Common Core’s Mathematical Practice 2, Reason abstractly and quantitatively, calls for students to make sense of quantities and their relationships in problem situations. They are able to translate a problem context into models, numbers and symbols that they can manipulate; they can interpret the meaning of numbers, symbols and units in their work in terms of what they represent in the original problem. This collection includes mathematical tasks and problems that provide opportunities for children to develop reasoning in a variety of ways, and a tool for creating visual models of problem situations. A companion Professional Development collection includes resources to help teachers and math specialists understand the intent and scope of MP 2 and plan for successful implementation.

ResourceTitle/DescriptionGrade Level

1, 2, 3, 4, 5, 6+

Thinking Blocks is an interactive Flash tool for modeling and solving math problems visually. Students represent quantities and relationships by placing blocks and braces on a work space and using the tools to resize and label them accordingly. Users can change the color of blocks, move them, copy them, divide them into equal parts, and separate them. A pencil tool and keyboard are also available. The site includes video tutorials demonstrating how to use the tool and how to model a wide variety of problem types. It also contains a bank of hundreds of word problems.

1, 2

In this problem students practice basic addition and subtraction skills along with logical reasoning to satisfy three interdependent conditions. Solvers use the clues provided to determine the number of eggs in each of three baskets. The Teachers' Notes page offers rationale, suggestions for implementation, discussion questions, ideas for extension and support, a PowerPoint presentation, and vocabulary cards (pdf).

2, 3, 4, 5

This problem requires a sound understanding of the fraction relationship between part and whole and can be used for finding fractions of numbers and quantities. Students are given the fractional amount of apples in a fruit bowl and the specific number of other fruit in the bowl in order to figure out how many apples are in the bowl. The Teachers' Notes page includes suggestions for implementation, discussion questions, ideas for extension, a link to a worksheet which provides student support, and a downloadable pdf of the puzzle.

K, 1, 2, 3

This problem promotes logical thinking and introduces learners to the trial and error (guess and check) problem solving strategy, especially with the interactive provided. In this problem children need to understand the difference between having a certain number of brothers and the number of boys in a family to answer the question, "How many children are there in the Brown family?" The Teachers' Notes page offers suggestions for implementation, discussion questions, ideas for extension and support, and a link to a down loadable worksheet for students to table their trials.

3, 4, 5

Solvers of this problem apply number sense and logical reasoning to determine the numbers of cows and sheep in each of five fields by using clues about how many cows and sheep can be seen by each animal. The problem includes questions for getting started, suggestions for implementation and differentiation, a printable student page, and sample solutions.

K, 1, 2, 3

This activity gives students a chance to make estimates and comparisons of the measures of length (height) and capacity. It presents an image of a tiny elf next to a normal size mug. Students are asked to estimate the Little Man's height and compare it with that of other objects. Next, they are asked to estimate the height and volume of the Little Man's mug. The Teachers' Notes page includes suggestions for implementation, discussion questions, and ideas for extension and support.

1, 2, 3

This problems is an opportunity to explore triangular numbers in the familiar context of decorating a birthday cake with a number of candles corresponding to a child's age. The problem lends itself to systematic strategies and multiple representations. The Teachers' Notes page offers rationale, suggestions for implementation, discussion questions, and ideas for extension and support.

3, 4, 5

This activity provides students with an opportunity to calculate area and perimeter of rectangles in order to determine the pricing system of objects with given costs (in British Pounds). The resource includes the problem, tips on getting started, a teacher resource page, and a printable student page.

This interactive puzzle provides an opportunity for learners to develop their algebraic thinking skills. The activity challenges students to find the unknown weights of three colored creatures called Wangdoodles. The weights of each possible pair are shown and these equations can be used to deduce the weight of each individual Wangdoodle. Sliders are available to assist in finding relative values and the question button provides advice on solving the puzzle.

3, 4, 5

This problem asks students to visualize a square drawn on a clock face and estimate and calculate its area. The problem can be solved without use of the Pythagorean Theorem. Ideas for implementation, extension and support are included along with printable sheets of clock faces.

3, 4, 5

This problem gives children an opportunity to explore patterns in a practical context and to generalize the results with a rule. Students investigate how many blocks would be needed to build an up-and-down staircase with any number of steps up. An interactivity in the hints shows the blocks transformed into a square pattern. The Teachers' Notes page offers suggestions for implementation, key discussion questions, ideas for extension and support.

4, 5, 6+

Brian Marks and Leslie Lewis maintain this blog dedicated to providing upper elementary and middle school teachers with rich, real-world problems intended to engage students in reasoning, critical thinking, questioning, and communication. The problems address the NCTM Process Standards and are aligned with the Common Core Standards and Mathematical Practices. Problems can be browsed by category or from a "bird's eye" listing. Activities (pdf) are free; paid members ($15/year) gain access to solutions, teaching tips, and other teacher support materials.

This is a set of 46 problems that focus on reasoning and problem solving skills. Successful solving requires careful reading of the problems. Math content topics include applications of basic operations, multistep problems, and comparing quantities and attributes.

K, 1, 2, 3, 4, 5

This is an archive of previous problems from "Math by the Month", a regular department of Teaching Children Mathematics Journal from NCTM for K-5 teachers that features activities organized by grades K-2 and grades 3-5 and usually based on a theme associated with the particular month. Activities and problems posed are classroom-tested with an inquiry or problem-solving orientation.