Desmos Graphing Exploration

Graphing Exploration 1

Graphing Exploration 1

I have been reading many blogs lately, some of which have lead me to Benezet’s experiment,  A Mathematician’s Lament, and reminded me that “Perplexity is the goal of engagement.”  As a result, I have adjusted how I am approaching the super-speedy-but-in-depth review of Algebra One I am doing this summer for one of my tutoring students.  I know, that sounds like an oxymoron and it definitely is.  I had planned a lovely, and still too short, 10 week course based on one of Geoff Krall’s Problem Based Curriculum Maps when I found out my student (I’ll call him Josh) was going to be gone much of the summer at various camps and on vacation.


I’m not sure how quickly Josh and I will cover the material this summer. He’s very bright, and with only one student I should be able to proceed much faster, while still achieving depth of understanding.  On the other hand, he also has many holes in his math fundamentals, which I deal with as they come up and which take time.  Still, I’m optimistic we’ll make it through enough of my planned curriculum that he’ll be reasonably well prepared for next year.  Also, I know Josh’s Geometry teacher reviewed selected algebra topics during the last month of class, and his Algebra II teacher will review concepts as well, so I’m not too panicked.  (Funny.  I can no longer think about Algebra II and Geometry without a wry smile.  If you aren’t also wryly smiling, read A Mathematician’s Lament – be sure to read all the way to page 25.  It’s worth it, and a little depressing at the same time.)

What to do?  My current approach to solving the speedy-but-in-depth challenge is to begin by helping Josh obtain a solid (and dare I hope, intuitive?) grasp of functions and relations, using the lessons outlined in the Algebra I curriculum map.  I also hope to finish the initial linear and quadratic functions sections and still have a couple of days to look over the Algebra II textbook so I can pre-teach some of the concepts.  I know, I know; summer is shorter than I think.

But back to the graphing exploration.  As part of my recent reading frenzy, I came across Fawn Nguyen’s cool Des-man lesson, in which she very creatively uses the Desmos online graphing calculator.  (So fun!)  I plan to work Fawn’s Des-man lesson in later, but for now I created a simple graphing exploration activity I will use to introduce Josh to the unit on functions.  Although my activity doesn’t start with a problem per se, I hope to get him intrigued (if not perplexed) about what makes linear, quadratic, and other functions change before we start a more in-depth conversation.

The activity starts with two graphing explorations, one for a series of linear equations and one for quadratics, and then has a blank template that can be used for as many student explorations as desirable.  In the first two explorations I plan to teach Josh how to use Desmos to enter equations, add a slider for variables, and record observations about the graph of each equation. (Okay, after about 3 minutes he will probably be teaching me how to use Desmos, but I have to start somewhere!)  At the bottom of each page are questions intended to provoke thoughtful reflection on his part (I can dream, can’t I?), or at least remind me of the questions I wanted to ask.

It’s easy to see when Josh is intrigued and engaged (or not), so I will get immediate feedback from a student perspective.  I’ll let you know how it goes.  Meanwhile, if you have feedback, comments, or questions, please leave a comment.  Thanks!

Assessments Update

Demise of Soccer Balls

Okay, this is becoming a little daunting.  I knew the fun pictures of soccer balls and ice cream cones I used for the Kindergarten assessment were going to be replaced by more number-dense problems, but I’ve just finished the third grade Operations and Algebraic Thinking domain and I already have 29 problems.  Yikes!  My philosophy of including each word problem variation may not survive third grade.  But how do I choose?  Will it be the Goldilocks solution: one easy, one hard, and one “just right?”  Or is there a better way?

I just looked back at my 3rd grade assessment to see how many word problems I have and noticed  many of my other problems have multiple parts.  For example:  Write a related multiplication fact for each division expression has 4 division expressions to solve.  So, my assessment is really much longer than 29 problems.  (My count: 11 word problems for 3.OA.3 and 7 for 3.OA.8.)

Oh, yes, and looking back at my notes from a month ago before I broke my foot (but that’s another story), I had already eliminated some of the word problem variations.  Here’s what I wrote: “Naïve to think I could do them all… hard and easy array problems, array versus area problems, measurement versus non-measurement, [and] compare problems.”  And that’s not even changing the location of the unknown in the problems, which makes a huge difference in complexity.


I think I will continue to create the assessments as a complete mapping of the CCSS-M (including all word problem variations) because it is a way for me to more deeply understand the standards.  However, it’s clear my original (and, apparently, naive) idea of creating a series of assessments that would illuminate concepts not mastered in previous math classes, and that would be practical to give students, is in need of some rethinking.

Dinosaur Math

Because I’m home with a broken foot, my daughter happened to overhear part of my latest tutoring session.  What she heard was my student abruptly asking, “Do you know how long ago dinosaurs became extinct?”  (This type of comment is very common during our sessions.)

Later, my daughter, who is 15, told me she was amazed I didn’t immediately shut down the dinosaur conversation.  I explained to her that listening to this student is an important part of teaching him.  Over several months we have built a relationship; we listen to each other.  As a result, I can challenge him to change how he solves problems, or to try something new.  It works.

It was only later I remembered another reason I listen (at least for a while) to these apparently random, off-topic utterances.  It’s because they frequently spring from a math-related source.  Take the dinosaur comment.  He wasn’t asking because he didn’t know how long ago dinosaurs became extinct.  Rather, he wanted to know if I knew because something about it was bugging him.  He wondered:  If 200 years ago scientists discovered that – 65 million years earlier – dinosaurs had become extinct, then why don’t we now say dinosaurs have been extinct for 65 million two hundred years?  (I chose to ignore the when-did-scientists-know-it question and instead focused on the concept of rounding.)

I used a whiteboard and wrote the numbers 65,000,000 and 65,000,100.  (Okay, I finessed the date a little and said, let’s just use 100 years.)  We talked about rounding and the fact that it would take a really long time before it would make sense to say anything other than dinosaurs became extinct 65 million years ago.  He seemed content and I was happy that I had answered what turned out to be a good question.

It was only a day or so later I realized I had missed an opportunity to make the concept more real.

What if, instead, I had said:  How long is 65 million seconds?  Next tutoring session we start with this one!

First Grade Assessments

1_Verbal_Assessment    1_Written_Assessment

You’ll immediately notice two things about the first grade assessments: They contain fewer pictures and MANY more word problems than the Kindergarten assessments.   First grade does not include the Counting and Cardinality domain or as many other standards that require pictures, such as classifying by color and recognizing shapes.  In the future, I might illustrate the word problems by including a picture of what the problem is about, e.g., use a picture of a toy car if the problem is about toy cars.  But for now, since this is a draft, I will leave it as is.

You may be wondering why there are so many more word problems.  If you read my Kindergarten Word Problems post, you know the Common Core State Standards for Mathematics document contains a table that describes four categories of word problems.  When you consider all cases and versions, there are 15 types of word problems listed.  (This does not cover all types of word problems, just the basic ones of adding, subtracting, and comparing.)  It seems to me a student cannot truly be proficient until she can solve every possible type of word problem; therefore, I have included 12 word problems in the first grade assessment.

Twelve?!?  Shouldn’t that be 15?  Hmm…perhaps I got tired.  I’ll have to fix that later.  In any event, this is a huge increase in the number of word problems (from 2 to 12).  As a result, I will most likely distribute them throughout the assessment before I use it.

For each word problem. I included a description that maps directly to the table in the Common Core standards.  For example, question 12 is described as “word problems within 20, comparing, smaller unknown, ‘more’ .”  If you are so inclined, you can use these descriptions to figure out which word problem types are not represented.

A Non-CCSS Question?

You may have noticed a question in the written Kindergarten assessment that did not map directly to the Common Core State Standards.  It was the last one:  What number goes in the box?  4 + 5 = ▢ + 6.  I added this problem as a result of reading a 1999 article entitled “Children’s Understanding of Equality:- A Foundation for Algebra” [1].  (See link in blogroll.)

What caught my eye in this article is this:  When 145 sixth graders were given the problem 8 + 4 = ▢ + 5, none of them answered it correctly.

Most of the students said “12” should go in the box, indicating they view the equal sign as “a signal to do something” rather than “as a symbol describing a relationship” ([1], p. 236).  So, students are thinking, “eight plus four equals twelve and then we want to add five to it,” instead of “what plus five equals eight plus four.”

I was surprised, until I realized I have seen similar thinking in the work of high school students. When solving a multi-step problem, a student will write and solve one equation, and then – as if forgetting about everything to the left of the equal sign – continue the solution by building onto the right side of the equation.  Of course, this makes the equation no longer true.  Here is an example I created to illustrate what I mean:

004 Equality Misconception

The beginning of the solution – using the formula for the area of a triangle – is just fine.  But multiplying the result by 6, while necessary to determine the area of the regular hexagon, makes the equation untrue.

I think these students know the difference and they are just being careless.  But … maybe not. Maybe they have fundamental misconceptions about equality.  That’s one of the things I hope to find out when I give these assessments.


[1]  Falkner, K. P., Levi, L., & Carpenter, T. P. (December 1999).  “Children’s understanding of equality:- A foundation for Algebra. Teaching Children Mathematics, 232 – 236.  Retrieved from

Kindergarten Word Problems


Yesterday, a friend commented about the word problems in my Kindergarten assessment and how unbelievable it was that Kindergartners were expected to solve problems of this complexity.  When I looked back at the assessment, I realized that I had only included two word problems and they were not, in my opinion, of the most difficult type.

I was aware of different types of word problems, but it wasn’t until I read Table 1 in the Glossary of the Common Core State Standards for Mathematics (p. 88) that I understood the distinctions. (, p. 88)  Not only are there four categories of word problems, (1) add to, (2) take from, (3) put together/take apart and (4) compare, but each category has different cases as well.  For example, in “add to” and “take from” problems the result can be unknown, the change can be unknown, or the start can be unknown.

Yes, my two measly word problems did not even scratch the surface.

But then I had another thought.  I had been focused on creating problems that were grade-level appropriate.  What was I expecting students to show me for a solution?

I tackled this and I came up with a drawing for each of my two word problems that is straightforward and makes sense to me.  For example here is my take from, start unknown problem:  3 children leave the party.  4 are still there.  How many were at the party before? As a solution to this problem, I drew the picture at the beginning of this post.  Not bad.  Makes sense.

The more difficult problems are the compare problems.  Here is an example of one of the more difficult problems taken from the Common Core State Standards for Mathematics: “Lucy has 3 fewer apples than Julie.  Lucy has two apples.  How many apples does Julie have?” (p. 88).  That was the fewer version of a compare problem with a bigger unknown.  Now, here is the more version:  “Julie has 3 more apples than Lucy.  Lucy has 2 apples.  How many apples does Julie have?”  See how much easier it is to understand the more version?

I tried but couldn’t figure out a simple drawing that would work for the fewer version of the compare problem.  I can solve it numerically, of course, but had trouble coming up with a good visual explanation.  So, I researched models for comparison word problems and came up with several resources that model this kind of problem with a pair of bars. ( has a video along with sample problems.  The Minnesota Stem Teaching Center has a good discussion of several types of word problem models, and a little more than half-way down the page is a discussion of comparison problems using bar models (

Based on the concepts from these resources, I created a drawing for the fewer type of comparison word problem that combines both pictures and bars.  It makes sense to me after-the fact, but doesn’t seem as intuitive to me as the other drawings.


Common Core State Standards for Mathematics, p. 88

I think this approach would have to be taught.  Which brings up another question:  Are Kindergartners or even first graders solving problems like this now?  It’s been several years since my daughter was in Kindergarten, but I don’t remember problems like this.  If Common Core standards in post-primary grades presume these skills, are students prepared?  If not, how are they going to catch up?

I have a feeling some of the Common Core word problems I’m creating may be a bit of a challenge for the students I work with.  As a result, that part of my assessment may be more a jumping off point for teaching how to recognize and solve different types of word problems than an assessment of what students know.  But that’s okay.  After all, my main purpose is to find out where the holes are and fill them – even if the holes are “new” ones that exist because of the emphasis on understanding in the Common Core State Standards.  To quote Martha Stewart:  “It’s a good thing.”

Kindergarten Assessments

K_Verbal_Assessment  K_Written_Assessment

Above are the two Kindergarten assessments I created.  Although my initial purpose for the assessments is to use them with older students, I attempted to make them appropriate for Kindergartners.  Full disclosure: I am a secondary math teacher, not an elementary teacher.  However, I have read Knowing and Teaching Elementary Mathematics by Liping Ma (fantastic book, I highly recommend it – thanks Jean M.!)   I have also been researching elementary math tasks so I can better help a 6th grade student with whom I am  working.  I’m sure I will learn more as I use the assessments.

Why two assessments?  Well, some standards need to be assessed verbally (e.g., counting to 100 or counting out loud).  Furthermore, I may want to see and hear the student’s performance.  Specifically:  How fluently is he adding or subtracting?  Where is he hesitating?  Even the written assessments are not intended to be completed in isolation; I plan to watch the student as he completes the problems.  Again, I will see where he is fluent and where he hesitates or skips a problem.  I will gain insight into any misconceptions as well as skills he may be lacking, which will help me decide the concepts I will need to  reteach.

Although the assessments may appear to be cast in stone, my intent is to use them more flexibly.  So, (and I’m thinking out loud here) after the student demonstrates she can count by 1 to 20 or 30, I would stop and ask her to count by 1 from some higher number to 100 (which actually incorporates the second Kindergarten standard, K.CC.A.2).  Ideally, I would then follow up with a question to see if she could explain the counting pattern.

Also, if a student is really struggling with a problem, I would intervene.  First I would start by asking some guiding questions to see if he could then figure it out.  I might change the problem to be simpler or let him skip it all together.  After all, these are not tests; I just want to know where the holes are.  It is paramount that I keep it positive, that I keep the student talking to me.  Teaching is as much art as it is science.  I need to know when to push and when to back off.  I need to know my students.

Still, there may be better ways to assess students on these standards.  If you have suggestions for how to improve the assessments, I hope you will leave a comment!  (I have already seen some assessments that elegantly group multiple standards in a single task, which are intriguing.  Although, as my purpose is to determine which concepts students have yet to master, it might be easier to assess each standard individually.  I’m still thinking about this.)

Finally, I consider these assessments to be “first drafts” and plan to modify or change them after I use them and see how they work.  What do you think?  Am I on the right track?

Start at the beginning.

Mathematics.  That word … it makes so many people cringe.  Not me.  I’m someone who always loved math, even though advanced math didn’t necessarily come easily.  So, it was natural that I looked to mathematics when my career seemed isolating and empty and I decided I needed a change.  But not just any change; I wanted to do something I was passionate about, something worthwhile.  I had been volunteering in middle school math classrooms and saw a need for helping students understand math.  Fast-forward a few years, and I have a master’s degree and a secondary math teaching credential.  I can’t decide – do I teach middle school or high school?  Or, do I go back to school so I can be a math specialist in elementary schools because that’s where the trains leave the tracks, if you know what I mean; that’s where kids decide “I can’t do math.”

Then, I’m sent in a new direction by a chance encounter and a phrase that resonated: teach students math all over again, starting with Kindergarten.  This seemed to encompass both my desire to help students comprehend math AND my personal need to know the entire curriculum – from Kindergarten through Calculus.  A broad mathematical understanding is necessary for helping students regardless of grade level, and I knew it would allow me to ascertain what skills and concepts students were lacking as well as how best to position them for the mathematics that lay ahead.

This blog is the account of my journey to understand (and understand how to teach) the entire K-12 math curriculum using the Common Core State Standards (CCSS) as both a framework and a jumping off point.  To begin with, I plan to create a series of CCSS-based assessments that, when given to students who need help, will illuminate missing foundation concepts (beginning  with Kindergarten) and provide a starting point for targeted assistance.  In the end, I hope to have a complete and useful set of assessments, plenty of online references, and a collection of tasks and lessons full of authentic and challenging problems.  Oh, and by-the-way, I’ll also have an intuitive and connected understanding of the whole K-12 mathematics arc.  This should be fun.  All I need now is someone to collaborate with.  Are you with me?  We start at the beginning.  We begin with Kindergarten…