Teacher: Well, how about if we use some concepts we've talk about in class before. Teacher: Can you come up to the board and show us what you are doing? Here is what I did: $\frac$ = $\frac$ = $\frac$ They all simplify to being the same concentrate, $\frac$, so, they are all proportional to one another. During a week, 10 hours may have been spent on homework while 35 hours were spent in school.
The proportion is still true because $\frac=\frac$. 2.0 L Answer: b DOK: Level 2 Source: Test Prep: MCA III Test Preparation Grade 7, Houghton Mifflin Harcourt Publishing Company, Attn: Contracts, Copyrights, and Licensing, 9400 South Park Center Loop, Orlando, FL 32819.
A lack of thought does not characterize most of these scenarios, but rather an incomplete consideration of the situation. Contrary to popular opinion, not all thinking is rational, at least as we would define rational.
Typically, poor decisions or other mistakes are a result of flawed or incomplete thinking, not the absence of thinking. Rational thinking is the ability to consider the relevant variables of a situation and to access, organize, and analyze relevant information (e.g., facts, opinions, judgments, and data) to arrive at a sound conclusion.
Student 1: I think Mix V will be the most orangey because it has 5 cups of concentrate, and no other juice has that much, so it has to be the most orangey. Student 2: Well, the denominator is the total, so the total would be 3 cups. But shouldn't we be able to use the fraction $\frac$? Student 1: Because when we compared all of the ratios, they were equivalent: 2 to 3 is equivalent to 4 to 6, is equivalent to 10 to 15.
Student 2: I think Mix S and Mix T will be the same because they each have only one more cup of concentrate than juice, so they should taste the same, shouldn't they? Student 2: Find out what percent of each is concentrate? In Mix S, though, there are 3 cups of concentrate AND 2 cups of water, so there's really 5 cups of ingredients in the juice. If we write them all as fractions we can see that even better. Note that this does not necessarily imply that "hours spent on homework" = 2 or that "hours spent in school" = 18.104.22.168.1 Add, subtract, multiply and divide positive and negative rational numbers that are integers, fractions and terminating decimals; use efficient and generalizable procedures, including standard algorithms; raise positive rational numbers to whole-number exponents.22.214.171.124 Use real-world contexts and the inverse relationship between addition and subtraction to explain why the procedures of arithmetic with negative rational numbers make sense.126.96.36.199 Understand that calculators and other computing technologies often truncate or round numbers.188.8.131.52 Solve problems in various contexts involving calculations with positive and negative rational numbers and positive integer exponents, including computing simple and compound interest.184.108.40.206 Use proportional reasoning to solve problems involving ratios in various contexts.220.127.116.11 Demonstrate an understanding of the relationship between the absolute value of a rational number and distance on a number line. 18.104.22.168 Solve problems in various contexts involving calculations with positive and negative rational numbers and positive integer exponents, including computing simple and compound interest.22.214.171.124 Use proportional reasoning to solve problems involving ratios in various contexts.For example, a recipe calls for milk, flour and sugar in a ratio of 4:6:3. Use proportional relationships to solve multistep ratio and percent problems. Others raise their hands to indicate they have either made it or helped make it. We need a breakthrough in the quality of thinking employed by both decision makers and by each of us in our daily affairs.We all, average citizens to world leaders, struggle to develop creative, workable solutions to pressing problems and issues.Indeed, several authors have defined intelligence, at least in part, as the ability to solve problems.For example, Sternberg (1996) writes: Successful intelligence as I view it involves analytical, creative, and practical aspects.Teacher: Today we are going to look at some different recipes for different juice mixes. Maybe putting the information into tables could help us. Here's the proportion we set up: $\frac$ = $\frac$ Student 4: So, we can solve and get 40 for x, just like the other group did.Each recipe is different from the others, so we are going to use our math skills and see if we can decide which juice would be the most orangey, and which would be the least orangey. If you look at the percent that is concentrate, we can see that Mix U has the least percent of concentrate, so it is the least orangey. So, we need 40 cups of concentrate and 60 cups of water.Solutions to significant problems facing modern society demand a widespread qualitative improvement in thinking and understanding. The news media are rife with examples of questionable responses or solutions to situations and events. [emphasis in original] The assertion that society and individuals can benefit from improving the ways we approach and consider some of life's toughest problems is hard to argue with.