# Problem of the Month: On Balance

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3 Problem of the Month On Balance Level A Debbie works at an apple orchard. She knows every so often there is a bad or rotten apple in the baskets of apples she picks. She knows that bad apples weigh less than good ones. There are a lot of apples in her basket so she wants to spend as little time as possible for checking apples. She thinks she knows a fast way to check. She has three apples and one is bad. How can she weigh apples just one time and still find out which one is bad. Making only one weighing, can you determine a way to know for sure which apple is bad? Explain. Problem of the Month On Balance 1 (c) Noyce Foundation To reproduce this document, permission must be granted by the Noyce Foundation:

4 Level B Debbie has a lot more apples to check. Now she has five apples. Four apples are good, all weighing the same. But the bad one either weighs more or weighs less than the others. List the process steps and decision you need to make in order to determine which apple is bad, and whether it is heavier or lighter than the good ones. A B C D E You know one of the apples is either heavier or lighter than all the others, what are the minimum weighs needed to determine which apple is bad, and whether it is heavier or lighter than the others. Show and justify your solution. Problem of the Month On Balance 2 (c) Noyce Foundation To reproduce this document, permission must be granted by the Noyce Foundation:

5 Level C Each type of fruit has a unique weight that is consistent between scales. Use the scales to determine the relationship between the different types of fruit. How do the weights of strawberries compare with limes? Problem of the Month On Balance 3 (c) Noyce Foundation To reproduce this document, permission must be granted by the Noyce Foundation:

6 Level D Suppose you were presented with nine apples (A-I). Eight apples are the same weight and the ninth, either weighs more or weighs less than the others. List the process steps and decision you need to make in order to determine which apple is different, and whether it is heavier or lighter than the others. A B C D E F G H I You are given any number of apples between 3 and 15 and you know that one of the apples is either heavier or lighter than the others, which weigh the same. For each set of apples, what are the minimum number times you need to weigh the items to insure that you find the one that weighs different than the others? Show the relationship between the number of apples and the minimum weighing needed. Problem of the Month On Balance 4 (c) Noyce Foundation To reproduce this document, permission must be granted by the Noyce Foundation:

7 Level E Each type of fruit has a unique weight that is consistent between scales. Use the five scales to determine the relationship between the different types of fruits. How do the weights of each fruit compare with each other? Which weighs the most? Which weighs the least? Suppose a strawberry weighs 3 oz., how many of one kind of fruit equal to another kind of fruit? Explain. Problem of the Month On Balance 5 (c) Noyce Foundation To reproduce this document, permission must be granted by the Noyce Foundation:

8 Problem of the Month On Balance Primary Version Level A Materials: A scale and three objects of different weights (maybe an apple, pear and orange), paper, and pencil. Discussion on the rug: (Teacher shows a balance scale) What do we use this object for? (Students volunteer ideas. If students don t know, then the teacher puts two different objects on the scale and asks the question). What do you think it tells us now? (Students respond with ideas) (Teacher asks) Okay, which object is heavier? (Students respond) How do we know? (Teacher asks) Which one weighs the most? Which one weighs the least? In small groups: (Each student pair has a scale and three different objects maybe a fruit, a piece of paper and a pencil. The teacher explains that they can use the scale to find the answer to these questions.) Which object weighs the most? Which object weighs the least? What do we know about the third object (fruit)? Put the objects in order from heaviest to lightest? (At the end of the investigation have students either discuss or dictate a response to this summary question.) How did you figure out which object (fruit) weighed the most? Tell me how you figured it out and how you know for sure. Problem of the Month On Balance 6 (c) Noyce Foundation To reproduce this document, permission must be granted by the Noyce Foundation:

9 Problem of the Month On Balance 7 (c) Noyce Foundation To reproduce this document, permission must be granted by the Noyce Foundation:

11 objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. MP.8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y 2)/(x 1) = 3. Noticing the regularity in the way terms cancel when expanding (x 1)(x+1),(x 1)(x2+x+1),and(x 1)(x3 +x2+x+1)might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

13 and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. MP.8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y 2)/(x 1) = 3. Noticing the regularity in the way terms cancel when expanding (x 1)(x+1),(x 1)(x2+x+1),and(x 1)(x3 +x2+x+1)might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

15 objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. MP.8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y 2)/(x 1) = 3. Noticing the regularity in the way terms cancel when expanding (x 1)(x+1),(x 1)(x2+x+1),and(x 1)(x3 +x2+x+1)might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

17 examine claims and make explicit use of definitions. MP. 7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 x 8 equals the well-remembered 7 x x 3, in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 x 7 and the 9 as They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. MP.8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y 2)/(x 1) = 3. Noticing the regularity in the way terms cancel when expanding (x 1)(x+1),(x 1)(x2+x+1),and(x 1)(x3 +x2+x+1)might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

19 and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. MP.8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y 2)/(x 1) = 3. Noticing the regularity in the way terms cancel when expanding (x 1)(x+1),(x 1)(x2+x+1),and(x 1)(x3 +x2+x+1)might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

21 will see 7 x 8 equals the well-remembered 7 x x 3, in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 x 7 and the 9 as They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. MP.8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y 2)/(x 1) = 3. Noticing the regularity in the way terms cancel when expanding (x 1)(x+1),(x 1)(x2+x+1),and(x 1)(x3 +x2+x+1)might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

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