Engaging Instruction:Rich Problems & Tasks

TLQP 2013-14

Thomas F. Sweeney, Ph.DThe 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 Focusin order to create instruction based upon the CCSSM and develop a foundation for curriculum revision.

Mathematics (California’s view)

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.

Standards for Mathematical PracticeMathematical ‘Habits of Mind’

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

+ 4 6

3

7+

4 10

7 13

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

http://map.mathshell.org/materials/tasks.php

Recall:Standards for Mathematical Practices1. Make sense of problems and persevere in solving them2. Reason abstractly and quantitatively3. Construct viable arguments and critique the reasoning of

others4. Model with mathematics5. Use appropriate tools strategically6. Attend to precision7. Look for and make use of structure8. 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 TaskWriting the date on the board, say 25th

Modified taskWrite today’s date using any operations and only 2’s and 5’s.

1. Modifying a task.

2

5

5 2 5 2 5

5 22 2 5 5 2 5 2 5 2

5 2

2. Ask the Answer!

I have an area of 24cm2. What does the shape look like?

This is 1 fifth. What does the whole

shape look like?

Your turnWork by grade level.Take an ordinary problem for your grade & make

it rich

orCreate an entirely new rich problem for your

grade.

Report.Repeat.

(Activity 5)

Rich Task SourcesMathematics Assessment Resource Service• http://map.mathshell.org/materials/index.php

Inside Mathematics• http://www.insidemathematics.org

Balanced Assessment (MARS tasks)• http://balancedassessment.concord.org

NCTM Illuminations• http://illuminations.nctm.org/

Nrich Project (Univ. of Cambridge)http://nrich.maths.org

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 athttp://www.illustrativemathematics.org/

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 BrownWhen 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 SocksBlack 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 42 + 22 = 20, 4, 16, 37, 58, 89, 145, 42…aha a circle of happiness!

19 becomes 12 + 92 = 82,68, 100, 1,1 ,1 …. Another happy number

Which positive integers are happy?

Sources• http://www.corestandards.org/ • http://www.parcconline.org/ • http://engageny.org/• http://www.illustrativemathematics.org/ • http://www.achievethecore.org/ • http://commoncoretools.me/ • http://insidemathematics.org/• https://www.teachingchannel.org/videos/• https:// www.OhioRC.org