Numbers+and+Operations

Numbers and Operations can be thought of as the  fundamental basics of math. PSSM addresses numbers and operations at the by intertwining all three goals, understand number, understand meaning, and compute fluently, with the noted importance placed on Number Sense, where students integrate their knowledge of these goals and apply them to basic operations. Specifically, students in the middle grades should be focused on attaining the ability to rationalize numbers and create connections between operations of those numbers. These interactives teach students how to perform operations with fractions, decimals and percents.


 * Numbers and Operations Evaluations **#1:

**//An interactive picture game using multiplication of fractions //** || 
 * **//__[|Fractions Mystery Pictures Game] __//**
 * **Which standard? **
 * **Understand Numbers **
 * **Understand Operations **
 * **Compute Fluently ** || * || //Rationale //: This online fraction game where students must figure out the solution to a brief story problem. The object is to get all the answers right to figure out what picture lies beneath the square tiles. Each tile has a number on it, which is an answer to the given story problem. ||
 * **What mathematical content is being learned (or intended to be learned)? **
 * **Fractions **
 * **<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">Multiplication ** || <span style="color: white; font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: auto auto 0pt;">* || //<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">Rationale //<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">: Students answer multiplication problems written in the form of taking some fractional part of some number. Students have to realize that the word “of” means to multiply the fraction and the whole number in order to uncover a tile. ||
 * **<span style="color: green; font-family: 'Times New Roman','serif'; font-size: 12pt;">Is the focus on instrumental or relational understanding? **
 * **<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">instrumental understanding (carrying out procedures) **
 * **<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">relational understanding (understanding the meaning of mathematical words and symbols; connections among ideas) ** || <span style="color: white; font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: auto auto 0pt;">* || //<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">Rationale //<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">: Students must understand the procedures for multiplying fractions by fractions and fractions by whole numbers. All questions are written in word form and students need to be able to relate the answer they find to the mystery tile picture. Students also need to connect clue words such as “of” to mean multiplication and check for reasonableness of their answers. ||
 * **<span style="color: green; font-family: 'Times New Roman','serif'; font-size: 12pt;">What role does technology play? ** || <span style="color: white; font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: auto auto 0pt;">* || //<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">Response //<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">: Technology allows students to get instant feedback on their fluency skills and gives them the extra motivation to continue practicing by trying to uncover the picture beneath the answer tiles. Technology also ||
 * **<span style="color: green; font-family: 'Times New Roman','serif'; font-size: 12pt;">What instructional function(s) does the resource serve? **
 * **<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">practice (i.e., practicing skills or knowledge already learned) **
 * **<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">direct instruction/explanation (i.e., explaining or presenting content to students) ** || <span style="color: white; font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: auto auto 0pt;">* || //<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">Rationale //<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">: Students practice their multiplication skills with fractions and see the relationship of multiplying fraction with real world applications, although very minimal (i. e. 1/5 of 65 lego blocks). At the top of the applet, examples are given using the standard algorithm for multiplying fractions and multiplying fractions by whole numbers. ||
 * **<span style="color: green; font-family: 'Times New Roman','serif'; font-size: 12pt;">What kinds of representations of the mathematics are used? **
 * **<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">symbolic (i.e., numerals, symbols) **
 * **<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">visual/spatial (e.g., circles or squares with lines to show fractions) **
 * **<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">concrete or real-world objects (e.g., images of base-10 blocks, puppies, or jars) **
 * **<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">dynamic (mathematical ideas represented through motion or sound) ** || <span style="color: white; font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: auto auto 0pt;">* || //<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">Rationale //<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">: In this game, students are exposed to symbolic fractions in the form of numbers. It is very visual, for as students uncover tiles, a mystery picture is embedded within. Students can make predictions about the picture and is motivation to keep students engaged in the game. Real world ideas are used in the text word problems such as beads, donuts, marbles, and recycled bottles. The applet is dynamic because as a student gets a mathematical problem correct, a tile is “removed” to uncover a pice of the mystery picture. Students must click on the word problem and the answer at the same time. ||

<span style="display: block; font-family: 'Times New Roman','serif'; font-size: 12pt; text-align: center;">

<span style="color: #400080; font-family: 'Comic Sans MS',cursive; font-size: 130%;">Numbersand Operations Evaluation #2:

//<span style="font-family: 'Times New Roman','serif'; font-size: 14pt;">An exploration in the relationship between fractions, decimals and percents // ||
 * //__<span style="font-family: 'Times New Roman','serif'; font-size: 24pt;">[|Fraction Models] __//
 * **<span style="color: green; font-family: 'Times New Roman','serif'; font-size: 12pt;">Which standard? **

<span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">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. ||  ||   || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">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%. ||  ||   || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">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. ||  ||   ||
 * **<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">Understand Numbers **
 * **<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">Understand Operations **
 * **<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">Compute Fluently ** || <span style="color: white; font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">Rationale //<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">:
 * **<span style="color: green; font-family: 'Times New Roman','serif'; font-size: 12pt;">What mathematical content is being learned (or intended to be learned)? **
 * **<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">Fractions ** || <span style="color: white; font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">Rationale //<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">:
 * **<span style="color: green; font-family: 'Times New Roman','serif'; font-size: 12pt;">Is the focus on instrumental or relational understanding? **
 * **<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">instrumental understanding (carrying out procedures) **
 * **<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">relational understanding (understanding the meaning of mathematical words and symbols; connections among ideas) ** || <span style="color: white; font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">Rationale //<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">:
 * **<span style="color: green; font-family: 'Times New Roman','serif'; font-size: 12pt;">What role does technology play? ** || <span style="color: white; font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">Response //<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">: While this concept could be taught without the use of technology, having students manipulate both the numerator and denominators in real time, and viewing how this change affects the decimal and percentage is essential to building the connections to these topics. The instructions are easy to follow and the models are easy to manipulate. The table function allows students to explore connections between the numbers they chose as well as examine how the numbers themselves are related. ||   ||   ||
 * **<span style="color: green; font-family: 'Times New Roman','serif'; font-size: 12pt;">What instructional function(s) does the resource serve? **


 * **<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">practice (i.e., practicing skills or knowledge already learned) **
 * **<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">learning through exploration (i.e., provides context in which students can see new relationships; come to new understandings) ** || <span style="color: white; font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">Rationale //<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">: Although students may be practicing modeling fractions, the interactive exporation of the connection between fractions, decimals, and percents allows students to create their own knowledge of this relationship. <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">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. ||   ||   ||
 * **<span style="color: green; font-family: 'Times New Roman','serif'; font-size: 12pt;">What kinds of representations of the mathematics are used? **

<span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">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). <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">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. ||  ||   ||
 * **<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">symbolic (i.e., numerals, symbols) **
 * **<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">visual/spatial (e.g., circles or squares with lines to show fractions) **
 * **<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">concrete or real-world objects (e.g., images of base-10 blocks, puppies, or jars) ** || <span style="color: white; font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">Rationale //<span style="font-family: 'Times New Roman','serif'; font-size: 12pt;">:

<span style="color: #400080; font-family: 'Comic Sans MS',cursive; font-size: 130%;">**Numbers and Operations Annotated Links:**


 * <span style="color: #000000; display: block; font-family: 'Times New Roman',Times,serif; font-size: 120%; text-align: center;">**Screen Shot** || <span style="color: #000000; display: block; font-family: 'Times New Roman',Times,serif; font-size: 120%; text-align: center;">**Name and Description**  ||
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">[[image:Circle_Game_.jpg width="262" height="136" link="http://nlvm.usu.edu/en/nav/frames_asid_122_g_3_t_1.html?open=instructions&from=category_g_3_t_1.html"]] || <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%; line-height: normal; margin: 0in 0in 0pt;"> The [|Circle Game] can be used to add and subtract positive and negative integers. Circle Zero, pictured to the left, requires students to add three integers within one circle so that each circle equals zero. Students use logical reasoning skills to complete each circle using addition or subtraction. When a successful circle has been completed, it changes color. There are four different circle games that students can choose from to determine the sum or difference. A new game appears with different integers to use to supply unlimited puzzles. It is an excellent tool to practice operations with integers and use logic to complete the circle. ||
 * [[image:Math_at_the_Mall_2.jpg width="254" height="132" align="center" link="http://www.mathplayground.com/mathatthemall2.html"]] || <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%; line-height: normal; margin: 0in 0in 0pt;"> In this [|Math at the Mall Game], students go on a virtual shopping spree that takes them all around the mall making different calculations with percentages and decimals. Students first choose a character and then can choose different paths to get through their adventure. If students need help at any particular establishment, they can click on the “Learn How” button to get a detailed explanation and the procedure for how to solve the problem. Although this interactive focuses on instrumental understanding, the reasoning and problem solving opportunities in a real world situation allows students to connect math to how they might use it outside of the classroom. ||