In the problem On Balance, students are engaged in tasks and puzzles that involve equality, inequalities, equations, and simultaneous constraints. The mathematical topics that underlie this POM are measurement, number sentences, equality, inequality, variables, inverse operations, and simultaneous systems.In each level, students must make sense of the problem and persevere in solving it (MP.1). Each problem is divided into five levels of difficulty, Level A through Level E, to allow access and scaffolding for students into different aspects of the problem and to stretch students to go deeper into mathematical complexity.
In this task, students are presented with a scale and three different objects. They are asked to compare the weights of the objects.
In this level, students are presented with a puzzle involving the weight of three apples and a balance scale. The task is to find which apple is lighter by only weighing with the scale once. They need to use logic and their knowledge of equality and inequality.
In this level, students are asked to directly compare three objects with the measurable attribute of weight in common and use it to describe which object is the lightest (K.MD.A.2). While the kindergarten standard only calls for comparison of two objects, students are told that two of the objects in this problem have the same weight. This logic problem is an application of their understanding of the standard.
In this level, students are presented with five apples, one of which is bad. They know the bad apple is a different weight than the other four, but are not sure whether the bad one is heavier or lighter. Students need to use the balance scale to determine the bad apple, using the least number of weighings.
In this level, students are presented with five apples, one of which is bad. Students must apply reasoning and logic (SMP 3) and draw on their knowledge of inequalities (6.EE.B.8) to determine if the weight of the bad apple is lighter or heavier than the other four apples.
In this level, students are presented with a drawing of balance scales showing how different kinds of fruit weigh in relationship to one another. The students use their knowledge of equality and equations to determine how the strawberries compare to the limes in weight.
In this level, students are presented with different kinds of fruit on a balance scale. Students must use their knowledge of equality and equations (6.EE.B.5, 6.EE.B.6) to determine the relationship of the weight of each type of fruit. This problem could be used to foreshadow the work with systems of equations (8.EE.C.8b, 8.EE.C.8c).
In this level, students are given a similar drawing to the one they had in Level B, but this time they have nine apples to weigh in order to determine the bad apple. The goal is to find the bad apple in the least number of weighings. The students are asked to make a generalization based on the number of apples being considered.
In this level, students extend their reasoning and logic (SMP 3) from Level B to determine the weight of one bad apple out of a batch of nine apples. Students draw on their knowledge of inequalities (6.EE.B.8) to determine if the weight of the bad apple is lighter or heavier than the other eight apples and are challenged to observe a pattern in their reasoning for any number up to 9 apples.
In this level, students are asked to consider five simultaneous equations represented in a drawing of fruit on a balance scale. The task is to compare the weights of the 6 different kinds of fruit and to find the weight of each, given that the strawberries weigh 3 oz.
In this level, students write equations to represent fruit on a balance scale (A-CED.A.2). They use these equations to solve a system of 5 simultaneous equations with 5 variables. This work extends the work students have been doing with systems of linear equations in 8th grade and Algebra I (A-REI.C.6 and 8.EE.8c). Students use reasoning and algebraic skills to find the solution to the system.
PROBLEM OF THE MONTH
Download the complete packet of On Balance Levels A-E here.
You can learn more about how to implement these problems in a school-wide Problem of the Month initiative in “Jumpstarting a Schoolwide Culture of Mathematical Thinking: Problems of the Month,” a practitioner’s guide. Download the guide as iBook with embedded videos or Download as PDF without embedded videos.
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