Exploring Number Lines

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One Saturday morning, I decided to try out an estimation task with my five year old daughter while we were eating pancakes. I gave her a piece of paper and drew a blank number line on it and labeled the ends with a 0 and 10.

I started by asking her to place where the 5 should go. I wanted to give her a landmark to see how that would shape her placement of the other numbers. Here’s what she did:

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I was curious whether the number line would prompt her to connect that 5 is half of 10 and should therefore go in the center. It did not.

Then I asked her to fill in the 9. I was trying to be strategic about first asking her to place 5 and then 9, so she would be more inclined to space the numbers throughout the number line. From there, I asked her to fill in the rest of the numbers. Here’s what she did:

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She quickly realized that her number line didn’t look right, and she wanted to fix it. So, she did another number line below and ran into trouble between 9 and 10.

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I asked her to do a third number line, and this is what she did next:

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You can see from this number line that she placed 5 at the midway point. She also corrected herself more than once to make the number line more accurate and linear.

Having kids experiment with number lines is an easy and fun way for them to think more deeply about number relations and estimation. There are some fascinating studies about how kids start with a logarithmic pattern of estimating (e.g. my daughter’s first number line, where the majority of the numbers lie towards the left of the number line) and with practice and age get more linear (evenly spaced).

Here’s a great research article by Dr. Robert Siegler (Carnegie Mellon Univ.) and Dr. Julie Booth (Temple Univ.) on children’s numerical estimation:  http://www.cs.cmu.edu/~jlbooth/sieglerbooth-cd04.pdf

Want to try this type of activity with your students? Here are some blank number lines you can cut out and give to your students.

Is four times always the same as two times doubled?

One of the trickiest part of teaching multiplication deals with the distributive property. Because of this, we made the distributive property the focus of Mt. Multiplis, our multiplication app.

The Common Core State Standards expect third graders to use the distributive property when multiplying (which you and I learned in middle school): “Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56” (CCSM.3.OA.B.5). That’s one complicated equation, and it’s hard to imagine that most 9-year-olds, who are just learning how to multiply, are able to make sense of that notation. So, we set out to see if we could not only make this complex content more approachable, but also fun.

At its core, the distributive property is about groups. It connects multiplication and addition by letting you break up a multiplication problem into smaller chunks. For example, when solving 7 x 8, you can think of it as 7 groups of 8. But, how does that help you multiply? Let’s say you forgot what 7 x 8 is, but you know 5 x 8 is 40. You can then think of 7 x 8 as 5 groups of 8 plus 2 more groups of 8, that’s 40 + 16.

In Mt. Multiplis students explore this property by using groups of wooden planks to build bridges. To fill a 7 by 8 area, students drag over 5 planks of 8 and then drag over another 2 planks of 8.


The best part of playtesting has been watching kids discover the applications of the distributive property on their own. While playing Level 5, which features a lot of 11 x __ problems, one fourth grader literally said, “Aha!” as he realized for the first time, why the 11s rule works. He explained to me that the 11s facts are easy (i.e. 5 x 11 = 55, etc), but that he didn’t know why the shortcut works, until playing Mt. Multiplis. Solving 11 x 7, he dragged over 10 groups of 7 to make 70 and then dragged over 1 more 7.

Another third grader was working through several 4 x __ problems. She would consistently drag out 2 planks and then use the Double-It card to make another set of 2 planks. She first saw the problem 4 x 6, dragging out 2 six-planks to make 12 then used the Double-It card to make another group of 12. She did not fully trust that the answer was actually going to be 24, but she entered it in anyway and was surprised when it was correct. Next, she solved 4 x 3 and then 4 x 8 in the same way. After solving 4 x 8, thinking about 16 doubled, she asked me, “Does that always work…with the 4 times problems…you can just double the 2 times?” Yep, it always works. And that’s some pretty sophisticated algebraic thinking. Another way to phrase her question: Does 4y = 2y + 2y? Yes, because of the distributive property.

To download Mt. Multiplis, go to: bit.ly/TeachleyApps

For a free extension activity to use with your students, click here.

 

Kara Profile Pic_smallKara

What Does Mickey Mouse Have to Do with Counting?

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One night before bed, in the spirit of Piaget, I decided to videotape my 2-year old daughter counting. As most parents know, counting is a fundamental part of children’s everyday mathematics experiences.

Here’s an example of my daughter counting. She’s starting to show understanding of certain aspects of counting and patterns. Pay close attention to the end of the video:

 

Before kids slide down a slide or begin a race, they chant, “1,2,3, go!” Kids learning to count often spend what feels like hours counting everyday objects, like fingers and toes, cheerios for breakfast or stairs they are climbing.  Children learn counting as if it were a song, a rhythmic chant.  Eventually, they begin to see patterns in the song that they can extend.

Young kids are often able to count higher than you might expect if you help them with a few key numbers, particularly the decades.  For example, a 2.5-year-old might count up until 19 and then get stuck on what comes next, but after you tell her it is 20, she can keep counting until 29.  If you tell her 30 comes next, she will keep going until 39, etc. In a past blog post, I wrote about how language makes our counting words particularly tricky for English-speaking kids, especially compared with their Asian peers. See that post here: http://bit.ly/1dnOJ4c

But knowing “the counting song” is just the tip of the iceberg in learning how to enumerate, or count objects.  Researcher Rochel Gelman from Rutgers University outlines 5 key principles that underlie the ability to enumerate.

  1. The stable ordering principle: This is what most of us think of when we say, “My child knows how to count.” It refers to knowing the counting sequence (or the “counting song”). But knowing the count sequence doesn’t necessarily mean you can count objects or use the sequence in any meaningful way.
  2.  The one-to-one principle: Each object to be counted gets one and only one counting word.  In other words, no double counting jelly beans and no skipping objects when you count.
  3. The cardinal principle: Knowing that the last number you say when counting objects refers to the entire set of objects, not just the last object. This is one of the hardest principles for little kids to grasp and has been the subject of much research.  The classic Piagetian task testing this principle is to ask a 3-year-old to count a set of blocks.  The child may be very good at carefully counting, “1,2,3,4” as he points to each of the 4 objects; however, if you then immediately cover the objects with your hands and ask how many objects he just counted, the child doesn’t know how to answer.  He needs to count them all over again, starting from 1 because he doesn’t realize that the “4” he just said refers to the set. One way to help reinforce this idea with kids is to have them always state the cardinal value after they count, i.e. “1,2,3,4. 4 blocks.”
  4. The abstraction principle: You can count just about anything (blocks, candies, ideas or all three mixed together)
  5. The order-irrelevance principle:  Items can be counted in any order.  This idea can be tricky for young kids, too.  If you line up blocks in a row, many kids will think you have to start from one specific end and not from the other end or even from the middle of the line.

Here’s an example of my daughter demonstrating this principle:

 

 

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Teachley Research at a Glance

At Teachley, we take research seriously (it’s in our tagline, after all).  Our products are developed with grant funding from the U.S. Department of Education, Institute of Education Sciences (IES) and the National Science Foundation (NSF). As part of these grants, we conduct ongoing research to evaluate the effectiveness of our apps in improving students’ mathematical abilities.

Design research. When developing our apps, we utilize an iterative design research process during which we build initial prototypes of the app and its features, such as levels and scaffolds, then give them to kids to play. We observe as they interact with the app, ask questions, and gather valuable feedback to inform the refinement of the app.

Efficacy research. To evaluate whether our apps impact learning, we conduct research studies during which we explore changes in students’ abilities before and after they play. To help synthesize some of our research efforts, we’ve put together two short briefs and link to them below.

Teachley Operations. Students who played our operations apps impaddimals-multiplis-ipad-2roved their fluency more than students who played traditional fluency games. Further, students using our Mt. Multiplis app were significantly more likely to use the distributive property when explaining how they solved problems. These results were also found when looking specifically at children who struggle in mathematics. Read the full Teachley Operations brief here.  

 

Teachley Fractions. Initial research on a prototype of our first fractions app, Fractions Boost found significant effects on students’ ability to estimate fractions on a number line. Read the full Teachley Fractions brief here.

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Click here to learn about piloting our apps in your classroom for FREE.

 

Teachley Operations has been funded at least in part with Federal funds from the U.S. Department of Education under contract numbers ED-IES-12-C-0046 and ED-IES-13-C-0044. The content of this publication does not necessarily reflect the views or policies of the U.S. Department of Education nor does mention of trade names, commercial products, or organizations imply endorsement by the U.S. Government.

 

Teachley Fractions has been funded at least in part with Federal funds from the National Science Foundation under contract numbers 1519618 and 1632238. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Tiggly joins the Teachley platform

Tiggly games now sync with Teachley

Tiggly makes interactive, educational iPad apps that your students will love! We are thrilled to announce that 4 of Tiggly’s apps now sync with Teachley Connect*. When you download any of the 4 free Tiggly apps below on an iPad that also has the Teachley Connect* app, the games will automatically sync with your Teachley class list for personalized gameplay. Download the apps and start playing today!

How to download: Download these FREE apps on the App Store: Addventure Number lineChef Addition, Chef SubtractionCardtoons.

 

Tiggly Addventure: Number Line

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For grades PreK-K, join Tiggly to play a number line adventure! This app helps your students get familiar with the number line, learn about number relations, and count in more than 10 languages. Learn about the number line, number relations, and counting in different languages including Spanish and Mandarin. Download here.

 

Tiggly Chef Addition

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For grades PreK-1, this math game helps your students learn addition and represent addition problems with drawings, verbal explanation, and math equations, all while helping Chef prepare his signature Spicy Hula Monkey Cake and over 40 other outrageous dishes. Learn about: early addition, representing addition problems with drawings, verbal explanations, and math equations, composing numbers less than or equal to 10 in more than one way, and understanding the meaning of plus (+) and equal signs (=). Created by Tiggly, a Teachley partner. Download here.

 

Tiggly Chef Subtraction

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For grades PreK-1, learn subtraction math concepts while experimenting in Chef’s super-duper secret kitchen laboratory to create the most silly and flavorful creations the world has ever witnessed! Learn about: mental subtraction, conceptually understanding subtraction as “taking away,” decomposing numbers less than or equal to 20 in more than one way, understanding the meaning of minus symbol (-) and equal sign (=), and representing subtraction problems with drawings, verbal explanation, and math equations. Created by Tiggly, a Teachley partner. Download here.

 

Tiggly Cardtoons

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For grades PreK-K, count, drag, match, and enjoy as the seemingly simple buttons you create come alive becoming part of a wildly imaginative cornucopia of storytelling. Tiggly Cardtoons will help your students learn basic math ideas such as one-to-one matching, counting, and equal sets while stretching their imagination and sense of wonderment. Features 25 imaginative stories focused on numbers, unique illustrations with textures taken from the real world, and guided counting gestures important for developing counting skills. Created by Tiggly, a Teachley partner. Download here.

 


Please note: Tiggly also makes physical manipulatives that seamlessly interact with the apps and are available for purchase separately. These manipulatives are not required to play the apps. The apps work by finger touch as well.

Teachley customers are eligible for a 10% special promotion on Tiggly connected manipulatives with the code TigglyTeachley. To claim your discount, email educators@tiggly.com.

 

*Teachley Connect requires a premium Teachley subscription. Sign up to pilot today!

Teacher Appreciation Sweepstakes Winners

 

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Congratulations to the winners of our Teacher Appreciation Sweepstakes!

 

Basic Account winners:

Erin Berthold

Cara Bartlett

Tammy Evarts

Julie Bormett

Amber Hoerner

 

Pilot School winners:

Cathie Herbers

Maurren Driscoll

Merle Goess

 

Premium Account winners:

Jami Zimmerman

Mrs. Owens

Crystal Malloy

Pam Gray

 

Introducing App Settings

App Settings

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At Teachley, we’re making using iPads in the classroom even easier and more efficient for you and your students. Teachley’s App Settings enables teachers to customize our apps right from the Teachley dashboard to meet each student’s needs.

 

Here are just some examples of ways you can customize the apps:

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Turn off the Speed Round in Addimals EDU Subtractimals EDU

 

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Select specific multiplication factors in Fact Flyer EDU

 

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Assign a pop-up quiz in Fractions Boost EDU

 

To see how App Settings work, watch this short video: bit.ly/2nDMrb5

 

App Settings work with the EDU versions of Teachley’s math apps (and select third party apps) and is available now for premium Teachley subscribers. Simply log into your teacher dashboard at www.teachley.com or in the Teachley Connect app. Not a Teachley subscriber? Sign up for a free pilot here.

 

 

 

Goodbye Pizza Pies (Hello Number Lines)

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Students in the U.S. have an enormous deficit in their understanding of fractions. For instance, half of the eighth graders in this country are not able to correctly order 3 fractions, even though this is a 4th grade standard. Let that sink in for a moment. This is content they should have learned four years earlier. Even worse, only 24% of eighth graders were able to figure out that the sum of  ⅞ and 12/13 is close to 2. What’s the take home lesson from all of this? The strategies we have traditionally used to teach fractions do not work. We need to take a different approach.

How have fractions been taught?

Just think back to your childhood. You’re in your 3rd grade classroom and you are starting a new unit called fractions. What are the images that come to mind? Pieces of a pie? Pizza?

But guess what? Leading researchers and mathematics experts, including Dr. Nancy Jordan from the University of Delaware and Dr. Hung-Hsi Wu from Berkeley strongly recommend against the pie model. They have recommended using the number line as a better way to teach fractions because it offers a more extendible understanding of fractions. The length model also lends itself to more real world applications (construction, architecture, sewing, manufacturing).

The pie model tends to reinforce the common misconception that a fraction is usually less than one. In reality, a fraction can be almost any point on a number line: 6/1, 24 ½ , 0/2, -¾ etc. Another enduring problem with the pie metaphor is that it makes fractions seem fundamentally different from other numbers. If students’ mental images of a fraction are wedge-shaped, what does it then mean to add, subtract, multiply and divide that wedge by other wedges? What is ½ of a pie divided by ¼? There are very few practical applications of fractions operations that involve circular shapes.

It’s much more useful to think about fractions on a number line. Take the tricky concept of dividing by a fraction. Using the measurement model of division with whole numbers, you can think about 16  ÷  4 as how many 4s you can fit in the length 16. Using that same model of division with fractions, ½ ÷ ¼ , you can think of how many ¼-length pieces fit inside ½. You can probably even think of a circumstance in real life when you’d need to divide by a fraction. If you have a ½ foot wood board and you need to divide it evenly into ¼ foot pieces, how many pieces could you make?

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To address the deficit in fractions knowledge, we applied for and received grants from the National Science Foundation to develop a suite of fractions apps that focus on the number line model. We recently released our first fractions app, Fractions Boost EDU. To pilot it for free,  click here.   

 

Read our blog post about Teachley: Fractions Boost EDU here. )

Introducing Fractions Boost!

Are your students confused by fractions?

If so, you are not alone. Fractions is one of the most difficult math topics for elementary students, yet couldn’t be more important. Research shows that developing a strong understanding of fractions is the foundation for future math success. For this reason, the National Science Foundation recently awarded Teachley a grant to develop a series of fractions games. We are extremely excited to announce the beta launch of our first game, Teachley: Fractions Boost EDU, now available for Teachley premium subscribers. Not a premium user? Sign up to pilot here.

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To download for FREE on your iPad, go to bit.ly/TeachleyApps.

Or, if using Apple’s VPP, bit.ly/BoostVPP

 

 

 

Teachley: Fractions Boost EDU is the school version of an exciting 3D racing game that helps 3rd-5th grade students gain conceptual understanding of fractions and represent them on a number line. Students race through a futuristic game world, driving through number line checkpoints and fraction tunnels.

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Students drive through racing checkpoints, estimating where fractions fall on a 0 to 1 number line. If they need help, their dashboard provides a series of supportive hints and scaffolds that help students develop an understanding of the meaning of the numerator and denominator. For example, for the fraction ⅔, students are prompted to “tap 3 times” to split the number bar into 3 equal segments and then “move 2 segments” to find the solution. This engaging game covers all 3rd grade fractions standards, and students learn to compare fractions with the same denominator or the same numerator and determine equivalence.

 

Fractions Boost also motivates students through social engagement. Students design and build tracks to challenge their classmates. In order to a share a track with the class, you must first pass that level yourself.

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Beta launch:
We decided to launch a Beta version of the app for our premium school customers as soon as possible so we could gather more feedback on the game before launching a full commercial version (expected later this spring). We’d love to hear how your students learn from the game, how we can enhance their understanding, and any other suggestions for improvement before we finalize the game design and professional audio recording. Not a premium customer but want to try it out? Sign up here to pilot Fractions Boost and our other math games for free.

 

 

 

 

 

FETC 2017 Teachley Giveaway

FETC 2017 giveaway

 

Hello teachers and tech coordinators!!!

 

In honor of this year’s FETC, we’re giving away a month access to Teachley’s EDU apps for FREE!

To get started:

  • Create a Teachley account at: signup.teachley.com using promo code: fetc17
  • Download our EDU apps and the Teachley Connect app at bit.ly/TeachleyApps
  • Login to your account in the Teachley Connect app with your email and password.*
  • Have students start playing any of our EDU apps.
  • Check your teacher dashboard at www.teachley.com or within the Teachley Connect app to see student progress, print their certificates, and access lesson plans and more activities.

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*Please note: It may take up to a day for your account to unlock the apps.

 

Questions? Reach out to us at info@teachley.com