Resources and Tools for Elementary Math Specialists and Teachers
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Building Understanding with Practice Standard 4


The resources in this collection present students with problems, lessons, and activities in which they will create or use a model in order to understand and communicate the mathematical concept (CCSS Practice Standard 4). Students learn to apply math to everyday life by working with concrete, pictorial, and symbolic representations of the problems encountered in real situations. Additional resources for educators can be found in a companion Professional Development collection for CCSS Practice Standard 4.
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Created:05-23-2013 by bethb
Last Post05-23-2013by bethb
ResourceTitle/DescriptionGrade Level

This problem is an investigation into combinations of a number of cubes. It is a practical activity which involves working systematically, and visualizing and relating 3D shapes to their representation on paper. Children are asked how many different towers are possible using seven cubes on a base of two of them. The Teachers' Notes page includes suggestions for implementation, discussion questions, ideas for extension and support, and printable handouts (word/pdf).

In this lesson activity students create visual representations for the data presented in the book, "If the World Were a Village" by David J Smith and Shelagh Armstrong. Students are challenged to create different representations and reason why one representation may be better than another. Suggested representations are pictographs, bar graphs, and tables. The activity includes the problem, teacher notes, hints, and solution suggestions.
4, 5, 6+

This problem provides an opportunity for students to reason about ratio and proportion in the realistic context of mixing a fruit drink from concentrate. The Teachers' Notes page offers suggestions for implementation, discussion questions, ideas for extension and support, and links to related problems (Blackcurrantiest is cataloged separately).

This interactive Flash activity gives students an opportunity to visually model and calculate six different types of fraction applications, all in the context of solving word problems. Types of problems include finding a fraction of a number, using a known part to find the whole or other part, and problems using the four operations. A video demonstration introduces the method, and then students work on problems. Teachers can track a student's progress throughout the problem sets.

This set of six interactive Flash activities gives students an opportunity to visually model and calculate the unknown quantity from ratios and other information, all in the context of solving word problems. A video demonstration introduces each method, and then students work on modeling and solving five problems. Students can check their modeling of the problem at intermediate stages. Teachers can track a student's progress throughout the problem set.

In this lesson from Illuminations, students explore and discover linear relationships. Linear patterns are identified, extended and described verbally, numerically and algebraically through three investigations. Using manipulatives and the linked applet, "Chairs", learners determine the number of chairs needed when the number of tables is known, and vice versa. Instructional plan, questions for the students, assessment options, extensions and teacher reflections are provided.

This unit consists of four lessons in which students explore several meanings and representations of multiplication, including number lines, sets, arrays, and balance beams. They also learn about the commutative property of multiplication, the results of multiplying by 1 and by 0, and the inverse property of multiplication. Students write story problems and create pictographs. The unit includes activity sheets, assessment ideas, links to related applets, reflective questions for students and teachers, extensions and a bibliography of children's literature with a multiplication focus.


This 3-lesson unit develops students' abilities to measure and model elapsed time in various ways. Students construct clocks, time line models, and they read and interpret schedules. The lessons employ hands on activities, interactive technology, and literature to enhance understanding. Student materials, assessments and solutions are provided.
3, 4, 5

This problem presents students with a situation in which the probability of an event must be determined. Students make a prediction about the probability of Kaia's father wearing the same tie during one week and simulate the event with an experiment to confirm their prediction with the experimental probability. Additional resources for the teacher are provided along with a printable version of the problem.

In this lesson students will develop strategies for adding and subtracting fractions as well as search for and generalize patterns while they investigate an everyday life fraction scenario involving a disappearing pizza. The lesson is based on the book "Fraction Fun" by David A. Adler.
3, 4, 5

In this PBS Mathline lesson plan, students investigate proportional reasoning through modeling, student sharing and questioning techniques. After reading "Oliver's Fruit Salad" to students, the teacher poses several problems about imaginary fruit salads that the students solve as a class. Using pieces of fruit shaped candy, students work in pairs and in small groups to solve other problems that build on the ones solved as a class. Finally, students share their solutions.

In this probability lesson plan students make predictions about the color of the bear they are likely to draw from a bag, and then draw and record the color of the bear they actually draw. Students complete this activity in cooperative learning groups and take turns removing a bear, without replacing it, and seeing if their predictions become more accurate. The lesson plan includes four student activity worksheets and extension questions and suggestions (PDF).

This packet of information for teachers is designed to be used with the Dad's Cookies Problem of the Week from the Math Forum collection for grades 3 to 5. The sample problem provides an exemplar of the type of problem teachers can use to promote students' mathematical thinking and engage their interest. The key concepts addressed in this problem are fractions describing parts of a whole and addition. The packet contains the sample problem, alignment to the Common Core State Standards, an answer check, possible solutions, teaching suggestions, sample student responses, a print-ready copy of the problem, and a problem-specific scoring rubric.

In this 8-lesson unit students use buttons to explore logical and numerical relationships that form the conceptual basis for understanding addition and subtraction operations. Topics include counting, ordinal numbers (and relative position), classification (attributes), relationships between numbers, addition of sets, commutativity of addition, sums to 10, fact families (including subtraction), three models of subtraction ("take away", comparative, missing addend), and bar graphs. Includes student activity sheets and a link to an online graphing applet.

In this 6-lesson unit, students explore 5 models of subtraction (counting, sets, number line, balanced equations, and inverse of addition) using connecting cubes. The lesson activities focus on the comparative mode of subtraction as children investigate the relationship between addition and subtraction, write story problems in which comparison is required, and practice the subtraction facts. The lessons include printable student activity sheets, a bibliography of children's counting books, questions for student discussion and teacher reflection, assessment options, extensions, and links to online applets (cataloged separately).

Teachers can use this interactive tool to help students build a conceptual understanding of what it means to multiply with fractions. The tool includes word problems and paraphrase applications of the word problems, visual models, and equations. It can be used to model the use of multiplication to find the product for both "groups of" and "parts of" problems. This page includes a video demonstration of the tool and sample lessons from the Conceptua curriculum. Free registration is required to use the tool. A paid subscription is necessary to access full curriculum and allow full student use.