|May: Equality for All||September: What Are the Chances?|
|February: Low Threshold, High Ceiling Activities||June: Sketchpad for iPad||October: Percents|
|March: Operation Sense||July: Patterns and Sequences||November: Turn, Flip and Slide|
|April: It's About Time||August: Representation||December:|
Each month we share a few links to exemplary math websites that we think kids and adults can benefit from. Most are cataloged in the Mathlanding project that Claire and Bob worked on over the last several years. There you can read reviews of the resources we picked or check out some of the nearly 2000 others. Mathlanding has tools and other features for educators, so you might mention it to school personnel that you know. The site is open to all and searchable.
January: Fact Fluency and Number Sense
We know it is important for children to know number facts. The most effective way to achieve that is through activities, games and problems that develop number sense, rather than through rote memorization. The following activities provide plenty of practice with numbers, but they also develop conceptual understanding, reasoning, and strategic thinking.
Jigsaws: This Flash applet helps students develop number sense by having them drag clusters of numbers to complete a rectangular grid. Levels vary in content and difficulty, and some feature missing numbers. Skills include counting from 1 to 20, counting from 1 to 100 in several different orientations, and multiplication facts 1-25 and 1-100.
Make Five can be played alone or with a partner. It provides practice with number facts and builds strategic thinking and pattern recognition. Players choose an operation (addition, subtraction or multiplication) and board size (3 in a row or 5 in a row). They then select fact pairs that match given target numbers while attempting to get 3 or 5 in a row in the grid, as in Tic-Tac-Toe. The game is a good balance of skill and luck.
Kakooma promotes fluency with addition and multiplication facts. Students complete a puzzle by finding the number that is the sum or product of two others within each grouping. The application accommodates a variety of abilities by offering choices of puzzle size and difficulty level.
KenKen helps develop whole number calculation skills, logical thinking and perseverance. Users complete a grid with the digits 1-4 (or 1-6) so that each digit appears exactly once in each row or column, while also forming a target number using a specific operation. This student version from NCTM's Illuminations service provides four new puzzles daily with a range of difficulty. For more challenging puzzles and additional features visit the official KenKen site or the the New York Times puzzle page. Beware – they are addicting.
February: Low Threshold, High Ceiling Activities
One of the many challenges teachers face is meeting the wide range of readiness levels among their students. Low Threshold, High Ceiling (LTHC) activities can help teachers (or parents) meet that challenge, or "differentiate instruction" in current educational jargon. Essentially LTHC activities can be approached at a fairly simple level, allowing some level of success by everyone, but also offer opportunities to dig deeper and do much more challenging mathematics. The NRICH project at Cambridge University describes these activities well in the article linked below.
Using Low Threshold High Ceiling Tasks in Ordinary Classrooms: This article describes the value and rationale for using LTHC activities.
Magic Vs: At a basic level this problem provides plenty of practice with addition and subtraction. More advanced children can investigate parity (even/odd) and form conjectures to test, which leads to sophisticated generalizations. Check out the Teachers' Resources pages of NRICH problems for suggestions for making the most of their problems.
Strike it Out is a simple pencil and paper game that can be played by anyone with basic number skills. It can be adapted to challenge more advanced learners. The real benefit comes from thinking strategically and generalizing a winning strategy.
I'm Eight: An open-ended challenge that is totally adaptable to any age or ability level. I know a second grade teacher who used this technique as part of her math routine, changing the number daily. The results were amazing as the year progressed!
Here are two more examples of LTHC tasks from the 2015 archive:
March: Operation Sense
Children with operation sense understand the meaning of of addition, subtraction, multiplication, and division. They recognize the effect these operations have on numbers and the relationships among these operations. They know how to apply these operations effectively in problem solving situations and in real life.
Beware of over-simplifying the effects of operations. Well-intentioned teachers (and parents) sometimes attempt to help children by saying things like, "Always subtract the smaller number from the bigger number," or "The answer to multiplication is always a larger number." While they may apply to whole numbers, memory tricks like these have to be unlearned when students encounter negative numbers and fractions. Example: The temperature outside was 12° F. It got 15° colder. How cold is it now? It must be 15 - 12 = 3 degrees, right? Wrong! Another caveat is to avoid reliance on key words, which can be very misleading. Better to focus on what's actually happening in the problem. For more on operation sense, see February in the 2015 archive.
Thinking Blocks: Of course the best way to develop operation sense (and most other mathematical understanding!) is through a problem context, which gives the numbers meaning. This website offers a variety of word problems and a visual tool for representing the quantities and their relationships, and hence for developing an understanding of how the operations solve the problems.
Expresso: This interactive game promotes understanding of the order of operations. The challenge is to pick operators, as quickly as possible so that each expression equals the target number. The game provides a wide range of levels of difficulty.
Which Symbol? gives learners practice in using the operation and equal symbols in ten number statements. Students drag symbols into empty boxes to make ten true number sentences (equations). I like the fact that it helps learners understand inverse operations and to look for alternate solutions.
Remainders Count: This interactive Flash game helps children understand remainders while developing reasoning skills and facility with multiplication and division facts. Playing against the computer, the user creates division equations from 3 randomly-generated numbers (1-6) with the goal of making the largest remainders. This game encourages children to think about division and what it means in a new way. What generalizations can be made about how to maximize remainders?
Crazy Counting Machine helps students develop fluency with addition, subtraction, doubling and halving. The applet presents a starting number, a target number, and four choices of operations (double, halve, add 7, subtract 3). The goal is to arrive at the target in as few steps as possible.
April: It's About Time
Learning to tell time can be difficult for children. Traditional analog clocks involve two different scales superimposed on each other. Dealing with elapsed time is even more challenging than simply naming the time shown by a clock face. It involves understanding how the two hands move in relation to each other and requires observing them move over time, either in real time or with a simulation. Geared clocks (often called Judy clocks) can be very helpful in developing this concept. Mark Cogan's Analog and Digital Clocks is an interactive online applet you might find useful, especially in working through the activities below.
Each of this month's offerings comes from Claire's favorite math website, NRICH at University of Cambridge in the UK. These activities include challenges with both digital and analog clocks (yes, I think it's important for children to be fluent with both!). Most NRICH problems and activities provide a Teachers' Resources page that explains the rationale for the problem, as well as suggestions for how to introduce, implement, and differentiate it.
What Is the Time? gives children practice in telling time from analog clock faces in five-minute intervals. The activity provides a printable sheet of the clocks and of the times in words. Simply matching the clocks with the words would be a good exercise.
How Many Times? asks students to find times on a 24-hour digital clock in which the digits appear in consecutive order, counting forwards or backwards. In addition to reinforcing understanding of digital time, it encourages working systematically. How do you know you have found all instances? Of course, you could adapt the question for a 12-hour clock. The Teachers' Resources page links to a page of blank clock faces that could be used to make up new challenges.
Times asks some engaging questions about how numerals are displayed on a digital clock. Spatial awareness and symmetry come into play as children imagine rotating or reflecting the numbers. Children (or their parents) can make up their own kinds of special numbers.
Wonky Watches: This problem does not depend on either kind of clock display, but rather on children's reasoning skills and their understanding of the passing of time.
Two Clocks helps solidify children's understanding of what each hand of an analog clock tells. It asks questions that can be answered with clocks that are missing either the hour or minute hand and also addresses elapsed time.
Making Maths: Make a Pendulum is an engaging investigation into how a pendulum behaves. Good record keeping will lead to greater understanding and insights.
May: Equality for All
I'm not referring to political slogans in this pre-election season. Through my years of teaching it became apparent to me that many children do not understand the mathematical meaning of the equals sign: that the expressions on either side of the symbol have the same value. Instead they believe that an equals sign indicates where to write an answer. The concept of equivalence is a critical one throughout mathematics, and this misconception has implications for students' work with numbers and also for their success with algebra. The resources below marked with * are articles that go into detail about the causes of the problem and its impact on children's mathematical development. The other resources are intended to help students develop an accurate understanding of equality.
* The Meaning of the Equals Sign: This web page from the state of Victoria, Australia, discusses the importance of students understanding the meaning of the equals sign in number sentences (equations). It illustrates common misconceptions and their causes and proposes strategies and activities that help teach and reinforce equality concepts.
Can You Balance? Primary students can explore the concepts of equality and inequality by selecting the set of cubes that will equalize the balance. While this is closed-ended, I like the fact that it connects numbers with their visual representation, provides the appropriate equality/inequality symbol, and offers hints when the child fails to achieve a balance.
Number Balance: This open-ended interactive Flash applet helps children develop operation and number sense, facility with number facts, and understanding of equations. Users designate single-digit whole numbers or integers and operations on both sides of an equation and test for balance.
Which Symbol? This problem is designed to help young learners use the symbols plus, minus, multiplied by, divided by and equal to, meaningfully in ten number statements. Users drag two operational symbols to empty boxes to make a true statement. It also helps learners understand inverse operations and encourages them to look for alternate solutions.
Pan Balance – Numbers: This Java tool helps to strengthen students' understanding of equality and the concept that equality is a relationship, not an operation. It is very open-ended and adaptable to a variety of levels.
* Children's Understanding of Equality: A Foundation for Algebra: The authors of this 5-page article present anecdotal and research evidence that misinterpretation of the equal sign as an operator results in difficulty with the concept of equality. They describe one teacher's efforts to advance the notion that the symbol represents a relationship and several reasons why that is important for students' further mathematical development.
* Students' Understanding of the Equal Sign Not Equal, Prof Says: This article summarizes the research of Texas A&M faculty Robert M. Capraro and Mary Capraro comparing students' interpretation of the equal sign internationally and its relationship to achievement in mathematics.
June: Sketchpad for iPad
To use this month's featured activities, you will need an Apple iPad and a free app called Sketchpad Explorer. Based on the award-winning software The Geometer's Sketchpad®, the Sketchpad Explorer app allows you to interact with, and investigate, any mathematical document created in Sketchpad (.gsp format).
Sketchpad maintains an extensive library of .gsp files that cover a wide range of mathematical topics and grade levels. The ones we're highlighting develop children's number sense and logical reasoning, as well as provide plenty of practice with basic arithmetic. Each of the first four below, available from the Dynamic Number website, include videos demonstrating how the activity works. After installing Sketchpad Explorer on your iPad, open the pages below in your iPad browser and look for "Sketchpad Files" on the right side of each page to download the files. Choose "open in Sketchpad Explorer." (You do NOT need to purchase The Geometer's Sketchpad.)
Cross Number Puzzles: Students solve a collection of puzzles in which they add, subtract, and use logical reasoning to find missing numbers in a 3 × 3 grid. Slide the red dots to enter the number. The activity provides four different levels of difficulty. (K- Gr 4)
Arranging Addends – Target Sum Puzzles: Children use logical reasoning skills to place numbers in circles and arrange the circles so that the numbers they enclose equal a given target sum. Sketchpad generates random challenges for the students, or they can create their own problems to share. (Gr 3-5)
Mystery Sums, Part One: Dragging pairs of variables (A, B, C, and D) across the divider will reveal their sum. By using guess and test or informal algebraic reasoning, you can deduce the value of each letter. Children can create and share their own challenges with each other. (Gr 4-8)
Sneaky Sums – Unknowns in a Grid: This game of logic develops children’s early algebraic reasoning skills as they deduce the secret values assigned to four geometric shapes arranged in a 4x4 grid. Students can view the sum of the shapes in any row or column of the grid and use that information to determine the numerical value of each shape. Students can create their own challenges for others to solve. (Gr 4-8)
These last two are available on a different website and do not have videos. Look for the Download button to access the .gsp files on your iPad.
Mystery Codes: Students use logical thinking and their knowledge of addition, multiplication, and place value to crack codes in which the numbers from 0 to 9 are represented by the letters from A to J. (Gr 4-6)
Circles and Squares: Two Unknowns: Students solve puzzles involving two unknowns, and learn about the use of symbols and variables as a way to represent numbers. (Gr 4-5).
July: Patterns and Sequences
Being able to recognize and describe patterns helps us make predictions and solve problems. Children are taught in early grades that patterns are things that repeat. The first two activities below ask children to determine a visual repeating pattern and extend it. It is usually helpful to describe or "read" the pattern out loud.
Color Patterns displays a pattern of beads. The user selects colors in the correct sequence to continue the pattern. It acknowledges each correct solution and encourages the user to fix incorrect ones.
Attribute Trains: A bit more challenging, users first identify which attribute of the pieces is used to create the pattern (color, shape, or number) and what constitutes the repeating group before extending it. The applet provided immediate feedback for false moves.
The next group of activities involve sequences. A sequence is a set of numbers arranged in a specific order. Rather than repeating, they either increase or decrease according to a rule. The next two are based on arithmetic sequences in which there is a constant difference between terms.
Domino Number Patterns: Children analyze given pairs of dominoes to determine which domino would go in the middle to complete a pattern.
The 4 Digit Sequencer: Enter a starting number and a step increment. Then mentally carry 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.
Code Games for Kids involves several types of sequences. A pattern is displayed on a safe lock. Players drag number tiles into the gaps and if successful unlock the safe and move onto a different challenge. There are two practice rounds and then ten codes to crack.
The most interesting and practical use of patterns is when the numbers represent something real, as in the following two activities. The challenge is to generalize a rule so that we can answer questions without constructing each successive iteration of the context.
Up and Down Staircases: Children investigate how many blocks would be needed to build an up-and-down staircase with any number of steps up. Children may build staircases with blocks or draw them, but the goal is to discover a general rule. It can be a way to introduce square numbers.
Chairs: How many tables need to be pushed together to give everyone a seat? There are two types of tables to choose from and two different table arrangements.The user can select Exploration mode, in which the number of chairs needed for a particular arrangement of tables is displayed; or Guess, in which the user is able to construct an arrangement and then predict the number of chairs.
The National Council of Teachers of Mathematics, in their Principles and Standards for School Mathematics, states that instructional programs should enable all students to:
Representation refers to both process (capturing a mathematical concept or relationship in a form that fosters understanding) and product (communicating mathematical thinking to others). Children can use a variety of tools and models to represent their thinking: tallies, words, objects, pictures, diagrams, tables, graphs, number lines, and number sentences (equations). It's important that the representations children choose make sense to them, but as they become ready, they can be encouraged to use increasingly more efficient and abstract models, e.g., moving from counting on fingers toward using numerals and symbols. Good representations help children organize their thinking and help to "uncover" the math.
Wooden Legs is a document intended to support teachers using a problem I created for The Math Forum. In addition to the problem posed at the bottom of page 1, I included several ways I imagine students might represent and solve the problem (pp 2-3) and several examples of actual student work, some more successful than others and ranging from very concrete to quite abstract (pp 4-6). I hope this resource helps illustrate the meaning of representation and how it applies to both solving a problem and communicating the ideas to a reader.
The following three resources are tools for creating representations:
Number Line helps students to visualize number sentences and create models for addition, subtractions, multiplication, and division. The number line can be adjusted to represent multiples of numbers from one to one hundred. Look for the info button at the lower right for instructions. Also available as an iOS app.
Rectangle Multiplication uses arrays and three different computational models (Grouping, Common, or Lattice) to help students visualize and understand the process of multiplication of whole numbers. Drag the sliders to adjust the two factors and then observes the resulting changes in the rectangular array and in the respective algorithm and product.
Create A Graph: This Flash applet allows students to create a variety of graphs: line graph, pie chart, bar graph, area graph and x-y plot. Each type provides a variety of layout and design options. Users enter data and labels and choose data parameters. Completed graphs may be printed, saved, and/or emailed. The accompanying tutorial provides general information about graphs and explains how to use the applet.
These problems offer opportunities to explore a variety of representations:
Noah: Multiple solution possibilities encourage students to represent and record findings in different ways and to be systematic in their strategies.
A Flying Holiday involves the ideas of compass point directions, time, and the more difficult concept of constant speed.
Match the Matches asks solvers to interpret different representations of data and match them with statistical descriptions.
September: What Are the Chances?
Probability is a challenging topic for young children who are still developing concepts of randomness and combinations. Primary students can begin with informal discussions about the likelihood of certain events happening. If I am in Dallas in July, how likely is it to snow? Very likely, somewhat likely, unlikely, impossible? Why do you think so?
Before they are expected to learn formal procedures and formulas, children need plenty of hands-on experience experimenting with different models. Fortunately technology provides opportunities to simulate multiple trials in a very short time. Whether tossing real dice or virtual ones, we need to trust in the fairness of the die (that it is constructed in such a way as to make each outcome as likely as every other one) and in the randomness of events (that each result is independent and not biased by any previous result). These ideas run counter to many children's intuition and "magical thinking," e.g., that by crossing their fingers they can influence a roll.
Bobbie Bear introduces students to the topic of combinations, a basic concept in probability. Users determine the total number of possible outfits they can create with the given shirts and pants. They may simply explore by placing the clothes on Bobbie, or make a guess and then test it. How can you tell that you have found all possible combinations? The number of shirt and pants choices is adjustable.
Virtual Coin Toss is a good follow-up to experiments with a real coin. By increasing the total number of tosses, children begin to understand that it takes many tosses before the experimental results begin to resemble the theoretical expectations.
All Change can be played on paper or with the interactivity. In each of the three challenges, the idea is to complete the grid in as few throws of the die as possible. It helps develop the concept of randomness as well as strategic and analytic thinking. How do the results of the three games vary and why? Would it be possible to fill the grid with fewer rolls?
Odds or Sixes? In this problem students determine whether a game is fair and why, based on the probability of the outcomes. An interactive simulation allows data to be collected quickly, or students may conduct the experiment with a real die, which helps them develop record-keeping skills.
Chances allows children to understand the distribution of sums that result when rolling two dice. Why are some sums more likely than others? By experimenting with different numbers of rolls, they see how increasing or decreasing the number of rolls affects the final outcome. After virtually rolling the requested number of times, the applet displays the result of each roll as well as a bar graph of the distribution.
The Twelve Pointed Star Game: Two or more player analyze possible outcomes and determine their likelihood in order to create a winning strategy. It encourages record-keeping and analytical thinking.
Adjustable Spinner: Change the number of sectors and increase or decrease their size to create any type of spinner, and then conduct a probability experiment by spinning the spinner many times. How do the experimental results compare with the student's expectations?
Marble Mania: Students explore probability by simulating up to 500 random draws from a "bag" of marbles. Users determine the number of each color marble in the bag and the number of marbles drawn at once; the computer makes the draws and displays the data in three ways: frequency table, bar graph, and circle graph.
Percents are ubiquitous in everyday life, especially with regard to probability, which was the topic of September's Super Math Websites. Understanding percents is critical in analyzing data and solving real world problems.
The National Council of Teachers of Mathematics recommends that during grades 3-5 children develop an understanding of the equivalence of fractions, decimals, and percents and the information each type of representation conveys. With these understandings, and well-established skills for whole number computation, children can develop strategies for computing with familiar fractions and decimals. "By studying fractions, decimals, and percents simultaneously, students can learn to move among equivalent forms, choosing and using an appropriate and convenient form to solve problems and express quantities." [NCTM, 2000] After that they'll be prepared to study formal operations with these numbers in the middle grades.
We want children to understand that fractions, decimals and percents are all part of the same number system that also includes whole numbers. I've chosen the resources below because they offer a variety of visual models to help students develop conceptual understandings of percents, and to reinforce connections with fractions and decimals. Many use common benchmarks (e.g., 1/2, 0.75, 40%) to establish the relative size of these numbers, a good strategy in the early stages.
Percent Grids develops the concept of percents as parts per 100. Three modes are available: "explore" shows what different percents look like on a decimal grid; "show" asks students to highlight a given percent on a grid; and "name" shows a highlighted grid and asks students to name the percent. Instructions and teaching ideas are available through the links at the top of the page.
Download helps students estimate percents of a unit length using the familiar context of a computer download ribbon. On Start a red bar begins to move across the download bar. The user clicks Stop to approximate the given target percent and is given feedback. After practice with some common benchmarks, users choose their own download targets.
Comparing Fractions and %: Users compare and match fractions and percents using one of four visual representation options. One can choose either the fraction or percent first and then view its equivalents in alternate formats.
Equivalence Demonstrator provides fraction, decimal, and percent labels to be placed correctly on 3 parallel and scaled (0-2) number lines. This helps children compare common benchmarks and understand their equivalent forms.
Decimals, Fractions, and Percentages: This instructional page develops students' understanding of how fractions, decimals, and percents are related. An interactive applet displays visual and numerical representations of all three forms. As users change one form, they see the equivalents in the other two forms. The page provides procedures for converting between forms (for those who are ready) and 12 questions to check for understanding.
Fraction Models allows users the ability to explore different representations for fractions and how they relate to mixed numbers, decimals, and percentages. Users adjust the numerator and the denominator in order to see a visual representation of the fraction. The visual representation can be viewed as a length, area, region, or set model. Users can record the equivalent forms in a table. Instructions and exploration questions are provided.
November: Turn, Flip and Slide
This month's interactive resources offer the opportunity to explore the three basic geometric transformations: Rotation, Reflection, and Translation (familiarly known as turning, flipping and sliding respectively). After any of these changes, an object remains congruent to the original one, i.e., it still has the same size, area, angles and line lengths.This type of geometry has many applications in how we navigate through everyday life.
The first five applets are Java-based and come from the National Library of Virtual Manipulatives (NLVM) at Utah State University. Each one includes tabs for Instructions, Parent/Teacher info, and, once you've played to your heart's content, challenging Activities.
Triominoes gives children practice in manipulating shapes on the screen, which will come in handy in the activities. By sliding, flipping and turning, these triangles can be made to tessellate – to fit together without gaps or overlapping to cover a surface like pieces of a puzzle. The challenge is to connect the sides of the triangles so that the dot colors of connecting sides match.
Transformations: Rotation: Explore the rotation transformation both informally and within a coordinate system. Students select and compose shapes and manipulate the center and angle of rotation to see the effect on a rotated image.
Transformations: Reflection: Select and compose shapes and manipulate the axis of symmetry (the mirror) to see the effect on a reflected image.
Transformations: Translation: Select and compose shapes and manipulate the ends of a translation vector (arrow) to see the effect on a translated image.
Pentominoes: Choose from 12 pentominoes to place and combine on the drawing board. Each pentomino contains 5 square units connected to form a unique shape. Children can flip, slide, and turn each pentomino to explore informally, or to work on a set of five activities.
Shape Tool: This interactive tool allows a user to create many geometric shapes. Squares, triangles, rhombi, trapezoids and hexagons can be created, colored, enlarged, shrunk, rotated, reflected, sliced, and glued together. The Instructions tab explains each tool; the Exploration tab offers a tessellation challenge.
It's time to apply some of those tranformation skills! These sets of 25 interactive challenges gives practice solving problems, using logical thinking, spatial orientation, and movement along a path. Learners must control the movements of one or more ducks or the stream of sugar granules to achieve an unstated goal. Learn on the fly, and be patient, for loads of fun.
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