- example, students in the elementary grades are often taught to add a zero to the end of a whole number when multiplying by 10 . However, this "rule" does not hold when multiplying decimals (e.ge, \( 0.65 \times 10=6.5 \) rather than 0.650; Karp, Bush, \& Dougherty, 2015). Misconceptions related to ratios and proportional reasoning focus on prerequisite knowledge about multiplication and fractions. So, what are some common misconceptions, such as Cara's misconception, that interfere with the ability to understand ratios and proportional reasoning and to generalize those understandings to algebraic thinking? example, a recipe calls for 2 cups of flour for every 1 cup of sugar. How many cups of flour are needed if a recipe is increased to 3 cups of sugar? Students can think 1 cup of sugar for 2 cups of flour, 2 cups of sugar for 4 cup of flour, and 3 cups of sugar for 6 cups of flour. Therefore, the amount of flour now needed is 6 cups, which maintain the same relationship as the original one given (1:2). Students who think ratios are additive would have mistakenly thought that 3 cups of sugar is an increase of 2 cups from the original 1 cup. They would then add to the original 2 cups of flour 2 more cups to get 4 cups needed flour (see illustration. Figure 1). For ratios to be

To address the misconceptions related to ratios and proportional reasoning, let's break down the problem step by step, focusing on the example provided about the recipe. ### Known Conditions: 1. The original ratio of flour to sugar in the recipe is 2 cups of flour for every 1 cup of sugar. 2. The recipe is increased to 3 cups of sugar. 3. We need to determine how many cups of flour are needed to maintain the same ratio. ### Step-by-Step Solution: 1. **Understanding the Original Ratio:** - The original ratio of flour to sugar is \( \frac{2 \text{ cups of flour}}{1 \text{ cup of sugar}} \). - This can also be expressed as \( 2:1 \). 2. **Scaling the Ratio:** - To find out how many cups of flour are needed for 3 cups of sugar while maintaining the same ratio, we can set up a proportion. - Let \( x \) be the number of cups of flour needed for 3 cups of sugar. - The proportion can be set up as follows: \[ \frac{2}{1} = \frac{x}{3} \] 3. **Cross-Multiplying to Solve for \( x \):** - Cross-multiply to solve for \( x \): \[ 2 \cdot 3 = 1 \cdot x \] \[ 6 = x \] 4. **Conclusion:** - Therefore, if the recipe is increased to 3 cups of sugar, the amount of flour needed is \( 6 \) cups. ### Common Misconceptions: - **Additive Misconception:** Some students may think that because the original recipe calls for 2 cups of flour for 1 cup of sugar, they can simply add the increase in sugar (from 1 cup to 3 cups, which is an increase of 2 cups) to the original amount of flour (2 cups) to get 4 cups of flour. This misunderstanding arises from treating ratios as additive rather than multiplicative. - **Misunderstanding of Ratios:** Students may not fully grasp that ratios represent a relationship between two quantities that must be maintained. They might think that increasing one part of the ratio necessitates a simple addition to the other part, rather than scaling both parts of the ratio proportionally. ### Generalization to Algebraic Thinking: - Understanding ratios and proportional reasoning is foundational for algebraic thinking. Students need to recognize that relationships between quantities can be expressed algebraically, and that operations on these quantities must respect the established ratios. Misconceptions in this area can lead to difficulties in solving more complex algebraic problems, where maintaining relationships between variables is crucial. By addressing these misconceptions through clear examples and reinforcing the concept of proportionality, educators can help students develop a more robust understanding of ratios and their applications in algebra.