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Name:_______________________________ 6th Grade Math- Unit 7 Measurement: Two-Dimensional This unit bundles student expectations that address length and area in order to investigate measurement relationships. Prior to this unit, in Grade 5 Unit 06, students connected the area model used to represent multiplication with the concept of area as a measure. During this unit, students explore measurement relationships within formulas for length, including perimeter, circumference, and area. In addition, students investigate pi to describe the relationships between the diameter, radius, and circumference of a circle. After this unit, in Grade 6 Unit 08, students will continue to explore measurement concepts to include standard measurement conversions, capacity, weight, time, temperature, and volume. In Grade 7 Unit 08, students will estimate measurements and solve application problems involving the length and area of polygons and other shapes as well as the volume of prisms and cylinders. --------------------------------------------------------------------------------------------------------------------------6.4B: The student uses letters as variables in mathematical expressions to describe how one quantity changes when a related quantity changes. The student is expected to: Use tables of data to generate formulas representing relationships involving perimeter and area. -----------------------------------------------------------------------------------------------------------------------------------------------------6.6C: The student uses geometric vocabulary to describe angles, polygons, and circles. The student is expected to: Describe the relationship between radius, diameter, and circumference of a circle. -----------------------------------------------------------------------------------------------------------------------------------------------------6.8A: The student solves application problems involving estimation and measurement of length, and area. The student is expected to: Estimate measurements (including circumference) and evaluate reasonableness of results. -----------------------------------------------------------------------------------------------------------------------------------------------------6.8B: The student solves application problems involving estimation and measurement of length, and area. The student is expected to: Select and use appropriate units, tools, or formulas to measure and to solve problems involving length (including perimeter), and area. Perimeter Definition for Perimeter: the distance around a two-dimensional shape. Memory Aid for Perimeter: Formulas for Perimeter: Square: P = 4s P Perimeter s side length Rectangle: P = 2l + 2w P Perimeter l length w width *You can also find the perimeter by adding all of the side lengths together.* Perimeter Practice 14 ft 86 in. 16 in. 48 yd 39 mi 24.2 km 32 km 154 cm Area Definition for Area: the amount of space inside the boundary of a flat (2-dimensional) object Memory Aid for Area: Different Ways to Find the Area: 1) Counting the Square Units: 1 6 11 16 21 2 7 12 17 3 4 8 9 13 14 18 19 22 23 24 5 10 15 20 25 2 25 units A=______________ 22 1 2 3 4 5 6 1 7 8 9 10 11 12 13 14 15 16 17 18 23 2 3 19 20 21 22 23 24 4 5 6 25 26 27 28 24 7 8 9 10 29 30 31 32 11 12 13 14 15 24.5 33 34 35 36 16 17 18 19 20 21 2 36 units B=______________ 2 24.5 units C=______________ 2) Using a formula: Formula for area of a Square: A = s2 (Area = side squared) s Formula for area of a Rectangle: A = lw OR A area l length w or h w width A = bh A area b base h height l or b Formula for area of a Parallelogram: A = bh (Area = base x height) h h b 𝒃𝟏+𝒃𝟐 𝒉 Formula for area of a Trapezoid: A = 𝟐 b1 A Area b1 base 1 b1 base 2 h height h b2 Formula for area of a Triangle: A = h b 𝒃𝒉 𝟐 (Area = base x height ÷ 2) Area Examples Square: A = s2 A = (4)2 A=4x4 4 cm A = 16 cm2 -----------------------------------------------------------------------------------------------------------------------------------------------------Rectangle: A = lw OR A = bh 8 yd A = (17)(8) A = (17)(8) 2 A = 136 yd A = 136 yd2 17 yd -----------------------------------------------------------------------------------------------------------------------------------------------------Parallelogram: 6m 5m 6 in. A = bh A = (9)(5) 9m A = 45 m2 -----------------------------------------------------------------------------------------------------------------------------------------------------Trapezoid: 8 in. A = (b1 + b2)h 2 7 in. 7 in. A = (8 + 10)6 2 A = (18)6 10 in. 2 A = 108 2 A = 54 in.2 -----------------------------------------------------------------------------------------------------------------------------------------------------Triangle: A = bh 2 5 km 5 km A = (6)(4) 2 A = 24 2 6 km A = 12 km2 4 km Area Practice 36 2 cm 14 m2 72 2 yd 99 2 yd 71.5 2 ft 22.5 m2 21 m2 105 2 m 110 2 ft Parts of a Circle Definition for Radius: a straight line that is the distance from the center to the edge of a circle Memory Aid for Radius: Definition for Diameter: a straight line going through the center of a circle connecting 2 points on the cirlcle Memory Aid for Diameter: Relationship between the Radius and the Diameter: The radius x 2 = the diameter The diameter ÷ 2 = the raidus Definition for Chord: a straight line connecting 2 points on the circle Memory Aid for Chord: Parts of a Circle Practice diameter chord center radius radius chord center diameter chord center Circles Definition for Circumference: the distance around the edge of a circle. The perimeter of a circle. Memory Aid for Circumference and Area of a Circle: Formula for Circumference: C = 2𝝅r OR C = 𝝅d C Circumference C Circumference 𝝅 Pi = ~3 𝝅 Pi = ~3 r radius d diameter Formula for Area of a Circle: A = 𝝅 r2 Area = 3 x radius x radius Circumference and Area Practice 1) AB = 13 cm XY = 11 cm 66 cm Circumference=___________________ 363 cm2 Area=____________________ ----------------------------------------------------------------------------------------------------------------------------------- 2) ZX = 10 in. WK = 4 in. YW = 6 in. 30 in. Circumference=___________________ 75 in.2 Area=____________________ ----------------------------------------------------------------------------------------------------------------------------------3) XM = 2 m YX = 8 m NL = 5 m 30 m Circumference=___________________ 75 m2 Area=____________________ ----------------------------------------------------------------------------------------------------------------------------------4) WY = 16 ft RX = 14 ft 48 ft Circumference=___________________ 192 ft2 Area=____________________ ----------------------------------------------------------------------------------------------------------------------------------- 5) BA = 21 mi XC = 18 mi 108 mi Circumference=___________________ 972 mi2 Area=____________________ Table Practice Table Practice Volume Definition for Volume: The amount of 3dimensional space an object occupies. Formulas for Volume: Cube: V = s3 Volume = side x side x side s Rectangular Prism: V = lwh Volume = length x width x height s s h l w Volume Practice 24 mi3 512 cm3 360 km3 44,268 ft3 64 cm3 1,386 in.3 Perimeter units Area units Volume 2 3 units 1