Box-and-whisker plots are a brand new concept for my 6th-graders, and when students are first introduced to them, they seem a little scary. However, with some structured directions, students catch on very quickly.
I break down the box-and-whisker plot into 5 steps, in order to plot the 5 points needed to create the box and whiskers: 1) Order the data set from least to greatest. 2) Identify the smallest and largest values; place those points on the number line (above the number line). 3) Identify the median and place that point on the number line. Students need to remember that if there is an even number of numbers in the data set, the median will be the mean of the two middle numbers - even though they've found median in the past, many students tend to need this reminder.
4) Find the first and third quartiles and place those points on the number line. This step can be tricky for students...I've found it to be the step that most often throws off their box. Here's what I've noticed: when there is an odd number of numbers in the data set, it's a little easier for students - the median is not included in the upper and lower halves of the data, so they are ok with circling the halves on either side of the median and then finding the median of each half, as in figure 1.
However, when the data set has an even number of numbers, students need to remember that the median is between numbers and that exactly half of the numbers are in the upper half of the data and half are in the lower, as shown in figure 2.
The common mistake that students make is to assume that since the two middle numbers (in this case 6 and 7) were used to find the median, they aren't part of the upper and lower halves. This throws off their 1st and 3rd quartiles. I've found it's important to spend extra time on the different scenarios possible in this step. I've also found that drawing that middle line to represent the median is a very helpful visual.
5) Draw the box and whiskers:
a) Draw a box, with the end lines going through the points of the first and third quartiles. b) Draw a vertical line through the median. c) Draw lines from the box to the least and greatest values. I created this notes/fold it up a couple years ago, to help guide my students as they began creating box-and-whisker plots on their own. I created two versions - a blank one for students and a completed one for me:-) Using the blank version, I walk the students through their creation of the box-and-whisker plot. And when we're done, they have the notes for their independent work. The only folding part of this fold it up is the section at the very bottom that has the definitions of the vocabulary. Students will need to cut the vertical lines along the bottom, fold them up, and then write the vocabulary words on the outside of the flaps (quartiles, first quartile, third quartile, variation, interquartile range). When students miss the instruction due to absence, I give them a copy of the completed version on their return.
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I love playing ping pong! I played it a lot as a kid and I play occasionally as an adult....we have a table in the basement:-) I would never claim to be a SERIOUS player, but I'm not bad!
I was playing with my daughter the other day, and it occurred to me that playing ping pong is a great way for younger children to practice their addition facts and some multiples of 5 (good for older kids too, if they don't know these facts very well). Now, this idea is based on the "serving rules" that we used when I was growing up. It appears (after I searched for info) that these are not the official rules any more, but since I'm not a professional, I'm ok with playing by the unofficial rules! The way we played is that the server switches every 5 points, and we played to 21 points. So, here's where the math comes in....when you're playing, you need to know when to switch who's serving, so you need to know what adds up to the multiples of 5. When the score is 5-0, 4-1, or 3-2, serving switches. To switch servers at 10 points, players need to know that the score would be 10-0, 9-1, 8-2, 7-3, 6-4, or 5-5. When serving switches at a total of 15 points, the score possibilities are 15-0, 14-1, 13-2, 12-3, 11-4, 10-5, 9-6, 8-7. At 20 points, the score would be 20-0, 19-1, 18-2, 17-3, 16-4, 15-5, 14-6, 13-7, 12-8, 11-9, 10-10. The repetition of these facts throughout many games can really help kids learn them. Over the years, I have noticed that students (in general) seem less aware of, and less automatic with, the digits that will add to 10. Playing ping pong is a great way for kids to practice these facts without thinking that they're practicing math (math in real-life!). This is great for parents to do with their kids, but also - a mini ping pong table in the classroom sounds like fun!!
I know the first thoughts many people have are the ability to work well with preteens and a degree in education. However, those are just the basics.
I’ve discovered to be successful as a middle school teacher, you need a certain skill set. While I learned some of this in school, most of it I learned through experience. After all, working with middle school students for years teaches you a few things. 1. PersistenceHonestly, I think this is a key skill for every single teacher, no matter what age group they teach. Students aren’t always happy to be in class or eager to learn. This means I have to be persistent and keep working with my students, no matter how stubborn they might be. Of course, sometimes persistence also means taking the time to figure out why a student’s having problems. 2. PatiencePatience and persistence go hand-in-hand. Naturally, working with a group of preteens that are just starting to deal with all kinds of new hormones is going to take patience. This is the age where kids start to act out more and push boundaries. It’s easy to just lose your temper, but I’ve developed more patience than I ever thought possible. Trust me, it’s well worth it. 3. Engaging Teaching StyleOne of the main purposes of my site is to promote a more engaging and active teaching style. Students learn better when they’re engaged versus just sitting and listening. I’ve found the more I let students interact during a lesson, the more they remember later. 4. AdaptabilityI know it sounds cliché, but every day is different as a middle school teacher (yes, for any teacher:-). As I mentioned, students are going through numerous changes during this time. I think one of the most important skills for teachers to have is adaptability. Being ready for any situation is crucial. It also means you don’t let your students take you by surprise (if possible, haha). 5. Social Awareness And EmpathyAs middle school students deal with new emotions and social situations, middle school teachers need to be socially aware and empathetic. In fact, empathy actually helps to engage students. I’ve found it makes it easier to relate to students and figure out what they need. With the increasing amount of bullying, teachers also have to be socially aware of what’s going on in students’ lives and look for any signs of problems. Formal education is important, but cultivating the above skills takes you from a teacher to an incredible educator and role model. Here's to the skills of middle school teachers (and all teachers, of course)!!
I know lots of people use beach balls in the classroom, but I haven't used them in such a long time that I thought I'd share my excitement about finally getting around to getting new ones! I have a little bit of a beach theme in my room this year, so that motivated me to get some beach balls again. I ordered a pack of 12 and am writing different math skills practice on them - so far I have multiplication facts, exponents, fraction/decimal conversions, and common measurement conversions. I have 12 beach balls to fill with math, so I need to decide on more topics. I think I'll do square roots, division facts, math vocabulary...I need to keep thinking:-)
Our math classes aren't that long, but I figure I can squeeze in 5 minutes at the end of class once or twice a week to toss the beach balls around for some quick facts. With so many different beach balls, I could even differentiate and have 3 groups tossing at a time, depending on their needs!
Do you use beach balls - if so, how? This post is from my old blog, and was written in April, 2015, but I thought it was worth transferring here and sharing:-)
Today, as my students were working on a color by number in math class (which I thought was a fun, different way to practice math), one of them asked "How does coloring help with math?" The question was asked with a "there's no reason I should have to do this" attitude. I explained that it helped with motor skills and helped one to use the brain in a different way, and that exercising the brain in different ways could help in all things that require thinking (not just math). I don't think he really appreciated my answer:)
So, I decided to do a little research, to see what I could find. Most of what I found (not a super-long time of searching, because I didn't have that much time!) was related to the benefits of coloring for young children (and did relate to math skills) and for adults. Here are a few things that I found, as coloring relates to adults: According to the Huffington Post (10/13/14), coloring benefits adults (and I would assume children as well) because it "generates wellness, quietness and also stimulates brain areas related to motor skills, the senses and creativity." In addition, psychologist Gloria Martinez Ayala states that when we color, we activate different areas of our two cerebral hemispheres. "The action involves both logic, by which we color forms, and creativity, when mixing and matching colors. This incorporates the areas of the cerebral cortex involved in vision and fine motor skills [coordination necessary to make small, precise movements]. The relaxation that it provides lowers the activity of the amygdala, a basic part of our brain involved in controlling emotion that is affected by stress."
According to PenCentral, coloring benefits adults in helping them to maintain fine motor skills -this requires extra work by your brain to coordinate your actions and muscle control in your hands and arms. Coloring can help delay the loss of fine motor skills as people age. Coloring may also help fight cognitive loss, especially
if challenging pieces are completed every so often. I didn't necessarily find research to answer my student's exact question, but what I found was quite interesting! If anyone knows of other articles or published research to support the role of coloring in improving math skills, please let me know!
There’s no denying that middle school is a difficult time in kids’ lives. I remember my middle school years (actually junior high years - we didn't have middle school in our district back then:-) and while they say the teen years are the worst, my tween years felt like the worst instead.
I see students struggling through so many transitions. It’s up to us as teachers and as parents to better understand the problems facing middle school kids today, so we can help. 1. BullyingMost bullying happens, or at least starts, in middle school. In fact, at least 25% of students in the U.S. say they’ve been a victim of bullying. I remember bullying growing up, but it’s become a much worse issue today. When students are having to constantly deal with bullying in school and online, it makes it difficult for them to concentrate or even care about coming to school. Sadly, the solution isn’t clear, but we just have to be there to help build students’ confidence and prevent bullying whenever we can. 2. Varying Growth RatesMiddle school is that incredibly awkward time, as I’m sure many of you remember. I know none of my friends ever seemed to be on the same page when it came to physical growth, maturity, emotions and all the other changes puberty brings. Adolescence is a time for change, but all those hormones create issues. While it’s not a new issue, today’s middle school students seem to have more self-esteem issues than ever before; every change affects their confidence, which affects their learning experience. 3. Finding The Value In EducationWith all the hormonal issues, bullying and even problems at home, sometimes middle school kids don’t think education is important. They tend to find the worst role models, such as athletes who dropped out of school or the latest pop stars. I remember thinking I already knew everything at that age, but when today’s kids find everything online, they honestly don’t think school’s important at all. 4. Relying Too Much On TechnologyWith that thought, I love technology, but today’s middle school kids use it as a crutch. They rely on it to the point that they’re distracted and would rather just look for an easy answer instead of learning how to do something. It’s also part of the reason I like to use engaging games to get them learning while having fun. 5. Finding BalanceOf course, middle school kids have more pressure on them today to join clubs, play sports, volunteer and do well academically to start preparing for college. This is on top of a growing social life and finding the balance between kid and a teen. It’s not always easy, but luckily, we can help guide them to find better balance. This post is transferred from my old blog - I can't believe it's been nearly three years since
I wrote it (Jan 13 of 2015)! I'm glad to say that I'm just as dedicated to my workouts as I was when I wrote this:-)
I don’t know about you, but I love my workouts. They do so much for me, mentally and physically, and I really miss them when things come up that cause me to skip them. I miss them so much that I often get up at 4:30 am to be sure I get some workout in, just in case my day ends up having other plans for me.
Why is it important for teachers to work out (besides the usual health reasons)? These are my top 5 reasons: 1) Exercise is a great stress reliever. How many days do you come home stressed out over the events of the day? Still thinking about things that kids (or parents, or administrators) did that got you worked up? Do you bring that stress home with you or leave it at school? I know I have trouble leaving it at the door, but when I can go jump on the treadmill or the elliptical and pound that stress out, my mind definitely becomes more free. 2) Working out can renew your energy level for the rest of the day. This is especially true if you can do it right after school. I usually have some grading (or planning) that I bring home, or I need to help my daughter with homework during the evenings; when I take that exercise time right after school, I get recharged for the evening. 3) It helps you avoid snacking. I don’t know about you, but when I get home, I feel like eating EVERYTHING I can get my hands on, (and I'm often tempted to choose foods that aren't very healthy!) When I work out, I don't have the same snacking urges. 4) Creativity Research shows that exercise helps memory and stimulates creativity. It's a great time to run lesson/activity ideas through your mind; somehow that extra physical activity gives your brain the boost to make those lessons more engaging/exciting/interesting! 5) Sleep better! Extra physical activity helps me sleep better. What teacher doesn't need to get good sleep? Who else loves their workouts??
I’m a firm believer that one of the best ways to learn is by playing games. It’s just more fun and students don’t even realize how much they’re learning.
While any game that helps kids learn is a winner in my book, I have some wonderful middle school games and activities I’ve created to use at home or in the classroom. Truth or Dare Math (and ELA) GameRemember playing Truth or Dare when you were younger? I’ve brought the concept to the classroom and incorporated math and language arts concepts (and Google classroom!). Students can choose a Truth, which is a one-point questions about the concepts. Or, they can boost their score faster with a more challenging Dare question. It’s one of my favorites among middle school games and it really gets the students excited. Decimal DiceI’ve always loved the dice game Yahtzee!, so I decided to create my own little spin on it. Students love rolling the dice and creating fraction pairs. The challenge comes when they have to convert those fractions into decimals or whole numbers. It only takes a few turns before students learn the rules. It’s an incredibly engaging game for small to large groups. Footloose Task Card GamesWhy should learning math concepts be boring? I’m always looking for and creating middle school math games to get students more engaged. With Footloose Task Cards, students answer various types of questions about math concepts - sometimes they are basic knowledge questions, sometimes they're word problems, and sometimes they're quite challenging. It's easy to differentiate using these cards:-) Students move around to get new Footloose cards each time they complete one, and they write all their answers and work on their Footloose grids. It's a great way to keep students practicing and moving - and it's amazing how quiet they are during this time! Math Color By NumberColoring for adults is one of the biggest trends at the moment, and it's become a great way to help students practice math concepts:-) I’ve put together a fun bundle that uses the color by number approach to make it more fun to learn and practice probability, algebraic expressions, prime factorization, combining like terms and more. While I do offer each separately, the bundle’s a great resource to have on hand as practice for a variety of math concepts for 6th and 7th graders. I could list my own middle school games and activities, and those of others, for days. But for now, try out the ones above, check out the other activities I’ve created on Teachers Pay Teachers and keep coming back to the blog for more games and great resources.
How often have you taught fraction division to your students only to find them "flipping" the wrong number? You may have taught them to "skip, flip, flip," "invert and multiply," or "multiply by the reciprocal." You may have listed out the steps, or taught them a nifty song, but somehow they
still flip the wrong one or they forget to flip at all.OR they change a mixed number into an improper fraction and seem to subconsciously think that since they did something to that mixed number, the flipping had already occurred...and then they don't flip anything. Why does this happen? I'm going to say that it happens because they don't see the sense in it - it doesn't mean anything to them.
So, I have another way to teach fraction division - perhaps you've heard of it, or you use it. I never learned it this way as a child, but I like it and it makes more sense to some students. I learned this method when I had a student teacher a few years back. She was teaching the fraction unit, and when her supervisor came in to observe and discuss, she asked if I had ever taught fraction division using common denominators. Having only learned (and then taught) to multiply by the reciprocal, of course I said no.
The next time she visited, she brought me a page from a textbook that explained dividing fractions using common denominators. These are the steps: Step 1: Find common denominators, just as when adding and subtracting and then make equivalent fractions (students are already used to doing this - hopefully). Step 2: Create a new fraction with the numerator of the first fraction over the numerator of the second fraction...this is your answer. Done (unless you need to reduce)!I was shocked - it seemed SO simple!
Check out this example - it's a simple one, for starters:
5/6 divided by 2/3. 1) Find the common denominator of 6 and 3, which is 6. This gives you 5/6 divided by 4/6. 2) The first numerator (5) becomes the numerator in the answer. The second numerator (4) becomes the denominator. Then reduce.
Let's look at another one, with mixed numbers:
1 and 4/7 divided by 1 and 3/4. 1) Convert the mixed numbers to improper fractions, which gives you 11/7 divided by 7/4. 2) Find the common denominator of 28 and make equivalent fractions. This gives you 44/28 divided by 49/28. 2) The first numerator (44) becomes the numerator in the answer. The second numerator (49) becomes the denominator. No reducing, in this case.
I've shown both methods to my sixth-graders. Some really like it. Others stick to the flipping method - but I don't know if this is because they like it better or because it was the first way they learned it.....most of them had been taught something about fraction division in 5th grade.
As far as teaching multiplying by the reciprocal - if students are going to use it, I think it's important that they understand WHY it works. It may be tough for them to understand, but if they learn the common denominator method first, the proof may then make more sense to them. I found a great article on the NCTM website that uses the common denominator method to prove why multiplying by the reciprocal works - check it out!
Recently I made two math wheels, to use to teach both methods of dividing fractions -taking notes will be more fun!
What do you think? Do you see any advantages or disadvantages to teaching fraction division using common denominators?
I love teaching fraction multiplication--particularly multiplication of mixed numbers. Why? Because I have fun explaining why multiplying mixed numbers DOESN'T work a certain way.
Inevitably, when we start multiplying mixed numbers, some students want to multiply the fractions by the fractions and then multiply the whole numbers by the whole numbers. And I can see why they might think that's ok - after all, when you add and subtract, you deal with the whole numbers and fractions separately. Sometimes, I think they don't want to be bothered with making improper fractions, because it's "easier" to just do 2 x 3 and then 3/4 x 1/2, haha.
So, every year, we end up having this discussion about why that just doesn't work. I enjoy showing/explaining that multiplying 2 3/4 by 3 1/2 means that ALL parts of 2 3/4 must be multiplied by ALL parts of 3 1/2. On the board, we make a list of the problems that would need to be completed: 2 x 3, 2 x 1/2, 3/4 x 3, and 3/4 x 1/2.
- Once we complete those problems, students can see 6 and 3/8 in that list of products (which would be the answer if they multiplied fraction by fraction and whole number by whole number), but there's also an additional 1 and 9/4 in there that are part of the final answer.
Now that we have all four products, we go through the process of adding them all together (finding common denominators, equivalent fractions,etc) and then reducing.....quite a bit of work to get to the answer:-)
Then we compare that to what we get when we convert the mixed numbers to improper fractions. The detailed example of completing four multiplication problems and adding the products proves not only that converting to improper fractions is necessary, but also that it's a lot faster! So,
Tip #1 is to show students WHY what they're doing isn't correct...show what the multiplication really means.This may also mean bringing out the graph paper and showing what 3/4 groups of 1/2 looks like, etc, in addition to doing the computation.
Canceling, or "Crossing Out"Tip #2 - Teach students to cancel, or "cross out" (or whatever you might call it), and show them why it makes life a little easier.I love teaching this aspect of fraction multiplication. It's hard for some students to grasp at first, but when they repeatedly see that if they don't cross out, they have to reduce at the end of the problem (with larger numbers, like 168/12), they start getting excited about finding how much they can cross out. In recent years, I've found that students aren't learning this in earlier grades as often as they used to--for many, the discussion we have in my classroom is the first time they've encountered it. Once I teach them the idea of reducing first, and we explore why it works, there are some that still want to stick with what they learned in earlier grades and reduce only at the end, while others get super-excited about the concept of making the numbers they're working with smaller at the start. I may be wrong on this, but it seems that the students who embrace it first are those who know their multiplication facts better and can more easily find the relationships between the numbers in the problem....a student who knows that 15 and 24 can both be divided by 3, for example, is more likely to go ahead and divide them by 3 than the student who can't see it because they can't remember/don't know what 15 and 24 are divisible by. Multiplying Fractions and Mixed Numbers WheelTip #3 - Give students a graphic organizer to help them remember the process. Some will need this and some won't, but it's handy to have in their binders to reference throughout the year. I recently created a fun math wheel, which is a great way to have students take notes about the concept, practice it, and then add their own personal, artistic touches.Do you have any special methods you use to teach the multiplication of fractions and mixed numbers? |
To find more posts you might like, check the Blog Table of Contents.## AuthorHi, I'm Ellie! I've been in education for 25 years, teaching all subject areas at both the elementary and middle school levels. |