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!

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.

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!

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.


“How does it feel to be white?”

In the past few months several leading math organizations (NCTM, NCSM, CMC) have released joint statements talking about the conversations of Math Equity.

From the NCTM paper, one quote stuck out at me regarding teacher education perspectives:

Providing all students with access is not enough; educators must have the knowledge, skills, and disposition necessary to support effective, equitable mathematics teaching and learning.
In other words, while I suppose you could have students read Flatland and then connect that to social injustice etc, that’s not the point here. In December Dan Meyer wrote about the problem of the proliferation of tall good looking white guys at education (of which I don’t think I fit into two of those descriptors, but close enough).

In college a friend of my roommates came into our dorm and casually asked, “How does it feel to be white?” when he saw my computer set up (nice looking case, big monitor), and I didn’t quite know how to react. I was taken aback –  I tried to justify his comment in my head – if I’d bought it new I can see that – but I hadn’t. I’d worked extra money, made a bags of skittles last a week just to save the extra buck and things like that for years. It took me a while to fully understand what he meant. After all, I’d worked really hard to buy that computer more than just financially – but hours learning about Linux, about hardware and how to best optimize things.

I cared for my computer almost much and probably more than my car. My parents told me in 8th grade that if I saved up at least $1,000 for a computer, they would match it. They thought this would take a few years as my allowance at the time was I think $20 a month for snacks and small trips – I had $130 in ‘savings’ at the time I remember. That summer and all throughout my freshman year I took extra small jobs whenever I could, even taking over my brothers chores to double the amount of income I could make. So by my sophomore year I had the money and carefully went about choosing what I wanted. I settled on an AMD-based Gateway system with all the trimmings. This computer would last me about 6 years through upgrading everything except for the case. I added a 17″ flatscreen monitor my sophomore year of college which was about $250 but looked more expensive.

What I eventually realized about his comment was that it wasn’t the amount of money having a nice computer took, it was the priorities in my life that let me spend money on that. It was the fact that because my parents were able to provide for my basic needs so something like a computer – which at the time wasn’t really needed for any job and something as nice as that wasn’t needed for school per se. It was that I’d chosen to spend that money knowing that I’d be able to get more money later. And when I was in middle school tinkering with spare parts and putting them together, having a dad that could help explain or point me to the right places, and even drive me to another city to get the needed parts (yes, this was in the days before amazon and ebay).

As a math teacher who taught predominately in lower-income areas, I couldn’t pretend to know exactly what kids were going through, or experienced, or even what daily life was like. I’ve never struggled with not having enough money to buy food or at least couldn’t put it on a credit card if I needed too (been there in my early teaching days!). But I could listen with empathy, keep in mind that their parents may not be able to help them, and give students opportunities. Through Tri-This! Inc I was able to help take kids to the snow often for the first (only?) time, go camping, travel up and down California and complete triathlons. Through math I was able to explain things to them and encourage them to college – several of my students even ended up at Fresno Pacific University my alma mater!

Being white is not a negative thing – it’s who we are. Because we are born into white privilege – and we are – doesn’t mean we can’t be that much more compassionate and strive for empathy. I cannot be the same type of figure in students lives that my sisters and brothers of color can be, but I can just be who I am – a mentor who strives for compassion, integrity and shows students unconditional acceptance and love.


I’ve been writing this post off and on since about November 2016, and I’m still not sure what I’m trying to say I guess. Justification for me to be slightly offended at the comment? Guilt or embarrassment about working hard for it? Not sure. What I do know is that little comment 15 years ago helped give me perspective whenever I did have physical possessions that were important to me, but not important to other people for very good reasons.

#TT4T Book Study – Turn Weakness Into Strength

Note: This is an ongoing book study started by Chase Orton about Tim Ferris' Tools of Titans. Intro Post

I was reading this week about how different people have taken what were considered weaknesses and turned them into their greatest strengths. Take Arnold Schwarzenegger talking about his heavy accent even after being in the United States for several years:

Arnold was able to use his biggest “flaws” as his biggest assets, in part because he could bide his time and didn’t have to rush to make rent. He shared an illustrative anecdote from the Terminator set: “Jim Cameron said if we wouldn’t have had Schwarzenegger, then we couldn’t have done the movie, because only he sounded like a machine.”

He took what critics told him was preventing him from further roles and made it something that became his trademark.

As teachers we often face the same thing. I’ve read books (can’t remember from where!) where they talk about some teachers on teams maybe aren’t great teachers, so make up for it by being really nice people. Or maybe the teacher who is always involved in fundraising and supervising sports never comes to the professional learning community collaboration time. For myself personally, I found that while I understood the math well, I wasn’t very good at things like recognizing kids birthdays, or even sometimes recognizing when discipline wasn’t quite what I needed it to be in a classroom. In other words, I was so inside my head about the lesson I would miss the bigger story of what was going on in my classroom.

This flaw was pointed out to me by the principal who became my friend that shaped my teaching career the most, Jeremy Ward at Computech Middle School. He had a good way through feedback in Google Docs and in person about being honest about what he saw. Organization was the root cause of my struggles – often I’d be fumbling from class to class for markers or materials that I would miss the cues from kids coming into class.  So over time I was able to become a minimalist teacher, and with less, “stuff,” I became more effective.  Likewise, I was able to point this out to both students and other teachers who may have struggled with being able to focus on the task at hand.

What are your weaknesses that you have turned into strengths?

#TT4T – The Damage Done In Not Waiting

I’m reading Tools of Titans: The Tactics, Routines, and Habits of Billionaires, Icons, and World-Class Performers after a gentle nudge from Chase Orton, whom I’ve gotten to know through CMC conferences recently.

There is a part in the first part of the book where it’s talking about learnings from Siddhartta Buddha. A merchant is asking what Siddhartha can give him if he can’t give him possessions. A short portion of the exchange is follows:

Merchant: “Very well, and what can you give? What have you learned that you can give?”

Siddhartha: “I can think, I can wait, I can fast.”

He can’t give money, he can’t give things. But he goes on to explain a bit – if he doesn’t have food, then he can fast.

Let’s take the same ideas and apply it to teaching mathematics:

I can think (Math Practices 7,8)

Tim Ferris extrapolates further that because he can think, he can make good decisions.

We can teach kids how to memorize things, or we can teach them the why behind the algorithms. We can give them a goal without support, or we can teach them a system of how to study and achieve in education and in life. Specifically I’m thinking about being able to give our students the foundational skills needed to really engage deeply in DOK 3 level problems – we know that we can’t immediately engage kids with a DOK 4. I once saw a chart from an administrator who thought the DOK levels were like a ladder, and that the end of every quarter should automatically see DOK 4 level problems… a blatant misrepresentation of the paradigm

I can wait (Math Practices 1, 3)

In the book, Tim expands – because he can wait, he can play the long-term game and not make short-term bad decisions.

This idea of waiting for students to understand things hit me pretty hard. When I moved from high school to middle school, I noticed that I was quicker to help the middle school students – probably because I felt they needed it. This was incorrect! I only discovered this fact when I recorded myself teaching over the course of several days and then watched the recordings on fast-forward to be able to spot trends.

I didn’t wait for kids to answer incorrectly or not. I was not giving kids a chance to struggle. This affects equity and as is talked about over and over again in Mathematical Mindsets, my classroom was not a safe place to learn by making mistakes and then being able to apply that knowledge in a new context. I made immediate changes to pre-write questions I knew I wanted to ask and adapt those questions from class to class as needed.

I can fast (Mathematical Practice 1)

Obviously we aren’t going to ask students to not eat here. But we can stop, “spoon-feeding,” them answers (see what I did there?!). Too often I would catch myself asking leading questions without even realizing it – but why should I be asking guiding questions at all when students should have the tools to self-diagnose.

What to do about it

Some ideas I had while writing this post:

  1. Use puzzles during warmup to help remind students that just as they can find different ways to solve a puzzle, they can find different ways to solve a problem.
  2. Have students explain their steps out loud to another student or explain it back to me using an app such as Recap. I would also often have students in a group be recording into an audio device and then the next day, have them play back parts of the audio to be able to hear themselves solving problems – their own insights to their thinking process was often quite interesting!
  3. Strategies/activities I have used to stimulate students perseverance include the Four Four’s activity, Grazing Goat, and having students find multiple ways of solving the same linear equation. When appropriate, three act tasks are also great to see results!

What tools/tasks do you use in the context of “Think, Wait, Fast”?


I’ve been having fun lately playing with puzzle games on my phone as I rock my daughter(s) to sleep. One is called Roll the Ball and another one introduced to me by Daniel Rocha is Flow Free.

I used to read books on kindle but sometimes it’s nice to have a little more stimulation. What I like about both of these games too is they offer a variety of ways to play the game.

Roll the Ball

My favorite mode here is when they will put stars in different parts of the tubes. It’s a fun way to engage my brain in thinking about different patterns and ways to solve things with or without a time or star challenge.

Flow Free

Flow Free is Roll the Ball’s bright, easier-to-play and faster brother. You start with dots throughout the screen and create paths to the other dot of the same color. I love it because it allows you to easily overwrite past paths. Daniel Rocha, whom I’ve only really met on Facebook/Twitter and we have a fair number of mutual friends – posted a great video of his kid playing and the conversations they were able to have about the experience of solving the paths. Hoping he’ll blog about that… my two year old was not interested 🙂

I would love starting off class every so often with these kinds of puzzles to get the students minds thinking, have some fun and of course occasionally remind them that solving a math problem can be very similar – we may take different ways to get to the answer but that doesn’t mean it’s invalid!

AIMS Center Colloquium Series Video on Puzzles as Models:

Puzzles as Models of Thinking from AIMS Center for Math and Science on Vimeo.

CMC North Recap

This past weekend at CMC North I was fortunate to not only be very engaged in social media work with CMC, but present as well! It was my first time presenting at CMC North so I was very excited. Specifically, my talk was on Math Practice 3- Construct viable arguments and critique the reasoning of others. I wanted my talk to start off with showing teachers how to get information into their kids in equitable ways so that the students would have the knowledge needed to construct said arguments – and then I would give them tools to create those arguments.

Things started off well enough but once I started trying to show some of the internet tools, too many people were online and the internet shut off. I lost about 1/4 of the audience due to that, which was understandable – next time I’ll make short screencasts of each tool in order to carry on with the presentation. It’s funny – I used to do that before presentations but didn’t this time because, “it’s never actually happened.”

However I wanted to take a few minutes of your time here to highlight some of the tools that people loved and why I think they’d be great for any classroom:

OpenEd: Formative assessment tool that provides targeted feedback and learning resources based on questions missed. Aside from the fact that it’s my day job, it’s nice to be reminded what a cool thing we have going. We have learning resources (videos, games) aligned to standards of all types (CCSS Math, ELA, NGSS, Social Studies, and fine-grained standards as well now that will be public soon.) that actually get judged over time based on how well students do on an assessment after interacting with the resource. For example, a kid watches a Khan academy video on adding unlike fractions, then takes and assessment on that standard). If they get 100%, the efficacy of the video increases. If they get a 0, it decreases. This leads to a library of resources that intentionally isn’t the largest out there, but is curated and we think is the best. Plus, our formative assessments do more than just test, they help teach.

H5p: I’ve written about H5p before, but it’s an interesting suite of assessment and interactive tools written in HTML5 so thus platform-agnostic. The tools I was showing off this weekend were the text tools that let students fill in the blanks – great for developing sentence frames for geometry examples, proofs or other areas where helping scaffold student understanding can come from text-based. Putting an image up on my screen for example and then allowing students to discuss/define the attributes of that shape or even move further along in a proof. Another great way that I’ve used tools like it before would be for the Interactive Video type – have students create a screencast describing how they solved a problem etc, and insert questions addressed to other students to help them along as well. I would grade the students questions ask to their peers as well as the content itself – we can often learn much more about what a student understands by the questions they ask then the answers they give.

Recap: When I was doing work with Fresno Unified’s Teaching Channel initiative, I became a big fan of the Swivl Company for their hardware and software to make capturing teachers teaching easier. So it’s only natural that their student-focused software is interesting as well. Essentially it makes it easy for teachers to ask questions, students to respond via a 30 second video, and the teacher can then comment further if desired. I haven’t had an actual class to try it in is the problem, but my students (who are teachers) at Fresno Pacific have loved it as I’ve introduced it to them. What i think I like most is the capturing of student voice when possible – no barriers really to get kids at least talking and then the mathematics discussion will come. If someone from Recap reads this – please reach out about an API agreement!

We had a great time of practice – some teachers already made sample H5p elements for their classrooms for the following week! My reviews pointed out some flaws in how I presented – I should have had portions pre-recorded for example so if the internet went out I’d have a backup – I was happy that people were honest as well. What I want to do next year is come up with a new presentation for next year that is focused on one particular math problem/three act task and then shows a variety of methods of assessment around it!