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

Well meaning, but bad for kids

I saw my friend Duane Habecker post this image as part of a thread about “math felonies”.

I later ignited a bit of a debate about math understanding on my personal facebook page, another post on Twitter etc.

The crux of the matter is a lack of content and pedagogical understanding. Often kids are taught to satisfy a grade levels understanding but later on it would have to be re- taught. Another example- the rules of working with radicals are pretty much the same as variables but rarely taught that way. Eg you can do 3rad(2) + 4rad(2) But not 3rad(3) + 4rad(2) because the bases are different… same as 3x + 4x and not 3x+4y. If I were designing curriculum I’d always have something saying “Think ahead too…” and “look back from When..” for every topic. Well, basically I’d integrate the coherence map.  🙂

As we work on building more conceptual understanding for students across ALL grade levels, this kind of thinking – really only meant to help kids-  will start to dissipate. I know it was a teacher that was really trying to help kids… and so thought of something FUN! But, it’s probably partly because he/she didn’t fully understand the process either.

Please, start with https://gfletchy.com/progression-videos/nixthetricks.cmachieve.org/coherencemap , teachingchannel.org for great resources that help target WHY and WHAT TO DO About misunderstanding how to teach kids certain techniques! A random example of how useful this is the map for 4.G.1

Note how it links to the High School Geometry standard G.CO.1

This is relatively easy information, but helps remind that high school teacher that it’s been a while since kids have seen these definitions, so might be good to review a bit more first.

Knowledge about the content of the standards and where they fit in the overall education career of a kid is needed. I am proud of the work I’m doing with ACT on the CASE standard and OpenSALT which will eventually be able to have this kind of visualization integrated for any competency framework as well!

New Role at ACT!

Monday November 6 will start a new era in my career as, “Lead Assessment Specialist,” at ACT. The title is similar to others I’ve had over the years (Lead Content Curator, Senior Manager, Teacher on Special Assignment…) but the work is a new, more technical direction! I will be known as the Business Owner (in Agile Development) for the OpenSALT tool within ACT. For the past year I’ve been working with the OpenSALT Consortium as the primary ACT representative, and performing SCRUM Master duties, etc. while overseeing the content curation pipeline and staff for OpenEd.

This past month OpenSALT was used as the tool to launch the ACT Holistic Framework. As ACT devotes more resources to the project more stakeholder communication and coordination is required. This will be the first job I’ve ever had not directly working with content/curriculum, which will feel strange – but on the other hand one reason I was chosen is because of my extensive content/academic standards knowledge. Basically I started out as a consumer of content and standards (teacher), then a trainer of teachers, and this is the next step on helping those who actually create the content and standards themselves at the technical level.

I will certainly miss working directly on OpenEd and the educational resources there! I was the first educator who had used OpenEd in my classroom in to be hired on full time in 2015. The details are still unclear a bit to me – Adam says Lisa encouraged him to ask me – but when I was initially asked to join the team, it took me by surprise. Regardless, I took a job requiring a skillset that far outmatched what I initially had (unbeknownst to me!), and ran with it. I am proud of the work we have accomplished at OpenEd and am excited to continue to watch the usage and application grow as integration with ACT products increases. If anything, having someone with fresh perspective on OpenEd’s current place in the market ecosystem is needed and I was probably holding it back.

Thank you Adam Blum and the entire OpenEd team for supporting me, helping me grow, and to ACT for this opportunity! I am looking forward to being under the leadership of Donna Matovinovic in Assessment Design and working even more closely with colleagues at ACT to empower students, educators and employers.

My Math Story: Part 3

A year and a half after starting this series, figured might as well finish it after I saw Robert Kaplinsky re-post his!

My previous two posts talked about how I started off loving math as a young kid, then started hating it and feeling I wasn’t good at it in middle/high school. Only to fall in love with it at the very end of high school with an AP Statistics course!

In college I finally saw the beauty of mathematics through courses like History of Math, Problem Solving, Number Theory and even Abstract Algebra.

When I started teaching, a lot of things happened at once. I student taught at a school that was famous for being a bit on the rough side, with kids from low socio-economic backgrounds. Due in large part to my previous work at Sugar Pine Christian Camps and Valley Teen Ranch, this was exactly the population I wanted to serve.

So like any young teacher, I wanted to make an immediate impact. First day of class I remember when that door shut and all of a sudden there was no master teacher, no one would come check on me just because I was, “new,”  – it was 22 year old me and all of the teenagers (my birthday is August 26 so usually right at the start of the school year). I had them do math from the beginning to get acclimated to what I wanted my classroom to be like – active and engaging. I gave a short speech early on in the school year that, “If you slipped through Algebra and convinced your teacher you should pass but you know you shouldn’t have, talk to me and let’s fix it.

Afterwards a tall skinny sophomore approached me and joked that he was one of those kids. So for the next several weeks we met on Wednesdays at 7:30 (almost an hour before school began) to go over times tables, review addition and subtraction, and algebra. It was life-changing for me, because I realized that I could actually make a difference in these kids lives with mathematics. He was missing many concepts… I remember still he could add mixed fractions for example but had trouble subtracting them. He saw everything as a new skill to be learned as opposed to connected to previous skills.

Throughout my time at McLane High School I taught Geometry, Algebra I, Algebra II, “Alg/Geo III,”(hybrid course), California Exit Examination Prep, and even a study hall that I turned into an occasional History of Math course. When I went to teach middle school, these experiences teaching the upper level content were invaluable because I knew intimately how the lower concepts related to the later learning. To be honest I wonder how anyone teaching the same grade level can do a great job of it without being exposed to actually spending time teaching students above and below their, “target grade level”. It’s one thing to know about the concepts coming up – it’s another to teach it to groups of students and through their learning it, learn it better yourself.

Take polynomials. Until I started teaching about them to students, their relation to Exponents and number groupings wasn’t as clear as it should have been. (Eg any number is simply a simplified polynomial in base 10.) Through teaching students, seeing their misconceptions as a feedback loop for my own understanding was invaluable.

This brings me to wha thas become the main point of this blog – Personalized Learning. Most definitions/marketing about it talk only about the benefits to the student. We’re missing an entire part of the equation here. Dan Meyer recently had a discussion about personalized learning and this is something that he mentioned – the drawbacks of computer-based instruction on the teachers by having less feedback from the students and their peers about what they don’t understand in a qualitative context.

I learned so much more from my students over the years then they probably learned from me. Algorithms can’t replace smiles when students understand the big ideas. It was what made teaching interesting, exciting and enjoyable every single day.

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!

First Day of School

My Colleagues in Fresno and Central Unified are starting school today – congratulations!

It’s now been two years since I have worked for a school district and three since I was in the classroom. My last ‘first day of school’ was 2013 – 4 years ago!

I usually tried to go straight to something that would set the tone for the school year – do some math! Often I started with the four 4’s game – where you can rearrange using exactly four four’s(or 8 or 6, etc) in any sequence in order to get different numbers with order of operations – exponents are excused from the 4 requirement to make it more interesting.


0: (4-4)*(4-4)

1: (4+4)/(4+4)

2: (4*4)/(4+4)

3: (4+4+4)/4

4: (4*4) / (4^1 * 4^0)

5: ((4*4)+4 )/ 4


Usually in high school kids started off confused but one thing I might do was show multiple ways to make 1 to get them started, and have it on posters around the room. Once one group got it others would go and ask how they did, which would spurn new ideas. I’d say only work on a number for 2 minutes and the goal was to get to 10 in the class period. Then I’d say for homework try to get 10 more number of any number. Often kids would come back with up to 100 almost filled in and really excited to be working on math in an open ended context.

My first year teaching I was watching my car wiper blades clean the windshield during a rain and figured that would be an interesting task, so bought several different windshield blades from Autozone across the street from McLane and brought them to class. It’s no understatement that being exposed to Dan Meyer and #MTBOS in 2011 or so while at the same time moving to a new grade level in middle school was revolutionary to my teaching. Moving to a new grade gave me a new reason to dive into the content and think of creative ways to teach it because there was no, “we’ve always done it this way.” I even used to write my own three act tasks which often weren’t that great or adapted ideas from memories of previous tasks.

So as school starts up, focus on relationships with your students. Focus on engaging them with rich content/low floor high ceiling math tasks. Affirm their struggles and point them towards a growth mindset. Don’t give pointless assignments. Use testing as a means to improvement.

Be inspired by your students.

ACT Leadership Cohort Trip 3 (Iowa City, Iowa)

This third leadership cohort trip was the best professional learning experience I have ever had. Considering I was in education for 9 years and have been leading learning experiences as well as attending conferences for the last three years at least, that’s a tall statement!

It wasn’t just the way the content was presented (a mix of discussion, lecture, panel, experiential, small group, active movements), but the content itself was deep, organized and immediately able to be applied to my day to day work as well as a resource for the future.

A day by day summary would be boring to read, but what I got out of it most was the focus on product development. Coming from education I have realized I am sorely lacking in many concepts of business and as I came to OpenEd later, much of the planning and conceptual thinking behind the product had either already been done, or as Content I wasn’t privy too the leadership brainstorming.


This graphic in particular(source unknown) was encouraging to see! As I’ve worked on OpenSALT it’s been eye-opening to see how much leadership thinks something will take when in reality it may take a lot longer (or shorter!).


The best part though by far was hearing about how ACT’s “horizon 3” division thinks up ideas and how to make them into actual products. The process is rigorous but designed to allow for numerous failures. Since this is a public blog I shouldn’t reveal the details, but let’s just say a lot of ideas come up and most of them don’t make it past stage 1. The ones that do go to another team for review and eventually they are turned into actual products. We had a chance to take ideas of our own and go through the same process in one day (normally the initial idea staging takes three days, as the presenters were keen to mention…). It was intense and full of great debate back and forth, but eventually we came up with a product that we think would serve a need based on market research. The next time we meet in October we’ll dive deeper into the product and see if it will actually be turned into something that the market needs and ACT and provide.

In my own career, I’ve gone from being a teacher and not thinking about products or the implications thereof to being deeply involved in making OpenEd a successful product that helps students and teachers around the glove -over half a million teachers and students have signed up and used the site over the years… and that number is only going to grow as we become the official remediation site of the ACT test.

I was immediately take away some of these ideas and use them with developing OpenSALT.

Note about #MTBOS

Dan Meyer isn’t trying to take away #MTBOS.

Rather, I get the sense from Twitter and his blog that he’s just suggesting the acronym-laden hashtag is a bit of a barrier to helping new teachers (or old teachers, new to twitter!) find resources they need.

My experience with the MTBOS started around 2010/11 after seeing Dan’s Ted Talk and starting to go to more tech-related conferences. I simply didn’t know before and felt all alone in trying to find more tech, inquiry based methods to teach my students even though I was part of a big school district in California.

There’s a bit of exclusivity, and there are people that are famous on MTBOS and perhaps not elsewhere in the math world… but those ‘celebrities’ are famous because they’ve contributed tangible content and strategies. (Shout out to Kaplinsky, Meyer, Stevens, Nguyen, Campos among many others.) MTBOS is more than just a hashtag with people spouting philosophical teaching adages… for the most part it’s sitll a community of math loving people talking about how they’d teach a concept, discussions of their understanding of concepts, and reflection on the entire spectrum of the math education world.

Since I left the classroom a few years ago, I’ve found Twitter and #MTBOS even more valuable. While I’m still helping math (and other) educators, my primary passion is still enabling high quality math education resources for all students.

No one likes not being a part of the ‘cool club’ and MTBOS is definitely that at math conferences. I feel like opening it up to something more generic would help.

But people want more participation, not just to, “get new members.” I blog regularly because of MTBOS even if not all of my blog posts are math related. I love talking with new teachers – I taught for 8 years… not forever but enough to be considered a veteran and can help newer teachers. If nothing else this debate has reminded me to regularly browse the hashtag and try to help!

Sugar Pine message of encouragement

I was given the opportunity last week to talk to Sugar Pine staff as part of their weekly devotional time, and chose to talk about FOCUS.

When I worked at Sugar Pine in 2002 and 2003, leadership awarded bricks based on character traits at the end of one of the summers.  I was awarded “focus,” because I was almost always focused on the kids. Thus, I used that word as an acronym for the message I wanted to deliver. I also included usually a biblical application or humorous/serious story for each of the themes to tie it all together and be somewhat more engaging than a 33 year old bald guy talking to 18 year olds.

Forgiveness: I once had a camper who was very, VERY excited to be at camp. He was so excited in fact he grabbed someone’s bug spray and sprayed it around the cabin like champagne after the world series. Unfortunately a kid next to him also got a mouthful. As he frantically indicated he could not breathe, I ran up the hill to the restroom and nurse’s station to wash out his mouth. The student said he forgave the “celebratory” kid and also thanked, “Brandon from saving my life.” The two had a moment of saying sorry and they actually became friends going through the scary moment together. While I have never sprayed bug spray intentionally in someones mouth, I have been involved in conflict with others. Often difficult relationships and situations end up forging the closest of bonds. We have to let down our guard and make forgiveness be the mantra not bitterness.

Outward : There’s an interesting story in Exodus 33 where Moses is asking God how others will know that God is with them. God says basically you’ll know because “I am God.” Well, Moses wants more and challenges with “Show me your glory.” Moses sees the end of God’s glory and his face shone like the sun to the people. I tried to tell them how so often working at Camp is a chance to be God’s love to people. To be drama-free, to smile, to take advantage of the small acts of kindness that can make a persons day and let them know they are loved.

Complacency in our jobs, our relationships, our lives. As camp staff, we could be interacting with students who didn’t want to go to camp but were forced too, we could be interacting with kids that have gone to church all of their lives but have never really known what it means to be a Christian… we don’t know and we shouldn’t act differently. Every day we should be treating kids as if it’s their only chance the whole summer to meet people who are filled with God’s spirit of love, peace, patience, kindness, gentleness…

Undeniable: gods role in our lives and the lives of our campers. Camp is great but not always real life. It’s easy to get distracted from what God is doing in our lives – whether we think it’s ourselves or other factors. But one thing I have always found is much like Elijah looking for God in the earthquake, then the fire – He is often found in a whisper (1 Kings 19). We need to make time to be still with no agenda.

Sarcasm kills relationships. This was and still is a bit of a catchphrase at camp! Yet over my years of working with people in close environments, so true! Especially for a guy like myself who is pretty serious most of the time, when I am trying to joke or be sarcastic – people don’t get it or outwardly show they see the truth in my statement. It’s not a good place to be – I’d rather just be truthful and not resort to aggressiveness hidden behind cleverness.

Thanks Chrissy for letting me come talk to what seemed like a great group of camp staff!

Two years of EdTech!

OpenEdACT logo

2 years full time at OpenEd have passed and I thought I’d write a bit about what it’s like to go from being a public school teacher to working in education technology, or EdTech as it’s commonly called! It’s been three years since I was a classroom teacher – so if I did go back, it would probably take a year to re-learn everything I was doing before!

I had been consulting for OpenEd for over a year before I came on full time, writing math assessment questions and curating resources. I had visited the office a few times so sort of knew what the office environment was like, and was definitely tempted by the idea of going for a run on my lunch break in beautiful Los Gatos!

The financial implications were perhaps less known than I would have liked- salary adjustment pretty well made up for the 3x different in rent, but what we didn’t take into account was the higher price of food and healthcare. On the other hand, even after we moved offices to 8 miles away I can still ride my bike to work and even in bad traffic it’s usually no more than a 20-30 minute drive to the office.

The hardest part to adjust to has been the lack of seasons of work. In teaching it’s easier to push through certain rough patches because Christmas Break is coming, or Spring break, etc. Of course summer was always great – even when teaching summer school, it was still better than a full day’s work. Now when I take a vacation it usually just means I have more work and less time to do things in and as a manager there are always issues that I’m thinking about to squash before they pop up as problems.

The advantage to the office lifestyle though is being able to go on a coffee walk with colleagues, talk to adults (sometimes a negative) instead of teenagers all day, etc.  In addition, we’re not as susceptible to faculty meetings, fire drills, no dances to supervise, etc. Distractions happen but can be managed by simply asking the person to not bother you or moving to a corner of the office away from other people.

The absolute best parts of working in edtech has been the learning! I started making a list of all of the things I’ve learned over the past two years and aside from being pretty boring (Github, some actual programming, technical standards, all about the Common Core and other taxonomies, etc). To not just be user of edtech but a creator and facilitator of it has been an amazing thing. It’s still cool to get emails from people that work for the likes of Google, Microsoft, Canvas, etc .I have always been pretty adept at computers, but came to this job and felt like I had to start all over in my knowledge!

This job at OpenEd/ACT truly engages me on a level of content, technical, and leadership development. When I started, I was simply the primary Math Curator and led the rest of the Curation team – those who are actively curating the videos, games, assessments and interactive resources on OpenEd.com so students can use them more easily. Now I do more leadership, less math, and more technical. I write dataclips in SQL to accurately make reports of the resources that OpenEd has for various partners and internal goal-setting. I use Geckoboard to track our goals and hold the team (and myself) accountable. I know Github well enough to be able to make my own branches of OpenSALT and merge it back into the development branch when warranted. This year I’ve been privileged to be a part of the ACT Leadership Cohort, even though it means time away from my beautiful family.

When OpenEd was purchased by ACT last year, at first I was nervous. Being purchased by a summative test company seemed like the worst thing possible for a student-centric educator like myself. Yet as I have gotten to know the people and company of ACT, I’ve been very impressed. They are professionals, driven towards creating a placement test that will help students know what they are capable of, and the Work Keys program has lofty goals as well to help give employees and employers knowledge about skills and capabilities. I am greatly excited about work I can make public soon around taxonomies of academic skills.

I couldn’t be more proud to be a part of ACT and excited for what OpenEd continues to do and expand. I have never met someone from another company who wasn’t in this business to make things better for students and teachers – even Pearson!