Four Ways That Self-Correcting Math Work
Can Benefit Students
I have been teaching for more than 20 years. If you have been teaching for a long time, then like me, you may have used a certain strategy/instructional tool for a period of time, and then for some reason, stopped using it....and then after another period of time you came back to it, and wondered WHY (or when!) you stopped in the first place!
That was me today. I had made 20 copies of my Footloose answer key and had the students correct their own papers (they had worked on the Footloose activity for part of yesterday's class and then finished during today's). I was surprised by the thoughts that went through my brain as they were correcting - the main one being - "When did I stop doing this?!"
I do have students check their homework answers with the answers shown on the board (sometimes), but I don't give them each a detailed answer sheet to use, and I rarely have them grade their own classwork.
Here are my re-discoveries related to students correcting their own math work. Some of these may be particular to the topic we worked on (writing algebraic expressions from phrases, phrases from expressions, and evaluating expressions given a value for the variable), and the fact that the answer keys were detailed (not just the answer), but I'm sure I'd observe the similar things when studying different topics as well:
1) Students asked me more questions when checking their work with my key. Since they were working at their own pace and checking individually, they seemed to be more comfortable with verifying whether or not their phrases were ok (I didn't have every possible phrasing option on my key). Students who wouldn't normally raise their hands to ask in front of the class did ask me questions during this time.
2) Correct work is modeled on the answer key. Because I had several options for phrases on my answer sheet, they had to read each one to see if theirs was on the sheet, giving them a little more exposure to correct options. I also had the steps for evaluating each expression, so they could go line by line and have those steps reinforced, as they compared the work to their own.
3) Students were finding their OWN mistakes, rather than me finding them. I heard things like, "I copied the problem wrong," "I said 3 x 3 was 6!" "Oh, I put division for product." And I realized, as I did years ago - it makes so much more sense to them when THEY see the difference between the correct work and the mistake they made, rather than ME finding it....do they really know why I circle a mistake that they made on their paper if they don't take the time to ask me? When they find the mistake, they know what happened. I don't need to make those types of connections and observations. They do.
4) Students are engaged - they enjoy having the key! It was fun to see them with their pens or colored pencils, pointing at their papers, question by question, making sure they were being accurate in their grading of themselves, and then being sure to write the correct answer accurately (I did make them write the correct answers, using pen or a colored pencil, so the change would stand out).
I don't know what prompted me to copy the keys to use today, but I'm so glad I did. It's wonderful to be reminded of forgotten/lost practices that help students to think just a bit more.
Have you re-discovered any strategies/practices recently?
Use Task Cards in a New Way, to Provide
Self-Differentiation and Promote Discussion
If you're like me (and so many other teachers), you know that task cards can be used in sooo many ways. From centers to Footloose (or Scoot) to exit tickets to entrance tickets to mini-quizzes - the list is long!
However, if you're like me in other ways, you're always looking for something new and different. This year, my "new and different" was to start using task cards to play Truth or Dare in math and language arts classes! To use them this way, some of the task/question cards need to be written as True or False questions, which can make the questions just a little trickier and lead to more in-depth thinking. I allow students to discuss the answers after the "official" answer is given, and depending on the question, students end up having great discussions!
The Dare questions are a little harder, require more calculation or perhaps more verbal explanation than the Truth cards, and so they are worth more points. (Truth cards are worth one point while Dare cards are worth 2 or 3 - I've even thrown in a 4-pointer here and there.)
What makes this game fun? Well, it's a little different - with the "dare" part in there. Students also don't always know how many points they're going to get to try, so that offers a little excitement. I like the fact that students can choose the type of question they want, so it allows for some self-determined differentiation...the choice gives the more hesitant students the chance to feel a little more confident.
After creating several paper and pencil Truth or Dare games, my wonderful friend Leah (Secondary Resources for Social Studies & English) suggested that I make a Google classroom version, and I'm so glad I did! It's so easy to use and there's little to no copying needed! (A little copying if I want students to write their work/answers on paper; no copying if I want to share the Truth or Dare game in Edit mode and have students type their answers.) Check out the 2-minute video below - it shows how the game works in Edit mode (there are one or two "slow to refresh" spots in the video, so please don't think it's not working:-)
Check out this video to learn more about the way the game is played with paper/pencil - in any subject!
I hope you can use this game idea-it can be used in any subject!
I'm really liking the math wheel idea, so I created a new wheel for fraction, decimal percent conversions:-)
How to use this resource (this information is also in the free download):
Around the outside of the wheel are the different conversion headings – you can use the wheel to introduce the conversions, filling in just the ones you are covering each day. Or, you can use it to review all the conversions at once. In either case, the wheel can be kept in students’ notebooks as a reference/study tool.
1) I like to begin with decimal to percent and percent to decimal. In the arrows in these sections, you’ll see x 100 and ÷ 100. It think it’s important that students understand that these are the operations being used for these conversions before giving them a shortcut, so I let them use calculators to complete the examples. Once the examples are complete, I ask the students to look for the pattern – what happens to the decimal point in each of these cases?
We decide on the “shortcut” rules together and then write them at the bottom of those sections.
2) The fraction to percent and fraction to decimal sections have the rules written already, so the examples just need to be completed.
I always relate fraction to percent to students grades. By the time we get to this topic during the year, students have been figuring out their grades for months (I never write their percentages on their assessments – they need to calculate
them). They know how to find their percentage if their quiz grade was 6/8 or their test was 48/52. However, sometimes they need a reminder that this official fraction to percent “rule” is the same thing they’ve been doing for months! I have them
write a little reminder in that section - “just like test grades!”
3) For percent to fraction, students need to remember that percent means “out of 100,” so the percent number will always go over 100. Then they must reduce.
4) I find that decimal to fraction is sometimes tricky for students. When they have trouble, I ask them to read the decimal number according to place value (“How do you say this number, using tenths, hundredths, or thousandths, etc.?”). Once
they speak it, they know how to write the fraction – 0.27 is 27 hundredths, which is 27/100. After completing the examples, we discuss the idea that the denominator will be whatever the last decimal place is (10, 100, 1000, etc.) and the numerator will be the digits in the decimal number. We write this rule as simply as possible.
5) Students then complete the 10 problems around the page. Above each number is the conversion to complete (F to P, P to D, etc.) They can then color the rest of the wheel background.
I had a great time coloring my answer key! These could make a fun decoration as well:-)
I hope you can use it!!
We know that graphic organizers can not only be helpful for organizing information, but they can also be helpful in creating visual cues that help students remember specific information.
This math wheel focuses on the topic of rounding decimals (obviously:-). When I have reviewed rounding decimals with my students in the past, they often remember whatever trick or saying they've been taught, but they often can't explain the math reasoning (therefore, I always save any sayings/tricks until after the math concept is understood, if I use them at all).
When using this math wheel, I start with the number lines - looking at the distance between 1 and 2, where 1.5 is, and visually draw attention to the fact that 1.6-1.9 are closer to 2 and 1.1-1.4 are closer to 1. The students write in the labels and then there's space for you, the teacher, to add several examples of your choosing.
Then I move to the benchmarks. You'll see on the completed version, I drew a small number line to create the visual of the space between 1 and 1.1, labeling 1.05 as the half-way point. The same thing could be done for the others, or examples of rounding can be added (like the one below 0.0005).
Students can then do the practice problems all around the page.Above each number is a T, H, or TH, to indicate the place to round to (tenth, hundredth, thousandth).
I have the students color their problems/answers according to numbers that rounded up (my example uses green) and numbers that rounded down (pink), which gives a quick, easy visual to see that they knew which way to round. A closer check will then tell me if their answers are actually correct:-)
(You can always let them just color the background later, for fun!)
Last,(if at all), I'll have them add a rule/saying to help them remember.....one that each student creates him/herself would be best.
I hope you're able to use this math wheel! Let me know if you have any questions:-)
Rumors is another great lesson from Mathline! This lesson allows students to explore exponential growth, in the context of spreading a rumor. In addition to the focus on math concepts, this lesson can also help students to understand how quickly rumors can actually spread....an important idea for middle schoolers to consider.
To begin the lesson, students are presented with the following scenario:
"Two students who were both born on December 21st, the date of the winter solstice, decide that it would be great not to have to attend school on that day. Therefore, they start a rumor that schools will be closed to celebrate the winter solstice. So, on December 1st, one of the students told two of her friends that school would be closed. On the next day, each of these students tells 2 students and on consecutive days, each of the new students tells 2 more students and so on. If there are 8,000 students in the school district, the question arises as to whether the rumor was started early enough for everyone to have heard it?"
Students can act out this scenario by having students form a human triangle, with Student A first, then the two students she told (students B and C), then four students representing the two that Student B told and the two that Student C told, etc (as far as possible, depending on how many students in the class). This will help students visualize the problem and understand how this rumor is being spread. The triangle also help students to understand the growth pattern.
The human triangle will only go so far, so students will then need to use their calculators or paper and pencil to find how many days it will take for the rumor to reach 8,000 people. I would recommend providing the students with a blank chart to give some structure to the students' work after they try the human triangle.
The chart below includes the first several days (the numbers for the entire chart can be found in the lesson).
In addition to understanding more about exponential growth, students can be asked to determine the algebraic expression to describe the number of new people to hear the rumor each day (2n), as well as the
expression for the total number of people (2n+1-1).
To read the full lesson and the possible extensions, check out the lesson here.
Do your middle school math students like to play math games? Mine do, but over the past few years I've noticed that many of them aren't familiar with some of the games I played when I was a kid, like Yahtzee, for example. So, as we started working on converting fractions and decimals, I decided to create a game to make practicing the conversions more fun AND give them some more game experience! I based it on the idea of Yahtzee:-)
Here's how it works:
Students roll four dice, and pair the dice up to create "target numbers" that are decimals or whole numbers.
For example, a student rolls 1, 2, 4, and 6. From these dice, the student may create any two of the following decimal (or whole) numbers:
½ = 0.5 4/1 = 4
¼ = 0.25 4/2 = 2
1/6 = 0.1666... 4/6 = 0.666...
2/1 = 2 6/1 = 6
2/4 = 0.5 6/2 = 3
2/6 = 0.333... 6/4 = 1.5
Once a player has chosen two target numbers, he or she finds the score by adding the dice that were used for each decimal. If the player chose to use 1 and 4 to get 0.25, he or she adds 1 + 4 for a sum of 5 to place in the score column. The second decimal choice uses 2 and 6, to equal either 0.333...or 3. The sum of 8 would go in the appropriate column as the score.
On the next roll, this student rolls 1, 1, 3, and 5. This student can pair 1 and 1, to get 1, and pair 3 and 5 to get either 0.6 or 1.666... The score for 1 is 2 (1 + 1) and the score for 0.6 is 8 (3 + 5).
In many cases, students' scores will be the same, but some of the decimals can be found with different combinations of numbers (1 and 3 = 0.333..., and so do 2 and 6, so students could score either 4 or 8). Some students will notice this sum difference and go for the combination that will give them the higher score.
The students have really enjoyed playing this game. They do need a few examples at the start, to understand exactly how the game works, so if you decide to try the game, be prepared to go through a few turns together.
Detailed instructions are included, and a complete answer key of highest and lowest possible scores for each target number are included as well. This is handy to quickly check student score cards as you check in on their games.
If you give it a try, please let me know how it goes!
Click to see on TPT
In my early years of teaching, I didn't always know what to say when students told me they didn't have time to do their homework (other than something like, "You must have had some time between 4:00 and 9:00!). There were all kinds of reasons - they had sports practice or a lesson, or they had to go to their brother's or sister's game/practice/event of some kind; or their parents took them shopping or out to eat. At that time I had one child (who was 2 when I started teaching), so I didn't have the experience from a parent's point of view of making sure I was getting my kids to their activities, getting done all the house-related things, and also making sure they were getting their homework done. This made it a little difficult for me to relate to the students' situations, but I tried to help them think about how much time they did have to do their work.
Being involved in activities definitely reduces time for schoolwork, but it
doesn't mean that schoolwork can't get done. Students can learn to manage their time, but they need to be shown how. There are many of us who, as adults, may not manage our time very well. And if a parent is not great at managing time, how will he or she teach their children to manage theirs? Even when adults are good at managing time, they don't always think to teach their children how to do what they do.
Because their parents might not talk about time management, I've spent many years teaching students (5th and 6th graders) how to find their available work time. I make these planner-type pages and have students fill in a sample week, so they can see where their available time is. When they fill in the practices, games, lessons, sibling practices, etc, they can then see what time is left in the day. If homework is assigned Monday and brother has practice, the student can see that they have a chunk of time from 3:30-6:00 (when they probably also eat dinner) and then 8-9:00. If homework completion can fit in those time slots, great! They can plan to use that time wisely. If it's not enough time, then they need to use another strategy to get things done. One of the fun parts of using the calendar/planner is the color-coding! When I used this for my own planning, I color-coded according to person (my son was green, oldest daughter orange, youngest purple, and I was blue:-).
If their chunks of time aren't big enough, students need to find other ways to complete their work. One of the strategies I share with students is to take backpacks and homework supplies in the car with them. When one of my three children had practice (they're all beyond this point now), the others brought any work they had to do. Sometimes homework was completed sitting on a blanket in the grass or sitting in the bleachers. Sometimes it was completed in the car while we waited. Do distractions occur when homework is done this way? Yes, they sometimes do. But, to me, using that time to work was better than losing an hour or two (or more, depending on travel time!) and then having to do everything after we got home (especially if we still had to have dinner!)
I also suggest that students try to study while they're driving to an event. They can read over notes and quiz themselves. If there are several people in the car, one person can quiz another. The student can quiz their parent as well, or explain information to mom or dad....this is a great way for a student to be sure his knowledge is solid.
I always suggest that students put upcoming tests on their calendars and then work backwards to schedule their study time....so they could label the driving time as study time. Projects should go on the calendar too, so students can again work backwards to fit in the necessary time to complete them.
The great thing about a week at a glance like this is that students don't have to depend on someone buying them a planner or printing out pages for them. They can write out their own schedule on their own paper and design it any way they'd like. Then they can post in it their room, on the frig, or keep it in a school binder.
As I mentioned, in the early days, I didn't quite know how to respond to students who didn't have time to do their work. But now, this is something I teach every year, to help avoid those "I didn't have time because...." statements :-)
At the end of the school year, I often have the students create "Memory Wheels." I laminate them and save them to put up at the beginning of the new school year. Creating these wheels gives students a chance to reminisce about the school year, and the wheels give the incoming students a chance to see what the "old" students thought was fun about their 6th grade school year. Students write a sentence or two in each section and then create an illustration.
I did put a few of these up at school today, for our new students to see on Monday.
The wheel could be used at the beginning of the year, as a "getting to know you" activity. The student's name would go in the center circle. The student would need to choose 8 things to share about him/herself, and then write a brief description of them and illustrate them. I haven't used the wheel in this way yet, but I think I like the idea:)
The wheel can be used for any type of project, at any time during the school year. In the past, I have used the wheels as a book report project: students choose main events from the book to feature in each section, they write a brief description of each event, and then illustrate each one. The title and author are written in the center circle.
The students use a template (that I had drawn) to help them create their wheel.
I have them use either oak tag or large white construction paper. I didn't measure the diameter of the circle (the wheels are at school now, while I'm at home), but I believe when I use the 12" by 18" paper, the circle just hits the top and bottom edges.
Now, I need to think of a way to use the wheel in math....maybe a geometry wheel? Any thoughts?
Wheel Templates on TPT
A few uses for content area:
When students finish their math tests and want to hand them in, I always ask, "Did you check your work?" Often the answer is "yes," but if it's not, I won't take the paper until the student does check over their work. How do they do this checking? Do you know what I'm going to say?
Here's what they do - they look at the questions, basically make sure that all the questions were answered, and again try to hand the test in. Unless they are clearly taught otherwise, many students seem to understand "checking your work" to mean "checking to see if everything is done." What does checking mean to you?
What does it mean to me? How do I think students should check work?
I teach them to use the following strategies. They aren't always thrilled by the extra time and effort this real checking takes, but they do find that it helps.
1) Redo every problem, on a separate piece of paper, without looking at the work that was already done (when they look at the work they already did, it influences them and they sometimes make the same mistake again). Then compare the original work and answer with the redo.
2) Use the opposite operation to check, if possible. If the problem was an addition problem, subtract one of the addends from the sum. If it was a division problem, use multiplication to check. You know what I mean:-)
3) Ask whether the answer is logical. If the problem was 2.56 x 7.91, an answer of 2.02496 (instead of 20.2496), is not logical.
4) Substitute the solution into the equation, if possible. If the solution is correct, the equation will be true.
5) If the test includes word problems, reread each problem carefully to be sure all of the information was understood correctly. Look for numbers that are written as words rather than digits.
Be sure the answer actually answers the question the problem is asking.
I made a How to Check Your Work reminder sheet for students to keep in their notebooks. You can download it if you'd like - there's a colored version and a black and white version, in case you want to print on colored paper. I hope you can use it!
I was looking through folders on my computer, and came across a document called "The Factor Game," and it occurred to me that in trying to think of some new things to do with my classes, I forgot to play the Factor Game this year when we started talking about factors!
I was so disappointed with myself. Of course, we can play it next week, or any other time, but I just couldn't believe that I had forgotten about it. I'm sure many people know of the Factor Game, or use a version of it, but for those who don't, here it is!
Players start with a simple game board, with the numbers 1-30 (in the past I've alternated between giving them a pre-printed sheet and letting them write their own numbers on their own paper; sometimes I have them play at the whiteboard).
Player 1 chooses a number, and marks it (they might circle it, square it, triangle it, color it etc). Player 2 then marks all of the remaining factors of Player 1's choice.
Player 1 receives points equal to the number they chose, and
Player 2 gets points equal to all of the remaining factors of that number.
Next, Player 2 chooses a number (and gets those points), and Player 1 identifies all of the factors of that number, receiving those points.
Play continues this way until all possible numbers have been used.
*Special rule - players can't choose a number that has no factors available for the other player. If they do, they lose their turn, receiving no points.
When I introduce this game to my classes, we play it on the whiteboard, and I always allow them to go first (the first time). Often, they will choose 30 as their number, since it's the biggest. When I go ahead and mark all the remaining factors of 30 (my points), they are shocked! After the first turn, they have 30 points and I have 42!
(I normally have 2 students record our scores on the board, while 2 others keep running totals on a calculator, but points can all be tallied at the end if you want).
The game continues, alternating turns, until all possible numbers are used, as shown in the "finished game" picture.
The second time we play as a group, I choose 29 as my first number, and they are disappointed to only get 1 point!
We continue to play the second game, and they get a better idea of what makes a good choice.
After our games as a class, I have the students work in partners, and they normally get one or two games played before we stop to discuss at the end of the period. We always talk about what the best and worst first choices are, and bring the idea of prime and composite into the discussion.
I have been playing this game with students for at least 15 years (I think), and every set of students has loved it! I found the game when I was involved with Mathline, so long ago. If you'd like to see the original lesson plan, with extension ideas, click here.
Have you played the Factor Game?
To find more posts you might like, check the Blog Table of Contents.
Hi, I'm Ellie! I've been an educator for more than 20 years, teaching all subject areas at both the elementary and middle school levels.