Content Objectives
Teachers will need a firm understanding of fractions, ratios, and proportions and the ways these concepts relate to cooking before teaching this unit. This section provides information and explanations to fortify existing knowledge and hopefully expand how they think about these concepts.
What are fractions?
A common explanation of fractions is a mathematical representation of parts of a whole or parts of a set. But in truth the typical can represent so much more. It can indeed indicate parts of a whole (slices of pizza) or parts of a set (how many red marbles in the bag). But it can also illustrate points on a number line, a ratio, a division problem, or an operator. In the special education classroom, students’ experiences with fractions will range and even if they have been exposed to all of these concepts, they may not have a firm grasp on any of them.
Let’s look at the basic structure of a fraction, . The line in the middle is the fraction bar. The number on top, represented here by a, is the numerator and generally is thought of as the ‘part’ being considered. The number on the bottom, represented here by the letter b, is the denominator and generally is the of as the number of equal parts or number of elements in the set. For example, of a pizza would indicate that the pizza was cut into four equal slices and we are taking or considering three of them. This illustration is an example of an area model and can be thought of when visualizing that pizza cut into equal slices. A set model is used when examining a situation like of the students are wearing green shirts today. The number 5 in the denominator position tells that there are five individuals (nothing is “cut” into 5 pieces) and the 2 in the numerator tells that two of them are wearing green.
Fractions can also be illustrated on a number line, but the thinking needs to be adjusted just a little. Fractions on a number line describe their position on the line and its distance from zero. When representing a number greater than zero, but less than one, it will be located between zero and one. That interval will be divided into equal sections, as indicated by the denominator. The numerator then tells how many of those section to count from zero. The number line can also be a valuable tool when demonstrating the relationship between fractions and decimals. The number 0.4 is read as four tenth and can be expressed as . On a number line, the interval between zero and one is partitioned into 10 equal parts and then the student travels four of those parts from zero.
There is a common misconception among many students when comparing fractions. The assumption is often made that a larger denominator denotes a larger fraction, such as saying is greater than because 8 is greater than 4. This concept can be easily demonstrated and the misunderstanding corrected by showing how the same size “whole” divided into more equal pieces results in all the pieces being smaller. In cooking, fractions are used for measuring ingredients, dividing servings, scaling recipes, and comparing ingredient amounts.
Quantities larger than one can be represented using improper fractions or mixed numbers. Improper fractions are fractions in which the value of the numerator is greater than the value of the denominator. A mixed number is the combination of a mixed number and a proper fraction. Many students will not realize that improper fractions and mixed numbers may represent the same values. This can be illustrated through the cooking when measuring and combining ingredients.
Improper fractions and mixed numbers can be represented with number lines, area models, fractions strips, measuring cups, recipe scaling, and repeated addition of unit fractions. A common misconception to look out for may surface when scaling recipes. They may see as impossible rather than understanding it as 1 and . Improper fractions may appear when recipes are multiplied or divided when scaling them for various groups.
Professional bakers tend to work with the weight of an ingredient like flour and discuss it as percentages rather than measuring cups which are often fractional values. Using weight makes them easier to compare and scale.[4] Many bakers will develop formulas by determining the proper ratios of ingredients before converting them to weights. This is a great discussion to have with students in algebraic reasoning to reinforce how we can write our own formulas to solve daily problems.
What is a ratio?
Ratios often get confused with fractions because they are also often represented as . But when used in this context, instead of representing parts of a whole, it is representing the comparison of two quantities. For example, 1 cup water and 2 cups flour is represented by the ratio 1:2 or also written as . In this example, the ‘whole’ actually has 3 parts (1 water + 2 flour). Cooking provides an example way to emphasize this distinction as the teacher can show the three individual cups, one with water and two with flour. There are many ratios in cooking for things like dough or sauces. Other ingredients may be added to influence taste or texture, but the general ratio usually stays the same.[5]
Changing the ratio of ingredients will change the final outcome when cooking. This is easily observable in the role of baking soda or baking powder. Both of these ingredients produce carbon dioxide, causing dough to rise. Too little and the finished product will be dense and flat. Too much and the finished product will have a bitter taste and may collapse.
The ratio of flour to water is another situation students may be familiar with, even if they don’t know it. Cakes, cookies, and pancakes all have ratios of flour and water but are very different foods because of not only additional ingredients, but the ratio of flour to water within them. Flour contains proteins that when combined with water, create gluten. Changing the ratio of flour to water causes changes to elasticity, viscosity, chewiness, and structure of the dough.
The water and oil in salad dressings will eventually separate due to their chemical properties, regardless of their ratio to each other. Water is hydrophilic (water loving) while oil is hydrophobic (water fearing). This effect can be altered by adding an emulsifier in a specified ratio to the other ingredients, such as an egg yolk which contains lecithin that is amphiphilic (having both water loving and water fearing properties).
What is a proportion?
Proportions describe the equality of equivalent ratios or fractions. They represent that while the overall quantities may change, their relationships to each other (or ratio) will not change. In cooking we often use proportions to determine a missing value when there is an unknown. For example, if the recipe says that it serves 6 but we have 12 people coming for dinner, every ingredient must be multiplied by 2. () They can also be used to find a missing value, such as could represent “If a recipe has a ratio of 1 part water to 2 parts flour, how much water would need to be added to four cups of flour?” Some students will remember that “butterfly method” they learned in middle school, but often will not understand the relationship. Students are taught to circle the diagonal pairs, creating a “butterfly” image. Then to multiply them, creating a one variable equation. So would become .
Proportions are not just about making recipes bigger or smaller. They are about keeping the relationship between the quantities. The amounts change, but their relationships do not. Much like the independent and dependent variables of functions we talk about in algebra and algebraic reasoning, the ratios of the ingredients remain the same.
Properly scaling a recipe preserves the chemical interactions between the ingredients. If students make mistakes, the interactions and the resulting food will change. This is a direct illustration of the chemistry of the ingredients and their relationships with each other.
Fractions, ratios, and proportions provide the mathematical foundation for understanding the relationships between ingredients in a recipe, while chemistry explains why those relationships matter. Throughout this unit, students explore how changing ingredient ratios affects physical and chemical properties such as gluten development, gas production, emulsification, and texture. Rather than learning these as abstract mathematical concepts, students apply them to authentic cooking situations where they can predict outcomes, test their ideas, and observe the results firsthand. By connecting mathematics and chemistry through a familiar, real-world context, students gain a deeper conceptual understanding of proportional reasoning while recognizing that mathematics is a practical tool for explaining and predicting the world around them.
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