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 • 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. • - See CCLS for Mathematics p. 55-56
Descriptive Modeling • PK-K: Modeling their environment / real world objects using geometric shapes and figures. • K-1: Modeling forms of addition and subtraction situations using concrete materials or diagrams. • 2-5: Place value models, multiplication models (equal groups, array, area), visual fraction models, and number line models. • 6-12: Tables or graphs of observations, geometrical and/or 3-D models of real world objects.
Analytic Modeling • 4-7: Use tape diagrams and/or equations to model relationships between two or more quantities. • 6-12: Use equations and/or functions to model relationships between two or more variables. • 9-12: Use probability models and statistical representations to analyze chance processes.
Modeling – A Closer Look • Summarize with a partner what the key difference is between descriptive and analytic modeling. • The key difference lies in whether the modeling simply facilitates informative visualization, or allows for new and deeper analysis.
The Modeling Process: G9–12 and Beyond Choose variables to represent essential features. Formulate the model (e.g., create the equations). Analyze the model (the relationships depicted) to draw conclusions. Interpret the results in terms of the original situation. Validate the conclusions by comparing them to the situation, then either improve model and repeat. Report the results incl. assumptions, approx. made.
Modeling – A Closer Look • How does knowledge of the modeling cycle and expectations for high school students inform our approach to modeling in grades PK–5?
Key Points – Modeling in the Standards • 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.
AGENDA Application in GK—M5 Modeling in the Standards Building Proficiency with Tape Diagrams Writing Word Problems and Real-World Problems
Using Tape Diagrams • 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?”
Forms of the Tape Diagram 8 ? 5 5 8 ?
Example 1: Sara has 5 stamps. Mark brings her 4 more stamps. How many stamps does Sara have now?
Example 2: Sara has 16 stamps. Mark brings her 4 more stamps. How many stamps does Sara have now?
Example 3: Sara brought 4 apples to school. After Mark brings her some more apples, she has 9 apples altogether. How many apples did Mark bring her?
Example 4: Matteo has 5 toy cars. Josiah has 2 more than Matteo. How many toy cars do Matteo and Josiah have altogether?
Example 5: Jasmine had 328 gumballs. Then, she gave 132 gumballs to her friend. How many gumballs does Jasmine have now?
Example 6: Jose has 4 paper clips. Harry has twice as many paper clips as Jose. How many paper clips does Harry have?
Example 7: Jose has 4 paper clips. Harry has twice as many paper clips as Jose. How many paper clips do they have altogether?
Example 8: William’s weight is 40 kg. He is 4 times as heavy as his youngest brother Sean. What is Sean’s weight?
Example 9: Jamal has 8 more marbles than Thomas. They have 20 marbles altogether. How many marbles does Thomas have?
Example 10: The total weight of a football and 10 tennis balls is 1 kg. If the weight of each tennis ball is 60 g, find the weight of the football.
Example 11: Two pears and a pineapple cost $2. Two pears and three pineapples cost $4.50. Find the cost of a pineapple.
Example 12: David spent 2/5 of his money on a storybook. The storybook cost $20 how much did he have at first?
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?
Example 14: Jim had 360 stamps. He sold 1/3 of them on Monday and ¼ of the remainder on Tuesday. How many stamps did he sell on Tuesday?
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?
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?
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?
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?
Key Points – Proficiency with Tape Diagrams • When building proficiency in tape diagraming skills start with simple accessible situations and add complexities one at a time. • Develop habits of mind in students to reflect on the size of bars relative to one another. • Part-whole models are more helpful when modeling situations where you are given information relative to a whole. • Compare to models are best when comparing quantities.
Standards Calling for Word Problems • K.OA.2 • 1.OA.1 • 1.OA.2 • 2.OA.1 • 2.MD.5 • 2.MD.8
Writing Word Problems and Real-World Problems • 1.OA.1 • 1.OA.2 • 2.OA.1 • 2.MD.5
Writing Word Problems and Real-World Problems • Write a word problem or real-world problem aligned with one of the following standards: • 1.OA.1, 1.OA.2, 2.OA.1, and 2.MD.5 • Exchange problems with a partner and solve using a tape diagram.
Key Points – Writing Word Problems • 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.
Key Points • 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.
Next Steps • How will you share your understanding of modeling with your colleagues? • How will you share your understanding of coherence through application with your colleagues? • How will your school address standards involving the use of tape diagrams with students new to the process?