Lesson on Multiplying Fractions Greater or Less Than One

Improper Fractions
Teaching math should be a lot of fun!

So, sometimes I find a random standard and think about lessons I could do to teach those. I find the exercises not only help inspire me as to why I’m working on a standards alignment tool, but also help me think about new ideas in general! Obviously this would be one lesson in a series teaching the concept and doesn’t cover the whole standard!

Lesson Title: Prove your Classmates Wrong! (Horrible title but… something…)

Time: About 40 minutes

Standard: 5.NF.5.b Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1.

Objective: Help students understand rules of multiplying by fractions greater than or less than 1.

ELD: Use visual clues and small group discussion to help students understand the rules of multiplying by fractions greater than or less than 1.

Technology: None.

Materials: Different colored notecards are a plus; clothesline or whiteboard will do. the numbers 0, 1, and 2 affixed for a standing number line activity.

Narrative: Hand students different colored cards in their table groups. Have group 1 write a fraction less than 1 and group 2 write a fraction greater than 1 and less than 2. Numerators and Denominators have to be less than 100 as well.

Iteration 1: A few kids come up to the board to show on the number line where their number would be.

Iteration 2: Have students multiply Group 1 and Group 2 fractions and see what happens – have those students then go up to the number line and explain what numbers were multiplied and if the end result number was less than or greater than the previous two numbers.

Iteration 3: Have students multiply with the SAME Group and a sample from each group comes up. Ask different students to summarize what they’re noticing.

Create three groups: Have students in the same fraction-type groups summarize their findings – ask what happens when fractions less than one are multiplied together, fractions greater than 1 are multiplied together, and make a third mixed group that tackles happens when a number less than 1 and a number greater than 1 are multiplied together.

After a few minutes have each group summarize their hypothesis in writing and.

Now rotate the written hypotheses within the groups. The new group is going to spend some time trying to find a counterexample to the hypothesis – kids love it if you say “TRY TO BREAK THE HYPOTHESIS!”.

After a few minutes of trying to find a counterexample, kids usually aren’t able too but this gives all students a chance to try a set of numbers that wasn’t theirs originally hopefully in a way that is motivating as well.

Now do one example on the board of each type of operation and have students see what is going to happen. Talk about the power of knowing what SHOULD happen as way to check their work… that is, if they are multiplying 3/4 by 1/2 and they get a number larger than 1, they know they did something wrong.

Closure: Write an example set of 3 questions on the board that cover various cases. Have students write those answers on the back as an exit ticket to hand in as they leave the classroom.


1 and 2/5 multiplied by 4/5

3/7 * 9/10

5/4 * 1/8

IMS Global Fall Quarterly, Ann Arbor Michigan

This was my second IMS conference but the first quarterly meeting. I knew it wouldn’t have the conference feel of Learning Impact from May but wasn’t sure what else to expect.

It was like drinking from a firehose a bit, but well worth the trip in both cost of money and time. I was most interested in the sessions involving CASE because I am as deeply involved in the OpenSALT project as one can be without contributing actual code (and I’ve started to do some of that too!).

The first session I went too was about the integration of gradebook services – OneRoster, CASE, etc. Now that CASE identifiers are being incorporated into all of the next version of the other IMS standards, it is important to be consistent in how it is called and referenced. Also a bit of a history lesson of previous educational tech standards was done which I found very interesting, having no previous context. The basic premise was to see how all of the standards interact on a structural level.

The CASE session Wednesday morning was a solid discussion going forward about how the CASE standard is well on its way to being well used, already with georgiastandards.org using it to publish their frameworks as well as other larger organizations coming onboard soon. One interesting piece of information was that we won’t be talking about updating the spec for a few months now, which makes sense to get everyone onboard with using it. CASE is more than just a way to store information about competencies, but eventually will also be a way to visualize their connections and finally have a clear path going forward when state standards are updated or created for the first time. CASE will allow for better OER adoption through more consistent standards identifiers at the digital level.

Probably the best session was the last – an explanation of a new way to search LOR’s (Learning Object Repositories). Aside from the fact my boss is the committee chair… I’ve been hearing about it and reading some of the documents before but this was really a walkthrough and discussion of all of the different fields and I think I understand more about it for the first time. It will really help people find the right resource at the right time efficiently.

Great times meetings at night with the CASE folks, OpenSALT team, and Karaoke with random IMS’ers as well!

An Example of What Standards Could Be

A couple of weeks ago Robert Kaplinsky posted about how future Math Standards writers might approach that monumental task.  In parallel, I have been working with the new CASE framework from IMS Global that wants to redefine how we view and share academic competencies (standards) on the Internet. In addition, there are various ways to define relationships between standards both within and outside of the specified framework. This has huge implications as many states have slightly changed the wording of the standards so that it’s not immediately obvious, but their standards are still mostly following the original published CCSS. It is our dream that this tool and format will truly enable interoperability of ed-tech platforms and OER content.

Robert’s post struck a tone as you can see that I’ve talked about several times on my site how standards can and should be better. I created an example sample of framework on our Open Source framework management tool called OpenSALT that shows some of the relationships that I thought would be interesting to people with a few Common Core Mathematics standards:


The Framework (click here – it will open in a new tab)

-Shows the relationship of a Common Core standard to it’s Missouri ‘cousin’

– Showing progression type relationships (precedes)

– Showing all of the various association views (Is Related To and Exact Match of will be the most interesting lists for crosswalks)


– Exemplars for the Standar


openSALT is not for the general user, but for State Departments of Education as they publish their standards – we implore them to not use PDF but instead something like this which can be quickly and easily read by… everybody. Contact Joshua Marks at PCG Education for more information if you’re reading this.

For teachers and districts, it could however be a useful too to create unpacked standards and be able to publish/user them on internal system software (the unpacked standards could still retain a connection to the ‘parent’ official standard through the “IsPartOf” relationship).

It is hoped that the next writers of standards could, when writing the standards, build into them these relationships and extra metadata such as more examples. Currently, if you want to know about the standards themselves you go here, if you want to read about the progressions of the concepts here OR here, and for a cohesive map yet another website.  Likewise, while some standards are clearly defined and demonstrable, others are quite vague on purpose… but that doesn’t help teachers trying to find the best content for their students.


The Open Education Ecosystem

The image on my Twitter profile (albeit sort of ugly because I made it in Google Draw for now) represents what I hope to see in the future of education. I didn’t come up with the idea but I’ve been playing with/thinking about FOSS in schools since at least 2001. More recently I read a book by Charlie Reisinger about what his school system has actually DONE in Pennsylvania with open source software!

The Open SchoolHouse is written for those who are in educationand probably should know a bit about technology like Linux, Apache etc to really get excited about what they’ve built over there in Penn Manor. Administrators would love to hear about it as well when he talks about the student-run helpdesk and other cost savings!

I took my current job at OpenEd because I was very interested in the idea of freely available, remixable educational resources – both creating and curating them. I’ve been excited to see more consumer devices being built on top of linux (chromebooks, android phones etc), and while Google Docs aren’t open source, they export to anything and satisfy the 5 R’s of OER albiet not the part of ALMA that talks about proprietary software because you have to give up privacy in order to use Google services.

So I made this diagram to represent and guide my thinking. I feel like we’re almost there already, but still don’t have open source alternatives in education to things like SIS’s, curriculum mapping tools (spreadsheets? no thanks…), and probably a few other things I haven’t thought about! oepoer-1


OER and Equity: Part 2

When I first got really into the ‘open’ movement nearly two decades ago as a Linux user, a huge part of it was for the issues of equality it promised. Free software meant more money could help students in poverty was the promise. Free educational resources, ideally teacher-created also held the same promise – replace expensive, proprietary textbooks with high quality, highly skilled content creators (Teachers).

So it has surprised me in the past few months to get to work on what I would have previously said, “that doesn’t concern me.” as a creator/sharer of educational content and not a programmer at heart – educational standards. I gave a conference presentation about this a few months ago at LibreLearnLab but became aware of an effort from IMS Global a few months later. It seems others in the ed-tech industry had already realized that a common language to describe the various fields that could go into an educational standard were lacking. For example, in the Common Core State Standards for Mathematics, there are a few things, “missing,” that could help both teachers, students, and ed-tech products:

  • Example problems to demonstrate the standardscreen-shot-2017-04-03-at-3-55-22-pm
  • Programatic progressions statements – ie this set of standards leads to this set. Yes, there are the progressions documents and it’s not too hard to figure it out, but not everyone can infer that the Expressions and Equations turn into the Functions strand in grade 8-high school unless they understand the pedagogy that leads them there – which I can tell you most engineers don’t care nor should they need to know this.
  • Type of statement (Cluster vs Standard vs higher-level headings) – sounds minor but when we’re talking about thousands of possible statements differentiating between these statements can really help. IE the math practices are standards not searchable by a keyword, but a human curator can recognize when those practices are apparent in a particular math task or learning resource.

In addition, the only ways to see possible connections between other frameworks such as the TEKS standards and CCSS are either costly correlation services or manual search. Even with that, there has not been a formally defined way to write the TEKS statements in common ways. Thinking about time, spending time translating statements and searching in the dark for resources is not the most equitable process.  Furthermore, at some point CCSS will be replaced by something else, and we will want some robust way to transfer the good curriculum activities/tasks to be in alignment (or not!) with those new standards. The NGSS introduce even more nuance with three dimensional standards that aren’t a simple tree anymore…








How Open Competency Frameworks Will Help K12

The advantage of machine-readable standards may not be apparent in their application to the Open Education movement. As standards change by jurisdiction or eventually new standards themselves, all of the hours spent categorizing and tagging resources may be able to be transferred over if the new standards have overlap. In addition, as opposed to monolithic PDF documents that States and Organizations normally publish their standards in, with an open competency framework those standards would be easily searchable and usable for content creators looking to find resources to help students.

The more metadata the standards have already on them – including notes, examples, rubrics as to what addresses that standard, connections to other standards and more – the better and more specific the OER content can be, which will give more choices of robust, rubric-evaluated standards-aligned curriculum to schools. Usually when standards are published they are not revised for 10-15 years which used to be the norm, but the world changes too fast now. Standards in an electronic format that automatically link to the updated format can become dynamically better (with major revisions every other year perhaps to address gaps as they arise or enhance wording, etc). Yes we need consistency, but just as once a student publishes a blog or other live-published feature we would expect some degree of revision based on feedback, so too should academic standards be open to revision and updating.

What is exciting to me – a curriculum guy – is seeing how the technical backend will enable better, more robust standards and description of the relationships between the standards down the line. As open educational resources are refined and revised, it is important that the standards they point too are not only the most accurate but also as informational as they can be.

A “Blank Page” Google Docs Add-On

I was talking with my online friend Tim Brzezinski the other day about the great work he does with Geogebra and how to accelerate even more synergy between the interactive element and Google Docs.  He is experimenting with linking to google doc exploration guides from within Geogebra applets. I wondered what it might look like if we embedded the Geogebra applet into a sidebar so created a simple, blank sidebar add-on to embed the Geogebra applet. What I found  was that it was far too small (300 pixels set width) to really be usable, but it was a good little project nonetheless.

To use this in practicality, click the link below, Make a Copy to your own drive from the File Menu, then select Tools, Script Editor.

You will see something like the image below – click on over to Sidebar.html and edit the HTML code (a few examples down below)screen-shot-2017-02-02-at-3-28-56-pm


Google Scripts Project


Example HTML that embeds a youtube video

<!DOCTYPE html>
<base target="_top">
<iframe width="300" height="100%" src="https://www.youtube.com/embed/ghfcrjTpZE0" frameborder="0" allowfullscreen></iframe> </body>

The main change I made is of course the width to 300 and the height to be 100%. (Sample Document) 








Map on the Side








Admittedly the method isn’t very pretty, and before I publish to the Chrome store I’ll make it so people can just paste the embed code into the frame and it will then automatically embed it (thus it will be able to be changed for each new doc…), but what are some thoughts for how this could be used?

#ShareYourLessons – Youtube

Introduction Post

The Problem

Youtube is by far the most popular video platform on the Internet with over a billion users. But a standard youtube license quite simply isn’t good enough. When you upload to youtube you are given a choice of these two licenses (and one has to navigate to the ‘advanced’ tab to find it):screen-shot-2017-01-04-at-2-57-42-pm The problem I have with the Standard Youtube License being the default is that others can’t remix the content which is one of the five R’s.

The Solution

Since 2007 Youtube has allowed users to remix others content to make their own original content. Go to the Youtube Video editor (https://www.youtube.com/editor ) then click the CC button.


You can add multiple videos to include, trim for size, add your own text, change the audio settings… and it doesn’t even have to be your own video!

As the image shows, you can also add your own free music tracks to the videos!

While I’m a huge fan of students using tools like WeVideo and others, why not use Youtube itself for basic editing/discovery? The native platform is almost always the best. Imagine the possibilities of students with youtube accounts now able to make their own mix-type style videos to share with friends and teachers. More education content could be created faster and with more fidelity then waiting for Khan to get around to recording more videos, that’s for sure!

Here’s the problem – if I search for just “math” on youtube, I get about 6,910,000 results. However if I filter for Creative Commons-licensed videos, I only get 279,000 or 4% of the total videos out there.

Share It Better!

  • When you upload your video, use a CC license
  • Write a good description – many education videos on youtube have terrible or NO description and thus are harder to find. Please write good descriptions!
  • Always include contact information in your description as well. As I watch videos on youtube I hate having to track down someone through their channel, then website, then contact page…
  • Make sure no background music etc that you use is copyrighted. It will get taken down.
  • Share your content on social media! Twitter, Facebook, you can even just take a screenshot to share it on Pinterest and Instagram! People are hungry for quality content…


output remixed video

#ShareYourLessons – Intro

This is the first in a series of posts designed to help teachers share their lessons and learning better.

I’ve gotten quite involved in the OER(Open Educational Resources) movement the past couple of years, building on a love of open source software in the late 90’s and early 2000’s. While the idea of OER is getting to be more commonplace in higher education, it’s still relatively new in K-12. Even within the US Dept of Ed’s #GoOpen movement, there is a lack of understanding of what it means to be open source.

A common definition that I will use here comes from David Wiley over at opencontent.org/definition :

  1. Retain – the right to make, own, and control copies of the content (e.g., download, duplicate, store, and manage)

  2. Reuse – the right to use the content in a wide range of ways (e.g., in a class, in a study group, on a website, in a video)

  3. Revise – the right to adapt, adjust, modify, or alter the content itself (e.g., translate the content into another language)

  4. Remix – the right to combine the original or revised content with other material to create something new (e.g., incorporate the content into a mashup)

  5. Redistribute – the right to share copies of the original content, your revisions, or your remixes with others (e.g., give a copy of the content to a friend)

Mr. Wiley also uses the ALMA framework to determine if a resources is sufficiently accessible.  Essentially, if something is published in a file format like PDF that may make it not able to be edited, then it would be difficult to call it truly ‘open’.

Some common content types (*my own analysis which may be flawed!)

Retain Reuse Revise Remix Redistribute Blog Post
Youtube Video Can't download natively IF CC-licensed Yes - but only by owner if not CC-licensed IF CC-licensed Yes - shareable Here
Google Doc YES YES YES with caveats around proprietary technology (google sign-in) YES with caveats around proprietary technology (google sign-in) YES
Word Doc YES YES YES with caveats around proprietary technology YES with caveats around proprietary technology YES

I’ll be taking a look at some of these content types and how teachers can actually share with fidelity in the coming weeks!