Place Value, Fractions, and Algebra: Improving Content Learning through the Practice Standards

Background and Rationale

The Common Core standards attempt to find a balance between getting the ideas and getting the answers in mathematics. Through the course of this unit I intend to strike this balance with actions rather than creating a checklist of 'mechanical practices' that I am following. The unit will be designed to fill in the gaps of knowledge on fractions and proportions that my students have a serious problem with solving. To achieve this, there is an underlying problem that needs to be addressed. Having taught in my district for five years, I have noticed that many of the value systems that we (the adult professionals) live by simply have no value to our students. I have noticed that many people tell our students that they need to learn and acquire knowledge to succeed. It is only this year that I realized that my students live in a culture that does not value education and higher learning. Upon researching, I realized that what my students are experiencing has been reported world-wide as a problem of students from lower socioeconomic conditions. Dr. John Ogbu, in his report titled "Minority Education and Caste: The American System in Cross-Cultural Perspective" ¹, eloquently explains that students from impoverished backgrounds need to be taught in a way that allows them to relate to the value systems which they hold dear, rather than in a way that has been designed by a society which they perceive has never accepted them. Dr. Ogbu expresses the need for us to understand the lived experience of our learners ², and I wholeheartedly agree. This unit aims appeal to the concerns of my students, while developing basic ideas they need to learn to increase their quality of life.

My chemistry students are often lacking some of the basic understandings of arithmetic rules, which severely limits their abilities to comprehend the material in chemistry. From molar ratios of reactions to stoichiometry, it is critical for all of my students to understand fractions, ratios, and proportional relationships in order to succeed in chemistry. While conducting research for this unit, I realized that there was a deficit of information for chemistry teachers who, in the pursuit of filling in knowledge gaps for their students, wanted to teach mathematics fundamentals used in chemistry. The math of basic chemistry (which is fundamental math) needs be taught as an engaging and exciting unit with a purpose to which students can relate. From this unit, I hope to increase my students' quality of life, as well as their belief that learning chemistry can directly relate to their life and their family; I hope to help my students gain the mathematical foundations to protect their house.

Here is the strategy of the unit. My students are often the caretakers of their elders and younger siblings, and as such, are often responsible for determining how much and when to give over the counter (OTC) medications to their family members. This situation presents an opportunity to create an innovative unit that will be personalized for each student in my class. The overarching goal of this unit will be to have each of my students create an OTC dosage chart customized for each family member. Students will record and measure each family members' age and weight. Students will then prove their understanding of the mathematical concepts taught in this unit by:

1. Calculating the appropriate OTC drug dosages for each family member,
2. Justify their methods used in their calculations to their peers, and
3. Review and verify their peers' mathematical calculations.

The resulting information will then, after also having been checked by the teacher, be placed on a poster in an easy to read format so that it can be used as a reference guide. Students will be able to take the poster home to ensure that their family uses the appropriate OTC medications, along with the correct dosages.

The Food and Drug Administration (FDA) regulates the standards of OTC drugs. Drug companies determine the maximum recommended starting dosage (MRSD) by testing the efficacy of active ingredients ³. Active Ingredients are drug compounds whose chemical properties have a direct correlation to the "diagnosis, cure, mitigation, treatment, or prevention of…disease…or to affect the structure or function of the body…" ⁴. The FDA's Center for Drug Evaluation and Research (CDER) approves a human equivalent dose (HED) in the form of the Amt. of Active Ingredient to be delivered in a range of time, also written as (Amount (Amt) of Active Ingredient/time). Or "amt. of active ingredient every time hours" ⁵

The HED for adults is calculated based on a human whose age is greater than 12, and who has a normalized mass of 60.0 kg; all adults are assumed to require the same amount of active ingredient over time. For children, the actual mass range of the child is considered, as there are significant physiological differences; thus, (Amount (Amt) of Active Ingredient/time) needs to consider the child's actual weight ⁶. The FDA requires that all Active Ingredients of OTC drugs be clearly identified in the label of the drug. The strength of a drug is written as (Amount (Amt) of Active Ingredient)/(Dosage Delivery Type); common dosage delivery types include capsules, tablets, pills, or in the case of liquids, milliliter (mL) of liquid. The adult recommended dosage of drug indicates how much (n) of the dosage delivery type should be delivered, based on a multiple of the ratio of Dosage Delivery Type over a certain time period, where Dosage = n x (Dosage Delivery Type)/(time period) ⁷. For children, dosage labels for OTC drugs are given in the format of age and weight ranges. Experts agree that if possible, the weight range that the child's actual weight falls in should be used first.

In order for a student to calculate the appropriate dosage, they must understand OTC drug dosage terms, concepts, and units as well as the math of stoichiometric conversions and proportional relationships. To do this, they must understand some key rules of arithmetic that this unit will focus on developing. To prepare for multiplying and dividing fractions, this unit will strive to have students understand repeated subdivision, reconstitution, and renaming. To understand repeated subdivision, reconstitution, and renaming, students must understand the arithmetic of fractions with a fixed denominator. This will be developed using the Common Core approach of first developing the idea of a unit fraction, and thinking of a general fraction as a multiple of a unit fraction. The key to all this work with fractions is the recognition that, in the real world, numbers come with units attached, and careful attention must be paid to the unit to which a number refers. This principle is also basic to stoichiometr ⁸.

My students lack a deep understanding of fundamental math concepts. They may understand the mechanics of basic math, but they are lacking the deeper understanding of the arithmetic ideas to apply math in practical contexts. Linear models and area models will be used whenever possible to represent each of the concepts in the unit. All images should be created to scale as much as possible when creating classroom sets. While the FDA has more complex systems for calculating accurate dosages to use with each OTC drug, this unit will focus on teaching students how to read, interpret, and mathematically manipulate information from OTC drug labels. The end product will not only be applicable for the students, but will be a great starting point for many of the essential stoichiometric conversions that will be necessary in General Chemistry.