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Information for course "Unit 1 - Area And Surface Area"
Description: Students began to learn about two- and three-dimensional shapes in kindergarten, and continued this work in grades 1 and 2, composing, decomposing, and identifying shapes. Students’ work with geometric measurement began with length and continued with area. Students learned to “structure two-dimensional space,” that is, to see a rectangle with whole-number side lengths as composed of an array of unit squares or composed of iterated rows or iterated columns of unit squares. In grade 3, students distinguished between perimeter and area. They connected rectangle area with multiplication, understanding why (for whole-number side lengths) multiplying the side lengths of a rectangle yields the number of unit squares that tile the rectangle. They used area diagrams to represent instances of the distributive property. In grade 4, students applied area and perimeter formulas for rectangles to solve real-world and mathematical problems, and learned to use protractors. In grade 5, students extended the formula for the area of rectangles to rectangles with fractional side lengths.

In grade 6, students extend their reasoning about area to include shapes that are not composed of rectangles. Doing this draws on abilities developed in earlier grades to compose and decompose shapes, for example, to see a rectangle as composed of two congruent right triangles. Through activities designed and sequenced to allow students to make sense of problems and persevere in solving them, students build on these abilities and their knowledge of areas of rectangles to find the areas of polygons by decomposing and rearranging them to make figures whose areas they can determine. They learn strategies for finding areas of parallelograms and triangles, and use regularity in repeated reasoning to develop formulas for these areas, using geometric properties to justify the correctness of these formulas. They use these formulas to solve problems. They understand that any polygon can be decomposed into triangles, and use this knowledge to find areas of polygons. Students find the surface areas of polyhedra with triangular and rectangular surfaces. They study, assemble, and draw nets for polyhedra and use nets to determine surface areas. Throughout, they discuss their mathematical ideas and respond to the ideas of others.

Because grade 6 students will be writing algebraic expressions and equations involving the letter x and x is easily confused with ×, these materials use the “dot” notation, e.g., 2⋅3, for multiplication instead of the “cross” notation, e.g., 2×3. The dot notation will be new for many students, and they will need explicit guidance in using it.

Many of the lessons in this unit ask students to work on geometric figures that are not set in a real-world context. This design choice respects the significant intellectual work of reasoning about area. Tasks set in real-world contexts that involve areas of polygons are often contrived and hinder rather than help understanding. Moreover, mathematical contexts are legitimate contexts that are worthy of study. Students do have an opportunity at the end of the unit to tackle a real-world application (MP2, MP4).

In grade 6, students are likely to need physical tools in order to check that one figure is an identical copy of another where “identical copy” is defined as follows:

One figure is an identical copy of another if one can be placed on top of the other so that they match up exactly.

Language: English
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Language: English
Professors: Elena Webster, Gia Laarz, LaShika Curtain, Paul Dube', Shane Maisonet, Perry Williams, Tonya James
Units: 1
Lesson content
Language: English
Professors: Elena Webster, Gia Laarz, LaShika Curtain, Paul Dube', Shane Maisonet, Perry Williams, Tonya James
Units: 1
Lesson content
Language: English
Professors: Elena Webster, Gia Laarz, LaShika Curtain, Paul Dube', Shane Maisonet, Perry Williams, Tonya James
Units: 1
Lesson content
Language: English
Professors: Elena Webster, Gia Laarz, LaShika Curtain, Paul Dube', Shane Maisonet, Perry Williams, Tonya James
Units: 1
Lesson content
Language: English
Professors: Elena Webster, Gia Laarz, LaShika Curtain, Paul Dube', Shane Maisonet, Perry Williams, Tonya James
Tests: 1
Lesson content
Language: English
Professors: Elena Webster, Gia Laarz, LaShika Curtain, Paul Dube', Shane Maisonet, Perry Williams, Tonya James
Units: 1
Lesson content
Language: English
Professors: Elena Webster, Gia Laarz, LaShika Curtain, Paul Dube', Shane Maisonet, Perry Williams, Tonya James
Units: 1
Lesson content
Language: English
Professors: Elena Webster, Gia Laarz, LaShika Curtain, Paul Dube', Shane Maisonet, Perry Williams, Tonya James
Units: 1
Lesson content
Language: English
Professors: Elena Webster, Gia Laarz, LaShika Curtain, Paul Dube', Shane Maisonet, Perry Williams, Tonya James
Units: 1
Lesson content
Language: English
Professors: Elena Webster, Gia Laarz, LaShika Curtain, Paul Dube', Shane Maisonet, Perry Williams, Tonya James
Tests: 1
Lesson content
Language: English
Professors: Elena Webster, Gia Laarz, LaShika Curtain, Paul Dube', Shane Maisonet, Perry Williams, Tonya James
Units: 1
Lesson content
Language: English
Professors: Elena Webster, Gia Laarz, LaShika Curtain, Paul Dube', Shane Maisonet, Perry Williams, Tonya James
Units: 1
Lesson content
Language: English
Professors: Elena Webster, Gia Laarz, LaShika Curtain, Paul Dube', Shane Maisonet, Perry Williams, Tonya James
Units: 1
Lesson content
Language: English
Professors: Elena Webster, Gia Laarz, LaShika Curtain, Paul Dube', Shane Maisonet, Perry Williams, Tonya James
Units: 1
Lesson content
Language: English
Professors: Elena Webster, Gia Laarz, LaShika Curtain, Paul Dube', Shane Maisonet, Perry Williams, Tonya James
Tests: 1
Lesson content
Language: English
Professors: Elena Webster, Gia Laarz, LaShika Curtain, Paul Dube', Shane Maisonet, Perry Williams, Tonya James
Language: English
Professors: Elena Webster, Gia Laarz, LaShika Curtain, Paul Dube', Shane Maisonet, Perry Williams, Tonya James
Units: 1
Lesson content
Language: English
Professors: Elena Webster, Gia Laarz, LaShika Curtain, Paul Dube', Shane Maisonet, Perry Williams, Tonya James
Units: 1
Lesson content
Language: English
Professors: Elena Webster, Gia Laarz, LaShika Curtain, Paul Dube', Shane Maisonet, Perry Williams, Tonya James
Units: 1
Lesson content
Language: English
Professors: Elena Webster, Gia Laarz, LaShika Curtain, Paul Dube', Shane Maisonet, Perry Williams, Tonya James
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