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Real World Examples For Teaching Fractions on a Number Line

Number lines go wrong in SO many ways. Let us count the ways:

  1. Students struggle to read number lines when labels are missing on the tick marks.
  2. When drawing number lines students do not evenly space the tick marks.
  3. Students count the tick marks instead of the spaces and are often off by one increment.
  4. When a number line is open (without labels or tick marks) it’s difficult for students to imagine a benchmark number.
  5. Not all number lines go up in increments of 1 like most students think.
  6. Number lines are so abstract that students are unsure of what they might represent.

So then for fun…let’s throw FRACTIONS on the number line, that won’t be difficult at all!

All kidding aside, number lines are incredibly important. We use number lines in our lives ALL the time. When we drive, a road is just a gigantic number line. We listen to our GPS and we know about how far 1/4 mile is to our next turn. We use number lines when we measure length, or even as we measure capacity in a liquid measuring cup. I use a mental number line when I’m trying to figure out how much sleep I’ve gotten the night before.  We use number lines in real life without thinking about their mathematical significance, and yet often in many books/math curriculum you will see a number line without any connection to real life with it. No connection whatsoever!

So maybe 4th graders aren’t old enough to drive, but they sure can pretend!  Since they could already identify benchmark fractions on a number line, could they use this idea to learn about equivalence? Could we find two points on the road that could be called two different things?

teaching-fractions-on-a-number-line

So we started our engines, the 4th graders quickly realized that our number lines represented a mile, and that a mile could be marked in different ways. They even talked about how they’ve seen signs for 1/2 mile or 1/4 mile on the highway.

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After working for a bit, they began to see that this mirrored the previous day’s lesson of fraction equivalence with fraction strips.  The connections were forming, and even better they were splitting number lines in two ways without us having to present it that way.  Best of all, they made connections to why 2/8 was the same as 1/4, they were just cut into twice the number of pieces.  This was an awesome link to why equivalent fractions are related to multiplication and division.

Don’t have toy cars to use? Here are some other ways to make number lines real life for kids (all of which can be so precise that fractions are useful when reading the line):

  1. A tiny number line: You are watching an ant crawling on the sidewalk. How far has it gotten?
  2. An eating number line: A snickers bar is being eaten, how much has been eaten if you start at one end and start chomping?
  3. A measurement number line: How tall is the doorway to the classroom?
  4. A time number line: How long can you hang from the playground bars?
  5. A reading number line: How many pages have been read of your book if you stop in the middle of page 39?

When number lines are playful, and connected to real life they are not nearly as scary. Give them a try!

Start driving!

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The Significance of Understanding Equivalent Fractions

I am going to tell you a secret about myself that I’m ashamed of still to this day, and then I will lose all credibility and you will likely unsubscribe from this blog forever and ever.

In the 4th grade, I received a “D” in math during the quarter that we studied fractions.

I know. It’s shocking.

A math coach and math INTERVENTIONIST helping children understand math who received a near failing grade in the 4th grade?!

Before you unsubscribe, let me explain…

It’s no secret that math up until the last few years in the United States has always been about memorization of procedures and algorithms.  If you were really great at memorizing every possible type of math problem, this likely worked for you.  It worked for me actually, I received A’s in math all the way up until the 4th grade. When things went wrong it was devastating for me as a student. I had gone to one school in 3rd grade and then switched to another school in 4th grade.  When I moved, I missed something along the way about fractions…but at the time I didn’t know what it was. Suddenly I was having to add and subtract fractions (numbers I had never seen written that way before) with unlike denominators and it made absolutely no sense to me. I still remember thinking as assignments were passed back to me full of red marks, that I couldn’t believe that I was bad at math, and that I was bad at fractions. The assignments (and the tears) piled up, and I received a “D”.  Shame.

When I look back and think about what I was missing, it’s clear to me. I had never explored or even known that there was such a thing as an equivalent fraction. That I didn’t recognize that it wasn’t a whole number for one thing, and that I didn’t know that two fractions could be the SAME number.  How could two numbers be the same? 7 was 7…21 was 21. How could 1/2 be the same number as 4/8?

So when I teamed up with a 4th grade teacher at my school, we decided to REALLY spend a bunch of time on fraction equivalence.  And we decided to make it as real world as we could. So we put up an inquiry statement, knowing that they have had some experience with equivalent fractions already in 3rd grade…and told them they had to use tools to PROVE what they were thinking.

equivalent-fractions-teaching-tips

Then we let them loose with measuring cups and sand, fraction tiles, fraction towers, cuisinaire rods, diagrams of pie charts and asked to see what they might notice.

It was really kind of awesome. As we walked around, we asked them what they noticed about the numbers.  They began to figure out the relationship between the two numbers without us even saying a word about it!

We came together and shared:

equivalent-fractions-anchor-chart-teaching-tips

We talked through which ones were equal and which ones weren’t. We added some more onto the chart and found out how we can actually use multiplication or division to decide equality if we didn’t have the tools with us.

There were definitely some misconceptions in the room as we worked. Those were noted and then cleared up as we continued on with the rest of the unit throughout the week. I’ll share more lessons soon.

 

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Getting UN-STUCK: A Test Prep Strategy

Testing season is upon us. Yuck.  We all have experience with standardized tests, for some of us starting all the way back in elementary school.  I will admit it…at times when I was tired of taking the test, knew it wasn’t for a grade, or was simply stuck, I just filled in the bubbles. Haven’t we all done that? I heard stories of classmates filling in bubbles in the shape of a fish, or a smiley face, or…

Here’s the thing. There’s no escaping measurement. We are measured in our adult lives at our physician’s office, at our dentist, and by our creditors when we want a loan (just to name a few). In many ways we face measurement, allowing others to see our choices and skills in number form. Surely there are pros and cons to this, but it’s a reality for all of us starting at a young age in our schools.

So what do we do about standardized testing? Well, it’s not about TEST PREP only.  It’s about laying a solid foundation of learning for your students, year after year.  It’s about providing them with rich tasks, and most importantly teaching them how to change their inner thoughts when encountering difficulty.  (See Carol Dweck and Growth Mindset if you’ve somehow missed it.)

I’ve put together this thinking map to show what happens in my brain when I am stuck.  It’s the actions I take to get myself out of those initial negative thoughts when something doesn’t make sense. Your personal thinking map may be totally different.  I want students to know that perseverance is in actions. So I tried it out in a few 4th grade classrooms last week to see if students might be interested in seeing my thought patterns when I’m stuck. I had the teacher give me a “tough” problem and I modeled how I’d work through it, even though I had those negative thoughts.

test-prep-sample-questions-elementary-standardized-tests2

 

Then they tried it with a Fraction Reasoning Puzzle.  At first, the initial reaction when I gave out the puzzle was kind of like a stunned silence. It was so many words to read, with a strange diagram at the top to figure out…kids were STUCK. Then they started to read it, and read it again, and maybe even once more.

test-prep-sample-questions-elementary-standardized-tests

Murmurs of “Oh! I get it!” started to ripple through the room. One student even wrote down her justifications for her thinking.

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When students got closer to being done, I asked them to walk around and look at others.  The room buzzed with them thinking it through and talking out loud. There were some pretty heated discussions. Some students thought that many of the statements were false because there was no point 0 or point 1 labeled on the line.  At one point one student was SO certain that one of his statements was false (when others thought true), he asked to defend his thinking to the entire class.  He stood up in front of everyone and said, “It’s completely false because there is no such thing as the end of a line, a line goes on FOREVER.”

It was so powerful to see them work through difficulty on their own!  You can find free Reasoning Puzzles by clicking on the image below if you’d like to give it a try in your own classroom:

math-test-prep-standardized-testing

 

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2 Winners!

For the sake of privacy, I’ll announce both the Amazon gift card winner and TpT gift card winner by their first name only. If your first name is Andrea, check your email! You won the $25 Amazon gift card. If your name is Faith, check yours! You won the $10 TpT gift card.

OK, whew! Gimmicks over.

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Gimmicks Are NOT My Thing…

I made a promise to myself when I started, to always blog about teaching tips, and things that would help the greater good of the teaching profession.

I’m not into gimmicks and buy one get one sales or anything of that sort. But when I got an email that Teachers Pay Teachers was giving away one thousand $10 gift cards, only for the purpose of giving away to a follower, I decided to enter my name to see if I could win one for you. Why wouldn’t I pass along the love?

Now would you believe it…I ACTUALLY snagged one of those little gift cards to give to you. So I’ll make it easy for you to try to win. Fill out this form:

Click me! Click me! Click me!

I’ll put your name into a random generator and choose a winner. The thing is, their sitewide sale begins tomorrow, Feb.7th – 11:59pm Feb. 8th, 2017…so I need to unload this thing fast.  I’ll leave it open for 24 hours! You don’t have to do anything except type in your name, email and why you love me.  Just kidding! Just your name and email will do.

If you haven’t seen this ridiculous random generator called fruit picker, check it out by clicking below. It’s really strange…but kids love it.

fruit-picker-random-name-generator-classroom-tools

There, a teaching tip for you after all!

 

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3 Ways to Make Meaning of Number Sentences

I told her I had a problem and asked her to solve it. (This intervention student shall remain nameless due to the fact that the internet is an infinite digital footprint, and I believe some day she will be a famous scientist and doesn’t need to read this story about herself online.) My problem?  I have 45 books that I want on three shelves, equally please.  I like when books are evenly spaced out. It’s my math brain I guess.

She sat there, solving this division problem right in front of me for a few minutes. She wrote the number sentence and acted it out, even got the correct answer.  Conceptually she was SOLID.  But the second I asked her what the 45 stood for, I got the deer in headlights look. I waited…nothing.  Then, I asked her what the 3 stands for in her number sentence…no idea.

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The very next hour I was in a kindergarten room.  This problem was up next.  Students were solving with drawings. They wrote number sentences to match their drawings.  When I asked “What does the 3 mean?” No idea.  Their understanding of what the 3 could mean became incredibly fuzzy.

math-practice-standards-number-sentences3

I was not shocked to see this. This is a daily occurrence at every grade level with students working at every level.  This is a fundamental problem with Math Practice Standard #2 (Reason abstractly and quantitatively.).  I’ve realized that once the students pull the numbers OUT of the problem, they aren’t thinking about what they mean as you try to connect that back IN to the problem. Remember all of your teachers saying “Label your answer!”? I believe they were onto something.  It is not enough to say “Label your answer!”. We won’t get over this barrier until we start asking what ALL the numbers in the number sentence that you wrote represent.

3 ways to deepen this understanding, and practice it over and over:

  1. Ask: “What does that number stand for? How do you know?” The simple act of going back INTO the problem after solving it will deepen their understanding.
  2. Write words along with the number sentences for a while, until they begin to see how they can move back and forth between the words in the problem and their number sentences. math-practice-standards-number-sentences
  3. Give a number sentence all alone. Ask them to write a story. It’ll be pretty hilarious at first. Apparently my student had potatoes on the brain. Notice, I’m not asking for a story problem, but a story. That means the story will not have a question or an unknown at the end of it. This helps them make sense of ALL the numbers in the number sentence.math-practice-standards-number-sentences1

This is something that we must all commit to to help students make sense of mathematics, make sense of problems and to make math less abstract. I would love to hear any other suggestions you might have to strengthen our little mathematicians in this area.

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Win a $25 Amazon Gift Card…Yes, Really

OK, I believe strongly in feedback and using it for my own personal improvement. I’m really looking for ways to find out more about my audience, and to improve my practice. So along with getting good, honest feedback, I decided to give something away.  I mean, who doesn’t love to win stuff???

Would you please fill out my short survey by clicking HERE? It’ll be open until midnight central time on January 31, 2017. From the responses I’ll randomly choose one awesome winner.

Or you can click on this great photo of my cat to take you to the survey. Cause she’s awesome. She really wants you to win.

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