Part and Whole
In the problem Part and Whole, students explore rational numbers and solve problems involving symmetry, congruence, determining equal area, sub-dividing area models, reasoning about measurements, and generalizing about fractions. The mathematical topic that underlies this problem is the understanding of rational numbers through different representations. Students explore fractions through area models using symmetry, congruence, measurement, and mathematical notation.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 explore cutting different shapes into two equal parts using a line of symmetry.
In this level, students view different geometric figures and determine whether they can partition the figure into two identical pieces. Their task is to use symmetry to answer questions about the same shape and equal area parts.
This level addresses Common Core standard 4.G.A.3 by having students identify line-symmetric figures and draw lines of symmetry.
In this level, students are given a picture of flat geometric shapes made out of clay. Students are asked to find a way to make a straight line cut in order to divide the figure into two parts of equal amounts of clay.
This level addresses Common Core standard 3.G.A.2 by having students partition shapes into parts with equal shares. This problem does extend this understanding to include the concept learned from standard 2.G.A.3, which states that the student should recognize that equal shares need not have the same shape. In several cases, a student will need to decompose the given shape and manipulate its parts in order to prove that they are equal in size.
In this level, the students are presented with a rectangular map. The map is divided into six different regions of various sizes. The task for the students is to determine the fractional part of each region in terms of the whole rectangular area.
Students will address Common Core standard 3.MD.C.6 by determining the area of each piece by counting the unit squares. This problem does go beyond the expectations of this standard and is a lead-up to 6.G.A.1 by requiring students to reason that the triangular pieces are equal to half a square unit and can therefore be combined to create the equivalent of one square unit. Students will also need to use understanding developed from standard 3.NF.A.1, which has students identify the fractional quantity from a whole number of parts for each region. The problem exceeds the expectations of this standard again by referring two fractions with denominators greater than 8.
In this level, students analyze a triangular region to once again find the fractional parts of the whole. The students are then asked to design their own map with sub-divided regions.
In Level D, students use their knowledge of finding areas of polygons (6.G.A.1) to analyze a triangular region to find the fractional parts of the whole. The students apply this thinking to design their own map with sub-divided regions.
In this level, students are presented with an investigation to find five different unit fractions with a sum of 1. Students determine if there is more than one set of five unit fractions that sum to 1. If so, they must determine a general method for finding other sets. They also explore other sized sets of unit fractions that can be found to sum to 1. Then students are given an algebraic identity to verify using rational expressions, and then use this identity to generalize the work they did numerically earlier in the task.
In this level, students use rational number operations to find five different unit fractions with a sum of 1 (7.NS.A.1D). Students determine if there is more than one set of five unit fractions that result in a sum of 1. If so, they describe a general method for finding other sets. Students verify an algebraic identity involving rational expressions (A-APR.D.6, SMP 3). Then, students use this identity to explore other sized sets of unit fractions that can be found that result in a sum of 1. Both the numerical reasoning and algebraic reasoning in this task require students look for and make use of the structure of the rational expressions they write (SMP 7) and look for and express regularity in repeated reasoning as they deconstruct 1 into the sum of unit fractions (SMP 8).
PROBLEM OF THE MONTH
Download the complete packet of Part and Whole 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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