“Open Middle” WorkshopEncourage productive struggle in our math class!

Thinking > Doing

Try this one on for sizeMake the largest expression you can by filling in the digits 0-9 in the boxes below, with no repeated numbers. Be ready to share your answers!

Credit: www.openmiddle.com

Open Middle ConceptDan Meyer coined the term. Problems with an open middle:

● Have a “closed” beginning, meaning we all start with the same initial problem.

● Have a “closed” ending, meaning we all want the same answer.

● Have an “open” middle, meaning how we all get to the answer can be completely different.

More info: http://www.openmiddle.com/about/Beginning

End

Attributes of Open Middle Problems● They often have many ways of solving them, beyond just

the standard algorithm.

● They often involve optimization, finding the largest or smallest answer, closest to 0, etc.

● They may appear simple (low entry point) but become much more complex as you try to solve them.

● They generally are not as formal or complex as a performance task, which may require a lot of prerequisite knowledge to complete.

Uses for Open Middle Problems● To introduce a topic. Build student intuition before

diving into a particular unit.

● Build fluency in a more fun/engaging way even without a context.

● Give students a chance to experiment/formulate like actual mathematicians do.

● Ask students more open questions for their warm-ups, exit tickets, and/or math journal entries.

Examples from Early Childhood K-2Kindergarten (Number & Operation in Base 10): Directions: I have 2 ten-frames that have counters on them. One is full and one is not. What is the largest number I could make? What is the smallest number I could make?

1st Grade (Number & Operations in Base 10: Fill in the blanks with numbers that make the equation true. How many different equations can you make?

4 + 5 = ____ - ____

2nd Grade (Measurement & Data): Make 47¢ three different ways using quarters, dimes, nickels, and pennies.

Grades 3-53rd Grade: Create 5 fractions using the digits 0-9 exactly once, and place them all on a number line.

4th Grade: Using the numbers 1-9 no more than one time each, fill in the boxes to make a fraction as close to 1 as possible.

5th Grade: Use the numbers 1-9 no more than one time each to fill in the boxes and create a true statement.

Middle School

Fill in the boxes with the digits 1-9, with no repeated values, so that the sum is as

close to 1 as possible!

Find three numbers that add to 10. Place

them in the boxes above so that the

result is the largest number possible!

Using only the digits 1-9 without repeats, fill in the boxes so that the value of x is as close to zero as

possible!

High SchoolAlgebra: Fill in the boxes with any numbers that make the equation true. How many correct answers are possible?

Geometry: Find 4 coordinates, using the digits 1-9 no more than once, so that the coordinates form a parallelogram. Explain how you know it is a parallelogram.

(___, ___) (___, ___) (___, ___) (___, ___)

Try It Out!Think of a type of problem your students always have trouble mastering. Think of a way to make that type of problem more “open.” Share your ideas with someone next to you!

3-Act Math Tasks

Act 1: Formulate the Question

Act 2: Guess, formulate, work

Act 3: The answer

Start small with a problem rewrite - “Real-World” problems aren’t usually very real!

McDougal-Littell Geometry Textbook

Cereal Box Problem● Why do you think cereal

boxes are all pretty much the same size and shape?

● Can you take the cardboard given to you, and create a cereal box of any size?

After students have had some time...

● What is the total area of cardboard you used?● If cardboard costs $0.07 per 100cm2, how much would your

box cost to make? How much cereal does it hold? Which group created the “best” box? Credit: Erica Zieren, CWCHS Math Dept Head

Dan Meyer Textbook Rewrite

http://blog.mrmeyer.com/2008/the-math-textbook-i-would-buy/

His solution: Remove all helpful tools (graph, numbers etc) so that students can realize they need them. Ask one simple question so students

can develop the steps.

Which ski lift is the steepest?

You try it! Pick the textbook problem in your grade band OR a problem from a unit you are teaching/about to teach, and discuss a re-write with the person next to you.

● What real-world context could I use with this that would connect with my students? Is the one given the best one or most engaging?

● What would a real-world problem actually be, and how would someone solve it?

● What directions can I give so that I can be the “least helpful” and not give anything away?

● How can they take it a step further?

K-2

Peter had 41 marbles. Mike gave Peter some more marbles and then he had 65 in all. How many marbles did Mike give Peter?

3-5

https://www.mathworksheets4kids.com/activities/4th-grade.php

Middle School

Glencoe Pre-Algebra Workbook Problem, section 2-2, (Not sure what edition)

High School

Connected Mathematics

Questions/Hurdles/Ideas - Discuss with Partner● What hurdles do you foresee when using this type of

problem?

● What ideas do you have for implementation of these problems in your classroom? Where could you see yourself using this the most?

● Do you still have any lingering questions about “open middle” problems?

More Resources for “Open Middle” Type Questions● Compilation of 3-act tasks by Dan Meyer

https://docs.google.com/spreadsheets/d/1jXSt_CoDzyDFeJimZxnhgwOVsWkTQEsfqouLWNNC6Z4/edit#gid=0

● Open Middle Website - www.openmiddle.com

● Which One Doesn’t Belong - wodb.ca

● Estimation 180 - www.estimation180.com

● LearnZillion’s K-8 Curriculum - https://learnzillion.com/resources/64474-math-full-lesson-plans-k-8

About Me - Alison Childers● My blog: shootforthemoonmath.wordpress.com

● Email: achilders@carmischools.org

● Twitter: @mrschilders314

● School: Carmi-White County High School, 618-382-4661

○ Algebra 1, Foundational Math, Pre-Calculus

○ Math Team Sponsor

○ Color Guard Sponsor

● Other:

○ LearnZillion Dream Team 2014 and 2015

○ Google Certified Educator - Level 1 (soon)