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Essential Questions for students (objectives): How can you write a thorough mathematical explanation when solving problems? How does using precise geometric vocabulary assist in communication?
Supplies: Sets of Tangrams, Reasoning graphic organizer, problem-solving graphic organizer, Explaining Your Work rubric
Instructional format: Whole group and/or small groups
Time needed: 60 minutes
Vocabulary for Word Wall: attribute words – congruent, acute, obtuse, parallel, isosceles, equilateral, rectangle, triangle, perpendicular, rhombus, square, parallelogram
Prior Knowledge/Possible Warm-up Activities: Review of attribute vocabulary. Look at each shape of a Tangram puzzle – students describe each shape using as many vocabulary words as possible. Compare and contrast multiples shapes using key vocabulary.
Description of Lesson:
Whole group lesson –
1) Explain to students that they are going to be listening to a story that has a lot of cause and effect relationships. Read Grandfather Tang’s Story to the class. Ask the class to discuss what cause and effect relationships that they heard. Pass out the story graphic organizer. This time when you read the story students are going to listen for an event that occurs and the reasoning why the event occurs and write it down on the graphic organizer. Students need to listen for 5 examples, so they don’t have to write them all down. I would stop at the first example in the story and model an example: EVENT: Chou changed himself into a dog, WHY? Because Wu Ling had changed himself into a rabbit and dogs are more ferocious than rabbits (and chase rabbits) and Chou wanted to out-do Wu Ling.
2) Have students share their examples in partners, but direct them to look for FULL explanations in the WHY? column. Poor example: Event: Wu Ling changed himself into a goldfish, WHY? Because Chou was a crocodile. I would re-read those pages and ask what part of the reasoning is missing? They could add that Wu Ling was caught in the crocodile’s mouth as a turtle, so he became a goldfish to slip between the crocodile’s teeth and still swim away under water. Have students provide their quality examples for the class and have the class weigh in on if the reasons for why were fully described. Point out reasoning in the Process Standards (Mathematical Practice number 1). Students may need to reference the text on their own.
3) The purpose of the story and the following exercise are to prepare students to write out complete explanations for math problems. Pass out a word problem to the class and model together how you write a complete solution (use problem-solving handout). It isn’t enough to just write the steps to solve a problem, but you need to write WHY you chose to do that particular step. For example, if you wrote I multiplied 38 x 4, you would need to write – I multiplied because the problem said that there were 4 equal groups of 38 which means that you have 4 sets of 38 and that is found by multiplying.
4) Give out a new problem and have students work on it partners. Have partner sets exchange solutions and provide feedback using the “Explaining Your Work” rubric on whether the reasoning was explained fully. Ask partners to share their best reason that they wrote in their problem to the entire class.
5) It isn’t necessary for students to always write out problems in their manner, but when real problem solving is going on, it is important for them to fully explain their work and reasoning so that many different ways to solve can be understood and explored. That is what true mathematicians do!
Small group/Center lesson –
1) As a whole group, explain Tangrams puzzle (see prior knowledge activity). Read Grandfather Tangs Story showing the pictures of the tangrams made into animals.
2) Center #1 – Looking at the book, have students pick one or two shapes made by Grandfather Tang and describe (and write down) how the shapes are arranged and related to one another. For example: The crocodile – All shapes except the parallelogram are arranged in a straight line to make the body. The head starts with one of the large isosceles triangles sitting with the long side as the base. Next to it on the right is the smallest isosceles triangle lined up that it’s long side is laying on the large triangle. It’s right angle is pointing up and to the right. Etc.
3) Center #2 – Students work in partners. One of the students creates a shape (it can be an animal) with the tangrams. They put a large divider between them, so that they can’t see each other’s creation (a large book or cardboard divider works well). The student who made the creation describes their creation using geometric words to their partner, while their partner tries to build it. The student who made the creation is the only one that can talk in an unlimited manner. The student making the creation can only say the following: please repeat and can you say it another way. When they are done, they can move the divider and see how they did. When they are finished they should reflect and write down – what directions were the most helpful? What words were used the most and helped the building?
***For younger grades – students can use tangrams for decomposition/composition of shapes. Can you use 2 or 3 shapes to make a rectangle?
Interdisciplinary connections –
1) ELA – read Chasing Vermeer by Blue Bailliett as a novel study. The book uses tangrams as part of the mystery that is being solved. In addition, students can examine the logic and reasoning that the main characters use to solve the mystery. They can use the event and why graphic organizer with the story.
2) Science – cause and effect work in science in most areas – natural disasters, predator/prey, habitats, etc. Students can read a science article and look for cause and effect with the event and why graphic organizer.
3) Social Studies – What cause and effect relationships happened in your state history? What were the results of those situations.
Assessment (Acceptable Evidence): For the group problem-solving lesson use the “Explaining Your Work” rubric. Students should score themselves and one another when problem-solving. Small group – reflective questions – what vocabulary helped the most? What directions made the information clear?