Opportunities for Growth in Practice Standard 8

The resources in this collection provide activities in a variety of math content areas that offer students rich problem solving experiences and afford opportunities to develop the habits of looking for and expressing regularity in repeated reasoning. Some resources provide tasks that call for repeated calculations and lend themselves to generalizations; others encourage pattern recognition and shortcut calculations. Some tasks require students to evaluate different strategies and monitor their own solving process. Educators looking for ways advance their own expertise in these areas should draw upon the companion PD collection for Practice Standard 8.

Collection Discussion

Replies: 2
Last Post05-28-2013by bethb
K, 1, 2, 3

This problem provides an opportunity for students to practice counting and to represent and record findings in different ways: "Noah saw 12 legs walk by into the Ark. How many creatures did he see?" Multiple solution possibilities encourage students to be systematic in their strategies. The Teachers' Notes page offers suggestions for implementation, discussion questions, and an idea for extension.
K, 1, 2, 3, 4, 5, 6+

This flexible, interactive Flash applet allows students to explore number patterns and to develop number sense and fluency with addition, subtraction, and multiplication. The student or teacher enters a starting number and a step increment. Both values may be up to 4 digits and either positive or negative. The user then mentally carries out the sequence and enters the resulting 10th and 11th terms. The first 9 terms are color-coded in groups of 3 and may be shown or hidden one group at a time. Users have the option of hiding or showing the starting number and/or the increment.
2, 3, 4, 5

This interactive Flash game gives students practice in telling time on an analog clock while also developing strategic thinking. Students take turns advancing the clock by 1/4, 1/2, or 1 hour. The winner is the player who moves the hands of the clock to exactly midnight. This activity can merely familiarize students with the clock and time intervals, or it can challenge older students to find the strategy that always wins.

This unit of four lessons highlights different aspects of studentsâ€™ understanding and use of patterns as they analyze relationships and make predictions, as discussed in the Algebra Standard. In this cluster of activities, students use two interactive math applets (both catalogued separately) to learn about repeating and growing patterns. In the first part, students explore a two-square pattern unit and in the second part, students investigate repeating patterns with pattern units of three, four, and five squares. In Part 3, students analyze repeating patterns of colored cubes and lastly in Part 4, students create growing patterns of colored cubes and compare them to repeating patterns.
1, 2, 3, 4, 5

This web page provides instructions for a basic form of Nim, an ancient game that develops strategic thinking. Starting with a total of 7 counters, two players take turns removing one or two at a time. The player who takes the last counter wins. The game of Nim is very adaptable. This page includes questions that encourage players to generalize a winning strategy as well as a link to another page describing several of variations.

In this interactive activity a user identifies two pairs of equivalent fractions for a given random fraction or one of the player's own and the user creates their representations by dividing and shading either a square or circular region. The fractions are shown as locations on the number line and their equivalency is demonstrated when they are at the same point. The user has the ability to construct a table of equivalent fractions. Instructions and exploration questions are given.
3, 4, 5

This interactive applet helps students develop mental arithmetic skills and strategies for learning multiplication facts. Frog asks the user to type in a known multiplication fact and then asks the user to complete a related fact by using one of several strategies, including commutativity, doubling a factor, halving a factor, multiplying one factor by 10. The student's fact and Frog's related one are recorded in a window to help students discover relationships and internalize the strategies.

This problem encourages children to identify and describe a pattern and to extend the pattern into a general rule. Using an applet, learners try to discover the number of garlic cloves being planted if the arrangement into various rows always finds that there is one left over. The Teachers' Notes page offers rationale, suggestions for implementation, discussion questions, and ideas for support and extension.

This interactive Flash game helps students understand remainders while developing reasoning skills and facility with multiplication and division facts. Playing against the computer, the student creates division equations from 3 randomly-generated numbers (1-6) with the goal of making the largest remainders.

This problem demonstrates the power of the 100 square (Hundred Chart) in helping children to recognize number properties and in beginning to reason carefully and form conjectures. Students look at sums of numbers located in a certain configuration within squares of the grid. There is a link to a printable hundred chart.

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.
3, 4, 5

In this investigation, students visualize and compare volumes in solids composed of unit cubes and look for patterns in the measurements. They work systematically to organize and analyze the results. Ideas for implementation, extension and support are included.

This card game provides practice in saying and writing the function rule, given an input/output table. One student has the rule which the other student has to guess. The guesser records an input number in the table and the rule person must apply the rule and tell the output number. Student pairs repeat this until the guesser correctly identifies the rule by saying and writing it. Students then switch places and repeat the activity. There are three different levels of rules for play: simple addition & subtraction, multiplication or two-step rules.

In this interactive game users develop decimal sense by dragging combinations of spare train track pieces, labeled from 0.1 to 1.0, to repair gaps in the track. Players can use each piece only once per round, which encourages strategic thinking as the game progresses.

This activity provides students with an opportunity to recognize arithmetic sequences and at the same time reinforces identifying multiples. The interactivity displays five numbers and the student must discover the times table pattern and the numerical shift. On Levels 1 and 2, the first five numbers in the sequence are given and on Levels 3 and 4, the numbers given could be any five numbers in the sequence. The Teachers' Notes page offers rationale, suggestions for implementation, discussion questions, ideas for extension and support.
4, 5, 6+

This interactive Java puzzle helps students develop decimal number sense and systematic analysis skills. The applet displays 7 interlocking circles and a set of 14 numbers (in tenths), 5 of them already placed in the circles. Solvers arrange the remaining numbers in the empty sections of the circles so that the three sections of each circle sum to 3. Parent/Teacher information offers instructions and implementation suggestions.

This interactive Java applet provides learners the opportunity to explore concepts of volume, its conservation, and relative capacities of containers. The applet will simulate pouring liquid from one tank (prism, cylinder or cone) into another. Students are asked to predict how high the level will be in the second tank. Instructions and teaching information are included.

This interactive resource allows students to determine the maximum number of possible braille codes there are and promotes the need to to work systemically. The Braille system uses dots either raised (bumps) or not, arranged in three rows by two columns. Students work out the answers to guided questions, complete tables and feedback is provided.
3, 4, 5, 6+

In this game students use basic transformations (sliding, flipping and turning) to make one or more shapes coincide with a congruent shape on a Cartesian plane. Players are challenged to complete the matching in the fewest possible moves.