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  • Staying With Students

    I was very intrigued by this article in the New York Times in October. Dr. Adam Grant, an organizational psychologist, describes a very interesting educational practice that I think merits some attention. One of the many things I love about tutoring is that I can work with the same student over many years. This allows me to get to know each student very well, which means I get to know the student's strengths and weaknesses well. I also typically get to know the hobbies of the child, which I can incorporate into our sessions. For example, one of my students likes basketball, so I regularly create word problems for him that use basketball: How many points do you score if you make X 3-pointers and Y 2-pointers? Can we find a total point score that would not be possible to make combining 3- and 2-pointers? If the other team is winning by Z points, which combinations of 2- and 3-pointers could our team make to win? What kid of probability of making a shot would you like? What does it mean to make a shot 3 out of 5 times; is that better than 4 out of 6 times? Why? Unit rates: if a player makes 30 points in 21 minutes, what's the player's score per minute? It is much harder for a teacher in America to get to know students well and give students personalized attention, especially with so many children in a classroom and with getting new students every year. This article spelled out some commonalities found among students who have achieved significant gains in math and reading performance. "In North Carolina, economists examined data on several million elementary school students. They discovered a common pattern across about 7,000 classrooms... Those students didn’t have better teachers. They just happened to have the same teacher at least twice in different grades. A separate team of economists replicated the study with nearly a million elementary and middle schoolers in Indiana — and found the same results." This practice of moving up with the same students is called "looping." The article continues and explains that "Finland and Estonia go even further. In both countries, it’s common for elementary schoolers to have the same teacher not just two years in a row but sometimes for up to six straight years. Instead of specializing just in their subjects, teachers also get to specialize in their students. Their role evolves from instructor to coach and mentor." This educational practice makes sense for the students, the teachers, and the parents. A rapport can be built and trusted. "Most parents see the benefit of keeping their kids with the same coaches in sports and music for more than a year. Yet the American education system fails to do this with teachers, the most important coaches of all. Critics have long worried that following their students through a range of grades will prevent teachers from developing specialized skills appropriate to specific grade levels. Parents fret about rolling the dice on the same teacher more than once. What if my kid gets stuck with Mr. Snape or Miss Viola Swamp? But in the data, looping actually had the greatest upsides for less effective teachers — and lower-achieving students. Building an extended relationship gave them the opportunity to grow together." As our students continue to struggle with standardized tests and the various pressures of school, I wish that more schools would consider thinking outside the box. Looping sounds like an educational practice that doesn't cost additional money and makes sense for students. It would require some organization and willingness of teachers to possibly expand their expertise. In my humble opinion, it may at least be worth trying it for a few grade levels for a few years and measuring results.

  • A New Daily Math Puzzle - DIGITS

    Were you also addicted to Wordle for a while? Or maybe you still are? I like puzzles a lot and I was definitely hooked on Wordle for a chunk of time, playing it daily. My dad and I came up with our strategies and we liked working together on it when we were visiting each other. Puzzles can bring people together to reach a common goal. I think puzzles are also a good way to let yourself take a break from typical daily life and challenge your brain. Sometimes you can learn something from puzzles, too. I must admit that I learned a few new vocabulary words after playing Wordle. A "trope" is a word or expression used in a figurative sense, for example. Who knew?! Not me! Well, move over, Wordle! There's a new puzzle in town from The New York Times. It's called Digits and once again, I'm hooked! This one is a math puzzle. I like this puzzle because it makes you think about numbers and operations. And boy, do I love asking my students to think about numbers and operations! Using estimation skills can also help you figure out which solution may work. I build estimation skills into my math sessions often because it is a great way to build number sense and a great way to check calculations quickly. For example, if a child is still learning long division and is still unsure of the method, with a problem like 2,219 divided by 7, a child with good estimation skills would be able to look at the problem and estimate that the answer will be around 300. This way, if the child gets 317 after following the steps for long division, the child will feel more sure about the answer because the estimated answer was 300. If the child gets an answer of 32 after the long division, then there's a way to know that something went wrong with the long division. Mistakes happen. We're all human. Estimating helps us know when to check our work. The Digits puzzle asks 5 different puzzles each day. For each one, you get a target number and you need to reach it (or get as close as you can to it) using the six numbers given. You're allowed to use all of the numbers (but do not have to use all of them) and all the operations as needed. I can tell they choose the numbers very thoughtfully. For some puzzles, there are many different tempting things to try. I also appreciate how they allow you to try things to see how it would work and backtrack if needed. I always encourage my students to try things. Here are two of the five puzzles from yesterday: Sometimes, the way I solve one puzzle may be different than the way another person solves the same puzzle. This is yet another reason to love these! Why do something one specific way when there are many ways to approach it? After solving the puzzles, you get a fun summary page, such as below. It shows the operations that you chose to use for each puzzle. Thank you, to The New York Times! I suggest that students play with Digits over the summer to keep up their math skills, or even build their math skills. They may even have some fun with it! Reach out with any questions you may have. =+-=+-=+-=+-=+-=+-=+-=+-=+-=+-=+-= Melissa Agocs Online Math Tutoring and Enrichment www.learnwithme123.com

  • How to Subtract with multiple zeros

    Did you once struggle with subtraction when there was a lot of borrowing involved, especially when there were multiple zeros in the first number? Maybe you now have a child who struggles with it. You're not alone! I've worked with students who were assigned subtraction handouts with a painful number of questions to complete. This can be grueling, especially if the standard algorithm feels like a lot of steps to memorize. I have seen many students struggle with the standard algorithm for subtraction, especially with problems like this one shown. Let's try a different way to solve this kind of problem. Why not!? It is important for students to realize that many times, there are different ways to answer the same problem. It's important to remember that subtraction can be thought of as the difference between two numbers. Or, if the two numbers were on a number line, it could be thought of as the distance between the two numbers. For the numbers shown above, 6,000 minus 926 can be thought of as the distance between 6,000 and 926, shown below. If we're interested in the distance between the two numbers, if I shift each number down the number line equally (let's say by 1 each), then the distance between the two numbers wouldn't change. See below. Since the distance hasn't changed, the answer to the subtraction problem hasn't changed. But what HAS changed is that if I write the problem using the new, red numbers, it's much easier to use the standard algorithm! There is no longer a need to borrow! See below. We would have gotten this same answer of 5,074 if we subtracted 6,000 - 926 also. If the standard algorithm is stressful for you or your child, you can use this strategy. It's a smart way to use math to your advantage! Here’s a video showing another example. 🤗🎉💫 Reach out with any questions you have. =+-=+-=+-=+-=+-=+-=+-=+-=+-=+-=+-= Melissa Agocs Online Math Tutoring and Enrichment www.learnwithme123.com

  • A Useful Tip on Percents- What is 17% of 50?

    Quick! What's 17% of 50? What if I told you it's the same as 50% of 17? Many of us were taught that to find 17% of 50, we need to convert 17% into 0.17 and then multiply 0.17 x 50. This would be correct and lead us to the answer of 8.5. However, let's consider some other ways to think about the question because math typically has many ways to solve the same question. Sometimes you may find one of the other ways more efficient. Let's break down the math behind solving the question 17% of 50: Finding 17% of a number is the same as taking 17/100 or 0.17 of the number. It's going to end up making the number smaller because 17% of a number is a portion of that number. As I mentioned above, to find 17% of 50, you would multiply 0.17 x 50. Let's remember what 0.17 is: it's 17 x 0.01, or 17 hundredths of one. THE COMMUTATIVE PROPERTY OF MULTIPLICATION The commutative property of multiplication says that you can reorder numbers that are being multiplied without affecting the answer. In other words, if I ask you 4 x 9, that's the same as 9 x 4. Or, if I ask you 4 x 9 x 2, that's the same as 2 x 9 x 4 and 9 x 4 x 2, for example. So, back to the problem of 0.17 x 50. This is the same as 17 x 0.01 x 50 because 0.17 is equal to 17 hundredths, or 17 x 0.01. Instead of ordering the numbers as 17 x 0.01 x 50, I could use the commutative property of multiplication and choose to order them as 0.01 x 50 x 17. Doing it this way, I could interpret the problem as 50 hundredths of 17. 50 hundredths is equal to 50%, which many of us know is equal to 1/2. You can also think of 50 hundredths as 50/100, which is equivalent to 1/2. Either way, the problem is now asking for 1/2 of 17 or 50% of 17. The answer is 8.5. We saw this answer before by finding 17% of 50. Both are equal to 8.5! USING REASONING SKILLS to solve problems I don't know about you, but finding 1/2 of 17 feels a bit more friendly of a problem to me versus multiplying 0.17 x 50 with the standard algorithm. It just took a little reasoning to get the problem into something I liked a little better. As a side note, there is yet another way to think of the problem 17% of 50 using reasoning skills and without multiplying 0.17 x 50! Based on what percentages mean, we know that 17% of 100 is 17. Therefore, 17% of 50 would have to be half of 17% of 100, which means half of 17, which is 8.5. Of course, this helpful trick of 17% of 50 = 50% of 17 works EVEN BETTER when the numbers are even more friendly. For example, if the question asks to find 36% of 50, this is the same as asking 0.36 x 50 or 0.01 x 36 x 50 or 0.01 x 50 x 36, or 50% of 36, which is asking to find half of 36, so the answer is 18 and we're finished! Unfortunately, the trick doesn't always help. If the question is something like 19% of 37, swapping wouldn't really make the question any easier. It works great when you recognize a number that can be easily turned into a "friendly percent", such as 50, 25, 20, or 10 because 50% = 1/2 of a number, 25% = 1/4 of a number, 20% = 1/5 of a number, and 10% = 1/10 of a number. For example: 68% of 25 = 25% of 68 and that's the same as 1/4 of 68 = 17 95% of 10 = 10% of 95 and that's the same as 1/10 of 95 = 9.5 35% of 20 = 20% of 35 and that's the same as 1/5 of 35 = 7 Do you have any other ways of solving these questions on percents? I'm always open to hearing other approaches. Also, if your child is struggling with percents and can use some support, I'd be happy to help. Reach out with any questions. =+-=+-=+-=+-=+-=+-=+-=+-=+-=+-=+-= Melissa Agocs www.learnwithme123.com melissa@learnwithme123.com

  • If Your Child is Struggling in Math, You Are Not Alone

    These results came out about two weeks ago and the reporters were expressing doom and gloom for our children. As stated in this New York Times article, "This year, for the first time since the National Assessment of Educational Progress tests began tracking student achievement in the 1970s, 9-year-olds lost ground in math, and scores in reading fell by the largest margin in more than 30 years." The math and reading tests were given to over 14,000 9-year-old students nationwide. Results from 2022 were compared to those of the same age group from early 2020, before the pandemic started. But it's not all doom and gloom! There are ways to gain back the loss that occurred. Many educators and tutors are standing by, eager to help. If you can't afford a private tutor, check out my previous blog post. I have spoken with many parents since the pandemic started and I've heard from many of them that their children were doing well in math prior to the pandemic and then lost interest in it or started doing poorly in it after COVID hit. Students who already struggled with math before COVID continued to show low achievement in the subject, to no fault of their own. Many kids were affected greatly in many aspects of their lives by the pandemic and remote learning. They had to adjust to a new reality, and very abruptly. During that time, many of the children I worked with expressed anxiety to me and loneliness from missing their friends. Many kids could not or did not attend classes that their schools offered. If their academics suffered after COVID, there were reasons for it. If your child is needing extra support to catch up to their curriculum, there is no shame in that. If there are a few kids in the same class who can benefit from support, they can do tutoring together so that they don't feel singled out. Reach out for help or advice so that your child can feel confident and you can feel more relaxed about the education they may have missed out on during the pandemic.

  • What if I Can't Afford a Private Tutor?

    The school year has started in some states and will start soon in others. If you're a parent who knows your child can benefit from some academic support and you are not sure how to afford it, I have some tips for you. ONline tutoring helps! Tutoring is a wonderful way to support students as they learn. One-on-one tutoring and small-group tutoring sessions allow your child to get help and focus on the areas that specifically need work. Sometimes students are embarrassed or too shy to ask questions in class. Sometimes the teacher is overwhelmed with all the responsibilities of the classroom to help each child at the level needed. Having a tutor regularly meet with your child can boost confidence and understanding of what is being taught in class. The Brookings Institution, a non-profit public policy organization, published this article on tutoring, saying: "As educators will attest, tutoring ranks among the most widespread, versatile, and potentially transformative instruments within today’s educational toolkit. We found that tutoring is remarkably effective at helping students learn, with over 80% of the 96 included studies reporting statistically significant effects. Averaging results across the studies included in this analysis, we found a pooled effect size of 0.37 standard deviations. In other words, with the help of tutoring, a student at the 50th percentile would improve to the 66th percentile. In the field of K-12 education research where there is little agreement on what works, these findings are remarkable not only for their magnitude but also for their consistency. The evidence is clear that tutoring can reliably help students catch up." Since COVID-19, remote learning has become more common. Some parents may feel that in-person tutoring is better for their child. If that's the case, I won't push online tutoring. For those parents who are on the fence about it, consider these benefits of it: Health - Especially in times of a pandemic, there is no need to worry about someone meeting with your child in person and someone getting sick....even if it's just a cold! Convenience - If your child switches households during the week or is not at home for a session, it is still possible to have a session as long as there is an internet connection. Technology - Online tutoring enhances students' comfort level with computers, the internet, and typing. I've witnessed my own students pick up new skills quickly on the computer and even teach me a few things! Multimedia - Tutors can incorporate multimedia and interactive activities and games into the sessions, making the child more engaged and interested in learning. As on online tutor, I feel that I'm able to do everything with my students online that I'd do with my students in person. In fact, I feel that I have easier access to my resources as an online tutor since I can find them on my computer and upload them to the screen instantly. There were many times when I was tutoring in person and I wished I had a certain piece of paper, book, or image with me. This easy access to materials benefits your child when I am working online. but how can my child get a tutor if I can't afford one? There are ways to find an online academic tutor who is eager to help. Especially since the COVID-19 pandemic, there has been increased federal and state funding to "close the gap" and provide support to students who need it. Responding to the needs of U.S. schools faced with the risk of learning loss resulting from the COVID-19 pandemic, the federal government passed the American Rescue Plan Act (ARPA), which provides $123B for K–12 public schools, the largest single stimulus funding ever directed to K–12 education. These funds must be used by September, 2023. Low-income districts may receive up to $8K per student; high-affluence districts less than $1K per student. One type of online tutoring that has been shown to help students is called high-dosage tutoring. High-dosage tutoring consists of: Intensive tutoring that occurs one-to-one or in very small groups on a sustained, frequent basis, during the school day An intentional use of additional time with a specific focus on building prerequisite knowledge and skills while also integrating new learning that is part of the grade-level curriculum. There are companies that provide high-dosage tutoring to schools, but your school district has to know about it first! Parent advocacy is very important and can make a difference. If you'd like your child to receive high-dosage tutoring provided by your school (and free to you!), make sure you speak to your principal or superintendent about it. There are many companies offering high-dosage tutoring. Here's a list of some of the companies that your school administrators can look into: Air Tutors Littera Education Varsity Tutors Catapult Learning Carnegie Learning I have personal experience as a tutor with Air Tutors and I can say that working with the kids during their school day is fun for them and gets them excited about the subject. They receive individualized help, get to use computers, and improve their academic skills. Students can ask questions, work on classwork or homework, and help each other over the platform. All of the tutors I interact with from that company are motivated to help and love what they do. With Air Tutors and Littera Education, the tutors work with each school and its needs and use whichever curriculum aligns with their classroom instruction. With Catapult Learning and Carnegie Learning, the companies use their prescribed method of learning. Your child's education is important. Speak to other parents in the district and see if you can gather interest in building a program that would be funded by government money. As always, reach out with any questions. =+-=+-=+-=+-=+-=+-=+-=+-=+-=+-=+-= Melissa Agocs www.learnwithme123.com melissa@learnwithme123.com

  • Fun Number Trick Magic to Engage Our Brains

    I love magic! I admit it. I can watch a magician for hours and feel like a kid, mesmerized and wanting to know how the trick worked. Magic tricks are possible with math, too. I love doing number tricks with students because I can tell their minds start to wonder. They become really interested in figuring out how to stump me or can't wait to do the trick to someone else. But I always want them to also try to understand why the trick works. Tricks can be a fun application of math rules. I will go over one math trick today. It is a good trick for elementary and middle school students who know how to multiply multi-digit numbers. Math TRICK Using Multiplication To follow the steps to this trick, the person must know how to multiply multi-digit numbers. What does that mean? The person needs to know how to do problems such as 29 x 67 or 56 x 34, for example. In this video, you'll see the four steps the "magician" asks someone to do (on the left) as well as each step of the trick after picking a number (on the right). Math Trick with Number 37: Perhaps as an adult you know why this works. Perhaps not. Either way, I encourage you to try the trick with your child a few times with different numbers. A pattern will emerge after it's performed a few times. See if it ignites a conversation or a curiosity to figure out why it works. If you've given it some thought and you're wondering how it works, let me reveal the magician's secret: Multiplication is like addition in that the order in which you add numbers together doesn't matter and the order in which you multiply numbers together also doesn't matter. For example, for addition: 7 + 3 + 5 = 3 + 5 + 7 = 5 + 7 + 3 = 15 All of these are equal to 15 despite the different ordering of the numbers. In the magic trick, I asked you to multiply your number by 3. Then, you took that number and multiplied it by 37. When I picked the number 6, it looked like this: 6 x 3 x 37 = 666 However, because of the commutative property of multiplication, the order of the numbers doesn't matter and I can rearrange the numbers into a different order and still get the same answer. So let's do that: 3 x 37 x 6 = 666 BUT WHY does it equal 666? Let's break it apart: The first two numbers to get multiplied are 3 x 37. What is 3 x 37? 3 x 37 = 111 A-ha! 🤩 111 multiplied to any single-digit number between 1-9 will result in that digit repeating in the answer, such as 111 x 6 = 666. If I had chosen 4 to start with, it would have ended up being 111 x 4 because 3 x 37 = 111 and 111 x 4 = 444. Once the trick is understood, it's usually really fun for the child to be the magician and ask someone else to do the trick. Having the math knowledge can feel empowering! After the trick, it's possible to spark some other math conversations: Do you think there are other factors of 111? How would we be able to tell that 111 is divisible by 3? (a great example of divisibility rules) Name some other big numbers that we can be sure are divisible by 3. Is 111 a prime or composite number? Can any of the other operations use the commutative property? If so, which one/s? Reach out with any questions at melissa@learnwithme123.com. =+-=+-=+-=+-=+-=+-=+-=+-=+-=+-=+-= Melissa Agocs www.learnwithme123.com melissa@learnwithme123.com

  • 5 Skills to Master Before Starting Algebra

    algebra can be exciting! Algebra 1 can be exciting and fun for students because they start to feel very "mathematically mature" when using variables (letters) with their math. It is a wonderful thing to see students embrace algebra. I've seen many students use and recite formulas with variables proudly because they feel like they have been introduced to knowledge that is part of a more adult world. Algebra 1 is the gateway to more advanced math. It is important that students have the skills needed for Algebra 1 so that they can indeed thrive in the class and be motivated to do well in future math classes. While I've taught students who excel in algebra, I've also tutored many students who start Algebra 1 without having the necessary foundational skills. This situation unfortunately makes them struggle and dislike the subject, instead of finding it exciting and reveling in their newfound math skills. If you have a child who is starting Algebra 1 this fall, now is the time to make sure certain key skills are in place for the class. Let's make sure algebra is embraced and not disliked! Take a look at my free resource that lists 5 skills that should be mastered before starting Algebra 1. I will go through each of these skills below. The 5 recommended Skills 1. Exponents Students are introduced to exponents before starting Algebra 1. Exponents are used to represent repeated multiplication. Make sure your child understands: how to use exponents with whole numbers, fractions, and decimals that any number raised to the 0 power is equal to 1 scientific notation, which uses positive and negative exponents that if a number is raised to a negative exponent (for example, 10 raised to the -1 power), it is really getting multiplied by a fraction, which makes the number smaller when expressions with exponents can be combined, such as: 2. multiplication and the distributive property Algebra 1 uses the distributive property with variables frequently. For this reason, it is important that your child feels comfortable with the property when using it with only numbers first. Students need to recognize that it provides a way to multiply numbers together. In the example provided below, students can use the Order of Operations and solve 6+5 first to get 11 and then multiply 4 by 11 to get the answer of 44. Alternatively, they could get the same answer by applying the Distributive Property and distributing the multiplication of the 4 across the 6 and the 5, then adding those products together to get 44 (see below). It would also be a good idea to make sure your child knows how to break down a product using the distributive property, for example: 8 x 14 = 8(10 + 4) or 8(7+7) or 8(11+3) or 14(6 + 2) or 14(3 + 5).....or other options I also suggest that while practicing multiplication techniques, that your child knows factors of numbers well. That means that if a number like 30 is given to your child, the factors of 30 are known: 1, 2, 3, 5, 6, 10, 15, 30. This comes in handy for factoring in algebra. 3. fractions, decimals, and percents I have spent time with many students on converting percents, fractions, and decimals. This is a critical part of middle school math and I find that it as a topic that is confusing for many of them. One of the new things introduced to students in Algebra 1 is the way to write expressions and equations with variables. For example, the multiplication sign...poof....goes away! If we want to express "6 times y", we simply write "6y." If we need to divide 4x by 4, we write a line underneath "4x" and write the "4" underneath the line to signify that "4x is getting divided by 4." This will make a lot more sense to a student if it's already understood that fractions are used to represent division. Once you arrive to Algebra 1, it is expected that you have the flexibility to work with numbers. Students may be asked to solve an equation that includes decimals and fractions, for example. This will present a problem to anyone who doesn't know how to convert one to the other. Word problems will also be presented frequently, sometimes with prices and percents, and there will be a need to translate the facts into an equation that makes sense. A great game to practice these conversions is BINGO, which I recommended in this blog post. 4. operations using positive and negative numbers Another key skill to master before starting Algebra 1 is how to use all of the four operations (add, subtract, multiply, and divide) with positive and negative numbers. Algebra problems will combine positive and negative numbers and students will have to solve equations knowing how to work with these kinds of numbers. For example, does your child know that: -6 - 13 = -6 + -13 = -19 1/2 - 3.5 = -3 (-5)(-12) = 60 (-3)(5) = -15 60 / (-1/2) = -120 -6.3 / -2 = 3.15 These are just a few examples of problems that use operations with positive and negative numbers. I've seen confusion with the first kind of problem with many students, when a negative number is subtracted from another negative number. Sometimes students think this follows the rule of "two negatives make a positive", when in fact the answer becomes more negative. Sometimes it helps to have students realize that subtracting is the same thing as adding the opposite of the number. The opposite of a number is the same number, but with the opposite sign in front of it. So if a number is positive, its opposite is negative and vice versa. 6 is the opposite of -6 and -6 is the opposite of 6. For example, if the problem is 7 - 5, the numbers are positive 7 and positive 5 and positive 5 is being subtracted from positive 7. I could also think of this problem as 7 + (-5) because -5 is the opposite of 5. Likewise, if presented with 10 + (-4), it can be switched to a subtraction problem to say 10 - 4 because 4 is the opposite of -4. The same holds true when the first number in the problem is a negative number: -20 - 5 is the same as -20 + (-5). The subtraction problem was turned in to an addition problem where the opposite of 5 was added. This may make it clearer to some students that since you're adding -5 to -20, the answer will be more negative. 5. graphing on the coordinate plane Graphing is a huge part of algebra and more advanced math. Students who understand how to graph points on the coordinate plane will have a much easier time when they become introduced to linear equations and more advanced topics. Make sure your child can identify the the origin (point (0,0)), the x-axis, and the y-axis correctly. Then make sure your child can correctly graph points in all four quadrants, for example: The point A (3, 1) in Quadrant I The point B (-3, 1) in Quadrant II The point C (-3, -1) in Quadrant III The point D (3, -1) in Quadrant IV It is also important to be able to write the coordinates of any point if given the picture of it. In other words, if presented with the picture above and asked the coordinates of Point D, it would be no problem to say, "(3, -1)." In summary, there are many critical skills to master before entering into the exciting world of Algebra 1. Let's make sure students have the necessary skills needed so that they can appreciate the beauty of algebra! Reach out with any questions or if you think I may be able to help prepare your child for what's ahead. =+-=+-=+-=+-=+-=+-=+-=+-=+-=+-=+-= Melissa Agocs www.learnwithme123.com melissa@learnwithme123.com

  • Tons to Pounds - How Would You Solve This?

    As I was hiking the other day, I couldn't resist snapping this photo (below) for the exact purpose of asking others how they would solve the conversion question: How many pounds are in 17 1/5 tons? This is an example a question that can be answered in many ways. As a math tutor, I always encourage my students to solve a problem using the approach that makes the most sense to them. The truth is that there are multiple ways to add, subtract, multiply, and divide. If a student is taught to only use one way to do any of the four operations, it is a sure way to discourage mathematical thinking and reasoning. Instead, it only encourages the student to mimic a strategy and not think on his/her own. When I was learning math in elementary school, I was only taught the standard algorithms for addition, subtraction, multiplication, and division. Only when I was older did I start to feel confident enough in my math skills to try to solve a problem using something other than the standard algorithm. Straying from the standard algorithm does not mean you are "breaking math." Instead, you're applying what you know and reasoning your way through the problem. I will write future posts on this topic. number sense and numeracy In an article published by the National Council of Teachers of Mathematics, Hilde Howden describes number sense as a “...good intuition about numbers and their relationships. It develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms. Since textbooks are limited to paper-and-pencil orientation, they can only suggest ideas to be investigated, they cannot replace the "doing of mathematics" that is essential for the development of number sense. No substitute exists for a skillful teacher and an environment that fosters curiosity and exploration at all grade levels." I like to include estimation in my tutoring sessions for this reason, especially when looking at answers to make sure they are correct. It is helpful to recognize when an answer simply doesn't make (good number) sense. If a student gets an answer incorrect during a session, I ask the student to take a step back, look at the answer, and see if it "makes sense." It's even a good thing to do when an answer is correct. Estimation can help identify possible correct answers and eliminate incorrect answers on a multiple-choice exam. Solving the same problem different ways and learning how to communicate strategies with others develops numeracy. It is like building a muscle that takes time to develop. By communicating and sharing answers, an awareness slowly builds that allows us to choose a more efficient strategy for future problems. As Cathy Smith states in her article in Mathematics in School, "It is mathematically empowering - and interesting - to use an appropriate method for a problem." I have been working on my numeracy skills as an adult and can really see how approaching problems in different ways makes me appreciate math in a whole new way. How many pounds are in 17 1/5 tons? I posted this question to my relatively new tutoring Instagram account and asked for the reasoning that people used. I also provided three solutions that I could think of that day. The truth is that I may have used three different solutions the following day . The fact that I may use one strategy one day and a different one the next day shows that I can be flexible with my thinking. This is part of what makes math beautiful to me. Math is a language and we may say something one way one day and slightly differently another day. Children should feel that they can also be flexible with their mathematical thinking. When we allow them to do so, they feel more confident and able to understand the math they are doing. Being able to experiment with numbers, have different relationships with them, and communicate strategies to others builds good number sense and numeracy. These are skills that can be carried throughout future math classes and life. There are six strategies presented below to answer the question "How many pounds are in 17 1/5 tons?" Some were shared from my Instagram post (thank you for your comments!). There are many additional ways that it could be solved. If you have an approach that isn't shown here, I'd love to hear it! Leave a comment here or send me a message at melissa@learnwithme123.com including your strategy to the problem. References: Howden, H. (1989). Teaching Number Sense. The Arithmetic Teacher, 36(6), 6–11. http://www.jstor.org/stable/41194455 Smith, C. (1999). Pencil and Paper Numeracy. Mathematics in School, 28(5), 10–13. http://www.jstor.org/stable/30215425 =+-=+-=+-=+-=+-=+-=+-=+-=+-=+-=+-= Melissa Agocs www.learnwithme123.com melissa@learnwithme123.com

  • A Fun Summer Math Idea: Design a Dream Bedroom

    (Note: This project uses 4th-, 5th-, and 6th-grade common core math standards.) I think all math teachers at some point have heard a child ask, “When am I ever going to use this in real life?” I can agree with the point of view that it is fully possible to go about daily life without knowing advanced math. However, I think it’s important to point out to students that many times, we can apply mathematical reasoning and skills to help us do things efficiently in life. Some examples of when math can help us in life include: Making a budget and tracking it so that you don’t go over your budget Taking measurements for furniture to see if something will fit Knowing how to use ingredients proportionately, depending on the recipe and how much you need Calculating sales tax, a sale price, or adding a tip Keeping track of performance using statistics Calculating when you should leave the house to arrive somewhere on time Working out - For example, if a swimmer wants to swim 800 yards for a workout and will be using a 25-yard pool, how many laps would the swimmer have to complete to complete the workout? If you have a child who is creative and likes to construct things, it can be fun for the child to use math skills while also being creative. This blog post shares a lesson plan that is a perfect way to target the creative side while using math skills. I am happy to share resources from other educators that I have experience with and recommend. This activity uses the first two bullet points from above and is a project-based learning lesson that gives the child the opportunity to construct a dream bedroom. There are many project-based learning activities for math. This is one that I recommend and it applies 4th-, 5th- and 6th-grade common core standards. Project-based learning is a fantastic way to connect concepts to real-life situations and present an answer when asked, “When am I ever going to use this in real life?” In this lesson, students will apply arithmetic skills, calculate area and perimeter, and use geometry nets to construct the room they would like. They also get to decorate it, which inspires their creative side! This project has enough parts to it that it can be spread out over several days or weeks, so I think summertime would be a great time to work on it. It will require some parental participation. I should also note that it costs $4.75. After purchasing it, you get a downloadable or printable PDF with instructions, math questions (with answer keys) and templates to use for building the room. It is a long document, but it's not necessary to print the entire thing. For example, it includes a version using feet and a version using meters, which gives you the option of which measuring system you'd like to use and print. The PDF includes instructions for a teacher, but the parent can play that role over the summer. I suggest that you are comfortable with these areas in math before helping to guide the student/s: Scale factor (For example: how to read a map when 1 inch = 1 mile or 1 cm = 1 km) Calculating area, perimeter, and surface area Keeping a budget (An option is to tell the child that the room must be designed for under $X, where X is a number you decide.) Geometry nets (This is optional....more below on nets.) Students will learn to: learn about scale factor and apply it to drawing a floor plan learn how to draw blueprint symbols to identify where they will put their door and window(s) use furniture dimensions to calculate the area and perimeter of their furniture use furniture dimensions to draw their furniture to scale on their floor plan calculate the area of their bedroom floor and the surface area of their bedroom walls and use this information to make decisions about flooring and wall treatments draw a final floor plan to show where all of their furniture will be in their room The instructions say that building the 3D model of the bedroom is optional, but I highly recommend doing it as long as your child is comfortable with scissors and glue. Putting the dream room together is quite satisfying for the kids and the best part is that they have something to keep afterwards. They usually like that part a lot. If they make any mistakes while constructing it, it is possible to print out new pages. I also like that it provides different cost sheets based on whether the child knows how to add numbers with decimals or not. So....what is a net in geometry, anyway? 🧐 I know I was never taught anything about nets when I was in school! Geometry nets are commonly taught now in schools. A geometry net is a term used to describe what a 3D shape would like if it was opened out and laid flat. Working with geometry nets uses spatial skills. This video is a good explanation of the nets of a cube. Here are some other examples of nets, including some graphics, some worksheets with answers, and some printouts to practice making 3D shapes from nets. Project-based learning is usually done in groups, so if you have multiple children of a similar age or if your child has friends in the same grade, this project could be done together, with each of them completing their own bedroom. This allows the kids to see that there are different ways to tackle the same project. It also can bring up discussions. While they articulate what they are doing and why they’re doing it, it can help bring clarity and depth to their understanding. As Judy Zorfass and her colleagues describe, “‘Thinking aloud’ requires students to talk through the details of the problem, the decisions they have made as they try to solve the problem, and the reasoning behind those decisions. Struggling students, in particular, can benefit from slowing down and articulating the problem-solving process, because it gives them time to focus on the key parts of the problem. This helps them to more fully comprehend the problem before they try to solve it.” If you decide to purchase this lesson and work on it, let me know how it goes! As always, reach out with any questions. The link to the assignment can be found here: https://bit.ly/3Og6Ldt =+-=+-=+-=+-=+-=+-=+-=+-=+-=+-=+-= Melissa Agocs www.learnwithme123.com melissa@learnwithme123.com

  • 5 Ways to Use Math Over Summer Break

    I've seen it all too often. Kids begin to grasp concepts in math during the school year, then return to school after the summer break and have lost a lot of the progress they made. Here are some tips you can do over the summer to help ensure that the math"summer slide" isn't too painful in the fall. FOR ELEMENTARY SCHOOL CHILDREN If your child has learned any of the FOUR OPERATIONS (ADDITION, SUBTRACTION, MULTIPLICATION, DIVISION), incorporate them into daily life however you can. Some ideas include: 1. USE FOOD - You can cook or bake together with your child. If you need to double a recipe, this is a perfect way to practice addition and multiplication. Make sure to include your child in figuring out how much of each ingredient you need to include. You can also ask questions such as, "How do you think this would taste if we add more sugar than the recipe suggests?" to get your child to think proportionately and realize that if one ingredient deviates from the plan, the recipe will taste differently. 2. MAKE A NUMBER - There are all sorts of ways to create a number. What do I mean by that? For example, if I say, "Make 20," I can arrive at 20 in any of the ways shown below. Encourage your child to come up with as many ways as possible to create that same number. You can pick a different number every day. You can keep track of all the numbers you created on a sheet of paper and keep it visible, perhaps on your fridge. This can be something your child can be proud about. If this becomes boring or too easy for your child, you can always tweak it to make it harder by asking to create larger numbers or requiring 3 or more numbers to be used (see my examples in the last row) or creating the number without using a certain digit or number at all (for example, make 20 without using the number 10). If your child knows how to multiply and divide, yet chooses to always make the numbers using strictly addition and subtraction, try to nudge your child to use multiplication and division also. Extra points for kids who use multiple operations in the same equation (for example: 6 x 4 - 4 = 20)! FOR MIDDLE SCHOOL CHILDREN If your child has learned FRACTIONS, DECIMALS, AND PERCENTS, the summer is a perfect time to reinforce how these numbers can be converted from one to the other. This would be great for middle school and pre-algebra students. Some ideas include: 3. Play BINGO! - I have this BINGO game and have played it many times with students. It is a great way to use skills and make it fun at the same time. I have seen kids get really into it! Make sure they have a whiteboard or paper and pencil nearby when they play and encourage them to do the math when it's necessary to do a conversion. To play, a number is called out loud, for example, "one-fourth." The number "one-fourth" may be written on the BINGO card as "1/4" or it may be on the card as "0.25" or perhaps as "25%." Each player covers a number on the BINGO card with a chip if it's said out loud and written on the board in any one of its forms: fraction, decimal, or percent. The goal is to place five chips in a row on your board. Of course, the better you understand how to convert decimals, fractions, and percents, the better your chances are of winning. This game is a great way to nudge a child to try to find a number's equivalent. For example: "Are you sure you don't see "one-fourth" on your card? I think it's there, but not as a fraction." It may be hard to find this physical game, but I have also discovered this site that allows you to create and print your own free math BINGO printable board. 4. Fraction Talks - There is a wonderful free resource from Math for Love called Fraction Talks, which provides instructions to guide a child through recognizing fractions in colorful pictures. The various pictures vary in complexity. This exercise can be extended to students who have learned decimals and percents by asking them the proportion of each color in each picture as a decimal and percent. Once your child gets the hang of how it works, perhaps your child would enjoy making and coloring an original picture for you to figure out! 5. Compare Numbers - If you're out shopping with your child, it's a great way to point out sales and percents. If your child is familiar with percents, ask which would cost more: the sale of 45% off or 1/2 off? Ask why. Explaining why an answer is correct helps form a deeper understanding of the concepts and how they work. Plus, if your child arrives at the answer using a different approach than you do, this is a wonderful opportunity to talk about how math problems can be solved in multiple ways. There are also these puzzles from Open Middle that ask the student to come up with several ways to plug in numbers to make these fraction-to-decimal comparisons work (Puzzle 1 and Puzzle 2). Each can be printed out or you can easily remake them on a piece of paper or whiteboard. If you're looking for other engaging activities to compare numbers, I highly recommend other offerings on Math for Love or Open Middle, especially for elementary and middle school students. There are many ways to use math in everyday life. If you don't have enough time to play games, try to ask questions while shopping or cooking and try to get your child to create numbers in different ways. Reach out to me with any questions or for additional ideas. Let's keep our kids engaged and ready for the fall! =+-=+-=+-=+-=+-=+-=+-=+-=+-=+-=+-= Melissa Agocs www.learnwithme123.com melissa@learnwithme123.com

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