Making Sense of 5 x 3/8

I wrote this blog post last year and somehow never finished it:

My youngest son is in the fourth grade.  He is a good student and loves math, but I wasn’t sure if he was truly understanding fractions.  I decided to ask him how he would solve 5 x 3/8.  He started by trying to use an algorithm (which he did not understand) but luckily saw the mistake and was able to reason about the answer to determine that the algorithm did not produce a reasonable answer (The entire video is at the bottom of this post).

After he was unsuccessful with the algorithm he just looked at me and said,  “but 9 is more than 5” with a puzzled look on his face.  He knew something went wrong but he wasn’t sure how to figure out what.  I asked him if there was a different way he could represent the problem to make sense of it and he said he could draw a picture or a diagram.  I said, okay what would that look like.  He struggled a little with trying to visual 3/8 so I asked him what 1/8 would look like.  He went to a number line.  However, when he labeled his number line, this is how he labeled it:

First attempt

I asked him to show me where 1 was and he quickly erased a line and was able to tell me that one was the same as 8/8.  I asked him to show me where zero was and he corrected the number line to look like this:

Second attempt

He then was able to plot 3/8 on the number line like this:


When I asked him what two 3/8 would look like, he was stumped.  I said, where would you put the dot to show another 3/8?  He was able to produce this number line and we discussed that this was two groups of 3/8.  He proceeded to extend the number line to show five groups of 3/8 which landed him at 15/8.

2 groups of 38

I asked him if he had a way he could teach this to someone else who didn’t know how to complete this problem and he produced this scenario and drawing:

Five times 38

Skylor: “There was a chocolate chip cookie and eight children.  They only had one chocolate chip cookie and they all wanted it and they needed an equal share.  So they broke the cookie into eight pieces and then let’s say five of the kids wanted to save theirs for later because they had just had food and they weren’t that hungry. 

So only three children had their cookies at that time.  And then I will shade in the cookie pieces that those children ate. Those pieces would be ⅜ of the cookie. But if there were five cookies, then you would take this and there would be five of these (five cookies all split up so the children could have their pieces of the cookies).  

Five of these ⅜ would be ate at one time. So that would be…let me just draw that…so that is five split up cookies. And then three children had their share of the cookies (that represents ⅜ of the one cookie). There was five cookies so then there was that much. These were the pieces that were ate, but the total of those pieces that were ate were 3, 6, 9, 12, and 15.  

Me: “So 5 x ⅜ equals…”

Skylor: “So five times ⅜ equals fifteen eighths.”

We know how important it is to allow students to make sense of a mathematics, yet many times we feel rushed to teach.  If I would have tried to “teach” Skylor the algorithm for this problem, would he have been able to internalize the concept?

If I would have switched to teaching mode and given him steps to complete or a specific diagram to draw (my diagram) what might have happened?

Will Skylor be able to make sense of the algorithm now?

As a Math Recovery Intervention Specialist, one of the most important things I learned was the importance of wait time and the practice of questioning instead of telling.  When we stop leading students thinking and start allowing them to make sense of their own learning (MP1) it is amazing what they can accomplish.

3 Act: Dart Throw

One of my youngest son’s favorite party games is the balloon dart throw.  Each year for his birthday party I come up with some way to incorporate the game.  I fill them with party favors such as mini lego sets or minifigs and his friends get to keep the prize and put it together when they pop them.

When I was brainstorming next year’s theme, I realized this would make a great task!  I have filmed several of these for different grade levels.  This is a great one for 1.OA.A.1, 1.OA.B.4, 1.OA.C.5 for adding within 20 and missing addends.  It also could be used for the second grade fluency standard, 2.OA.B.2.

Act 1

Possible Question: How many balloons were popped?

Act 2

12 total balloons

I threw 10 darts and missed three times.  On my ninth throw I popped two balloons with one dart.

OR less challenging: there are 4 balloons left.

Act 3

3 Act: He Loves Me…He Loves Me Not…

It’s time to stop and smell the flowers!  This post is dedicated to Math Practice Standards 1-3! Of course, it does hit some content standards (as well as more practice standards), but let’s focus on constructing viable arguments with this one…just for fun!

Hopefully this light-hearted throw back video won’t offend anyone as I had so much fun reminiscing about all the flowers my friends and I went through as kids playing this game!

Act 1:

Possible Question:  What will it end on?  He loves me or he loves me not?

Act 2:

35 petals

Starts with “He loves me…”

Act 3:

I’d love to hear strategies used by students to figure this out so feel free to tweet them or comment here!  Have fun and take time to stop and smell the flowers!

Scaffolding Instruction Through the CRA Approach

When I was an instructional Numeracy Coach, one of the common trends I noticed was that many teachers were not providing concrete experiences for students to learn mathematics and were relying heavily, if not solely on abstract teaching methods.  Although our district curriculum resource came with tubs of manipulatives for students to use, many teachers had never even opened the packages and they sat collecting dust in cabinets and book rooms.  This led me to reflect on why that might be and to do some digging to determine how to get these resources in the hands of students.

It occurred to me that many teachers simply didn’t know how to use the manipulatives themselves to scaffold learning and so they simply taught the way they knew how.  Another reason was that they felt it took too long to get out the manipulatives and put them away so they opted for less mess and management.  Both of these were understandable reasons and as an educator, I knew that I had to provide experiences for teachers to see the value in these powerful tools.  I was on a mission to model how to use  manipulatives when I went into rooms for coaching, but knew I wanted to provide them a resource to refer back to in the end.   I created several tools such as resource sheets for tools for mathematics and scaffolding documents for addition and subtraction and multiplication and division, but these tools never got to the heart of what I wanted to provide for teachers.

It wasn’t until I met with Jennifer Ritter at Republic Public Schools that I realized how I would begin organizing these resources for teachers.  Jennifer shared a trouble-shooting document that some of her teachers and specialists had begun for a third grade standard organized with the CRA method of number acquisition; Concrete ->Representational ->Abstract.  I loved the idea of organizing standards this way and have been pondering for months how to best organize the examples into a resource for teachers.

See below for a modified version of what Jennifer and her team started.  I hope to continue to collaborate with them to finish this project, but would love to open this project up to all interested in contributing.

This project is going to take time and lots of it.  The videos will be linked over time, but I plan to start by creating ideas for each area at each grade level.

The process will look something like this:

  1. Create a Google Slideshow for each grade level that includes each standard.
  2. The standard will have the unwrapped expectations for the Missouri Learning Standard as well as a table with Concrete, Representational, Abstract columns.
  3. Under each column will be ideas for scaffolding students using manipulatives and diagrams.  For now I will be simply putting the unwrapped standard under the abstract column, but I will eventually replace this with stems for the grade-levels that do not have those provided in our Item Specifications for Missouri.

Please feel free to offer suggestions and revisions if you use these documents by emailing me or replying below so that we can make these valuable resources for teachers.

Here is an example of the first grade resource I have started for our teachers:

The great thing about Google Slides is that there is an option to export individual slides as .png images or a .pdf so I can link them directly to our other standards documents.

Please realize that this is a work in progress and will be refined over time.  I wanted to get this out there though in case anyone was looking to create a similar resource.

Feel free to contact me via email at or on twitter to @MsCastillosMath.

Doodling the C’s – Getting Started

This is so awesome!

Experiments in Learning by Doing

How do we practice Information Age skills?  Which of the C’s do we actively engage with, share in the-struggle-to-learn with others, and intentionally insert into daily practice?

Creativity and innovationCommunicationCritical thinking and problem solvingCollaboration, …

At Trinity, a small cohort of faculty meet at either 7:15 a.m. or 3:30 p.m. to learn more about sketch noting.  We call it #doodling #TedTalkTuesday (or #TEDTalkThursday).  We meet, watch a TED talk, and doodle.  We share our work and offer each other feedback.

But, how do we differentiate for faculty unavailable at these times? In other words, how can we leverage technology to learn and share together?

Challenged by members of the Trinity Faculty to exercise creativity and critical problem solving,  I have started developing the following prototype to attempt to offer a solution to this identified need.

Screen Shot 2014-11-02 at 2.58.06 PM

At the end of these eight 75-minute sessions, participants should…

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How Do I Change Math Class Tomorrow?


As Dan Meyer put it, “math class needs a makeover.”  It breaks my heart to hear students say things like, “I’m not good at math” or “I hate math!”

In my opinion, one of the first things that needs to change is the focus on correct answers and the need for speed.  This leads students to believe that the only way to be good at math is to be right (quickly).  The other practice that needs to be eliminated is the language we use as teachers such as “that’s right” or “you’re so smart.”  

You’re So Smart

You’re so smart is one of the most damaging things you can say to a child.  What is smart?  What do we value and what are we showing them that we value?  Being smart needs to mean learning as a means in itself.  Or rather, the journey of learning.  This is not…

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Rethinking Geometry Instruction: How to Think Holistically About Standards

Each year I continue to see teachers frustrated when they get their middle of year testing results back for their class because Geometry is often one of the lowest strands.  Over the years I have puzzled about why this is but have realized that most textbooks (ours included) waits to teach Geometry until the last couple of units.  So many teachers who rely heavily on the textbook do not teach Geometry until right before the state test.  Often this means students get told about geometry instead of getting the opportunity to experience it through hands on application and discussion.

So why do we wait until the last minute to teach geometry?

I would like to propose that we teach geometry throughout the year by incorporating geometric ideas into instruction of other standards.  I would argue that too often we teach skills in isolation and don’t connect the mathematical ideas.

In Kindergarten, instead of giving students random objects, have them count shapes such as triangles, squares, rectangles, circles and as they develop the idea of cardinality have them confirm their count by labeling the objects.  For instance, if they count nine triangles, they would say “there are 9 triangles.”

Provide multiple types of triangles so that they can develop the idea that triangles come in all sizes and orientations and that counting does not have to be an object of a particular size or congruency.

Put out squares and ask students to get 18 squares or lay out circles and say how many circles are there?
Give students a geoboard and ask them to make a shape with 6 sides.  This is a great conversation starter and gives another context for counting while offering the opportunity to discuss shape names and properties.
When asking students to compare objects, ask them to compare a group of circles and a group of squares and ask which one has more?
Or provide a low floor, high ceiling task like this:
I have some shapes.  The total number of sides of all of my shapes is 21.  What shapes do I have?
Build arrays of multiple shapes and then ask how many sides are in the array? How many corners (or vertices).  Or build an array out of 3-D objects and ask how many faces are in the array.
Use the geometric subitizing cards by Graham Fletcher to have number talks about attributes.

Have students engage in conversation to solidify understanding and practice vocabulary with WODB sets like this one:
Look at your grade level standards for geometry.  Are there opportunities to teach geometric concepts through operations and algebraic reasoning?  Data?  Fractions?
Leave comments here or on twitter to share your ideas!