Essential Questions:
Lesson Plan C.7a: Using Tiling to Find Area
Lesson Seed C.5: Covering the Shapes
Lesson Seed C.7c: Using the Distributive Property to Find the Area
Lesson Seed C.7d: Area Rectilinear
Unit Overview
Content Emphasis By Clusters in Grade 3
Progressions from Common Core State Standards in Mathematics
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Lesson seeds are ideas that can be used to build a lesson aligned to the CCSS. Lesson seeds are not meant to be all-inclusive, nor are they substitutes for instruction. When developing lessons from these seeds, teachers must consider the needs of all learners. It is also important to build checkpoints into the lessons where appropriate formative assessment will inform a teachers instructional pacing and delivery.
Geometric measurement connects two critical domains in mathematics, geometry and number, each domain providing support for the other. Students will expand their knowledge of the structure and properties of multiplication and addition to finding area, but it must be based on a firm conceptual foundation of area. The shift of coherence is strong in these two domains. The unit should be taught making specific connections to the operations or as extension of the unit for each operation. Measurement is the process of assigning a number to a magnitude of some attribute shared by a class of objects such as length, width, area, etc . . . In this unit, students recognize area as an attribute of two-dimensional regions. They expand their knowledge of using arrays to represent multiplication and division to conceptually develop the understanding of area. Students first measure the area of a shape by finding the total number of same-size units required to cover the shape without gaps or overlaps. A square with sides of unit length is considered the standard unit for measuring area. Through multiple experiences using tiling of various shaped rectangles to find the area leads students to understand that instead of titling, the area can be calculated by multiplying the length and width of the rectangle. Students, who first conceptually understand the concept of area, develop the formula for calculating area so that the concept of area makes sense to students. The practice of tiling to show the area of a rectangle can be found using the distributive property as a reasoning strategy. Students expand their work with arrays (in multiplication) to recognize that a rectangular array can be decomposed into identical rows or into identical columns and count to find the total. They also expand their understanding of decomposing shapes into other regular shapes, to find the area of rectilinear figures. Students additive thinking helps them find the area of these types of figures by decomposing them into non-overlapping rectangles and adding the area of the non-overlapping parts. This technique is used to solve real world problems with area.
The student will be able to:
The Partnership for Assessment of Readiness for College and Careers (PARCC) has awarded the Dana Center a grant to develop the information for this component. This information will be provided at a later date. The Dana Center, located at the University of Texas in Austin, encourages high academic standards in mathematics by working in partnership with local, state, and national education entities. Educators at the Center collaborate with their partners to help school systems nurture students' intellectual passions. The Center advocates for every student leaving school prepared for success in postsecondary education and in the contemporary workplace.
Student may:
Interdisciplinary Connections:
Interdisciplinary connections fall into a number of related categories:
Sample Assessment Items: The items included in this component will be aligned to the standards in the unit and will include:
Topic
Standards Addressed
Notes
School Mural
3.MD.C.7
3.OA.A.3
3.OA.B.5
SMP 4
SMP 7
This two-part task is visually appealing and accessible to all students, using drag-and-drop technology to connect a rectangular array with multiplication concepts and properties. The task involves mathematical modeling as students move between concrete and symbolic representations (MP.4) and mathematical structure as they use the Distributive Property to decompose multiplication equations and decompose area models into manageable pieces to make sense of the situation (MP.7)—thus making it a “practice-forward” task.
An innovative feature of this task is that it elicits modeling and reasoning about structure, yet is entirely machine scorable without losing any important data about students’ mathematical thinking. The drag-and-drop technology allows students to experiment with their ideas before submitting their answer.
Finding the Area of a Polygon
3.MD.C.7d
3.MD.C.6
The purpose of this instructional task is for students to find the area of figures that can be decomposed and then recomposed into rectangles.
Halves, Thirds, and Sixths
3.FN.A.3b
3.GA.A.2
3.NF.A.1
The purpose of this task is for students to use their understanding of area as the number of square units that covers a region (3.MD.6), to recognize different ways of representing fractions with area (3.G.2), and to understand why fractions are equivalent in special cases (3.NF.3.b). Determining the fraction of the area that is shaded for rectangles A-D in part (b) is increasingly complex. Rectangles E, F, and G show that there are many ways for ^{1}/_{2} of the area to be shaded blue, which implies that there are many ways to represent the fraction ^{1}/_{2} with area. Rectangle H requires students to see the equivalence of two fractions, neither of which is a unit fraction. Students get a chance to demonstrate what they have learned in part (b) by generating their own representations of fractions in parts (c) and (d).
Interventions/Enrichments/PD: (Standard-specific modules that focus on student interventions/enrichments and on professional development for teachers will be included later, as available from the vendor(s) producing the modules.)
Vocabulary/Terminology/Concepts: This section of the Unit Plan is divided into two parts. Part I contains vocabulary and terminology from standards that comprise the cluster, which is the focus of this unit plan. Part II contains vocabulary and terminology from standards outside of the focus cluster. These “outside standards” provide important instructional connections to the focus cluster.
Part I – Focus Cluster
measurement quantities:examples could include inches, feet, pints, quarts, centimeters, meters, liters, square units, etc.
area: the number of square units needed to cover a region. Examples:
tiling: highlighting the square units on each side of a rectangle to show its relationship to multiplication and that by multiplying the side lengths, the area can be determined. Example:
rectilinear figures: a polygon which has only 90 and possibly 270 angles and an even number of sides. Examples of Rectilinear Figures:
properties of operations:
Here a, b and c stand for arbitrary numbers in a given number system. The properties of operations apply to the rational number system, the real number system, and the complex number system.
decomposing: breaking a number into two or more parts to make it easier with which to work. Example: When combining a set of 5 and a set of 8, a student might decompose 8 into a set of 3 and a set of 5, making it easier to see that the two sets of 5 make 10 and then there are 3 more for a total of 13. Decompose the number 4; 4 = 1+3; 4 = 3+1; 4 = 2+2 Decompose the number ^{3}/_{5} ; ^{3}/_{5} = ^{1}/_{5} + ^{1}/_{5} + ^{1}/_{5}
composing: Composing (opposite of decomposing) is the process of joining numbers into a whole number…to combine smaller parts. Examples: 1 + 4 = 5; 2 + 3 = 5. These are two different ways to “compose” 5.
Commutative Property: In both addition and multiplication, changing the order of the factors when adding or multiplying will not change the sum or the product. Example: 2 + 3 = 5 and 3 + 2 = 5; 3 x 7 = 21 and 7 x 3 = 21
Associative Property: in addition and multiplication, changing the grouping of the elements being added or multiplied will not change the sum or product. Examples: (2 + 3) + 7 = 12 and 2 + (3 + 7) = 12; (2 x 3) x 5 = 30 and 2 x (3 x 5) = 30
Distributive Property: a property that relates two operations on numbers, usually multiplication and addition or multiplication and subtraction. This property gets its name because it ‘distributes’ the factor outside the parentheses over the two terms within the parentheses. Examples:
expression: one or a group of mathematical symbols representing a number or quantity; An expression may include numbers, variables, constants, operators and grouping symbols. An algebraic expression is an expression containing at least one variable. Expressions do not include the equal sign, greater than, or less than signs. Examples of expressions: 5 + 5, 2x, 3(4 + x) Non-examples: 4 + 5 = 9, 2 + 3 < 6 2(4 + x) ≠ 11
estimation strategies: : to estimate is to give an approximate number or answer. Some possible strategies include front-end estimation, rounding, and using compatible numbers. Examples:
Part II – Instructional Connections outside the Focus Cluster
product: the result when two numbers are multiplied. Example: 5 x 4 = 20 and 20 is the product.
partitioning: dividing the whole into equal parts.
partitioned: the whole divided into equal parts.
share: a unit or equal part of a whole.
inverse operation: two operations that undo each other. Addition and subtraction are inverse operations. Multiplication and division are inverse operations. Examples: 4 + 5 = 9; 9 – 5 = 4 6 x 5 = 30; 30 ÷ 5 = 6
fact families: a collection of related addition and subtraction facts, or multiplication and division facts, made from the same numbers. For 7, 8, and 15, the addition/subtraction fact family consists of 7 + 8 = 15, 8 + 7 = 15, 15 – 8 = 7, and 15 – 7 = 8. For 5, 6, and 30, the multiplication/division fact family consists of 5 x 6 = 30, 6 x 5 = 30, 30 ÷ 5 = 6, and 30 ÷ 6 = 5.
fluently: using efficient, flexible and accurate methods of computing
fraction of a region: is a number which names a part of a whole area. Example:
Shaded area represents ^{4}/_{24} or ^{1}/_{6} of the region.
Free Resources:
Math Related Literature:
References: