5 Tips for Middle School Math
The math homework dilemma – to give or not to give (IF you have the option in your district)? How much to give? To go over it all or only review some of it? What will be most helpful to your students?
Maybe your experiences have been similar to mine: I’ve adjusted my practices from year to year, sometimes spending a lot of time reviewing homework, but other times spending little; some years giving homework related to the lesson, other years giving homework that was basic skills practice; some years lots of problems, other years just a few. There seemed to be pros and cons to each.
Thinking about this topic yet again, I decided to look for some research to see how we can help students get the most out of the homework we assign.
Using Your Time Effectively and Efficiently
Having the perfectly-run math class....that's been my goal, year after year. Somehow, in middle school, it has consistently tried to evade me!
In other posts, I've shared that I taught elementary math for years, and always had an hour for math class. That hour gave me the time I wanted to have good warm-ups every day (sometimes taking up half the class with one particular problem that led to additional discussion/extension!); the hour gave me the time to go over homework the way I wanted to. And it still gave me time for a new lesson and practice.
But when I got started teaching math at the middle school, with "44"-minute periods, that was all over. (They aren't really 44 minutes - the students get no time between classes for switching, so switching time comes out of the 44.)
This is a post I wrote back in 2013 (now revised), on my other blog, so the observation I refer to was quite a while ago now...how time flies!
I was observed by one of my assistant principals today (a Friday). After 20 years, I don't get super-worried when I'm going to be observed, but I still feel a little anxious. Today, I decided to have the students complete a problem solving activity and then start a "Footloose" activity, even though they wouldn't finish....Footloose normally takes about 40 minutes, so I figured they could do about half and then finish on Monday. (I do this fairly often, to give students flexibility in their work time - they can take as long as needed to complete problem solving, but if they get done quickly, they can move on). Things went so well during the observation...AP commented that there was so much going on in the room, and that the kids were so engaged! I was happy:)
During the class, students worked on group problem solving, (which they have done previously, with other math skills). These particular problems involved comparing and ordering fractions. Our procedure was as follows:
1) Each group received a different sheet with a problem "situation" and 3-4 questions about that situation. (I have five different sheets so that we can do the problem solving several different days with the same concepts, if needed and if time allows).
2) Each group read their situation and each of the questions together.
3) Each student spent 5-7 minutes, thinking/working individually to solve the questions, writing their work on their own recording sheet.
4) When students completed their individual thinking time, they compared their ideas (and answers if they had them), discussed any differences in thought, and worked to agree on final answers.
5) The final answers (with work) were written onto a group answer sheet to hand in.
When we did this type of group problem solving the first time (with decimal problems), we spent about 5 days on the problem solving, with each group working on a different problem sheet each day. The students really like the problem solving, partly because they are able to talk out their answers with each other. It's great to hear their communication about math and how they are able to point out the steps a group member needs to complete or the concepts that they may have missed.
Today, it was great to hear them say "Oh, we're doing this again. I like this!" My AP commented that he listened to hear what they were talking about, to see if they were focused, and he could hear one student explain to another how the work that they had done was different from another student.
The problem solving took about 15 minutes, and then as each group finished their problem, they moved on to Comparing and Ordering Fractions Footloose. This is a great game for keeping students engaged, but moving! Students start out with one card and a sheet of paper with 30 blank "blocks" in which to write answers to the questions on the cards. Each card has a number on it, and students record the answer to each card in the same number block as the number on the card. After answering the question on the card they start with, students put the card on the chalk ledge and pick up another card with another question to answer. Students continue answering and returning cards until they have answered all 30 questions. Students work so quietly when they are doing this activity! My AP said it was like "night and day" when they switched from the problem solving to Footloose - they were talking about the p.s., but as soon as they started the Footloose, it was sooo quiet.....and I didn't have to say anything for it to be this way - it just happened.
As I mentioned, I don't really get worried when an observation comes around, but it was great to hear the positive feedback for these activities that I create for my students!
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?
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!!
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 either 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 (or whole number). 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. If the second choice used 2 and 6, to equal either 0.333...or 3, then 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 (or 1.666) 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 have a score of either 4 or 8). Some students will notice this sum difference and go for the combination that will give them the higher score....bringing in the possibility of using some strategy, for those higher level thinkers.
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.
You can create a score sheet like this on your own, or go to TPT and use what I've created. 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!
Have I mentioned that I love Jo Boaler’s books and site, Youcubed.org? Well, I do! She shares so much fantastic research and so many wonderful ideas.
So, I was reading her book Mathematical Mindsets this week, and read about the “array game” (called How Close to 100), which I’ve seen all over Pinterest and thought was very cool. I tried it with my classes last year during a little bit of down time, and they liked it. I hadn't really thought of using it this year, but last week I noticed the baggie of polyhedral dice that I've had for a looooong time and thought it would be cool to use the dodecahedron dice for the array game. With these dice, the students could use numbers up to 12, rather than 6.
To set up their game, students each outlined a 20 by 20 area on their own graph paper. They took turns rolling their dice and creating arrays to represent the multiplication problem they had rolled. It was very interesting to observe the way students arranged their arrays. Some started in the corner and worked their way out, while others started on one side and worked their way across. Some made the arrays touch, if possible, while others left a row between each one. Some just drew their first few arrays anywhere and then discovered that they didn't have a lot of room to fit additional ones. The "winner" was the student with the fewest number of boxes left (some did get to zero left). The students really had fun with this!
Of course, some finished their games earlier than others. In these cases, I asked students to create arrays that used different numbers than the numbers they rolled, but represented the same area. For example, if they rolled 12 and 5, their arrays could be 10 by 6, 15 by 4, or 20 by 3 (not 30 by 2, we discussed, because the grid is only 20 by 20). If they rolled a number that couldn't be represented by a whole-number array, they could then use an irregular shape, or a triangle - anything they could find the area of. It was interesting to see how some students got stumped when they tried to draw an irregular shape to represent a number like 81.
Most students enjoyed this twist (we continued it the next day so they all got to play this version), but a few complained that it made their heads hurt! That's ok...I know they were really thinking and growing mathematically!
The next extension for early finishers (only a few) was to use the icosahedron (20-sided) dice, and have students create area models to cover their grids and find the answer to the multiplication problems. This required a larger grid, so I had them tape 2 pieces of graph paper together and create 20 by 40 grids. Using the icosahedron dice gave a mix of 1-digit by 1-digit, 1 by 2-digit, and 2 by 2-digit problems to model and solve. Most students didn't get very far with this before we ran out of time, but I think this is a great way to them to visualize what multiplying by a two-digit number means. I'd like to revisit this one!
I'm so glad I thought about using those polyhedral dice!
Have you used polyhedral dice in your math classroom? If so, please share how!
Hi, I'm Ellie! I've been in education for 25 years, teaching all subject areas at both the elementary and middle school levels.