Grade 5 Unit 5 Addition and Multiplication with Volume and Area (5 Weeks)

Transcription

1 Grade 5 Unit 5 Addition and Multiplication with Volume and Area (5 Weeks) Stage Desired Results Established Goals Unit Description Students will utilize the work done in the fraction unit to explore how the area changes when a rectangle is scaled by a whole or fractional scale factor. Measuring volume once again highlights the unit theme as a unit cube is chosen to represent a volume unit and used to measure the volume of simple shapes made out of rectangular prisms. The Mathematical Practices should be evident throughout instruction and connected to the content addressed in this unit. Students should engage in mathematical tasks that provide an opportunity to connect content and practices. Common Core Learning Standards 5.NF.4b: Find the area of a rectangle with fractional side lengths by tiling it 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.MD.3: Recognize volume as an attribute of solid figures and understand concepts of volume measurement. a. A cube with side length unit, called a unit cube, is said to have one cubic unit of volume, and can be used to measure volume. b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units. 5.MD.4: Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. 5.MD.5: Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication. b. Apply the formulas V=l w h and V=b h for rectangular prisms to find volumes of right rectangular prisms with whole- number edge lengths in the context of solving real world and mathematical problems. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems. 5.G.3: Understand that attributes belonging to a category of two- dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles. 5.G.4: Classify two-dimensional figures in a hierarchy based on properties. Common Core Standards of Mathematical Practice. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. ESL Language Standards Standard : Students will listen, speak, read, and write in English for information and understanding... Identify and use reading and listening strategies to make text comprehensible and meaningful.

2 .3 Select information appropriate to the purpose of the investigation, relate ideas from one written or spoken source to another, and exclude nonessential information..5 Formulate, ask, and respond to various question forms to obtain, clarify, and extend information and meaning..7 Present information clearly in a variety of oral and written forms for different audiences and purposes related to all academic content areas..9 Convey and organize information, using facts, details, illustrative examples, and a variety of patterns and structures..6 Apply learning strategies to acquire information and make texts comprehensible and meaningful. Bridge Guidance: 5.NF.4 carries over from the previous unit, and students should have had experience with elements of 5.G.4 categorizing 2D shapes based on their attributes from prior years. Big Ideas. 2 and 3-dimensional objects with or without curved surfaces can be described, classified, and analyzed by their attributes. 2. Some attributes of objects are measurable and can be quantified using unit amounts. Content (Students will know.) A. The area of a rectangle can be found by multiplying length x width. Area is represented as units squared (u 2 ) (5.NF.4) B. A cube with side length unit, called a unit cube, is said to have one cubic unit of volume, and can be used to measure volume. Volume is represented as units cubed (u 3 ) (5.MD.3) C. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units. (5.MD.3, 5.MD.4) D. The volume of a right rectangular prism can be found by computing the area of layer and multiplying it by the height (l x w x h = b x h) (5.MD.5) Essential Questions. How can we analyze the attributes of 2D and 3D shapes?. How does analyzing shapes help us to classify them?. Why is volume represented with cubic units and area represented with square units? 2. What is volume? How do we measure volume? 2. How are area and volume related? 2. How can addition and multiplication help us compute volume efficiently? 2. How can rectangular prisms be decomposed to help us understand volume? 2. Does volume change when you change the measurement material? Skills (Students will be able to ) A. Determine the area of rectangles with both whole number or fractional lengths using the formula: A rec = length x width A2. Find the area of rectangles by tiling with unit squares and prove that that area is the same as if using the formula A3. Represent area using an array. Relate this array to the formula. B. Recognize that the volume of an object can be found by counting the number of unit cubes you can pack inside it (without gaps and overlaps) C. Measure volume by counting unit cubes (cm, in, ft, and improvised units) D. Build various rectangular prisms out of cubes, layer by layer to construct the relationship between volume of the whole and the area of one layer D2. Use the formula for volume of a prism to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems. D3. Use the associative property of multiplication to solve for volume in a number of ways

3 Ex: (2 x 9) x 2 = 2x (9x2) 08 x 2 = 2 x 8 E. Recognize volume as additive. Volumes of solid figures composed of two non-overlapping right rectangular prisms can be found by adding the volumes of the non-overlapping parts. (5.MD.5) F. Understand that attributes belonging to a category of two- dimensional figures also belong to all subcategories of that category. (5.G.3 5.G.4) E. Find volumes of solid figures composed of two nonoverlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems. F. Classify two-dimensional figures in a hierarchy based on properties. F2. Argue that a shape can have multiple names as long as it meets the criteria (it has the properties) of those other shapes Terms/ Vocabulary: 2-dimensional, 3-dimensional, volume, area, quadrilateral, quadrangle, parallelogram, rhombus (rhombi), rectangle, square, triangle, pentagon, hexagon, octagon, right angle, prism, criteria, hierarchy, right rectangular prism, properties, classify Stage 2 Assessment Evidence Initial Performance Task: Measuring Around the School Final Performance Task: Hall of Gems Other Evidence Teacher observation, conferencing, teacher designed assessment pieces, student work, exit slips, journal entries Stage 3 Learning Plan Everyday Mathematics/Impact Mathematics Lessons The following lessons may support some of the CCLS & essential questions outlined in this unit map: 5MD.3: 9.3, 9.4, 9.8, 9.9, ,.3, Project 9 5MD.4: 9.3, 9.8, 9.0, 0.,.3 5MD.5: 9.8, 9.9, 0.,.,.3..5,.7, Project 9 5G.3: 3.4, 3.7, 3.8, 4., 5., 5,6, 6.8, 7., 8.3 5G.4: 3.7, 3.8, 4., 8.3 5NF.4: 8.5, 8.6, 8,7, 8.8,.7 Additional Resources: Georgia Unit 6 Content Frameworks: 2 Dimensional Shapes df Georgia Unit 7 Content Framework: Volume and Measurement df North Carolina Unpacked Standards: all 5 th grade Common Core Standards broken down into content and skill requirements Quadrilateral Quest (can be used w/ the Georgia investigations on rectangles and quadrilaterals): This website helps students practice classifying quadrangles based on a bulleted list of properties. HOWEVER- it never names the quadrangles (for example * 4 sides * pair of parallel sides - the site never says that s called a trapezoid. You might want the students to name the category presented on each page based on the criteria.

4 Dynamic Rectangle: from NCTM can be used with the parallelogram and rectangle lesson from the Georgia unit Utah Education Network Curriculum Guides, activities, lesson plans, videos: 5.NF.4b: 5.MD.3, 4, 5: rdid=74 5.G.3, 4: rdid=753

5 Name: Date: Grade 5 Unit 5 Initial Task: Measuring Around the School. Leo and John drew some shapes: Leo says they are all rectangles. John says they are all parallelograms. Who is correct? Why? Justify your thinking using mathematical vocabulary. 2. Examine the shapes. Circle all of the rhombi. Explain how you know they are rhombi.

6 3. PS 23 is planning on building a new basketball court. They need find its area, so they can order enough asphalt to pave the basketball court. Basketball Court Jenny says you can find the area by adding 5 + ¼+ ¼ + ¼ + ¼ + ¼ + ½ + ½ + ½ + /8 a) Using the diagram, prove whether she is right or wrong. b) Using multiplication, show Jenny a different way to compute the area of the basketball court. 4. Anna is stacking cube shaped cushions for the meeting area. Each cushion is cubic foot. Look at the diagram below. What is the volume of the stacked cushions in cubic feet?

7 5. Circle the expression that you can use to find the volume of the right rectangular prism below: a) 5 x 6 b) 2 x 3 x 5 c) 5 x 2 d) both A and C e) both B and C f) All of the above g) None of the above How do you know that your answer to #5 is correct? Explain your thinking: 6. The school is getting a full cleaning for spring break. Class 4B packed the classroom library into boxes that were cubic foot each. They packed a total of 60 boxes. They found an old bookcase to store the boxes in. Is the bookcase large enough to hold all of the boxes? How do you know? Explain your thinking with numbers and words.

8 7. In science class Maya and Jessica are trying to find the volume of a rectangular prism with a length of 7 inches, and width of 4 inches, and a height of 2 inches. Maya says you can find the volume by multiplying 84 x 4. Jessica says you can find the volume by multiplying 28 x 2. Using the associative property of multiplication, explain why both students are correct.

9 Grade 5 Unit 5 Initial Assessment Scoring Guide Measuring Around The School Scoring Guide Points Section Points. (5.G.3, 5.G.4) Leo and John are both correct. (pt) student must explain using properties why. For example: They are all rectangles because they all have opposite sides that are congruent and parallel, 4 sides, and 4 right angles (pt). They are also parallelograms because all rectangles are parallelograms (they have 2 pairs of parallel sides). (pt) Note: It s not enough if students write They meet the criteria for both rectangles and parallelograms without explaining why? 3 2. (5.G.4) a) Shapes, 4, and 5 are classified as rhombi. or points for all 3 rhombi and NO incorrect responses. point for 2 correct responses and NO incorrect responses b) All the shapes have 4 equal sides and 4 equal angles (or similar, mathematically correct response) 3. (5.NF.4b) Jenny says you can find the area by adding 5 + ¼+ ¼ + ¼ + ¼ + ¼ + ½ + ½ + ½ + /8 a) Use the array to prove why she is correct: 3 2 (or other correct response) b) 3 ¼ x 5 ½ =7 7/8 sq yds. OR 3/4 x /2 = 43/8 =7-7/8 sq yds. (give one point if they have the correct process, but make a small computation error) 4. (5.MD.3) The volume is 32 ft 3 4 x 4 x 2 = 32 (or students can count each cube) 5. (5.MD.5) Correct answer is f - all of the above. The volume of the prism is 5x3x2 = 30u 3 because volume = l x w x h or area of base x h 5x6=5x2=5x3x2 or other correct response 6. (5.MD.4, 5.MD.5) Volume of left prism: 6x 2 x 3 = 36ft 3 Volume of right prism: Students have to figure out the missing measurement for small section s length of 4 ft, then use that information to find the total volume 4 x 4 x 2 = 32 ft 3 Correct explanation: = 68 ft 3 68ft 3 > 60ft 3, therefore no, it s not big enough. 7. (5.MD.5) The problem can be solved as 4 x 7 x 2. Due to the associative property of multiplication, we can group these numbers any way. (4x7) x 2 simplifies

10 to 28 x 2, while 4 x(7 x2) simplifies to 4 x 84 Showing all of the steps in a clear way and explaining with words earns a full 2 points, if they are partially correct, you may give point Total Points 7 7 Novice Apprentice Practitioner Expert 5 and less pts

11 Name: Date: Grade 5 Unit 5 Final Assessment Task: Hall of Gems The Gem exhibition at the Museum of Natural History is undergoing renovations.. A scientist wants to display gems with faces shaped like parallelograms. a) Circle the gems that the scientist wants. b) Explain how you decided what gems the scientist needs for her new exhibit. c) The scientist finds one more gem. Its top face is a parallelogram with 4 right angles. Draw 2 examples of what the gem s face might look like. 2. A hands-on display of cube shaped gems is set up for school visits. What is the volume of the display in cubic units? Volume =

12 3. The bottom of a display case is being tiled in black to help the gems stand out. The tiles are square inch each. The length of a display case is 8 ½ inches and the width is 4 ½ inches. Rob, a volunteer, says 8x4 = 32 and ½ x ½ = ¼ so, the area is 32 ¼ square inches. a) Using the array, prove WHY Rob is wrong with any method. b) Write a multiplication equation that will give Rob the correct area for the display case and solve to find the correct area below: Equation: 4. The museum is checking their inventory of gold. Some gold powder is packed into a small case that has a length of 32 mm, a width of 0mm and a depth of 8mm. In order to find how many cubic millimeters of gold there is, one scientist multiplied 320 x 8. To double check, another scientist multiplied 32 x 80. Explain, using the associative property, why both expressions will give them the correct volume of the case.

13 5. The museum is ordering a new display case to hold some of its most valuable gems. To display the necessary gems, the case should have a volume between 70 and 80 cubic feet. Should they order case A or case B and why? Show ALL of your work (not to scale) Display A: Display B: 6. A new microscope arrived in a box that looks like this: a) Which expression is equivalent to the volume of this box? A) 7 x B) 5 x 8 x 7 C) 35 x 8 D) both A and B E) both B and C F) All of the above G) None of the above b) How do you know that your answer to part a is correct?

14 Grade 5 Unit 5 Final Assessment Scoring Guide Hall of Gems Scoring Guide Points Section Points. (5.G.3, 5.G.4) a) Student correctly circles the 3 rd and 5 th gem 4 b) Those gems have faces shaped like parallelograms because they are quadrangles (or have 4 sides) whose opposite sides are equal and parallel. (if the student mentions some of this, you can award point) c) Student correctly draws 2 parallelograms with 4 right angles: Any combination of rectangles or squares (must have 2 correct shapes to get the point) 2. (5.MD.3, 5.MD.4) Student correctly finds the volume of the cube to be 9 cubic units either by multiplying or by counting 2 3. (5.NF.4b) a) Student proves why Rob is wrong on the array by decomposing areas and adding them back up. For example: ¼ = 38 ¼ as shown below: 2 4 (or other correct response) b) Student writes the equation: 4 ½ x 8 ½ = 38 ¼ OR 9/2 x 7/2 = 53/2 = 38 ¼ (award point if the process is correct, but a small computational error was made) 4. (5.MD.5) Student correctly explains that the associative property of multiplication says that when I multiply 3 numbers, I can group them differently and still get the same answer. The first scientist grouped (0 x 32) first then multiplied by 8 whereas the second scientist multiplied 32 by (0 x 80) or a similar explanation. Student may also prove using numbers only for full credit. 5. (5.MD.5) Student correctly explains that the museum should order Case B because Case B has a volume of 78 ft 3 (5 x 4 x3= 60 ft 3 plus 6 x 3 x =8ft 3 ) and 78 is between 70 and 80. They should not order Case A because the volume of case A is only 54ft 3 (2 x 5 x 3 = 30 ft3 plus 2 x 4 x 3 = 24 ft 3) (5.MD.5) a) Student correctly circles E both B and C b) Student explains that they know their answer is correct. For example: shows that both A and C produce the same volume and follows the formula V = l x w x h Total Points

15 Novice Apprentice Practitioner Expert pts