Problem of the Week, #12 Mixed Operations

Last school year, I started posting a problem of the week, but got a bit off track with the craziness of life! I got to Week 11 last year, so this year I'm picking up with Week 12 and will do my best to keep them coming :-)

This week's problem includes a percent calculation, multiplication and division (some decimals involved) and interpreting the quotient. Of course, there are many ways to solve problems, and your students may find a different way to solve than the way I did!

Click here to download.

Or here!

To access all of the past Problem of the Weeks, click here!

Have a great week!


Math Dates 9/19-9/23/16

This week's math dates are pictured here!
If you've been using the math dates for a few weeks, this week might be a good week to make a "mistake" in one of the days and see if your students notice -  keep them on their toes!

I included several fractions with exponents in both the numerator and denominator this week. If your students aren't familiar with exponent rules, they might make some observations and discover a rule for themselves by the end of the week:-)

If you have trouble reading the dates here on the blog, you can also find them on my Instagram and Facebook accounts.

Have a great week!


How Can You Help Change a Negative Math Mindset?

How often have you started your year with students  saying that they hate math? I don’t know about you, but I find that it happens every year…and when I ask why, students often can’t say....so,
I don't necessarily believe that it's math that students "hate" (though they think so). Instead, I believe it's something that happens or has happened in math classes that they dislike.

My goal is to help students like math class enough that they’ll open their minds to the possibility that math can be fun, interesting, and even helpful.  This can be a challenge when some of them have had a negative math mindset for years.
I’ve found that a few simple actions/habits on my part seem to lead to improved student attitudes and the mindset that math might actually be a “likeable” subject. It's not necessarily the way I explain concepts (although I like to think I do that well:-), but more my attitude of acceptance that I think helps to change their attitudes.

Here are a few ideas I believe have helped my students develop a like (if not a love) of math.

Using Pentominoes
1) Encourage students to ask questions and then be willing to take the time to answer them. Students often hesitate to ask questions, especially in front of the whole class, and especially at the beginning of the year. I start my year with an activity that encourages students to work together right from the start, to try to foster a comfort level that will encourage them to ask questions in class.
I constantly remind students to ask questions when we introduce and discuss concepts. 

At the beginning of the year, students will typically start asking questions during group work time, and I take the time to answer, even if it takes me a little longer to get to other students. As the year progresses, and students see that I really do want questions and that I really will answer those questions, they become more comfortable asking questions in class. The more they ask, the more they learn. Yes, it takes extra time sometimes, but it’s worth it to me….I want them to develop deeper understandings and that happens through questioning.

2) Allow students to talk to each other about math. We know that many students love to talk! Why not encourage them to talk about math? Students often come to my class having had very little chance to talk about math with others, and they are surprised at how often I ask them to do so. They always have a talking purpose – often it’s discussion of the warm-ups they did for homework….what were the answers, how did they solve, why do they have different answers? Other times they work on problem solving together….what do they know, how would they approach the problem, where are they stuck? Talking about math is so important and they come understand that they can share their ideas freely.
3)Ask why. I ask why all the time. To begin with, students often think that me asking why means they are wrong, so they change their answers. But it doesn’t….it just means I want to know why they think what they’re thinking....I think this makes them feel valued. The more I ask them to justify their thinking, the more able they are to do so, and the more they like to explain….even if it’s not “right”- they know I'm not judging, I'm just listening.  I love having them go to the board to illustrate their "whys." Some of them are super willing to do so at the start of the year, while others take a while. But by the end of the year, they feel comfortable explaining their whys.

4) Tell stories about math  in real life. Last year, I told my students about the night my son (who was working as a server in a restaurant) got a $60 tip. I knew the total of the bill, and we were working on percents at the time, so we figured out what percent tip that was….I think it was somewhere around 50%! They were so interested, because it was about a real person. It was a short story, but those little tidbits really add to"math interest."

5) Be accepting of students’ thinking, explanations, and mistakes. This might be the most important thing you can do. Sometimes students are on the right track, and sometimes they aren’t....but making mistakes helps to grow the brain, so it's ok if they aren't on the right track. Yes, some math problems have one right answer, but when students are simply told “No, that’s not right,” they may shut down and tune out….feeling embarrassed about making a mistake or sharing "wrong" thinking. Peter Sims, writer for the New York Times, says that successful people “feel comfortable being wrong.” Students need to realize that they ARE doing some correct, valid thinking even if it leads them to the wrong answer. It can sometimes be challenging to take time in class to find the parts of student thinking that can be built on, to lead to the right answer. But as a math teacher, that’s part of my purpose…take students from where they are, ask them questions, share thoughts, accept their mistakes, understand what they’re thinking....and expand it or redirect it to help grow the concept in their minds. According to Jo Boaler, “One of the most powerful moves a teacher or parent can make is in changing the message they give about mistakes and wrong answers in mathematics.” When students observe you accepting and building from others' mistakes, they become comfortable sharing their own ideas.

Making Triangles
6) Let them explore math concepts! I know it’s hard to take time to explore, especially when your math periods are short and you may feel pressure to “cover” material within a certain amount of time. But letting students explore math and play with math is so valuable! In Jo Boaler’s book, Mathematical Mindsets, she references brain research and the idea that, “If you learn something deeply, the synaptic activity will create lasting connections in your brain, forming structural pathways, but if you visit an idea only once or in a superficial way, the synaptic connections can “wash away” like pathways made in the sand.” She also references a Park & Brannon study that found that “...the most powerful learning occurs when we use different pathways in the brain...” Giving students time to explore math allows them to explore those pathways and think more deeply. This can only benefit them and build their foundation for the topics you'll teach.

At some point during the year, most of my “haters” stop hating math. They realize that maybe it’s not so bad, and they become willing to have conversations about math topics. They become willing to ask those questions in front of the class and explain their thinking about those tricky problems  - even when they don’t know if they’re correct. They become willing to take risks because they know they won’t be judged or harshly told “sorry, that’s wrong.” And when they learn that their thinking wasn’t quite on track, they don’t feel judged or stupid...and their minds stay open to the learning. 

For me, one of the best parts of teaching math is watching the metamorphosis of a child, from one who “hates” math to one who willingly goes to the front of the class to illustrate and explain his or her math thinking.


Integers Fold it Up Freebie!

Click to download!
Wow, the summer has just flown by (as always)!

Among other things this summer, I've been working hard on creating new units for 6th grade math. I've made an effort to include different "colorable" sections on many of the pages, and all of the units include a couple different Fold it Ups that I've created or tweaked over the years. I just love the Fold It Ups, as I'm sure many of you do.

This one is from my Rational Numbers unit - (it's also in my Fold It Up book, but the labels on that one are black rather than "colorable").
I especially like this one because it gets glued onto a second page, which has a number line. This allows for some flexibility when teaching about negatives, positives, and opposites. The number line can be used during the introduction of the concepts, to label vocabulary on the number line....make it a visual. Or, it can be used during review, when preparing for a test or quiz....students can label when self-quizzing (or quizzing each other)!

Or click here!
The download also includes notes for the inside:-)

Hope you can use it!

Font in the graphic by KG Fonts.


Fun with Math Dates, 2016-17

It's almost August! I'm always amazed at how quickly the time goes!
This is just a quick post to share what I'm planning with "math dates" this year. Last year, I got such a positive response when I posted ways to write the dates as mathematical expressions. So I decided to get a little more organized with it this year, and plan the week of dates on Sundays, instead of thinking of them one day at a time.

Every Sunday, I'm going to post the week's dates on Instagram and Facebook, so if you'd like to use them, you might want to follow me one of those platforms (if you aren't already). I'll post the dates here sometimes as well, but not as regularly.

I loved using expressions as the dates last year - it kept students fresh with their exponents/exponent rules, exposed them to roots (we don't really study these yet in 6th grade), and encouraged them to figure out their own expressions.

Do you already use something like this in your classroom?

**Fonts in images by KG Fonts

WHY the Butterfly Method When Adding and Subtracting Fractions?

The "butterfly method" for comparing, adding, and subtracting fractions -WHY teach this? I don't
understand why this method is being used.

I understand that it works and that it can make finding an answer easier for students (for simple fractions), but I am discovering definite drawbacks to this method. I don't want to seem negative, but I don't like to see students adding and subtracting fractions this way! Let me explain-

I used to have students come to me, having learned the butterfly method for comparing fractions.....they often had no idea why it worked....they knew the trick better than they knew how to find a common denominator...they didn't seem to understand that the products from the cross-multiplying were the actual numerators they would get if they made certain equivalent fractions (figure 1); they just knew it worked.This bothered me, because I believe they should understand why things work. So I always made sure to explain why the method worked.

But this year, I had students tell me they were taught to use the butterfly method to add and subtract fractions (cross multiply and add those products, then multiply the denominators together -figure 2). But again, they didn't really have a conceptual understanding of WHY it works. It seems to me that many students are being taught "hacks" like this, to make learning fraction operations "easier and fun." In reality, they aren't learning what it means to add or subtract fractions. (And, really, why wouldn't we want them to see that 6 is the LCD in the problem in figure 2? Why would we want to them to use a larger denominator and then have to do more reducing...?)

I recently gave students problem solving that required them to use all fraction operations. Since adding and subtracting is in the 5th grade curriculum (and I teach 6th), I did just a brief review of adding and subtracting fractions before students worked on these problems - to see what they remembered. This is when I found out that many of them had been taught the butterfly method, among others.

In the problem solving, students had to add 5/6, 2/3, 7/12, and 7/10. And here's where the butterfly method totally fails the students who have learned to rely on it, not only because they don't understand why it works, but because it becomes so cumbersome! They couldn't use the butterfly method to add 3 or 4 fractions at a time, so they added two fractions at a time. Instead of finding a common denominator for all 4 fractions, they found a new common denominator each time they added on the next fraction. For example, they added 5/6 and 2/3, getting a denominator of 18, as in figure 2. Then they added 27/18 and 7/12, as in figure 3,  getting a denominator of 216. From there, they added 450/216 and 7/10, shown in figure 4. They ended up with a HUGE denominator that they then had to work really hard to reduce! It might seem crazy that they continued this process to get such huge numbers, but some of them did, because this was the method they learned. I was so shocked to see this....and this is the first year I saw this method used in this way.

When we as teachers (or parents) find certain tricks that work for simple math problems, we need to look ahead to what our students will experience in future years. We need to try these methods with more complicated problems, to see if they will still be effective. We need to think about whether the "hacks" teach them math concepts, or number sense, or number connections....or do they just teach short-cuts? I have no problem with teaching short-cuts once the conceptual understanding is there. But short-cuts before understanding is detrimental to our students. They are capable of understanding the concepts and we need to have faith that they can "get it" without the tricky methods.

In the video linked below, Phil Daro stresses the value of teaching mathematics in greater depth and avoiding "clutter" in the curriculum - one of his examples includes the butterfly method.
Don't Leave Out the Math 

Thanks for reading!


Table Tennis and Math

I love playing table tennis (but I really call it ping pong all the time)! 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!!

**Fonts in graphic by KG Fonts


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