Mr. and Mrs. Hesslink want to build a sandbox for their son Geoffrey. They bought 5 boards to create the sandbox frame. Each board is 8 feet long. Plastic must also be purchased for the bottom of the sandbox. However, they are not sure how much to buy. First they must determine the size the sandbox is going to be. What are all of the possible rectangular sandboxes that they could make using all of the wood they purchased? Which one do you think they should build? Why? How much plastic do the Hesslinks need to buy to cover the bottom of the sandbox?
This task was given to students as an assessment following a unit on area and perimeter. The students had experiences finding the area and perimeter of geoboard shapes and trains of pattern blocks. They solved Pick's Theorem and other problem-solving tasks related to area and perimeter. Similar problems were done in cooperative groups and discussed as a whole class. This task asks the students to determine the possible perimeters for 40 feet of wood, as well as calculate the area of the rectangles in order to determine the largest one. After finding the area, they then determine the amount of plastic needed. This task also helps students to see the relationship between area and perimeter. There is an underlying pattern in the dimensions possible that allows students to know when they have found all possible combinations. Students first need to determine the amount of wood they have to work with. Then most students begin to draw rectangles that use 40 feet of wood. Once they have found a pattern, they create a chart to help them organize the possibilities so that they know when they have found all possible rectangles. Next students will usually use their table or chart to find the rest of the rectangles so they are able to determine the one with the largest area. After choosing the sandbox they prefer and supporting their decision, they need to state how much plastic is needed.
One - two, 45-minute periods This task could be linked to a unit on playgrounds, toddler care or recreation. Students were invested in finding a solution because the task was an actual problem my husband and I needed to solve. Personalize this task for your own students. Substitute a different family and child's name that the students in your class are familiar with or use one from a popular television program with which the students can identify.
There are 10 possible rectangular boxes 1' x 19', 2' x 18', 3' x 17', 4' x 16', 5' x 15', 6' x 14', 7' x 13', 8' x 12', 9' x 11', 10' x 10'. Most students will choose the 10' x 10' because it is the largest in area. It will require 100 square feet of plastic.
Novice solutions will show no evidence of strategy and no evidence of mathematical reasoning. They might obtain a solution, but have no evidence to support it. There may be diagrams present, as most students begin with drawing diagrams as a strategy, but there will be little or no mathematical language.
Apprentice solutions may have an approach that would work, but may have flaws in their reasoning. They may not work systematically and therefore miss some of the possible solutions. They may neglect to find the amount of plastic needed or may make mistakes in calculations.
Practitioner solutions will have an approach that works and a correct answer for all parts of the problem. They will use appropriate math language throughout and have a well-labeled and accurate representation to communicate an aspect of their solution. Practitioners may also begin to recognize some simple patterns in their solutions.
Expert solutions will have an approach that may be sophisticated and will have correct answers for all part of the problem. The Expert will use precise math language and use representations as a tool in determining a solution. The Expert will also go beyond the task requirements and make mathematically relevant observations and connections.
|
