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Choosing Tasks that Encourage Students to Use Mathematical Practices

When you were working did you…

1. make sense of a problem and persevere in solving?

2. reason abstractly and quantitatively?

3. construct a viable argument 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 repeated reasoning?

Low Cognitive Demand Tasks Memorization

 – The task/solution Involves reproducing previously learned facts, rules, formulas or definitions or committing these to memory.

 Cannot be solved using procedures because a procedure does not exist or because the time frame in which the task is being completed is too short to use a procedure.

 Is not ambiguous. Such tasks involve the exact reproduction of previously seen material, and what is to be reproduced is clearly and directly stated.

 Has no connection to the concepts or meaning that underlie the facts, rules, formulas, or definitions being learned.

Procedures without Connections

 – The task/solution Is algorithmic. The use of a procedure either is specifically called for or is evident from prior instruction and/or experience.  Requires limited cognitive demand for successful completion. Little ambiguity exists about what needs to be done and how to do it. a.

Is not connected to the concepts or meaning that underlie the procedure being used.



Is focused on producing correct answers.

Requires no explanation or explanations focus solely on describing the procedure that was used.

High Cognitive Demand Tasks Procedures with Connections

 – The task/solution Focuses students’ attention on the use of procedures for the purpose of developing deeper understanding of mathematical concepts and ideas.

 Suggests explicit and/or implicit pathways to follow that involve the use of broad general procedures that have close connections to underlying conceptual ideas as opposed to narrow algorithms.

 Can usually be represented in multiple ways, including the use of manipulative materials, diagrams, and symbols. Making connections among the representations helps students develop meaning.

 Requires some degree of cognitive effort. Although general procedures may be followed, they cannot be followed mindlessly. Students are engaged in conceptual ideas that underlie the procedure and develop understanding.

Doing Mathematics

 – The task/solution Requires complex, non-algorithmic thinking  Requires students to explore and understand the nature of mathematical concepts, processes, or relationships.

 Demands students do some type of self-monitoring or self-regulation of their own cognitive processes.

 Requires students to access relevant knowledge and experiences and make appropriate uses of them in working through the task  Requires students to analyze task constraints that may limit possible solution strategies or solutions.

 Requires considerable cognitive effort and may cause some level of anxiety for the students as they are working through the problem.

Blocks measure 1 1/2 inches on each edge. A cube one foot high, one foot wide, and one foot deep is made with these cubes. How many little blocks are in the large cube of blocks?

The area of the floor of a rectangular room is 315 sq. ft. The area of one wall is 120 sq. ft. and the area of another is 168 sq. ft. The floor and ceiling are parallel. What is the volume of the room? Source: Wheatley, G. & Abshire, A. (2007). Developing mathematical fluency: Activities for grades 5-8. Tallahassee, FL: Mathematics Learning.


To create a cone, Cut out the circle below.

Cut through to the center on the radius line.

You will slide the radius line to different points on the circle to create cones. Try the different cones. Predict the cone you believe will have the greatest volume.

D E C B F G I H cones-spheres/


Complete the table and then plot your Radius and Volume on the graph below. Label your points with the appropriate letter. Which point has the maximum volume? _______ How does the data below help you better estimate the position for the maximum volume?

I J K L M N O D E F G H A B C Letter Radius Height Volume Make a second prediction for maximum volume.

Use your prior knowledge to solve for the height using the length (l) and the radius (r). Show all work. h = ___________________________ Using the formula for volume of a cone, find your maximum cone volume in cubic centimeters. Show all work. V = ______________________

What happens to the volume of a rectangular prism if one, two, or all dimensions are doubled? (begin with a 2 x 3 x 1) Source: Chapin, S. H. & Johnson, A. (2006) Math Matters: Understanding the math you teach, Grades K-8 (Second Edition). Sausalito, CA: Math Solutions.

Use a centimeter ruler to measure the l, w, and h of a block. Build larger and larger cubes using blocks. Record the number of blocks on any edge and the volume for each of these larger cubes. What patterns do you observe? Generalize to a formula for volume of a cube. Source: Hatfield, Edwards, Bitter, & Morrow (2008). Mathematics Methods for Elementary & Middle School Teachers

A swimming pool is 8 m long, 6 m wide and 1.5 m deep. The water resistant paint needed for the pool costs $6 per square meter.

How much will it cost to paint the interior surfaces of the pool?

Making tasks more challenging…

• Open Tasks – framed in such a way that a variety of strategies or solutions are possible • Parallel Tasks – sets of related tasks at different levels that are close enough in context that they can be discussed simultaneously Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York: Teachers College Press

Open Tasks

A shape has a triangle for a cross-section. What could the shape be?

One vertex of a triangle is at the point (1,2). After a reflection, one vertex is at the point (5,8). Name all three vertices of the original and final triangles.

Small, M. (2009).

Good questions: Great ways to differentiate mathematics instruction

. New York: Teachers College Press

Parallel Tasks

Option 1: A and B are the location of Arjun’s and Bob’s homes. They want to meet equally far from both homes. Where could they meet?

Option 2: Carol is joining Arjun and Bob. Locate a meeting place equally far from all three homes. A B C Small, M. (2009).

Good questions: Great ways to differentiate mathematics instruction

. New York: Teachers College Press

Changing Textbook Tasks