5 th Grade Mathematics


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1 5 th Grade Mathematics Instructional Week 20 Rectilinear area with fractional side lengths and realworld problems involving area and perimeter of 2dimensional shapes Paced Standards: 5.M.2: Find the area of a rectangle with fractional side lengths by modeling with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. 5.M.3: Develop and use formulas for the area of triangles, parallelograms and trapezoids. Solve realworld and other mathematical problems that involve perimeter and area of triangles, parallelograms and trapezoids, using appropriate units for measures. + Connections to other 5 th Grade Standards: 5.AT.3: Solve realworld problems involving multiplication of fractions, including mixed numbers (e.g., by using visual fraction models and equations to represent the problem). 5.NS.2: Explain different interpretations of fractions, including: as parts of a whole, parts of a set, and division of whole numbers by whole numbers. 5.AT.1: Solve realworld problems involving multiplication and division of whole numbers (e.g. by using equations to represent the problem). In division problems that involve a remainder, explain how the remainder affects the solution to the problem. Prerequisite/Foundational Standards: 4.M.4: Apply the area and perimeter formulas for rectangles to solve realworld problems and other mathematical problems. Recognize area as additive and find the area of complex shapes composed of rectangles by decomposing them into nonoverlapping rectangles and adding the areas of the nonoverlapping parts; apply this technique to solve realworld problems and other mathematical problems. 3.M.5: Find the area of a rectangle with wholenumber side lengths by modeling with unit squares, and show that the area is the same as would be found by multiplying the side lengths. Identify and draw rectangles with the same perimeter and different areas or with the same area and different perimeters. 3.M.6: Multiply side lengths to find areas of rectangles with wholenumber side lengths to solve realworld problems and other mathematical problems, and represent wholenumber products as rectangular areas in mathematical reasoning. 3.M.7: Find perimeters of polygons given the side lengths or by finding an unknown side length. 3.AT.1: Solve realworld problems involving addition and subtraction of whole numbers within 1000 (e.g., by using drawings and equations with a symbol for the unknown number to represent the problem). 3.AT.5: Determine the unknown whole number in a multiplication or division equation relating three whole numbers. 5 th Grade ISTEP+ Toolkit
2 Teacher Background Information 5.M.2: Find the area of a rectangle with fractional side lengths by modeling with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. This standard extends students work with area. In third grade students determine the area of rectangles. In fourth grade students continue this work with composite rectangles. The fifth grade standard calls students to continue the process of covering (with tiles). Grids (see picture) below can be used to support this work. Example: The home builder needs to cover a small storage room floor with carpet. The storage room is 4 meters long and half of a meter wide. How much carpet do you need to cover the floor of the storage room? Use a grid to show your work and explain your answer. In the grid below, I shaded the top half of 4 boxes. When I added them together, I added ½ four times, which equals 2. I could also think about this with multiplication ½ x 4 is equal to 4/2 which is equal to 2. Example: In solving the problem 2 x 4, students use an area model to visualize it as a 2 by 4 array of small rectangles each of 3 5 which has side lengths of 1/3 and 1/5. They reason that 1/3 x 1/5 = 1/(3 x 5) by counting squares in the entire rectangle, so the area of the shaded area is (2 x 4) x 1/(3 x 5) = 2x4. They can explain that the product is less than 4 3x5 5 because they are finding 2 of 4. They can further estimate that the answer must be between 2/5 and 4/5 because 3 5 2/3 of 4/5 is more than ½ of 4/5 and less than one group of 4/5. Process Standards to Emphasize with Instruction of 5.M.2: 5.PS.1: Make sense of problems and persevere in solving them. 5.PS.2: Reason abstractly and quantitatively. 5.PS.4: Model with mathematics. 5.PS.7: Look for and make use of structure. 5.PS.8: Look for and express regularity and repeated reasoning.
3 Teacher Background Information 5.M.3: Develop and use formulas for the area of triangles, parallelograms and trapezoids. Solve realworld and other mathematical problems that involve perimeter and area of triangles, parallelograms and trapezoids, using appropriate units for measures. Teacher Background Information: Students continue to understand that area is the number of squares needed to cover a plane figure. Students should know the formulas for rectangles. Knowing 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 the figure. This understanding should be for all students. Finding the area of triangles is introduced in relationship to the area of rectangles a rectangle can be decomposed into two congruent triangles. Therefore, the area of the triangle is ½ the area of the rectangle. The area of a rectangle can be found by multiplying base x height; therefore, the area of the triangle is ½ bh or (b x h)/2. The following site helps students to discover the area formula of triangles. Students decompose shapes into rectangles and triangles to determine the area. For example, a trapezoid can be decomposed into triangles and rectangles (see figures below). Using the trapezoid s dimensions, the area of the individual triangle(s) and rectangle can be found and then added together. Isosceles trapezoid Right trapezoid Note: Students recognize the marks on the isosceles trapezoid indicating the two sides have equal measure. Example: 1: Find the area of a right triangle with a base length of three units, a height of four units, and a hypotenuse of 5. Solution: Students understand that the hypotenuse is the longest side of a right triangle. The base and height would form the 90 angle and would be used to find the area using: A = ½ bh A = ½ (3 units)(4 units) A = ½ 12 units 2 A = 6 units 2
4 Teacher Background Information 5.M.3: Develop and use formulas for the area of triangles, parallelograms and trapezoids. Solve realworld and other mathematical problems that involve perimeter and area of triangles, parallelograms and trapezoids, using appropriate units for measures. Example 2: Find the area of the trapezoid shown below using the formulas for rectangles and triangles. Solution: The trapezoid could be decomposed into a rectangle with a length of 7 units and a height of 3 units. The area of the rectangle would be 21 units 2. The triangles on each side would have the same area. The height of the triangles is 3 units. After taking away the middle rectangle s base length, there is a total of 5 units remaining for both of the side triangles. The base length of each triangle is half of 5. The base of each triangle is 2.5 units. The area of one triangle would be ½ (2.5 units)(3 units) or 3.75 units 2. Using this information, the area of the trapezoid would be: Example 3: = units 2 A rectangle measures 3 inches by 4 inches. If the lengths of each side double, what is the effect on the area? Solution: The new rectangle would have side lengths of 6 inches and 8 inches. The area of the original rectangle was 12 inches 2. The area of the new rectangle is 48 inches 2. The area increased 4 times (quadrupled). Students may also create a drawing to show this visually. Models for developing formulas for the area of triangles, parallelogram and trapezoids.
5 Teacher Background Information 5.M.3: Develop and use formulas for the area of triangles, parallelograms and trapezoids. Solve realworld and other mathematical problems that involve perimeter and area of triangles, parallelograms and trapezoids, using appropriate units for measures. Here are a few suggestions, each leading to a different approach to finding the area of a trapezoid: Make a parallelogram inside the given trapezoid using three of the sides. Make a parallelogram using three sides that surround the trapezoid. Draw a diagonal forming two triangles. Draw a line through the midpoints of the nonparallel sides. The length of that line is the average of the lengths of the two parallel sides. Draw a rectangle inside the trapezoid, leaving two triangles, and then put those two triangles together. Process Standards to Emphasize with Instruction of 5.M.3: 5.PS.1: Make sense of problems and persevere in solving them. 5.PS.2: Reason abstractly and quantitatively. 5.PS.3: Construct a viable argument and critique the reasoning of others. 5.PS.4: Model with mathematics. 5.PS.7: Look for and make use of structure.
6 Name: Grade 5 Instructional Week 20 Assessment 1. A rectangular placemat is covered by 5 1 in. by 5 1 in. square tiles. The diagram below shows a model of the placemat. KEY = 5 1 inch What is the area, in square inches, of the placemat? A) 63 square inches B) 1 square inch C) 2 square inches 25 D) 63 square inch Mr. Lucas wants to cover a rectangular living room floor with 2 1 foot by 2 1 foot square tiles. The diagram below shows the dimensions of the rectangular floor. How many tiles will be needed to cover a 2 ft. by 2 ft. area of the floor? Living Room 8 feet 12 feet 1 A) of a tile 4 B) 4 tiles C) 16 tiles D) 96 tiles
7 3a. Determine the area of a rectangle that is cm wide and cm long. Show all work. Answer: cm 2 3b. How many 4 1 cm squares will be needed to cover the rectangle? Show all work Answer: squares
8 4. The area of triangle ABC is 36 in 2. The length of line segment AB is 12 inches. What is the length of line segment AC? C A B A) 3 inches B) 6 inches C) 12 inches D) 48 inches 5. Determine the area of the trapezoid. 14 units 5 units 6 units 8 units A) 110 square units B) 55 square units C) 38 square units D) 32 square units
9 6. Which of the following is true for the shape below. A) The area of triangle WXY is equal to the area of parallelogram WXYZ. B) The area of triangle WYZ is twice the area of parallelogram WXYZ. C) The sum of the area of triangle WXY and triangle WZY is equal to the area of parallelogram WXYZ. D) The area of parallelogram WXYZ is half the area of triangle WYZ. 7. B E C A F D AD is 25 units CD is 15 units If rectangle ABCD has the side lengths shown above, what is the area of triangle ADE? Answer:
10 Grade 5 Instructional Week 20 Assessment Answer Key Question Correct Answer Standard(s) 1 C 5.M.2 2 C 5.M.2 3a 5 5.M.2 9 cm 2 8 3b 154 squares 5.M.2 4 B 5.M.3 5 B 5.M.3 6 C 5.M M square units Content Rubric A score of two indicates a thorough understanding of the mathematical concepts embodied in the task. The response shows content related work executed correctly and completely. A score of one indicates a partial understanding of the mathematical concepts embodied in the task. The response contains errors in the content related work A score of zero indicates limited or no understanding of the mathematical concepts embodied in the task.
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