# Miles of Tiles

In the problem Miles of Tiles, students are engaged in tasks that involve puzzles of number relationships, equations, and simultaneous constraints. The mathematical topics that underlie this problem are measurement, number sentences, area models, variables, inverse operations, equations, quadratics, factoring, 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.

PRE-K
In this task, students will explore filling a frame with different sized tiles with no gaps or overlaps.

LEVEL A
In this level, students are presented with a picture frame and a set of tiles of differing lengths; the task is to find different arrangements of the set of tiles to make a decorative frame. Their task involves finding number sentences that will sum to the region surrounding the picture.

Though this problem deals with concepts beyond first-grade expectations, it can be used to address Common Core standard 1.OA.A.1. The students can use the concepts from this standard to add different lengths of various tiles to find the total length of each side. Students at this level will need some guidance on how to fill in the frame before this can be discussed.

LEVEL B
In this level, students are presented with an order 4 Magic Square. Students need to use number sense and a guess-and-check method to determine an arrangement of number tiles that makes the puzzle a magic square.

To solve this problem, students will use skills and strategies from Common Core standard 2.NBT.B.5. The problem requires students to be fluent in addition strategies within 100 in order to determine where to place each number in the square.

LEVEL C
In this level, students are presented with a patio made of uniform rectangular tiles. Only one dimension of the rectangular patio is given, and the task is to determine the dimensions of the uniform tile. This situation can be translated into systems of constraints with equal numbers of unknowns (8.EE.C.8b, 8.EE.C.8c).

LEVEL D
In this level, students are given a situation that involves a set of two non-commensurate square tiles. A rectangular tile is constructed with the length the size of the larger square tile, and the width the size of the smaller square tile. Students are asked to create rectangular designs using 6 large square tiles, 4 small square tiles, and an undefined number of rectangles. The task involves developing an understanding of an area model of polynomial multiplication.

Students will need to make sense of the problem using concrete models. Once they have found an arrangement, they can start to look for the structure in the arrangement (SMP.7). They should recognize that they can write expressions for the rectangular arrangements using the variable lengths of the squares. The area of the rectangular arrangements is the product of two binomials (A-APR.A.1). The terms in the product represent the number of large squares, the number of small squares, and the number of rectangles (A-SSE.A.1a). To find all the possible arrangements, they will use the structure of the resulting trinomial to find all possible factors with an unknown middle term (A-SSE.A.2). This problem encourages the connection between a geometric model of binomial multiplication and factoring (G-MG.A.3).

LEVEL E
In this level, students are asked to generalize the principles of the area model and its dimensions in Level D. Therefore, it is important for students to have completed Level D before working on this level. Students will use their understanding of quadratic relationships as well as products and factors to generalize the mathematics.

In this level, students generalize the ideas they explored in Level D. Students interpret the terms of a trinomial expression and the related binomial product (A-SSE.A.1a) and the geometric model of a rectangle (G-MG.A.3) to write a generalization of when a rectangular arrangement is possible. They use the structure of the trinomial (A-SSE.A.2) and the geometric model to decide when an arrangement is possible.

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