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Incorrectly grouping numbers below 10.

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A common theme running through the Common Core Standards for Kindergarten is being able to find pairs of numbers within base 10. Some standards include adding or subtracting numbers in base 10 and others are more specific by having students decompose numbers in a base 10 in different ways. Students that struggle with this may be incorrectly grouping numbers by stating that 6 and 3 make 10 when in reality it is 6 and 4.

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An applet to help with this misconception is the Illuminations Ten Frame on National Council of Teachers of Mathematics. This applet contains different games that the students can play including "How Many?" "Build" "Fill" and "Add." All of these games will assist students that have grouping misconceptions. The visuals of the 10 frame and the filler chips will guide students and provide more practice with addition, subtraction and grouping.

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https://www.nctm.org/Classroom-Resources/Illuminations/Interactives/Ten-Frame/

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According to Table 7.2 Van de Walle, this applet is almost perfect. It is free from all errors, prominently displays the authors and connects to the proper audience. There are no biases within the applet. The only thing I am unaware of is how often the site in updated. While the information on the site is always relevant and correct, it may need a bit of updating to be supported by all technological devices.

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I would asses my students' grouping skills by playing number blocks. The blocks would contain numbers from 1-9 and would need to find their partner to match the 10 block. This would allow me to see whether or not the students are able to find the correct pairs of numbers to create 10.

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Misinterpreting the meaning of the equals sign.

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Within first grade, the students are required to understand the meaning of the equals sign and relate them to addition and subtraction equations. Students that struggle with this topic may be answering questions incorrectly based on Carpenter's "Equality" article. Carpenter states that the most common misconceptions are "extending the problem," "the answer comes next," and "using all the numbers." All of these incorrect answers stem from not understanding what the equal sign means.

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An applet to help with this misconception is using a base-10 block website to compare numbers. The applet allows students to drag 1s, 10s, or 100 blocks into an area and then the same on the other side and state whether or not the numbers are equal. If used as a whole class activity, the teacher could write addition and subtraction sentences on the board while the students put the answers to the equations in the two area and state whether or not the equations are equal. This will help students that are struggling because it will break down different parts of the equations and visually show them where their mistakes are.

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https://www-k6.thinkcentral.com/content/hsp/math/hspmath/na/common/itools_int_9780547584997_/basetenblocks.html

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According to Table 7.2 Van de Walle, this applet's authority, content, objectivity and accuracy and nearly perfect. It could be updated a bit to make the currency more relevant and also create more sections to enhance the audience. This applet could be used for multiple different grade levels depending on how it is being used, but I believe it serves its' purpose very well.

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One way I would assess my students understanding of the equals sign is having them play a balance game. I would bring multiple balances for the students to work with while they would have to combine different expressions with their appropriate answers and put them on the balance. This would also go along with a worksheet where the students could record their answers. I would be able to evaluate their work in class while also collecting their worksheets for accuracy.

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Not having a strong enough foundation of addition and subtraction to be able to move onto multiplication.

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Within second grade, students need to create a strong enough foundation to be able to move from addition and subtraction to multiplication and division in third grade. Some students are not able to grasp the concept of repeated addition which is the foundation for multiplication. The standards state that the students need to be able to use rectangular arrays to find the total amount of objects and create equations from them.

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An applet to help understand addition arrays and creating a strong foundation for multiplication is an Education Place e-manipulative. This manipulative can be used for different things but is great for adding counters to create addition arrays and begin conversations for multiplication. Just like we did in our class, I would have my students take different numbers and break them down into different arrays like 9, 20, 30, and so on.

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http://www.eduplace.com/cgi-bin/schtemplate.cgi?template=/kids/mw/manip/mn_popup.thtml&filename=1cc_prim&title=Counters&grade=K

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According to Table 7.2 Van de Wall,
this applet is extremely useful. After using it in our class first hand, I know that it serves the purpose it is trying to very well. The authority, content, objectivity and accuracy are nearly perfect. Similar to most mathematics applet's the audience is not as clear because it can be used for multiple ages and grade levels.

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I would assess this by having the students complete different arrays and screenshot them to show their understanding and present it to the class. This would be a larger assessment that would include technology, presentation skills and math skills. I would allow the students to work in groups and present the information they've learned. Listening to the students explain what they have learned in their presentations would be a great way to see whether or not they are ready to move on to multiplication.

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Not understanding what the communicative property of multiplication is.

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The communicative property of multiplication states that no matter the order of the numbers in the equation, the product will still be the same. For example, 6 x 4 = 24, and 4 x 6 = 24. Students may have trouble with this because visualizing 4 sets of 6 and 6 sets of 4 doesn't always seem like it would be the same total amount. The standard states that students should be able to apply the properties of these operations to make multiplication and division simpler.

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I believe it would be a strong idea to use the same applet that was used when developing the skills of repeated addition with multiplication as well. It will further send home the concept that repeated addition is multiplication. I would reuse the Education Place e-manipulative to create arrays for 6x4=24 and 4x6=24 as one example. This visual will help students understand that it doesn't matter whether you have 6 groups of 4 or 4 groups of 6, there are still 24 total objects.

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http://www.eduplace.com/cgi-bin/schtemplate.cgi?template=/kids/mw/manip/mn_popup.thtml&filename=1cc_prim&title=Counters&grade=K

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According to Table 7.2 Van de Wall,
this applet is extremely useful. After using it in our class first hand, I know that it serves the purpose it is trying to very well. The authority, content, objectivity and accuracy are nearly perfect. Similar to most mathematics applet's the audience is not as clear because it can be used for multiple ages and grade levels.

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In order to assess my students on this standard, I would first have them use the applet to practice, then have them create their own arrays using objects of their choice in the classroom. I would then have them explain how it shows the communicative property of multiplication. Their wording for thrir explanation is the most important part of the assessment because students need to be able to explain the why and how of mathematics, not just the correct answer.

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Not being able to see the connection between factors and multiples.

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Factors and multiples are always concepts that trip students up. This relationship is also key to understanding the relationship between multiplication and division. The standard states that the students should be able to find all of the factor pairs for numbers up to 100 and understand that a whole number is a multiple of its factor. If students are lacking multiplication or division skills, this concept would be very difficult to comprehend and would need additional explanation.

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The applet I would use to help students practice with multiples is from Math Playground and is titled, "Pumpkin Multiples." Within this game, students choose a target number that will find multiples of. Once that number is chosen, pumpkins will fall from the sky with different numbers. The students need to choose only the pumpkins that are correct multiples to win the game. I would pair this applet with a worksheet where the students would keep track of all of the multiples they got correct and explain which numbers are factors and which are multiples.

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https://www.mathplayground.com/multiples.html

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This Applet is more fun than the others, but is still very educational. According to Table 7.2, the information is always accurate, but may allow the students to be bit less focused than the other applets. I still believe this is a strong applet because if the students are enjoying themselves, they may remember more of what they are learning.

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I would assess my students on factors and multiples by having them creating a 100 number chart and highlighting all of the multiples of different numbers in different colors. For example, all of the multiples of 2 would be in yellow, all of the multiples of 3 would be in pink, so on and so forth. I would also have them write and explain any overlap they see in the factors/multiples and state what that means.

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Not being able to recognize patterns between numbers.

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Understanding patterns and the relationships between numbers is essential to fully understanding mathematics and not just giving the correct answer. The standards for fifth grade state that students should be able to generate patterns from rules, identify a rule from corresponding terms and finally, graph the points created from the patterns. Students may struggle with this because it is a difficult topic to grasp the deeper meaning. Until this point, students are so used to being given a problem and finding the answer, whereas now, they are working backwards from the answer and need to find the problem.

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Once again, I am using an applet that we used in our class that I believed to be extremely helpful. The function machine on math playground is a great tool to practice finding relationships from numbers based on inputs and outputs. The students can choose either beginner or advanced levels and decide whether they or the computer will create the input. Once selected, the function machine will spit out an output and the student needs to find the relationship between the numbers. This tool would be greatly helpful for students struggling with finding patterns and functions because practice makes perfect with a skill like this. The more sets of numbers students see, the stronger they will be on this topic.

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https://www.mathplayground.com/functionmachine.html

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According to Table 7.2 Van de Walle, the information is from a reliable source, pertains directly to this misconceptions, and is always accurate. The only thing it could need work on is becoming more up to date. It looks a little outdated but the information is still extremely accurate and relevant.

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In order to assess my students on this misconception, I would play a round of "Guess My Rule," like we did in class. I would differentiate the problems by making some easier than others, but allow the students to have the time they need to be able to answer the problems to the best of their ability. I would then collect the worksheets and provide feedback on how my students are doing and determine whether or not they have grasped the concept.

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Misinterpreting the meaning of variables

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Variables become a huge part of the mathematics curriculum in sixth grade. The switch from operations and algebraic thinking to expressions and equations in the common core standards introduce variables, linear equations and other elements that are the basis of traditional algebra. While the standards are trying to develop algebraic thinking from the very beginning, sixth grade is when students begin to feel the pressure of learning algebra and may struggle with math. Variables are a huge part of mathematics and are involved almost every step of the way after they are introduced in sixth grade. Students need to fully grasp the concept of variables and how they can represent different things. One standard states that the students must be able to "Write, read, and evaluate expressions in which letters stand for numbers." This is a basic concept that will travel with the students through the rest of their mathematics careers and needs to be fully grasped.

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Incorrectly understanding and converting fractions, decimals and percents.

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One of the more difficult concepts in middle school math is understanding the conversions between percents, decimals and fractions. I always struggled as a student understanding that .5 was equal to 1/2 and 50%. The entire concept always confused me and I wish I received extra practice on this when I was in middle school. Not only do students needs to understand that mathematics that goes into conversions, but they also need to apply it to real world situations and understand how the phrasing changes the problem. For example, the standards state, "Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that "increase by 5%" is the same as "multiply by 1.05."

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Not understanding the rules of exponents.

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One applet that is extremely useful for working with variables and equations is the "Algebra Equations" applet from Math Playgrounds. I really enjoy this applet because it combines the meaning of the equals sign by having the balance representation while adding in the variables and numbers that make up an equation. As students use the applet, an equation appears at the top of the screen and the students have to show the equation using x blocks and number blocks. They put the left side of the equals sign on the left side of the balance and the right side of the equals sign on the right side of the balance. This first step will help students understand what and equation looks like and will make the balance uneven. The students then need to manipulate the blocks to make the balance even again by adding or removing number and x blocks. This applet is extremely visual and is a great representation of what the numbers and letters mean in an equation. It would be a great practice tool for students.

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https://www.mathplayground.com/AlgebraEquations.html

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According to Table 7.2 Van de Walle, all of the information in this applet is accurate and from a reliable source. While it may look like a game, it is extremely educational and will assist students in the misconception about variables. The information is also for a wide range of people from teachers to students to parents and families.

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In order to assess my students on their understanding of variables I would have them complete different things. The first thing I would do is create a flip book to help them clearly define variables and understand how they can be used in different circumstances. I would also have them use this applet in class to gain extra practice, and finally, I would assign different sections of IXL work for them to complete.

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The applet I would have my students work with to further understand this topic is the "Fractions, Decimals and Percents" grid by Visnos. The grid consists on 100 blank tiles and six cards that show percents, decimals and fractions. The student can then shade in a certain number of the tiles and the cards will show the fraction, decimal and percent for the shaded and un-shaded regions. This tool is wonderful for understanding the connection for the three forms because it provides all options so the students can see how they are equal and how they differ in form. I also like how it provides the fractions for the shaded and un-shaded regions so the students can see that as well. This would also be beneficial as a whole class activity because the teacher would be able to show either a fraction, decimal or percent and have the students find the other different forms for practice.

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https://www.visnos.com/demos/percentage-fraction-decimals-grid

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I believe this is one of the strongest applet's on this Padlet. It is extremely user friendly, the information is always correct, it directly relates to the misconception and is easy to use for all ages. According to Table 7.2, I am unsure how frequently the site is used, but I would use it regularly in my classroom. I believe it could serve multiple purposes and could help many different kinds of students.

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In order to ensure my students have grasped the concept of percents, fractions and decimals, I would play Around the World with them. Two students would begin the game and would compete head to head against each other with myself standing in front of them. I would hold up a value in either a percent, fraction or decimal form and ask the students to give it to me in another form. Whichever student answers correctly and the quickest moves on to the next round against another student. We could continue to play until one student makes it completely around the room. The students who are not playing can also write down the correct answers of other students' problems to gain bonus points. The students would hand in the papers at the end of the game and I would gauge the classroom as the game was being played.

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When I was middle school, math was commonly taught in a "drill and kill style" that many students did not respond too. Exponents and the rules of exponents was very commonly taught in that style and you either knew the rules or you didn't. Because of that style, it is something a lot of students in my generation struggle with and is something I would like to teach in a different fashion when I am a teacher. The eighth grade standards state that the students must be able to apply the rules of integer exponents and use them when solving equations. The first step in achieving this standard is understanding the rules of exponents.

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The Applet I would use with my students for practicing the laws of exponents is titled, "Exploring Laws of Exponents - Use It" from Math Interactives. This applet allows students to dig for dinosaur bones while using the law of exponents to do use. When entering into the applet, the students have to choose the "exponents" tab and then choose the lesson titled, "Exploring Laws of Exponents (Object Interactive)." The students go on a hunt for dinosaur bones and have the explain the different laws of exponents to solve problems and uncover different bones. I believe this applet would be extremely useful for students who are struggling with this misconception because not only is the math content helpful, but it also has a fun aspect to it. It is also extremely clear on how to solve the problems and will allow students to work through whatever they are having trouble with.

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http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.NUMB&ID2=AB.MATH.JR.NUMB.EXPO&lesson=html/object_interactives/exponent_laws/use_it.html

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Using Table 7.2 to evaluate this applet: the content, objectivity, and accuracy are almost perfect. The authority could be stronger because there is no recognition anywhere on the site. The currency and audience could be a bit stronger also because the applet is clearly out dated a little bit and is unclear exactly who their audience is. It is clear that it is made for students, but could be multiple age levels.

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In order to assess my students for this misconception, I would first create a guided notes worksheet that they would need to complete writing out all of the rules of exponents. I would then have them play around with the applet to practice their skills. Finally, the assessment I would use would be to have the students work together in groups to draw diagrams of what the different exponent rules mean. I would then have the students present their diagrams and explain how they represent an exponent rule.