They have different numerators and denominators, but their fractional values are the same. It means half of something. They represent the same part of the whole. Multiply both the numerator and denominator of a fraction by the same whole number. As long as you multiply both top and bottom of the fraction by the same number, you won't change the value of the fraction, and you'll create an equivalent fraction.
This problem can also be used for a variety of instructional purposes, including assessment, where the focus might be on assessing students' ability to use a variety of strategies, or as a task for a cooperative-problem-solving group, where the goal is to use as many different strategies as possible in solving the problem. Guess-and-Check: The guess-and-check strategy starts with an original guess for how many mangoes were in the bowl prior to the King's entry into the kitchen.
Students then use the structure of the problem to see if their initial guess works to solve the problem correctly. If their initial guess fails to work, they make another, it is hoped "better," guess and check to see if it works. They continue this process until they make a correct guess. Some students may make wild and unreasonable guesses, so teachers should point out how to make "reasonable" first guesses and discuss the importance of making a table to collect and organize the data.
Students might realize that an initial guess has to be divisible by 6 so that the King could take one-sixth of the mangoes. For example, a student might guess that 24 mangoes were in the bowl originally. When checking this guess, however, the student will find that it results in 4, not 3, mangoes at the end.
Since this outcome is too many mangoes, the student would revise his or her initial guess downward to 18, the next smallest multiple of 6. This number does, in fact, work. Not all students will necessary note the relevance of the initial guess's being a multiple of 6. An initial guess may be 14, suggesting that students are not aware of the relevance of divisibility by 6. For their guess of 14, students may get off track and do the following computation on a calculator:.
Draw a Picture: The easiest solution method to this problem is surprising in its simplicity. Start by drawing a rectangle to represent all mangoes in the original pile prior to the removal of any of them. Since the King took one-sixth of this pile, divide the rectangle into six equal strips and "remove" one strip. Notice that five strips remain, from which the Queen removed one-fifth, so this one-fifth is also represented by one of the original strips.
Continuing, when the first Prince removes one-fourth of what is left, the one-fourth is represented by one of the strips.
Similarly, the one-third, one-half, and 3 remaining mangoes are each represented by a strip. The draw-a-picture strategy may lead to some of your most interesting observations. Students may first draw six circles and shaded one to represent the one-sixth the King took.
They then would explain that the Queen ate one-fifth of what was left, so they would have shaded one of the remaining five circles. The process is continued until students have shaded the last of the original six circles drawn. Other students may draw a picture but divide a pie into six wedges.
In the image below, one student shaded one wedge, noted five remaining wedges, and shaded one of them. Then, multiply the two denominators. In the following intermediate step, the fraction result cannot be further simplified by canceling.
Rules for expressions with fractions: Fractions - simply use a forward slash between the numerator and denominator, i.
If you are using mixed numbers, be sure to leave a single space between the whole and fraction part.
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