Fraction Series, Week 3: Benchmarks and Estimation
In the fraction standards review part of last week’s fraction post, I briefly summarized the fraction skills addressed at different grade levels.
In 4th grade, we have “comparing fractions with different numerators and denominators by using common denominators or benchmarks.”
When we move to the 5th grade standards, we see benchmarking is also included in 5.NF.A.2:
Using Fraction Benchmarks
Using benchmarks to estimate is a powerful way to help students understand whether or not their answers make sense.
I'm not sure if your experience is the same, but over the years, this understanding of whether or not answers make sense was definitely an area of weakness for students )not just in the area of fractions).
So, if we can incorporate the reinforcement of benchmarks, estimating, and the understanding 'reasonable' it will be a huge benefit to our middle school students.
Let’s look at the 4th grade concept of using benchmarks to compare fractions.
When students are asked to compare fractions like 5/8 and 2/5, for example, the standard would like them to be able to look at 5/8 and think:
This is great logic. I have found, however, that some students don’t know how to find half of a number and even if they do know, they have trouble with half of an odd number. Have you found this too? That may be something for another post!
When it comes to using benchmarks in 5th and 6th grades and beyond, we might not be using them as much to teach comparing fractions; but we can teach using benchmarks (and finding half) in the context of estimating answers for adding and subtracting fractions (and eventually for multiplying and dividing).
Reviewing the Standards and Using Fraction Models
Welcome to Week 2 of the fraction basics series. Last week, we talked about basic meanings, a little bit about fraction models, and suggested a couple ways to work fraction basics into everyday instruction.
This week we’ll take a look at the progression of fraction skills through the grades before our students reach us in upper elementary and middle school math class. And then we’ll look at models:-)
Fractions in Grades 1-4
As upper elementary and middle school math teachers, we know our students have had fraction instruction – what fractions are, identifying fractions of shapes, finding equivalent fractions, comparing fractions, and some fraction operations; but we may not know exactly what students did in which grades.
So, for a 'quick' overview (based on the Common Core Standards), in the earlier grades (1st – 4th ) students should be learning/mastering the following:
1st Grade Math:
I think it's important to note that ‘using visual models’ is included in numerous places in these standards.
End of Year Activities for Upper Elementary and Middle School
Are you approaching the end of the school year?
Does it feel like the school year flew by and it's already the end of the school year? Or did it seem slow in coming this year?
Regardless of how quickly or slowly your school year went, the end of the year is a great time for students to look back and reflect on the time they spent at this grade level.
If you're looking for a few activities for the end of the year, I have a few ideas for you:-)
End of School Year Activity #1
1) Students can write letters to themselves. You can have students self-address their envelopes, and then you can save them and mail them to students in August.
Kids are so excited to receive actual letters in the mail!
AND, addressing an envelope is great practice for a life skill that isn't used quite as often these days.
For most of the years we did this, many of my students had never addressed an envelope before.
Welcome to this blog series about teaching fractions in middle school!
For the next several weeks, we’ll be looking at various aspects of fractions: basic meanings, fraction models, finding equivalent fractions, comparing fractions, benchmarks and estimation.
As an upper elementary or middle school math teacher, you may find yourself needing to review some of these fraction concepts before you can move on to the fraction curriculum for your grade level.
When this is the case (needing to review fraction concepts that aren't part of your curriculum), there usually isn't much time to squeeze in extra instruction...it can be tough!
We realize that some of our middle school kids just don't 'get' fractions, but our time is limited. I've got a couple ideas at the end of the post for how to weave in some of the 'fraction basics' in middle school.
But first, for this week, we’ll start with some of the foundational fraction vocabulary terms.
A few ways to use mini whiteboards and a great way to clean them!!
Do you use mini whiteboards in your middle school classrooms? I love them!
I've never purchased them, but I remember when I had them made, when I was teaching elementary school. Other teachers at my school and I went to the local Home Depot, bought the large white panel boards, and the Home Depot people cut them to size!
It's been more than 20 years since I had them made, but I believe we were able to get thirty-two 12 x 12-inch whiteboards from one panel, and it cost less than $20.
Then I used blue and green electric tape to tape the edges so students wouldn't get any kind of scratches from the unfinished surfaces.
After several years, I retaped them with colorful Duck tape, to make them a little more pretty:-)
Using Mini Whiteboards
1) Mini whiteboards are a great way to increase student engagement.
When each student has their own marker and whiteboard, they are more likely to participate - they like the color of the markers and the space to do just a couple problems.
Even those students who often hesitate to participate are more likely to engage when they're using their own whiteboard.
2) Mini whiteboards are fantastic for fast finisher activities.
When students complete a whole-class activity at different times (like Footloose activities), they can grab a whiteboard and do something like practice with Order of Operations flashcards (these are awesome, by the way... they make for great, quick, differentiated practice).
With the whiteboards, markers, and erasers readily available, the kids are perfectly happy to grab a whiteboard and marker and get practicing:-)
Algebraic Expressions in 6th Grade Math
Translating between words and math in 6th grade - sometimes this can be easy, but some phrases can definitely be tricky!
By the time we start 'officially' translating between words and algebraic expressions, we've already done some translating to numerical expressions through our daily spiral review.
To connect translating words to numerical expressions and translating between words and algebraic expressions, we take some more time to translate between words and numerical expressions. We use an organizer to list the key words that typically signal the different operations. (I use the same type of circle organizer I use for my "Memory Wheel" templates.)
We typically include the following math terms and phrases, as you can see in the organizer:
Great End of School Year Activity!
This post was taken from my old blog and revised...much more detail added:-) Originally posted in June of 2014!
Tessellations in middle school math - one of my all-time favorite activities! I especially love using tessellation activities at the end of the school year....art projects like these keep students engaged:-) We normally take a several days to work on tessellations...sometimes more, depending on how much time we have.
First off, what is a tessellation? I show students a few examples and ask what they believe a tessellation is. They typically come up with the idea that a tessellation is the repeating of a shape or shapes in a pattern. I usually have to add the idea that there are no gaps or overlaps in a tessellation.
Tessellations with Regular Shapes
We start with using regular shapes, so students understand the idea of tessellations in general.
We were fortunate to have pattern blocks of hexagons, triangle, squares, etc. in the classroom, so students could use those to trace and make their shapes as precise as possible.
After using regular shapes, we extend to creating Escher-like tessellations. I share several example of these with students so they can study them and see the different types of animals, objects, etc found in them.
How to Teach Decimal Addition and Subtraction
How do you teach adding and subtracting decimals in upper elementary or middle school math?
In 6th grade, my math students have typically come to me knowing the 'rules' for adding and subtracting decimals.
However, when the number of digits in the numbers they're adding or subtracting aren't the same, they don't necessarily line the numbers up the way they need to...even though they 'know' the rules. Why is this?
I believe it's because they really don't understand the point of 'lining up the decimal points.'
My belief is reinforced by student comments I collected one year as we began our decimal operations unit.
I asked my 6th grade math students to solve 35.2 + 7.489 and then explain why their answer made sense. These are a few of their responses:
Of the 120 students in my classes, only 8 said the answer made sense because "35 + 7 is 42" or because "I estimated" or "when we're doing addition, we know we end up with a bigger number."
I don't want to assume that students who didn't write this didn't think about those things at all, but to the majority of students, their answers "made sense" when they followed the rules - even if they didn't remember the rules correctly.
With a Peer Teaching Component
An Alternative to Daily Oral Language
I'm really excited to share my version of daily language review! I've used this method in 4th, 5th, and 6th grade language arts classes.
This daily language review incorporates peer teaching and was many years in the making:-) I published 24 weeks of this daily language review for 6th grade a few years ago, but didn't share much about it at that time!
So, here's the story:-)
About 20 years ago, after using Daily Oral Language in a variety of ways in my classroom, I decided that each student needed to become more engaged in our daily language instruction.
Many were already engaged...students took turns coming to the board to make corrections and identify parts of speech, but typically, only one student had a turn at any one time.
At that time, my goal was to make sure students understood the reasons for the corrections they were making. Unfortunately, if they tuned out for even a little bit, they missed some information.
So, using the Daily Oral Language sentences, I created a peer teaching method for our daily language, which I'll explain below.
** Note: this daily language review is not the only method I use to teach grammar, punctuation, spelling, etc, but it has been a fantastic supplement to provide continual reinforcement throughout the year.
Six Tips to Make the Grading Process Faster
Grading Papers Quickly and Efficiently
How often have you wished you could just snap your fingers and have all the grading finished??
I don't know about you, but grading papers is definitely an area where I needed to find some ways to make the process faster. Over the years, I found a few little ways to get my grading done more quickly and efficiently.
These days, the use of digital assessments can make some grading faster; however, you may not be able to (or want to) use digital assessments all the time.
So, for those cases when you're working with paper assessments, I've got six tips for you!
Grading Tip #1: Answer Sheets
1) If you're grading multiple choice/true or false (and aren't planning to do any partial credit), design your assessment so students will put their letter answer in a left-hand column.
Or, give them a separate answer sheet with answer spaces in a column.
This way, all the letters are in an organized column for you and you can quickly go down the column, comparing the answers to the answer key.
End of Class Routines
What are your end of class routines in middle school?
Do you have specific routines?
A few weeks ago I wrote about routines to begin class; today I'll touch on routines to end class in middle school.
I have to admit, my end-of-class routines often didn't go as smoothly as my beginning-of-class routines:-)
Best laid plans, you know....
When it went well, my routine at the end of class would include:
1) Writing homework on the board so students could add it to their assignment books.
I did accomplish this nearly every day:-) Writing the homework was critical. Our team teachers made a point of doing a quick check at the end of class every day, to be sure students were writing their homework down (for some students, this was an accommodation we had to meet - checking the homework book. So, why not quickly scan/check all students' books?)
2) Asking an 'exit' question.
I had a "ticket out the door' poster in the classroom that I numbered 1-30. Each student had a class number, so they'd answer the question on a post-it and then put the post-it on their number on the way out of class. I loved this method because:
Graphing Functions Practice with Footloose Task Cards (free download)
This Graphing Linear Functions post is from my old blog, written in May, 2015, but updated here:-)
Graphing functions hit us in May of this school year. Our state testing had wrapped up and we started working with function tables and graphing function equations, on a pretty basic level.
This post is my summary of how I approached practicing the basic idea of graphing linear functions with my 6th grade math students:-) Included is a description of the task card activity I used, which you can download for free at the end of the post.
Pi Day Ideas for Middle School Math
March 14, also known as Pi Day, is coming soon!
Do you have exciting plans for Pi Day activities in your math class?
Maybe you have some tried and true favorites, or maybe you're looking for something new.
Whether you teach 5th or 6th grade math, elementary math or higher levels of math, Pi Day is a day that can be celebrated and used to explore the concept of Pi - in so many different ways!
Pi Day Activity
Have you tried using donuts to have students discover the Pi relationship between the diameter and the circumference? Or maybe cookies?
I tried it with donuts recently.....can be a little messy, but fun!
In my pictures, you'll notice I had some donuts with icing and sprinkles....I'd suggest NOT using those if you want to minimize mess:-)
Materials for Pi Day activity:
2) Licorice 'strings' for measuring (if you have tape measures that would be best, but I never had those in my classroom, so I went with the string idea)
Beginning of Class Routines
What do the beginning of class routines look like in your middle school classes?
Do your students enter the classroom and know exactly what to do? Or, in this age of virtual learning, do they know exactly what to do when they log on....or even before they log on?
For a few of my first years of teaching, I didn't have specific classroom routines. Then I read The First Days of School and everything changed...I started incorporating specific tasks and routines for students to do/follow when they arrived in class.
For several of my middle school teaching years, my homeroom students were my first period class, which made getting 'into' class really easy - the routine for those homeroom students was to begin specific tasks as soon as they had completed their homeroom routines.
What Makes Math Fun?
"Math is boring."
"I wish math were more fun."
Do you ever hear comments like this from students?
Did you feel this way as a student?
Do students walk out of your math class saying, "That was a fun class?"
What Makes Math Fun?
Can middle school math class be fun?
Is math supposed to be fun?
These are good questions, but my real question is - what makes math fun? Are we talking about having fun practicing math or having fun learning math? Or both?
I'd argue that practicing math can't be much fun if a student doesn't understand the concepts. You can have the most exciting, engaging activity, but for the students who don't know how to solve the math problems, the activity is NOT fun; it's meaningless and frustrating...especially if they see other students having fun because they 'get it.'
Before practicing math can be fun, students need to understand the concepts.....at least partially! Maybe a fun math activity can help solidify the concepts, but there must be some understanding to begin with.
Note-Taking in Upper Elementary and Middle School Classes
How much do your math and ELA students love taking notes?!
What's your favorite method for taking notes with your upper elementary or middle school students?
Maybe your favorite math method is different from your favorite ELA method. Maybe they're similar:-)
I used to use fold it ups (or foldables) quite often in both math and language arts. However, my middle school classes were always 40ish minutes long and often, no matter how prepared I was, creating the fold it up just ate up too much of the class time.
Using Fold It Ups
Sometimes there wasn't enough time to add the notes before math or language arts classes was over.
And the fold it ups were kind of hard to keep organized. We tried using folders and envelopes, and in my last years of using them, I tried using a bound book of fold it ups. I took all my math fold it ups and organized them in the order we'd use them during the school year, added some blank pages in between them, and had them bound as a book for each math student. Then all the fold it ups stayed in the book. Some remained attached on their original page because they were never totally cut out; some were glued or taped onto the blank pages. This method of organizing the notes was the one that worked best for my math students.
I never did that with ELA...because I stopped teaching that class before I had this idea, lol.
But then, I started making math wheels....
Two Types of Digital Coloring Activities
Remote Learning Activities
There are so many distance learning activities available for your upper elementary and middle school math classes right now!
What will work best for your math students (or for your ELA students, or science students, etc, if you also teach other classes)?
What do your students like? But just as importantly, if not more importantly, what provides great practice of the math (or other) skills during this time of virtual learning?
Since I'm such a lover of color by number activities, I want to discuss two styles of digital color by number: the 'pixel art' mystery picture style and the 'fill color bucket' style.
Color by Number Activities Overview
Let's start with a quick overview of these color by number/color by answer types, in case you aren't familiar with them.
Pixel art mystery pictures:
1) Created in Google Sheets; could be downloaded and used in Excel.
2) Students solve, enter their answers in the cells, and color appears in the squares if the answer is correct. If the answer is incorrect, the squares may stay white or show an incorrect indicator, depending on how the creator designed it.
Loving Prime Factorization
Is it weird that I love prime factorization??
Every year of teaching math, I have come to appreciate prime factorization more and more! Maybe it’s because when I was a student (forever ago!), I didn’t learn how to use prime factorization to find greatest common factors, least common multiples or to reduce fractions. (I will admit to the possibility that I learned and forgot….but I truly think I didn't learn it!). In addition to missing this information as a student, I didn't find it in math teacher manuals until I'd been teaching for more than 20 years.
I'll share why I love it so much by explaining three ways to use prime factorization: to find GCF, LCM, and lowest terms for fractions.
Six jobs to Help you Supplement or Replace Your Teaching Income
Do you love your teaching job, but need extra cash? Maybe something to do on the side or during the summer?
Do you love teaching, but your circumstances require you to find an alternate career?
Or, are you hoping to leave the classroom, because teaching isn't what you thought it would be and you want a change, but aren't sure where to start?
I've been where you may be! I loved teaching and taught elementary and middle school for 24 years, but throughout those years, there were times that I wanted to be home with my own children, and that made me want to leave the classroom. And sometimes we needed extra money. AND I wanted to work for myself! So, from as long ago as my second year of teaching, I was always seeking other ways to earn money - especially ways that would let me work from home.
I tried several ways to earn extra money over the years, and I'll share the most successful of those here (yes, there were some unsuccessful ones too:-).
The jobs I share here are the ones that allowed me to make enough money to replace my teaching income and leave the classroom.
Emergency sub plans! Do you have them?
Will you need them?
This fall especially (2020), you may need to have those emergency sub plans ready. Perhaps your school district is requiring you to have a week, or even two weeks, of plans. Or perhaps you just want to be extra-prepared. What's the best way to approach emergency sub plans in middle school math?
If you're like me, you don't want your absence to cause your math students to 'get behind' in the curriculum...you don't want things to be put on hold or to stop in the middle of a unit. But, you don't always know who will be taking over your math class....will it be a math teacher or someone who has never taught math and isn't comfortable with math?
If you must have the plans set at the beginning of the year, you can't really include the specific content you'll be teaching when those plans are needed.
So what's the best approach for sub plans?
First off - if you do have to be out and there's a test (or some other really critical item) scheduled for your first or second day of absence, I'd keep that on the schedule. Do whatever you can to be sure students get to take the test as scheduled. Make sure you're prepped for the test a day or two before....copies made or links prepared, so no one has to scramble to take care of it that morning (or has to postpone it). After the test, your sub can move on to the emergency plans. (I only mention this because I was guilty of last-minute 'morning of' prepping too often. Eventually, I started prepping my classroom for the next day before I left in the afternoon....writing the agenda on the board, being sure all copies were done, etc.)
OK, on to some ideas...
Problem solving is always an area that students can practice!
I typically include problem solving sheets requiring students to show all their math work AND do some writing, to explain how they arrived at their answers.
Some problem solving suggestions include:
Do you teach divisibility rules in your math class?
I've always enjoyed teaching divisibility rules and my 6th grade math students have always seemed to have fun using them!
I've read different opinions about whether or not teaching divisibility rules should be a focus in math class, because they may be viewed as 'tricks.'
However, I think understanding and using them in middle school helps students develop number sense and number fluency.
Rather than being taught as a 'unit,' I think divisibility rules should be introduced and then referred to again and again in any applicable situation throughout the year. To make the continuous revisiting easier for students, I've always liked to have a resource for them to refer to throughout the year. We used to create fold-it-ups, but then I moved to using a Doodle Notes resource or a Math Wheel.
Any reference sheet is helpful so that when you ask a divisibility question, students can grab it (or look at it on a wall) to quickly refresh their memories, if needed.
You walk around the classroom, looking at students' math work, listening to students' conversations, chatting with students about their math work, correcting possible misunderstandings, and reinforcing correct thinking. This is a big part of your 'normal' teaching day, right?
But this probably won't be the scenario for your fall 2020 math class. Whether you're totally online, in a hybrid model, or face-to-face but need to social distance, providing feedback will look different.
Providing feedback in distance learning
Feedback is so important. When a student completes a task - a practice problem, responding to a reading passage, reading aloud, writing an essay - whatever they're learning - they need to know if they did it right or they need to know what to do differently. Then they can repeat and practice correctly.
When you're with your students, giving feedback is relatively easy. You can nod, give a thumbs up, give verbal feedback, etc. But when the learning is virtual, or you need to social distance, it's not so easy! You can't walk among the students, look at their work, and take the quick moment to correct a little misunderstanding. So providing feedback becomes even more critical with distance learning, when you can't see students' body language, facial expressions, etc, that indicate they don't fully understand the concept. Even on a Zoom call or Google hangout, it's tough to just see everyone, let alone notice everyone's cues!
How can you provide feedback to students in this situation? Or in a situation where you're in the classroom, but can't be close enough to students to check their math work in the same way? Self-correcting digital activities are one great way to provide that feedback.
Do your math students love using technology? Playing math games?
Do you love math resources that are easy to use? AND great quality?
The digital math activities on my site are perfect for those who answered yes to these questions!
As more and more teachers and students are using the digital games on my site, I've received a few commonly asked questions, so I'm taking some time to answer them here, for anyone who has the same questions, but hasn't asked:-)
Evolution of the Digital Math Activities:
The digital math activities page of this site has been active since August 2019. It started out with 14 Truth or Dare math games for only members to use. Over the past year, I've added seven free activities for anyone to play, as well as several new member activities (member activities are up to 28 and I'm adding more as quickly as I can).
I've migrated the activities to a stand-alone math activities site, called cognitivecardiomiddleschoolmathdigitalactivities.com (I'm still adding/updating new activities there.) On that site, I added 'upgraded' versions of the free activities to the member section - I added question banks and more exercises to them, so students don't get the same questions each time. So, while you may see the 'same' activity on both the Free and Member pages, the activities aren't quite the same.
On to the FAQs...
Color by number math activities and distance learning? You bet!
Digital color by numbers? Nope, not for me:-) I like the paper and pencil!
A Little Distance Learning History
As we know, distance learning and digital learning are not synonymous. Distance learning has existed for a looonng time. Early on, it was called 'correspondence education' or 'correspondence learning.' Students received assignments in the mail, completed them, and mailed their work back to their educational institution.
A few examples of early distance learning include:
Check out this infographic for more detail about distance learning history.
Distance Learning Doesn't Have to Mean Digital Learning
Why consider this history? With schools moving to distance learning in light of the COVID-19 pandemic, I think it's important to remember that even though we have the technology to provide digital learning experiences, distance learning doesn't have to mean just digital learning. Paper and pencil activities have their place in distance learning; in some cases, such activities can provide greater benefits than digital activities.
What type of math activity do you most like to have your middle school math students work on?
For me, it's almost always been problem solving. This could include word problems that apply specific math concepts, word problems that incorporate a variety of math concepts, logic puzzles, or word problems that focus on problem solving strategies (create a table, make an organized list, find a pattern, work backwards, draw a picture, etc). I love using problems that have more than one correct solution, so students can share the thinking that leads to different answers.
When we work on problem solving activities, I often have students work together, so they can model for each other and share/listen to each others' thinking and reasoning.
I wrote the "Party Planning" problem to give students practice with decimal operations and with solving problems with multiple solutions. To solve the problem, students worked with one or two partners to come up with combinations of foods that Reggie could buy for a party. To find their solutions, students needed to add decimals; multiply if they were going to include several of one item; and possibly subtract, if their total was over $50.
Student Conversation and Feedback
I loved listening to the kids' conversations as they worked on this problem. I heard comments like, "No one eats pretzels," or, "I'd choose candy and chips over pretzels," and so on.
The students had a few important questions for me, as they were pretty serious about this planning.
"Is this a "regular" party or like a sleep-over party, because the kind of food would depend on how long the party is."
"How big is the container of ice cream?"
"How big is the bag of candy?"
Hey there! I'm Ellie - here to share math fun, best practices, and engaging, challenging, easy-prep activities ideas!