Modeling – A Closer Look. Think of an example of modeling that you would share with a colleague who wanted to better understand modeling in the context of a math curriculum. Modeling – A Closer Look. - PowerPoint PPT Presentation
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N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Units
Modeling – A Closer Look
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• Descriptive modeling – using concrete materials or pictorial displays to study quantitative relationships
• Analytic modeling – using graphical representations, equations, or statistical representations to provide analysis, revealing additional insights to the relationships between variables.
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Units
The Modeling Process: G9–12 and Beyond
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1. Choose variables to represent essential features.2. Formulate the model (e.g., create the equations).3. Analyze the model (the relationships depicted) to
draw conclusions.4. Interpret the results in terms of the original situation.5. Validate the conclusions by comparing them to the
situation, then either improve model and repeat.6. Report the results incl. assumptions, approx. made.
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Units
Key Points – Modeling in the Standards
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• Application is the students’ ability to use relevant conceptual understandings and appropriate strategies and tools even when not prompted to do so.
• Modeling can be descriptive, using concrete materials or pictorial displays to study quantitative relationships, or analytical, using equations or statistical representations to provide analysis of the relationships between variables (quantities).
• A curriculum rich in application, including modeling, provides coherence from PK – 12 and beyond, to college and career.
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Units
Using Tape Diagrams
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• Promote perseverance in reasoning through problems.
• Develop students’ independence in asking themselves:• “Can I draw something?”• “What can I label?”• “What do I see?” • “What can I learn from my drawing?”
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Units
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Example 13: Alex bought some chairs. One third of them were red and one fourth of them were blue. The remaining chairs were yellow. What fraction of the chairs were yellow?
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Units
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Example 15: Max spent 3/5 of his money in a shop and ¼ of the remainder in another shop. What fraction of his money was left? If he had $90 left, how much did he have at first?
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Units
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Example 16: Henry bought 280 blue and red paper cups. He used 1/3 of the blue ones and ½ of the red ones at a party. If he had an equal number of blue cups and red cups left, how many cups did he use altogether?
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Units
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Example 17: A club had 600 members. 60% of them were males. When 200 new members joined the club, the percentage of male members was reduced to 50%. How many of the new members were males?
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Units
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Example 18: The ratio of the length of Tom’s rope to the length of Jan’s rope was 3:1. The ratio of the length of Maxwell’s rope to the length of Jan’s rope was 4:1. If Tom, Maxwell and Jan have 80 feet of rope altogether, how many feet of rope does Tom have?
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Units
Key Points – Writing Word Problems
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• Tape diagrams are well suited for problems that provide information relative to the whole or comparative information of two or more quantities.
• Visual fraction models includes: tape diagrams, number line diagrams, and area models.
• When designing a word problem that is well supported by a tape diagram, sketch the diagram for the problem before or as your write the problem itself.
N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Units
Key Points
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• Use of tape diagrams, as described in the progressions documents provides visualization of relationships between quantities thereby promoting conceptual understanding, provides coherence through standards from Grade 1 through Grade 7, and supports standards for mathematical practice.
• Proficiency in the tape diagram method can be developed in students and teachers new to the process through a natural development of problems and representations.
• Content knowledge directed by the standards and the progressions is required to provide coherent and balanced instruction.