7G-Solve+real-life+and+mathematical+problems+involving+angle+measure,+area,+surface+area,+and+volume

7.MP.1. Make sense of problems and persevere in solving them. 7.MP.2. Reason abstractly and quantitatively. 7.MP.3. Construct viable arguments and critique the reasoning of others. 7.MP.4. Model with mathematics. 7.MP.5. Use appropriate tools strategically. 7.MP.6. Attend to precision. 7.MP.7. Look for and make use of structure. 7.MP.8. Look for and express regularity in repeated reasoning. **7G 5** 7.MP.3. Construct viable arguments and critique the reasoning of others. 7.MP.4. Model with mathematics. 7.MP.5. Use appropriate tools strategically. 7.MP.6. Attend to precision. 7.MP.7. Look for and make use of structure. **7G 6** 7.MP.1. Make sense of problems and persevere in solving them. 7.MP.2. Reason abstractly and quantitatively. 7.MP.3. Construct viable arguments and critique the reasoning of others. 7.MP.4. Model with mathematics. 7.MP.5. Use appropriate tools strategically. 7.MP.6. Attend to precision. 7.MP.7. Look for and make use of structure. 7.MP.8. Look for and express regularity in repeated reasoning. || ===‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍**Learning Target/Task Analysis**===
 * ===**Common Core Standard**===
 * 7.G.4.** Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
 * 7.G.5.** Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
 * 7.G.6.** Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. || ===‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍**Anchor Standard/Mathematical Practice(s)**===
 * 7G 4**
 * ===‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍**Information Technology Standard**=== || ===‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍**Revised Bloom's Level of thinking**=== ||
 * 7.G.4 Students understand the relationship between radius and diameter. Students also understand the ratio of circumference to diameter can be expressed as Pi. Building on these understandings, students generate the formulas for circumference and area.**

The illustration shows the relationship between the circumference and area. If a circle is cut into wedges and laid out as shown, a parallelogram results. Half of an end wedge can be moved to the other end a rectangle results. The height of the rectangle is the same as the radius of the circle. The base length is 1/2 the circumference (2Πr). The area of the rectangle (and therefore the circle) is found by the following calculations:



Students solve problems (mathematical and real-world) including finding the area of left-over materials when circles are cut from squares and triangles or from cutting squares and triangles from circles.

“Know the formula” does not mean memorization of the formula. To “know” means to have an understanding of why the formula works and how the formula relates to the measure (area and circumference) and the figure. This understanding should be for all students.


 * Example 1:** The seventh grade class is building a mini-golf game for the school carnival. The end of the putting green will be a circle. If the circle is 10 feet in diameter, how many square feet of grass carpet will they need to buy to cover the circle? How might someone communicate this information to the salesperson to make sure he receives a piece of carpet that is the correct size? Use 3.14 for pi.

Solution: Area = Πr2 Area = 3.14 (5)2 Area = 78.5 ft2

To communicate this information, ask for a 9 ft by 9 ft square of carpet.


 * Example 2:** The center of the circle is at (5, -5). What is the area of the circle?

Solution: The radius of the circle is 4. Using the formula, Area = Πr2, the area of the circle is approximately 50.24 units2. Students build on their understanding of area from 6th grade to find the area of left-over materials when circles are cut from squares and triangles or when squares and triangles are cut from circles.


 * Example 3:** If a circle is cut from a square piece of plywood, the length of one side is 28 ft,how much plywood would be left over?

Solution: The area of the square is 28 x 28 or 784 in2. The diameter of the circle is equal to the length of the side of the square, or 28”, so the radius would be 14”. The area of the circle would be approximately 615.44 in2. The difference in the amounts (plywood left over) would be 168.56 in2 (784 – 615.44).

Solution: The ends of the track are two semicircles, which would form one circle with a diameter of 62m. The circumference of this part would be 194.68 m. Add this to the two lengths of the rectangle and the perimeter is 2194.68 m
 * Example 4:** What is the perimeter of the inside of the track?

“Know the formula” does not mean memorization of the formula. To “know” means to have an understanding of why the formula works and how the formula


 * 7.G.5 Students use understandings of angles and deductive reasoning to write and solve equations.**


 * Example1:** Write and solve an equation to find the measure of angle x.

Solution: Find the measure of the missing angle inside the triangle (180 – 90 – 40), or 50°. The measure of angle x is supplementary to 50°, so subtract 50 from 180 to get a measure of 130° for x.

Solution: First, find the missing angle measure of the bottom triangle (180 – 30 – 30 = 120). Since the 120 is a vertical angle to x, the measure of x is also 120°.
 * Example 2:** Find the measure of angle x.

Note: Not drawn to scale.
 * Example 3:** Find the measure of angle b.

Solution: Because, the 45°, 50° angles and b form are supplementary angles, the measure of angle b would be 85°. The measures of the angles of a triangle equal 180° so 75° + 85°+ a = 180°. The measure of angle a would be 20°.


 * 7.G.6 Students continue work from 5th and 6th grade to work with area, volume and surface area of two-dimensional and three-dimensional objects. (composite shapes) Students will not work with cylinders, as circles are not polygons.**

“Know the formula” does not mean memorization of the formula. To “know” means to have an understanding of why the formula works and how the formula relates to the measure (area and volume) and the figure. This understanding should be for all students. Surface area formulas are not the expectation with this standard. Building on work with nets in the 6th grade, students should recognize that finding the area of each face of a three-dimensional figure and adding the areas will give the surface area. No nets will be given at this level.

Surface area formulas are not the expectation with this standard. Building on work with nets in the 6th grade,students should recognize that finding the area of each face of a three-dimensional figure and adding the areas will give the surface area. No nets will be given at this level; however, students could create nets to aid in surface area calculations. Students understanding of volume can be supported by focusing on the area of base times the height to calculate. Students solve for missing dimensions, given the area or volume.


 * Example 2:** A triangle has an area of 6 square feet. The height is four feet. What is the length of the base?

Solution: One possible solution is to use the formula for the area of a triangle and substitute in the known values, then solve for the missing dimension. The length of the base would be 3 feet.


 * Example 3:** The surface area of a cube is 96 in2. What is the volume of the cube?

Solution: The area of each face of the cube is equal. Dividing 96 by 6 gives an area of 16 in2 for each face. Because each face is a square, the length of the edge would be 4 in. The volume could then be found by multiplying 4 x 4 x 4 or 64 in3.

Solution: The surface area can be found by using the dimensions of each face to find the area and multiplying by 2: Front: 7 in. x 9 in. = 63 in2 x 2 = 126 in2 Top: 3 in. x 7 in. = 21 in2 x 2 = 42 in2 Side: 3 in. x 9 in. = 27 in2 x 2 = 54 in2
 * Example 4:**Huong covered the box to the right with sticky-backed decorating paper. The paper costs 3¢ per square inch. How much money will Huong need to spend on paper?

The surface area is the sum of these areas, or 222 in2. If each square inch of paper cost $0.03, the cost would be $6.66.

Solution: Volume can be calculated by multiplying the area of the base (triangle) by the height of the prism. Substitute given values and solve for the area of the triangle V = Bh 3,240 cm3 = B (30cm) 3,240 cm3 = B(30cm) 30 cm 30 cm 108 cm2 = B (area of the triangle)
 * Example 5:** Jennie purchased a box of crackers from the deli. The box is in the shape of a triangular prism (see diagram below). If the volume of the box is 3,240 cubic centimeters, what is the height of the triangular face of the box? How much packaging material was used to construct the cracker box? Explain how you got your answer.

To find the height of the triangle, use the area formula for the triangle, substituting the known values in the formula and solving for height. The height of the triangle is 12 cm. The problem also asks for the surface area of the package. Find the area of each face and add:

2 triangular bases: ½ (18 cm)(12 cm ) = 108 cm2 x 2 = 216 cm2 2 rectangular faces: 15 cm x 30 cm = 450 cm2 x 2 = 900 cm2 1 rectangular face: 18 cm x 30 cm = 540 cm2 Adding 216 cm2 + 900 cm2 + 540 cm2 gives a total surface area of 1656 cm2.

===‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍**I can draw, construct, and describe geometric figures and describe the relationships between the figures.**===


 * (7G4) I can explain the formulas for area and circumference of circles.**
 * (7G5) I can write and solve simple equations for an unknown angle in a figure.**
 * (7G6) I can solve real-world math problems involving area, volume, and surface area of 2-D and 3-D objects.**

‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍**Essential Vocabulary**

 * supplementary, complementary, adjacent, circumference, volume, surface area, diameter, radius,** i**nscribed, circumference, radius, diameter, pi, Π, supplementary,**
 * vertical, adjacent, complementary, pyramids, face, base**

‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍**Differentiation**

 * peer tutoring, grouping, blogs, guided notes, foldables, justification of answers, inquiry based learning to accelerate/remediate, draw and label shapes when solving for area and volume, use real 3-D figures to manipulate when solving surface area, create 3-D figures out of paper to show the different views, create a blueprint of the classroom**

‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍**Intervention:**

 * [|DPI math resources]**
 * Building Blocks--Area, Volume, Surface Area**

‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍**Enrichment:**
===‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍**Instructional Resources**===

http://www.dpi.state.nc.us/curriculum/mathematics/middlegrades/grade07/ http://qta.quantiles.com/qtaxon/goal/29765/ http://qta.quantiles.com/qtaxon/goal/29766/ http://qta.quantiles.com/qtaxon/goal/29767/ http://kutasoftware.com/freeipa.html http://www.montgomery.k12.nc.us/18032092913393583/site/default.asp
 * Math Partners resources,**
 * shapes for students to cut out for scale, yardstick, 3-D figures, 3-D cutouts,**
 * http://www.mathgoodies.com,**

[[file:7_Math_Gfinal.docx]]
===‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍**Notes and Additional Information**===