The Purpose of Hand Raising

Here we goooooo!!!! I have just started the book, Hacking Questions: 11 Answers That Create a Culture of Inquiry in Your Classroom, by Connie Hamilton. Below is my first    ah-ha moment.

Raise your hand if you have ever asked a question to the class and…

  • The one hand (that always goes up) speeds through the air like Hermoine Granger’s in potions class…
  • Several hands go up and now you must choose…
  • No one’s hand goes up

Yeah, me too. And if you say you have never done this…


We have all done this. And, before reading the first chapter, I would still be doing this. It is what we did in school; it is how we have seen others questions students; it is how it has always been done. But just because that is our norm, should it be the norm for questioning students???

Let’s go back to the three hand-raising options. Here is what I have been thinking about and how this might roll in my class:

  • The one hand (that always goes up) speeds through the air like Hermoine Granger’s in potions class… So now everyone else can breathe a sigh of relief as she answers. Or if I don’t let her answer, she gets all impatient and sighs vehemently.
  • Several hands go up and now you must choose… I spend 4-5 precious minutes going through multiple responses and get some answers I wasn’t ready for and now I have to think on the fly about a positive way to respond.
  • No one’s hand goes up…I answer for the class and basically take the learning away from them.

Here is what hit me: “Disengagement is the enemy of learning. We unintentionally create the conditions for disengagement when we allow students to keep their hands down”    (p. 21).

Say whaaaatttt??? THIS was not my intent when I ‘cold-called‘ (the term in the book for hand-raising) on students. These are my questions I want to reflect on for this week as I move forward with this slap-me-in-the-face realization. I invite you to think about them as well:

  • What is my purpose for using ‘cold-calling’ in class?
  • What is the purpose of questioning in my class? What do I want to get out of my questions?
  • How am I asking students these questions? Are there ways in which I can ask questions and get more feedback and engagement?

I am hoping to get into some elementary math classes this week and reflect on these as I teach. I encourage you to comment below or email me your reflections. Let’s learn together!



The Purpose of Hand Raising

Lesson Study: Lines of Best Fit

I had the privilege of working with Integrated 1 teachers last week on statistics. We voiced our concern that students are often given the line of best fit for a data set rather than exploring how to create it themselves. We also wondered why their stats chapter was not right after exploring equations in slope-intercept form, as that how we wanted students to write the equation for the line they created.

Our goals for creating the lesson were as follows: 1) Students would review writing a linear equation in slope-intercept form from a graph, 2) Students would create lines of best fit for data plots on a graph, and 3) Students would write the equation for the line of best fit in slope-intercept form.

After searching several open resource sites, we settled on using some graphs from an earlier exploration to review writing equations of lines (please see the attached worksheet). We felt that these graphs would be easier for students to be able to read, and allowed students to determine the slope either by counting or by using their slope formula when given two points. We didn’t want the lesson to get stuck in this review section, so the cleaner the graphs were, the better. We strategically chose two with positive slopes, two with negative slopes, some with fraction values and one with a y-intercept of 0 so that we could frontload the necessary skills before moving to the new concept of creating lines of best fit.

The second component of the lesson was the Desmos activity, Fit Fights (see link below). We wanted to use the activity to build understanding for how lines of best fit are created. We used the following questions to drive the discussions:

  • What do you notice?
  • What do you think the gray lines represent?
  • Do the data points make a line if we connect them? Is this a linear function? Why or why not?
  • Do the data points gravitate towards making a line? Would it be a positive or negative (or zero or undefined for the graphs they created) slope?
  • If we drew a line to ‘best fit’ our data points, where might it go?
  • How can you tell if your line is a good fit for the data points?

The third component of the lesson was to relate the lines of best fit students created to writing equations of linear functions in slope-intercept form. We took snap shots of the lines of best fit from the Desmos activity, strategically choosing lines that had at least two whole number coordinates. We purposefully related the warm up to the lines of best fit, using similar language and questioning.

Our Take-Aways

  • Our students are successful in determining the slope by counting rise/run, but need to continue to work on using the slope formula when given two points.
  • The Desmos activity only had negative slope lines of best fit. Consider having students create data points that could have lines of best fit that have other correlations so they don’t overgeneralize and think it will always be a negative correlation.
  • Students struggle when the axis are scaled differently. Need to build in some warm-ups that continue to explore differently scaled axes.
  • We need to be clear in our questioning. Using questions such as, “Are we good? Do you get it?, etc.” does not give us formative information to move forward.

Below are some of the students’ work and what they felt they had learned. Overall, students were more engaged with this lesson than they have been in the past when learning this concept. We deflines of best fit 1initely want to use this lesson again, and hope you find it helpful as well!

lines of best fit 2







Desmos Fit Fights:

Lesson Study Worksheet (Shout out to Fontana High School!): Stats Lesson Study Worksheet


Lesson Study: Lines of Best Fit

Thinking Rationally With Your Tween

Students typically start exploring positive and negative numbers towards the end of elementary or early middle/junior high school. And it can be a bit weird. It is literally the opposite of what they had been taught for the last ten years in a number of ways (Puns intended.). Here are some ways you can support your child in their negative number journey to make it a positive experience (Dang, I am on FIRE!).

  1. Bring in finances. “In the Red” and “Black Friday” are references to business. When a company in “in the red”, they are in debt. They owe money. Traditionally “Black Friday” (The Friday after Thanksgiving) was the first day of the fiscal year companies got out of the red and posted a growth. Credit cards are another great place to explore debt and credit. A debt would be what you owe and would be represented as a negative number. A credit would be a positive number. Show them your mortgage and credit card statements and discuss terms such as “deposit, credit, debt, with drawl, etc.”. The stock market is a great place to discuss negative and positive fractions and decimals. Pull up the daily NYSE and discuss which companies have an increase (or positive) and which have a decrease (or negative) change that day.
  2. Football Season!!! This is a fantastic place to bring in integers (positive whole numbers and their opposites). If I gain six yards on a drive, how could I represent that change? (+6 or 6). What if the QB gets sacked? How many yards did they lose? (ex: -7). How far do they now have to go to get a TD? If your child is interested in football, use it to your advantage!
  3. Playing Cards. Below are a couple of games you can adapt to include negative numbers. I prefer to omit the face cards and only use numbered cards, but you can make Aces = 1 (and -1) and the face cards values after 10 (and -10).

War! Black Cards are positive values; Red Cards are negative values

  1. Shuffle and divide the cards evenly among players. Keep your cards in a pile face down. Everyone flips over their first card. Player with the greatest value wins all the cards for that round. Tie? Flip another card and whoever has the greatest value that round wins all the cards from both rounds. Whoever has all of the cards at the end (or the most cards when you get bored) is the winner.

Example: I flip over a red 9 (-9) and you flip over a black 2 (2 or +2). A gain of 2 is greater than a loss of 9 so you win the cards.

You can also play that the winner is the one with the smallest value.

  1. For students who are learning to add integers. Shuffle and divide the cards evenly among players. Keep your cards in a pile face down.

Everyone flips over TWO cards and finds the sum (add them). Whoever has the greatest (or least) sum wins the cards for that round.

Example: I have a black 2 and a red 4. 2 + (-4) = -2.

You have a red 4 and a red 2. -4 + (-2)=-6.

Since -2  is greater than -6 (a loss of 2 is better than a loss of 6), I would win.

  1. Go Fish! Black Cards are positive values; Red Cards are negative values

Shuffle and hand each player 7 cards. The rest are in a pile in the middle face down.  The objective is to be the first one out of cards.

How do you get rid of cards? By making matches of cards that have a value of 0.

Example: Jen has a 5 black (5 or +5). She says to Chris, “Do you have a negative 5 (5 red)?” Chris does, and hands the 5 red (or -5) to Jen. Jen takes the positive 5 and the negative 5 and lays the pair in front of her.

This is not an exhaustive list so I will be adding other fun ways to integrate math into your home conversations. Let’s make math a positive experience for our kiddos!

Thinking Rationally With Your Tween

Creating Structure for Context in Math

I was honored to facilitate lesson study with IM1 teachers today. Their students are struggling (due to high EL/SPED population) with solving word problems. I dug deeper, and we decided the struggle is really the first step: creating equations from situations.

We decided our goal as educators this year is to work on teacher clarity: making our lessons streamlined and very goal-oriented. If we know our goals for the lesson, then every move we make (every breath we take…) is for the goal. So how do we clarify translating context to equations?

We started from the end: the benchmark. We took a problem the students struggled with, and tweaked it several times, each time only altering only one component. Students had to work from the original version (which we used simple numbers to keep it accessible) for each new “version”. They discussed what changed from situation to situation and how that affected the prior equation.


Version 1: Troy works for an ice cream cart vendor. He receives $10 for taking the cart out for a shift, plus a commission of $2.00 for each item he sells. Troy worked a shift Saturday and earned $60.  How many items did he sell?

Version 2: Troy works for an ice cream cart vendor. He receives $15 for taking the cart out for a shift, plus a commission of $2.00 for each item he sells. Troy worked a shift Saturday and earned $60. How many items did he sell?IMG_8202

Version 3: Troy works for an ice cream cart vendor. He receives $15 for taking the cart out for a shift, plus a commission of $1.25 for each item he sells. Troy worked a shift Saturday and earned $60. How many items did he sell?

Version 4: Troy works for an ice cream cart vendor. He receives $25 for taking the cart out for a shift, plus a commission of $0.10 for each item he sells. Troy worked a shift Saturday and earned $52.90. How many items did he sell? (Problem from the benchmark.)IMG_8205

We used 3 scenarios. In each, we kept our questions as consistent as possible (again, clarity):

  • Which part is varying (changing)? How do you know?
  • Which quantity would be the coefficient? How do you know?
  • Which quantity would be the constant? How do you know?
  • (From version to version) What has stayed the same? What changed? How does the changed quantity affect our equation? Why?IMG_8208

Students were engaged, writing on their tables and willing to discuss with each other. They had many moments of “ohhhhhh” and “oops!” and learned quite a bit about the components of 2-step equations. They definitely need more time, and the teachers have committed to continuing the work as warm-ups or on modified days.

Oh! And did I mention this was a co-taught Special Ed class, with many English Learners?! Amazing!

So our major takeaways were: IMG_8207

  1. Know your goal!
  2. Keep your goal in mind when creating the tasks/lesson and questions for clarity and focus.
  3. Breaking the situations into translating and solving (working on a single component) allows students to focus and interpret.

Below is our ppt. Hope it is useful! Happy Math-ing!

Linear Equations in Context LS 8.27.19



Creating Structure for Context in Math

What’s Your Teaching WHY?

This morning at the airport (At 5 fricking o’ clock! I need to fire my secretary for scheduling this flight. Oh wait. That’s me!) I was answering emails and a timid voice interrupted my thoughts.

Excuse me. Are you a teacher?

This always makes my heart happy. As a middle school teacher, you often believe the kids won’t think twice about you once they leave your room. Why would they, with all the distractions the world has to offer?!!

So when a former student not only remembers you from looong ago, AND takes the time to share her experiences and life journey with you, you tear up just a little. You remember that these precious moments are WHY you were born to be an educator.

My WHY is simple. I love seeing my peeps have the “click” in math. I love learning about these humans, with all their quirks and unique personalities. I love supporting them and inspiring them to do great things. I just love THEM.

What is your WHY? Ponder, remember, and remind yourself of this through the year.

Have a great 2019-2020! I KNOW it will be a year to remember! 💕

What’s Your Teaching WHY?

How Do Our Beliefs in Math Affect Our Students?

I was honored to work with amazing teachers this week. We took a survey from NCTM (National Council Teachers of Mathematics) on our beliefs regarding student learning and our instructional practices in mathematics. This, in itself, led to amazing discussions about what we truly believe math IS and how we interpret that into instructional decisions within our classrooms.

But then we took it further. We got into groups and discussed not so much whether we agreed or disagreed, but whether it was a productive or unproductive belief in respect to student access and learning.  Here are two to consider:

There were fantastic discussions about these particular ones, especially for educators of EL and SPED. We also considered how parents might respond to these. Powerful conversations around access, flexibility in thinking, understanding conceptual and procedural mathematical ideas, and yes, fluency.

Here was the point. Our beliefs, whether productive or unproductive, affect our attitudes towards mathematics and the children we are blessed to teach. Those attitudes affect the actions we take. Who gets to answer which questions? Who gets the “tough” tasks and who has to keep doing drill and kill worksheets? Who gets to explore puzzles and who has to retake tests or do homework (because their home life doesn’t lend itself to being able to do it at home)? And those actions MATTER. They affect the results you will get from your students.

Belief chart

So as you gear up for this school year, consider taking the beliefs survey yourself. Even better, have your team take it and REALLY dive in to what beliefs are productive an unproductive. The more we reflect, the more we can grow and be effective at what we truly want; to teach students to love, learn and understand mathematics. Have a great year!

For the beliefs survey: 2017_06_19_Holstrom_Grady_2BeliefsSurvey


How Do Our Beliefs in Math Affect Our Students?

Equations and “Flow Charts”

A group of seventh grade teachers and I were trying to figure out how to move from concrete representations of solving equations (some used chips/cups and some used tape diagrams) to the more symbolic procedural (traditional) representation. While students were able to model the “moves” with the concrete, some still struggled to move from that to solving on paper.

I had recalled a method a dear colleague, Bruce Grip, had shown me years ago using a flow chart. We decided to try it out ourselves.

Starting with expressions, we discussed what the “moves” are when simplifying. Order of operations made a showing, and we moved through the flow chart. We decided this was a valuable use of time, as it built understanding of the structure of numeric expressions and fluency with integers (Which, let’s be honest; they need LOTS of practice with!). IMG_1854   IMG_1856


From there, we decided to bust out a single step equation. We started the same way we did with expressions, using x as our starting value. “What moves am I making to x in this equation?” We then built our flow chart. HOWEVER, rather than simplifying (as in the expressions), we know the value we want to get. So the flow chart looks like this:


To solve for the value of x, we need to work backIMG_1859wards through our flow chart. If I had added 2 to a value to get -5, then I need to subtract that 2 to figure out what I started with. We could then parallel the flow chart with the more traditional algorithm for the students.

Below are several of our examples, limited to the structures seventh grade explores for CCSS.

We also explored some “messier” problems, as shown here.IMG_1863 It illustrates the fluency with the distributive property piece of “When do I need to distribute and when is it efficient to divide out the factor first?”. We liked that the students could show both ways and determine which route to take.



Our big commitments to this flow chart method:

  1. Start with the concrete/visual. This is not a substitute for chips/cups nor the tape diagram. This is the next step for students who need it.
  2. Next year, use the flow chart when exploring simplifying expressions so we can build on that understanding for solving equations.
  3. Use friendly numbers (NUMBER CHOICES MATTER!!!) first to build understanding.
  4. Bring in some messier problems to seal the deal and discuss different moves they can make based on the given numbers in the equation.
Equations and “Flow Charts”