Piece It Together
In the problem Piece It Together, students use two-and three-dimensional geometry to solve problems involving polygons and polyhedra. The mathematical topics that underlie this problem are the attributes of polygons, linear measurement, angular measurement, spatial visualization, and geometric solids.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.
PRE-K
In this level, students are presented with the task of examining and comparing attributes of different pattern blocks. Their task involves making a list of attributes and finding area relationships between different pattern blocks (triangle, blue rhombus, hexagon, and trapezoid).
LEVEL A
This level has students recognize shape attributes (2.G.A.1) for each of the pattern blocks. The students then use area (3.MD.C.5) and perimeter (3.MD.D.8) concepts to compare the pattern blocks and identify different combinations of pattern blocks used to cover the hexagon block. It is important to note that the task asks students to consider the area of the triangle block as one area unit. While this is beyond the expectations of Common Core standard 3.MD.C.5, the concept of covering a shape with an area unit without overlapping still applies. This task can be extended to meet other grade-level expectations by having the students identify attributes such as number of parallel and perpendicular sides (4.G.A.2) and measuring the angles of each shape (4.MD.C.6).
LEVEL B
In this level, students are presented with situations that involve covering a two-dimensional design with different pattern blocks. The students are asked to find the ways the design can be covered with just one pattern block and the ways it can be covered using multiple blocks.
Students will experience using two-dimensional shapes to compose new shapes using pattern blocks (1.G.A.2). The task will also have students analyze how many of each shape can be used to cover the outline, which touches on area concepts introduced in standard 3.MD.C.5. Note that students will be using shapes other than squares to represent an area unit, but the concept of covering a shape with an area unit without overlapping still applies.
LEVEL C
In this level, students are given three different perspectives of a soccer ball. Using spatial visualization, the student must determine the number and type of each polygon that makes up the surface area of the ball. Two different soccer ball designs are presented.
Using spatial visualization, students may apply their knowledge of nets and 3-D objects (6.G.A.4, 7.G.B.6) to determine the number and type of each polygon that makes up the surface area of the ball.
LEVEL D
In this level, the student is told that a major league soccer ball is between 27 and 28 inches in circumference. Students use their knowledge of nets and surface area from middle school to determine the dimensions. Note: Students will need the answers from Level C to solve this problem.
Student use geometric shapes to model a soccer ball (G.MG.A.1) and use their knowledge of circumference to find the dimensions of the polygons that make up the soccer ball (G.GMD.A.1). While students study the volume of spheres in grade 8 and high school geometry, the formula for the surface area of a sphere is not required by the standards. For students to use the surface area of the sphere to solve the problem, they will need to be given the formula. With that approach, students can set up equations for the areas of the polygons, building on work from grade 7 (7.G.B.6). Finding the area of the polygons will require special right triangles and trigonometry (G.SRT.C.8).
LEVEL E
In this level, students are asked to design their own soccer ball and describe the dimensions of the surface area polygons and the attributes of the polyhedra they created.Students use their knowledge of nets and surface area from middle school to determine the dimensions. Note: It will be helpful for students to have completed Levels C and D prior to working on this level.
Students use geometric shapes to design a soccer ball (G.MG.A.3) and describe its attributes and dimensions. While students study the volume of spheres in grade 8 and high school geometry, the formula for the surface area of a sphere is not required by the standards. For students to use the surface area of the sphere to solve the problem, they will need to be given the formula. With that approach, students can set up equations for the areas of the polygons, building on work from grade 7 (7.G.B.6). Finding the area of the polygons will require special right triangles and trigonometry (G.SRT.C.8).
PROBLEM OF THE MONTH
Download the complete packet of Piece It Together 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.
SOLUTIONS
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