Harnessing the Power of the Purposeful Task Graham Fletcher gfletchy@gmail.com @gfletchy www.gfletchy.com
? 3 Questions
? 1 Billion Circles
1 Billion Circles • 100 circles : minute • 144,000 circles : day • 1,000,000,000 would take 6944 days • 19+ years with no sleep
? 2nd Question
Where does 1 billion go on the number line? 0 1 trillion
Where does 1 billion go on the number line? 0 1 trillion
? What is mathematical modeling What is modeling with mathematics
Today’s Goals • Understand the structure of 3-act task and see how they fit into the scope and sequence of a unit. • Investigate mathematical modeling (SMP #4) • Explore the importance of progressional understanding and how a good task can be used as formative assessment. • Multiplication • Understand the importance of an e ff ective closing and the role it plays in deciding our next move.
Procedural Conceptual Fluency Understanding Application
Demetrius has 17 Skittles which is 12 fewer than Alicia. How many Skittles does Alicia have?
Demetrius has 17 Skittles which is 12 fewer than Alicia. How many Skittles does Alicia have?
Demetrius has 17 Skittles which is 12 fewer than Alicia. How many Skittles does Alicia have?
Demetrius has 17 Skittles which is 12 fewer than Alicia. How many Skittles does Alicia have?
Demetrius has 17 Skittles which is 12 fewer than Alicia. How many Skittles does Alicia have?
17 12 fewer
17 12
17 12 W T F ?
17 12 W T F ? hat’s he ive
Current Research
65%
“ 65% of children entering primary school ” today will ultimately end up working in completely new job types that don’t yet exist. http://reports.weforum.org/future-of-jobs-2016/chapter-1-the-future-of-jobs-and-skills/
“ ” Early mathematics competency predicts later reading achievement better than early literacy skills. A. Szekely. Unlocking Young Children’s Potential: Governors’ Role in Strengthening Early Mathematics Learning (Washington, D.C.: National Governors Association Center for Best Practices, October 28, 2014).
You little plucker! John SanGovanni, NCTM ‘16
number You little plucker! John SanGovanni, NCTM ‘16
?
58 Packages
14
The Big Reveal
Graham had 58 packages of Skittles. Each package had 14 Skittles. How many Skittles did Graham have?
3-Act Tasks Act 1: • Real world problem or scenario presented • What do you notice? What do you wonder? • Make estimates Act 2: • Identify missing variables and missing variables to solve • Define solution path using variables Act 3: • Solve and interpret results of the solution • Validate answer
Most asked questions: • How often should we use 3-Act Tasks? • When should we use 3-Act tasks? How do they fit into the scope of a unit? • How long does one task usually take? • What if we don’t have the time?
Standards for Mathematical Practice 1. 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.
Standards for Mathematical Practice 4. Model with mathematics.
What is mathematical modeling? What is modeling with mathematics?
What ISN’T mathematical modeling • The use of manipulatives does not ensure that modeling with mathematics is taking place. • If the mathematics is not contextualized, modeling with mathematics cannot exist. • Modeling with mathematics does not mean, “I do, we do, you do.”
Model with Mathematics Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life , society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense , possibly improving the model if it has not served its purpose.
Model with Mathematics Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life , society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense , possibly improving the model if it has not served its purpose.
Mathematical Modeling in the Elementary Grades Contextualized Decontextualized Real-World Make assumptions & Identify missing Problem estimates variables Solve & interpret Define a solution path Validate Answer results using variables
Mathematical Modeling in the Elementary Grades Contextualized Decontextualized Real-World Make assumptions & Identify missing Problem estimates variables Solve & interpret Define a solution path Validate Answer results using variables
Mathematical Modeling in the Elementary Grades Contextualized Decontextualized Real-World Make assumptions & Identify missing Problem estimates variables Solve & interpret Define a solution path Validate Answer results using variables
5 The practices are: 1. Anticipating student responses to challenging mathematical tasks; 2. Monitoring students’ work on and engagement with the tasks; 3. Selecting particular students to present their mathematical work; 4. Sequencing the student responses that will be displayed in a specific order and; 5. C onnecting different students’ responses and connecting the responses to key mathematical ideas. MTMS: Vol. 14, No. 9, May 2009-5 Prac8ces for Orchestra8ng Produc8ve Mathema8cs Discussions
14 Skittles 58 Packages
Identify and name the strategy used, then place the student work in order in terms of efficiency (least to greatest)
Sequence the order students will share during the closing.
Identify and name the strategy used, then place the student work in order in terms of efficiency (least to greatest)
1-skip counting
2-doubling 1-skip counting
3-counting on 2-doubling 1-skip counting
4b-Partial Products 3-counting on 4a-Partial 2-doubling 1-Skip Products counting
4b-Partial Products 5-Partial Products 3-counting on 4a-Partial 2-doubling 1-Skip Products counting
4b-Partial Products 5-Partial Products 3-counting on 4a-Partial 2-doubling 1-Skip Products counting ?
4b-Partial Products 5-Partial Products 3-counting on 4a-Partial 2-doubling Where the Lattice goes 1-Skip Products counting
Which student work samples do you share? Why?
Area Model
4 (x + 3)
? 2 Questions
Surface Area of an Apple
Approximately 0.25”
2.25” 2.5”
2.25” 8.13” 2.5”
Redemption Time
Green Red Diameter 2.5” 3.63” Height 2.25” 3.25” Circumference 8.5” 11.5”
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