem, since we found the number of uniforms, we then need to multiply by the number of years that each uniform will be used, 6 years, to find the total number of students that will be able to use the uniforms: "number of uniforms available × 6 years = total number of students
When working with these problems, the units of the various components of each problem may need adjustment. As with the most recent example above, in the first step, we were finding the number of uniforms, which would indicate the units, uniforms, which represents a discrete value. However in the second step, the number of uniforms found in step-one of the problem need to be considered with units representing the number of uniforms available per year. This enables the multiplication with the time of 6 years to yield a result that represents the quantity of students that will be outfitted with the same uniforms after 6 years.
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