Engaging Instruction: Rich Problems & Tasks TLQP Thomas F. Sweeney, Ph.D The Sage Colleges

Session Goal Developing a clear picture of the Common Core State Standards by: Using rich problems to understand the Standards for Mathematical Practice Digging into the content standards through the Critical Areas of Focus in order to create instruction based upon the CCSSM and develop a foundation for curriculum revision.

MP + CAF + Standards = Instruction In order to design instruction that meets the rigor and expectations of the CCSSM, understanding the Mathematical Practices and Critical Areas of Focus are essential.

Critical Areas of Focus Critical Areas of Focus inform instruction by describing the mathematical connections and relationships students develop in the progression at this point. Found in CCSSM on the first page of each new grade

Grade Priorities in Support of Rich Instruction and Expectations of Fluency and Conceptual Understanding K–2 Addition and subtraction, measurement using whole number quantities 3–5 Multiplication and division of whole numbers and fractions 6 Ratios and proportional reasoning; early expressions and equations 7 Ratios and proportional reasoning; arithmetic of rational numbers 8 Linear algebra Critical Areas of Focus

Rich Problems: A Wealth of Benefits

A Problem or an Exercise? Problem The answer is not immediately known Requires persistence Engaging Feasible Valued Exercise Computation “problem” Solution process is recognizable Routine Contextual but not engaging

Let’s Warm-up! Activity 1: Complete these two puzzles Which caused more thinking?

Think of a three digit number and write it twice making a six digit number. Now divide it by 7, the answer by 11 and the answer by 13. What do you notice? Why does this happen? Your turn… Activity 2:375375

Rich Mathematical Tasks... Accessible to everyone Can be extended Let students do the thinking, speculating, conjecturing, proving, explaining, reflecting, reporting Are fun and enjoyable

Is it Rich? What are essential characteristics of rich problems?

What Makes a Problem Rich? Significant mathematics Mathematical Practices Multiple layers of complexity Multiple entry points Multiple solutions and/or strategies Leads to discussion or other questions Students are the workers and the decision makers Warrants reflection - Paired with discourse

Activity 3: Connect Task to the Standards for Mathematical Practice Individually work MARS task #3 Identify Standards for Mathematical Practice Share with a partner: – Solution(s) – What makes the problem(s) rich? – support 1-2 Mathematical Practices

Recall: Standards for Mathematical Practices 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

Incorporating Rich Problems in Instruction NOW (Activity 4) Pick one of the problems on the accompanying sheet and work on it with your table buddy.

Original Task Writing the date on the board, say 25 th Modified task Write today’s date using any operations and only 2’s and 5’s. 1. Modifying a task.

2.Ask the Answer! I have an area of 24cm 2. What does the shape look like? This is 1 fifth. What does the whole shape look like?

Your turn Work by grade level. Take an ordinary problem for your grade & make it rich or Create an entirely new rich problem for your grade. Report. Repeat. (Activity 5)

Rich Task Sources Mathematics Assessment Resource Service Inside Mathematics Balanced Assessment (MARS tasks) NCTM Illuminations Nrich Project (Univ. of Cambridge)

Next Steps Rich Problems Identify “rich” mathematical problems – Make sure to cite the source of the problem* Align this “rich” problem to: – Grade level – Critical Area of Focus – Mathematical Practice(s) Share with colleagues

Find rich tasks at Prepare to present one to the group.

Extra time: Sub Problem Grade 3-5

Extra time ?: Break this square into 11 smaller squares that don’t overlap and whose union is the original square. Check with numbers. Generalize your solution. Grades 5 – 8 & up

Extra time? Farmer Brown When Farmer Brown travels to town at 30km/hr he arrives an hour early. When he travels at 20km/hr he arrives an hour late. What is the question? What can I find out? Grades 7-8

The Puzzles How far was the return journey? How fast should he travel to arrive on time? How long did it take him to get to town? How fast should he travel to arrive 2hrs late? There is a separate Farmer Brown PPT with several solutions

Pure Logic No box is labeled correctly. Select one sock from one box and re-label them all correctly. Black Socks Black and White Sock Mixture White Socks

Happy Numbers Think of a whole number. Square the digits and add the results. This creates a new number. Repeat this process. If the sequence of numbers forms a cycle then the original number is happy. 42 becomes = 20, 4, 16, 37, 58, 89, 145, 42…aha a circle of happiness! 19 becomes = 82,68, 100, 1,1,1 …. Another happy number Which positive integers are happy?

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