Number and Operations Online Resources
THE STANDARD (NCTM - PSSM)
Understand numbers, ways of representing numbers, relationships among numbers, and number systems:
Understand meanings of operations and how they relate to one another:
Compute fluently and make reasonable estimates:
Understand numbers, ways of representing numbers, relationships among numbers, and number systems:
- work flexibly with fractions, decimals, and percents to solve problems;
- compare and order fractions, decimals, and percents efficiently and find their approximate locations on a number line;
- develop meaning for percents greater than 100 and less than 1;
- understand and use ratios and proportions to represent quantitative relationships;
- develop an understanding of large numbers and recognize and appropriately use exponential, scientific, and calculator notation;
- use factors, multiples, prime factorization, and relatively prime numbers to solve problems;
- develop meaning for integers and represent and compare quantities with them.
Understand meanings of operations and how they relate to one another:
- understand the meaning and effects of arithmetic operations with fractions, decimals, and integers;
- use the associative and commutative properties of addition and multiplication and the distributive property of multiplication over addition to simplify computations with integers, fractions, and decimals;
- understand and use the inverse relationships of addition and subtraction, multiplication and division, and squaring and finding square roots to simplify computations and solve problems.
Compute fluently and make reasonable estimates:
- select appropriate methods and tools for computing with fractions and decimals from among mental computation, estimation, calculators or computers, and paper and pencil, depending on the situation, and apply the selected methods;
- develop and analyze algorithms for computing with fractions, decimals, and integers and develop fluency in their use;
- develop and use strategies to estimate the results of rational-number computations and judge the reasonableness of the results;
- develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.
FRACTION MODEL
Which standard?
This model shows the different representations of fractions, decimals and percents. Although there are no computations, changing values of the fractions allows the student to explore the relationships between each type of number and creates a table for each simulation that allows students to see patterns within the relationships.
What mathematical content is being learned (or intended to be learned)?
There is a clear content of fractions and decimals in relation to whole numbers which I feel is a strength of this online technology source. Students in the middle level still seem to not understand that fractions, decimals, and percents are less than 1 and then what happens with a decimal of 1.0 which is equivalent to 100%. Also, this provides a great understanding of values greater than 100%.
Is the focus on instrumental or relational understanding?
While some instrumental understanding is needed to understand how the fractions change, most of the learning taking place in relational to not only fractions and mixed numbers, but also to the connection between fractions, decimals and percents.There is also a focus on representing fraction amounts which would help in understanding what a fraction is.
What role does technology play?
The technology makes representing the fractions easier. Sometimes the task of modeling fractions is difficult for students.(Ashlee) This site uses technology to allow students to manipulate fractions instead of having to write or draw fractions on paper. Instead, they can use the visual provided. In addition, the technology creates the decimal and percent equivalents and posts them in a table so that the equivalents are shown side by side.
What instructional function(s) does the resource serve?
Direction and meaning are not explained in this activity. Instead, it is left for the students to find and develop an understanding through manipulation of the fractions.
What kinds of representations of the mathematics are used?
There were four different visual representations given, as well as symbolic representation of the fractions created. The ability for this interaction to switch from different types of representations (pie-chart, sets, bars) is useful to and the ability to see the quick changes in numbers (fractions, decimals, and percents). This model uses this combination of visual/spatial representations along with concrete and real world objects to represent fractions using sets of blocks, circles, apples, stars, or butterflies in a visual model.
- Understand Numbers
- Understand Operations
- Compute Fluently
This model shows the different representations of fractions, decimals and percents. Although there are no computations, changing values of the fractions allows the student to explore the relationships between each type of number and creates a table for each simulation that allows students to see patterns within the relationships.
What mathematical content is being learned (or intended to be learned)?
- Fractions
There is a clear content of fractions and decimals in relation to whole numbers which I feel is a strength of this online technology source. Students in the middle level still seem to not understand that fractions, decimals, and percents are less than 1 and then what happens with a decimal of 1.0 which is equivalent to 100%. Also, this provides a great understanding of values greater than 100%.
Is the focus on instrumental or relational understanding?
- instrumental understanding (carrying out procedures)
- relational understanding (understanding the meaning of mathematical words and symbols; connections among ideas)
While some instrumental understanding is needed to understand how the fractions change, most of the learning taking place in relational to not only fractions and mixed numbers, but also to the connection between fractions, decimals and percents.There is also a focus on representing fraction amounts which would help in understanding what a fraction is.
What role does technology play?
The technology makes representing the fractions easier. Sometimes the task of modeling fractions is difficult for students.(Ashlee) This site uses technology to allow students to manipulate fractions instead of having to write or draw fractions on paper. Instead, they can use the visual provided. In addition, the technology creates the decimal and percent equivalents and posts them in a table so that the equivalents are shown side by side.
What instructional function(s) does the resource serve?
- practice (i.e., practicing skills or knowledge already learned)
- learning through exploration (i.e., provides context in which students can see new relationships; come to new understandings)
Direction and meaning are not explained in this activity. Instead, it is left for the students to find and develop an understanding through manipulation of the fractions.
What kinds of representations of the mathematics are used?
- symbolic (i.e., numerals, symbols)
- visual/spatial (e.g., circles or squares with lines to show fractions)
- concrete or real-world objects (e.g., images of base-10 blocks, puppies, or jars)
There were four different visual representations given, as well as symbolic representation of the fractions created. The ability for this interaction to switch from different types of representations (pie-chart, sets, bars) is useful to and the ability to see the quick changes in numbers (fractions, decimals, and percents). This model uses this combination of visual/spatial representations along with concrete and real world objects to represent fractions using sets of blocks, circles, apples, stars, or butterflies in a visual model.
MATH BASEBALL
Which standard?
Students add, subtract, multiply and divide whole numbers within parameters set by the player’s choice of difficulty, beginning with single digit operations and ending with four digit operations for the "super brain" level. The game decides if an answer to a problem will be a single, double, triple or home run based on the difficulty of the problem. If a student gets an answer incorrect, they receive an out and are allowed three outs per game. No hints are given if an answer is incorrect.
What mathematical content is being learned (or intended to be learned)?
This game is intended to help students understand the four basic operations when computing numbers and become fluent within these operations as they calculate different problems at different difficulty levels. Students may choose only addition, subtraction, multiplication or division, or any combination of the four operations to guide their fluency.
Is the focus on instrumental or relational understanding?
This game clearly uses instrumental understanding of the four basic operations and students must have prior knowledge of the procedures in order to play. Because there is the use of inverse operations, there is relational understanding as well, as in the connections being made in order to come up with various solutions.
What role does technology play?
This is a very simple interface where students simply type in the answer and click enter to see if they are correct. There are no boxes for individual steps to see where students might be succeeding or failing, and the emphasis is placed on getting the correct answer. Students can go against each other or play individually and not have the same questions. It can be tailored for individual student needs and skill level with a simple click. The characters in the game represent baseball players and offer a motivational effect as they run the bases as students get answers correct.
What instructional function(s) does the resource serve?
At this point students will have been taught how to carry out the operations. Students use this to practice the instrumental procedure quickly and learn to manipulate between operations paying attention to mathematical symbols to solve correctly. There is no pictorial presentation of the numbers/answers. Students need to know what to do mentally.
What kinds of representations of the mathematics are used?
Standard math problems are given to students using whole numbers and and the operation symbols. Students need to know what each one represents and how to carry out the problem if not already memorized.
- Understand Operations
- Compute Fluently
Students add, subtract, multiply and divide whole numbers within parameters set by the player’s choice of difficulty, beginning with single digit operations and ending with four digit operations for the "super brain" level. The game decides if an answer to a problem will be a single, double, triple or home run based on the difficulty of the problem. If a student gets an answer incorrect, they receive an out and are allowed three outs per game. No hints are given if an answer is incorrect.
What mathematical content is being learned (or intended to be learned)?
- Whole numbers
- Addition
- Subtraction
- Multiplication
- Division
This game is intended to help students understand the four basic operations when computing numbers and become fluent within these operations as they calculate different problems at different difficulty levels. Students may choose only addition, subtraction, multiplication or division, or any combination of the four operations to guide their fluency.
Is the focus on instrumental or relational understanding?
- instrumental understanding (carrying out procedures)
This game clearly uses instrumental understanding of the four basic operations and students must have prior knowledge of the procedures in order to play. Because there is the use of inverse operations, there is relational understanding as well, as in the connections being made in order to come up with various solutions.
What role does technology play?
This is a very simple interface where students simply type in the answer and click enter to see if they are correct. There are no boxes for individual steps to see where students might be succeeding or failing, and the emphasis is placed on getting the correct answer. Students can go against each other or play individually and not have the same questions. It can be tailored for individual student needs and skill level with a simple click. The characters in the game represent baseball players and offer a motivational effect as they run the bases as students get answers correct.
What instructional function(s) does the resource serve?
- practice (i.e., practicing skills or knowledge already learned)
At this point students will have been taught how to carry out the operations. Students use this to practice the instrumental procedure quickly and learn to manipulate between operations paying attention to mathematical symbols to solve correctly. There is no pictorial presentation of the numbers/answers. Students need to know what to do mentally.
What kinds of representations of the mathematics are used?
- symbolic (i.e., numerals, symbols)
- dynamic (mathematical ideas represented through motion or sound)
Standard math problems are given to students using whole numbers and and the operation symbols. Students need to know what each one represents and how to carry out the problem if not already memorized.
DIVIDING FRACTIONS BY FRACTIONS
Which standard?
Students learn division of fractions and must understand how the operation works. Students practice to gain fluency with division and fractions. It also helps students become fluent in simplifying of fractions.
What mathematical content is being learned (or intended to be learned)?
With this tool students need to understand what a fraction is, how to multiple, divide and then multiply and divide fractions and all skills are evident in this activity.
Is the focus on instrumental or relational understanding?
In order to correctly divide fractions, students must carry out the procedures given at the top of the applet. Students must also have a relational understanding of how to multiply fractions in order to use this tool. The ideas of multiplying, dividing, and fractions all have to be introduced prior to even teaching someone how to divide fractions.
What role does technology play?
The technology generates random division problems and students type in the answer. When the answer is incorrect, a box pops up and gives them the right answer without an explanation (i.e. 3/20 divided by 1/8 =6/5). It is very simple for students to figure out the mechanism and they are given a percentage of how many they got right. This practice interactive also gives students the opportunity to play against themselves with options such as countdown, 20 questions and give me time in the same interface with an added timer.
What instructional function(s) does the resource serve?
Direct instruction is evident in this activity, but will be very confusing if someone has never been introduced to dividing fractions. I believe that the main instructional function is to allow students to practice a skill that has already been taught.
What kinds of representations of the mathematics are used?
No pictures here. This is less than what a textbook might supply to help students through the process. It consists of all math lingo and symbols. It also is difficult for students to line up visualize numbers in numerator/denominator spot, let alone flip the second fraction.
- Understand Operations
- Compute Fluently
Students learn division of fractions and must understand how the operation works. Students practice to gain fluency with division and fractions. It also helps students become fluent in simplifying of fractions.
What mathematical content is being learned (or intended to be learned)?
- Fractions
- Multiplication
- Division
With this tool students need to understand what a fraction is, how to multiple, divide and then multiply and divide fractions and all skills are evident in this activity.
Is the focus on instrumental or relational understanding?
- instrumental understanding (carrying out procedures)
- relational understanding (understanding the meaning of mathematical words and symbols; connections among ideas)
In order to correctly divide fractions, students must carry out the procedures given at the top of the applet. Students must also have a relational understanding of how to multiply fractions in order to use this tool. The ideas of multiplying, dividing, and fractions all have to be introduced prior to even teaching someone how to divide fractions.
What role does technology play?
The technology generates random division problems and students type in the answer. When the answer is incorrect, a box pops up and gives them the right answer without an explanation (i.e. 3/20 divided by 1/8 =6/5). It is very simple for students to figure out the mechanism and they are given a percentage of how many they got right. This practice interactive also gives students the opportunity to play against themselves with options such as countdown, 20 questions and give me time in the same interface with an added timer.
What instructional function(s) does the resource serve?
- practice (i.e., practicing skills or knowledge already learned)
- direct instruction/explanation (i.e., explaining or presenting content to students)
Direct instruction is evident in this activity, but will be very confusing if someone has never been introduced to dividing fractions. I believe that the main instructional function is to allow students to practice a skill that has already been taught.
What kinds of representations of the mathematics are used?
- symbolic (i.e., numerals, symbols)
No pictures here. This is less than what a textbook might supply to help students through the process. It consists of all math lingo and symbols. It also is difficult for students to line up visualize numbers in numerator/denominator spot, let alone flip the second fraction.
MATCH THE FRACTION
Which standard?
In this online game, students are required to match a given fraction to a fraction on the number line. Students identify the fraction by clicking on the fraction given on the number line. Students can manipulate the number of slices on the number line to match the denominator and choose the correct fraction. They do not compute or perform any operations.
What mathematical content is being learned (or intended to be learned)?
Fractions are explored using a number line. Alignment of fractions on the number line is addressed.
Is the focus on instrumental or relational understanding?
The understanding for this tool is really basic. Students do not really need to understand what a fraction is in order to correctly match the fraction. They really only need to know what it means to match something. If relational understanding is present, it comes in the way students must connect that the denominator in a fraction means the total number of parts and the numerator means the number of parts out of the total number of parts.
What role does technology play?
This technology provides instant feedback on correct picture representation. It also allows for various fraction forms and quick practice of random fractions with both picture and number line representations.
What instructional function(s) does the resource serve?
Students practice recognizing fractions on a number line. No instruction is given, and no exploration is necessary as students are not given an opportunity to develop new relationships or understanding. Using the 'Introduction to Fractions' link below Match the Fraction, students do get an explanation and examples of fractions and simplifying. If we are to focus only on the Match the Fraction part, then students are learning through exploration. Matching and learning a visual amount to its number fraction counterpart.
What kinds of representations of the mathematics are used?
Both symbolic representation and graphical representations are use to demonstrate how to place fractions on a number line. There are visual/ spatial representations of fractions using a fraction bar.
- Understand Numbers
In this online game, students are required to match a given fraction to a fraction on the number line. Students identify the fraction by clicking on the fraction given on the number line. Students can manipulate the number of slices on the number line to match the denominator and choose the correct fraction. They do not compute or perform any operations.
What mathematical content is being learned (or intended to be learned)?
- Fractions
Fractions are explored using a number line. Alignment of fractions on the number line is addressed.
Is the focus on instrumental or relational understanding?
- instrumental understanding (carrying out procedures)
- relational understanding (understanding the meaning of mathematical words and symbols; connections among ideas)
The understanding for this tool is really basic. Students do not really need to understand what a fraction is in order to correctly match the fraction. They really only need to know what it means to match something. If relational understanding is present, it comes in the way students must connect that the denominator in a fraction means the total number of parts and the numerator means the number of parts out of the total number of parts.
What role does technology play?
This technology provides instant feedback on correct picture representation. It also allows for various fraction forms and quick practice of random fractions with both picture and number line representations.
What instructional function(s) does the resource serve?
- practice (i.e., practicing skills or knowledge already learned)
- learning through exploration (i.e., provides context in which students can see new relationships; come to new understandings)
Students practice recognizing fractions on a number line. No instruction is given, and no exploration is necessary as students are not given an opportunity to develop new relationships or understanding. Using the 'Introduction to Fractions' link below Match the Fraction, students do get an explanation and examples of fractions and simplifying. If we are to focus only on the Match the Fraction part, then students are learning through exploration. Matching and learning a visual amount to its number fraction counterpart.
What kinds of representations of the mathematics are used?
- symbolic (i.e., numerals, symbols)
- graphical (i.e., standard graphical notation such as Cartesian (X-Y) coordinate system, bar graph, pie chart)
- visual/spatial (e.g., circles or squares with lines to show fractions)
Both symbolic representation and graphical representations are use to demonstrate how to place fractions on a number line. There are visual/ spatial representations of fractions using a fraction bar.
FREE RIDE
Which standard?
This technology's primary focus is understanding fraction/ratio numbers.
What mathematical content is being learned (or intended to be learned)?
Using the premise of bike gears, fractions are represented as ratios of the front gear to the back gear. In order to move on a chosen path or random route, students need to be able to estimate fractions as well as add and subtract them if the estimate does not fall on the flag. The goal is to hit each marker along the path to pass the route or to have the combination of gear ratios equal the flag marker on the distance line.
Is the focus on instrumental or relational understanding?
Although students need to know the procedures for adding and subtracting fractions, this is not taught explicitly and is expected that students already know these procedures. Students also need to understand the procedural aspects of estimating when adding and subtracting fractions. Students connect this knowledge to a real-world situation of which bike gears to use on which course to get to the desired flag or marker. In the extension part of the investigation, students critically think about the relationships between distance and gears and are asked to calculate possibilities once the gears have been manipulated.
What role does technology play?
The technology allows students to see that how adjusting a fraction of a gear ratio would effect a bike and the distance it travels. This encompasses the visual along with motion. There are many levels depending on student need. It also allows students to keep trying until they get it right.
What instructional function(s) does the resource serve?
With little to no direct instruction, students explore the relationship between gear ratios, pedal counts and distance traveled. Throughout the game, students will practice the skills of adding and subtracting fractions. They use this information to traverse a bike route and hit exact markers. Through the exploration students discover the relationship between distance traveled and the ratio of gears.
What kinds of representations of the mathematics are used?
Symbolically, ratios are represented as fractions and numerals are used to decipher pedal counts based on the manipulations of the ratios. The bike route is represented graphically as a distance number line along each route. Pictorial representations of the bike, tires, gears and pedals are present to visualize the ratios. These mathematical representations are dynamic in that the bike moves, along with the gears, tires, and pedals when each ratio is modified and to indicate the distance the bike will travel based on the chosen ratio.
- Understand Numbers
- Understand Operations
- Compute Fluently
This technology's primary focus is understanding fraction/ratio numbers.
What mathematical content is being learned (or intended to be learned)?
- Fractions
- Addition
- Subtraction
Using the premise of bike gears, fractions are represented as ratios of the front gear to the back gear. In order to move on a chosen path or random route, students need to be able to estimate fractions as well as add and subtract them if the estimate does not fall on the flag. The goal is to hit each marker along the path to pass the route or to have the combination of gear ratios equal the flag marker on the distance line.
Is the focus on instrumental or relational understanding?
- instrumental understanding (carrying out procedures)
- relational understanding (understanding the meaning of mathematical words and symbols; connections among ideas)
Although students need to know the procedures for adding and subtracting fractions, this is not taught explicitly and is expected that students already know these procedures. Students also need to understand the procedural aspects of estimating when adding and subtracting fractions. Students connect this knowledge to a real-world situation of which bike gears to use on which course to get to the desired flag or marker. In the extension part of the investigation, students critically think about the relationships between distance and gears and are asked to calculate possibilities once the gears have been manipulated.
What role does technology play?
The technology allows students to see that how adjusting a fraction of a gear ratio would effect a bike and the distance it travels. This encompasses the visual along with motion. There are many levels depending on student need. It also allows students to keep trying until they get it right.
What instructional function(s) does the resource serve?
- practice (i.e., practicing skills or knowledge already learned)
- learning through exploration (i.e., provides context in which students can see new relationships; come to new understandings)
With little to no direct instruction, students explore the relationship between gear ratios, pedal counts and distance traveled. Throughout the game, students will practice the skills of adding and subtracting fractions. They use this information to traverse a bike route and hit exact markers. Through the exploration students discover the relationship between distance traveled and the ratio of gears.
What kinds of representations of the mathematics are used?
- symbolic (i.e., numerals, symbols)
- graphical (i.e., standard graphical notation such as Cartesian (X-Y) coordinate system, bar graph, pie chart)
- concrete or real-world objects (e.g., images of base-10 blocks, puppies, or jars)
- dynamic (mathematical ideas represented through motion or sound)
Symbolically, ratios are represented as fractions and numerals are used to decipher pedal counts based on the manipulations of the ratios. The bike route is represented graphically as a distance number line along each route. Pictorial representations of the bike, tires, gears and pedals are present to visualize the ratios. These mathematical representations are dynamic in that the bike moves, along with the gears, tires, and pedals when each ratio is modified and to indicate the distance the bike will travel based on the chosen ratio.
CONNECT 4 MULTIPLICATION
THE STANDARD (NCTM - PSSM)
This virtual manipulative requires students to practice their skills with factors and multiples and thinking of multiple moves (factors or a multiple) that is dependent on what the opponent is going to choose.
What mathematical content is being learned (or intended to be learned)?
The resource provides an opportunity for students to increase their fluency with multiplication and factors.
Is the focus on instrumental or relational understanding?
The focus is clearly designed for instrumental understanding and it is more like a “drill” to practice single or double-digit multiplication.
What role does technology play?
The technology provides a fun and interactive way to blend rote learning and understanding of factors and multiples. The applet provides a quick “visual” of the product that the student desired, and if the student will get immediate feedback once the applet colors in the square of the actual product, which is helpful if the student was incorrect while hoping to block her/his opponent’s next move.
What instructional function(s) does the resource serve?
This resource primarily serves a “practicing skills” function; purely a “drill” resource to increase fluency of a prior skill.
What kinds of representations of the mathematics are used?
- use factors, multiples, prime factorization, and relatively prime numbers to solve problems
This virtual manipulative requires students to practice their skills with factors and multiples and thinking of multiple moves (factors or a multiple) that is dependent on what the opponent is going to choose.
What mathematical content is being learned (or intended to be learned)?
- Multiplication
The resource provides an opportunity for students to increase their fluency with multiplication and factors.
Is the focus on instrumental or relational understanding?
- instrumental understanding (carrying out procedures)
The focus is clearly designed for instrumental understanding and it is more like a “drill” to practice single or double-digit multiplication.
What role does technology play?
The technology provides a fun and interactive way to blend rote learning and understanding of factors and multiples. The applet provides a quick “visual” of the product that the student desired, and if the student will get immediate feedback once the applet colors in the square of the actual product, which is helpful if the student was incorrect while hoping to block her/his opponent’s next move.
What instructional function(s) does the resource serve?
- practice (i.e., practicing skills or knowledge already learned)
- learning through exploration (i.e., provides context in which students can see new relationships; come to new understandings)
This resource primarily serves a “practicing skills” function; purely a “drill” resource to increase fluency of a prior skill.
What kinds of representations of the mathematics are used?
- symbolic (i.e., numerals, symbols)
- visual/spatial(e.g., circles or squares with lines to show fractions)
- dynamic (mathematical ideas represented through motion or sound)
RATIOS & PROPORTIONS WORD PROBLEMS
Click on image!
The resource provides an opportunity for students to understand the vocabulary and set-up of a “ratio” by using dot notation and matching the numbers in the ratio with the correct item/person. The practice sets teach students how to model simple ratio problems in which two amounts are compared. There are questions for each problem where students are asked to find one of the two quantities in the ratio, the difference between the two quantities or the total.
CRAZY TAXI M-12
Click on image!
A FUN, DYNAMIC online resource to practice multiples. The game begins with a sign on the side of the road which lets the player know what "multiples" their car should crash into. The game is completely interactive and the arrow keys move the car left or right, and speeds it down or slows it up. The space bar lets the car jump over obstacles. The player's car can move at a fast speed so the player needs to be a fast-thinker in order to decide which approaching car is a "multiple."