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Geometric Subitizing: A Different Kind of Number Talk | Questioning My Metacognition

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Geometric Subitizing: A Different Kind of Number Talk | Questioning My Metacognition

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These Geometric Subitizing Cards have been in beta the past two years.

I asked our buddy Joe Schwartz to take them for a test drive last spring and he reported back. Joe's all in.

How do they work? Let's play...

Last week I took them for another spin in some kindergarten classes. Here's two examples of how it went and we're only 2 months into the school year.

Last year I captured me playing cards with my girls. Here's me playing with my kindergartner.

Here's me playing with my 3rd grader.

Your turn.

Take these cards and give them a try. If you need some ideas how to use them you might want to check out this task card. I plan to share my thoughts about Geometric Subitizing in a future post but for now I just want you to go and play. The only thing I ask is that you report back and offer feedback.

How did it go?

How can we make them better?

Any other ways we could use them?

What am I missing?

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Geometric Subitizing: A Different Kind of Number Talk

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Download the Resource These Geometric Subitizing Cards have been in beta the past two years. I asked our buddy Joe Schwartz to take them for a test drive last spring and he rep...

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Where’s Poly? An Exploration in Geo-Dotting | Questioning My Metacognition

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Where’s Poly? An Exploration in Geo-Dotting | Questioning My Metacognition

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Where’s Poly? An Exploration in Geo-Dotting

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Download the Resource What’s geo-dotting? I have no clue but that’s what I’m calling this lesson. We started by asking, “What do you notice?” Our favori…

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Intuition, nudges, and other thoughts after Asilomar – Becoming the Math Teacher You Wish You'd Had

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Intuition, nudges, and other thoughts after Asilomar – Becoming the Math Teacher You Wish You'd Had

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'Tis the eve of FDOS (first day of school, duh), and I'm no longer nervous or anxious. I'm washing these laptop covers for my homeroom students only because that's what some of my colleagues did with their kids' covers. I will not let a kid taunt me on FDOS with, "How come Mrs. So and So gave her students clean covers?" Oh yeah? Well, did you ask if she'd adopt you for the next 180 days?

I committed all 7 sins at one time or another, but there's no shame involved - says a recovering Catholic - instead, it is a reflection of sorts.

Giving extra credit. I don't care where you teach, how old your students are, what your zodiac sign is, you're going to have at least one kid who'll ask for extra-credit "work" at the eleventh hour of the grading period. Don't do it. Say no and walk away because the tears might come streaming down his/her face and you have to ration the use of Kleenex. And you should be ashamed of yourself for giving students extra-credit points for bringing in copy papers, sticky notes, dry-erase markers, tissue boxes, doughnuts. Yes, you should send me some.

Giving timed multiplication drills. Maybe there's a well-documented success story behind this madness that I'm not aware of, but to me, it perpetuates the myth of faster-is-smarter. This practice raises self-doubt and affirms the why-should-I-even-bother mindset.

Giving out the equation. That's like giving away life's secrets to someone who flies to Paris to have lunch. Meaning, they don't need it, nor did they ask for it. Your students' conversations, their conjectures, their models - are all at the heart of a math class. To give away the equation is to passively (and aggressively!) dismiss our students' abilities to think for themselves. It's okay to eventually give them the equation in due time, just don't start with the equation. Imagine if I just gave my students the equations for slope and area of a circle.

Teaching from one source. No one source is that good. The creators of that source would be fools to not concede that point. It's like eating out at the same restaurant or boasting that you can make chicken 50 different ways. No you can't, and nobody cares. Let one or two sources be your structural outline, your mainstay, then supplement it with your favorite lessons or other teachers' favorite lessons. Remember, any well-crafted lesson outside of the textbook that you can bring in is your gift to your students. Tell them that. And with our prolific #MTBoS, you cannot afford not to supplement.

Talking, talking, you're still talking. I pretty much end every workshop with this reminder: The more you talk, the less your kids learn. I plan each lesson using this as my go-to guiding principle. Math is a highly social endeavor, so for the love of Ramanujan and Lovelace, please stop talking so much so your kids may talk! Every question you pose is an opportunity for your kiddos to ponder [quietly by oneself first] and share their thoughts with peers. Every question! If you fret that your kids don't talk in class, then I wonder about two things, 1) Do students feel safe enough to talk in your class? and 2) Is the question you're asking interesting/worthwhile/challenging to even bother? (I must have asked hundreds of lame, boring, worthless questions, but I'm not giving up. I practice and get better.)

Keeping up with the Joneses. That colleague whose hair and complexion are always perfect is just not as funny as you are. That teacher whose students all adore her probably owns a cat that wants to kill her. And that "amazing" teacher whom everyone talks about probably sucks at everything else in life! And he might be a compulsive hoarder of all things creepy! So, don't mind them. We're not here to compete with one another. We're here to make mathematics rock for our kids. There is one you and 24 hours in a day. Make time for yourself, make time for your family. We all have shitty days that rob us of our wits and sensibilities, but recognizing that and committing to having a better day tomorrow are worthy endeavors. Our students need us more than they care to admit.

Being an asshole. I already stated from the beginning that I'm guilty. No one wants to learn from someone who's mean and angry and bossy. When we try to establish authority in the classroom, we may inadvertently end up being perceived as this person. The meaner we get, the less students want to have anything to do with us, so the angrier we get. It's a vicious cycle, and everyone is losing. We're the adult in the room, charged with a magnificent duty to establish a learning culture, which will not happen if we don't behave like an adult. Children are said to be resilient, but they are also impressionable, and their impressionable minds are vulnerable - vulnerable to criticism, to shame, to false praises.

Let's pray for more patience, more kindness, more badass. Here's to us - and to a great school year!

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'Tis the eve of FDOS (first day of school, duh), and I'm no longer nervous or anxious. I'm washing these laptop covers for my homeroom students only because that's what some of my colleagues did with their kids' covers. I will not let a kid taunt me on FDOS with, "How come Mrs.

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Courtesy of Sarah Carter, a pair of high yield geometry routines I experimented with last month with our fifth graders:

Quick Build
I flashed this image on the SmartBoard for a few seconds, then took it down:

Using snap cubes, students had to build the structure from memory:

After all had built the structure, I asked them to share how they had visualized it in their minds, and their responses were quite varied. Some saw a 2 by 3 rectangular prism with a two-step staircase attached, others saw a three-step staircase with a 1 by 3 rectangular prism attached. Still others saw 3 levels: 4 on the bottom, 3 in the middle, and 2 on the top.
Then I asked about the structure's volume, and they generated lots of equations:

I liked this activity for several reasons:

It reinforced the idea of finding the volume of non-overlapping right rectangular prisms, which is a major grade 5 focus.

It reinforced order of operations and writing equations, another grade 5 standard.

It inspired a little number talk.

Also, the kids loved it! They want to do it again, but next time with, as one student put it, "Harder shapes." So I have this one loaded up:

This activity comes from Grayson Wheatley. Again, flash the image for a few seconds, then take it down. Ask kids to draw the figure from memory.

Another:

As with quick build, I asked the kids how they visualized:

"A square, with a box around it, with 4 trapezoids."

"A box in a box with 4 lines connecting the corners."

"A 3-D cube facing me."

"A room with a ceiling, two walls, and a floor."

"A pyramid with the top cut off."

Lots of great vocabulary was generated:

Number talks, counting circles, estimation180, subitizing; we've done a good job infusing these routines into our instructional practice. That's a balance weighted heavily on the number sense side. These two routines, which emphasize spatial reasoning and geometry, are a needed and important counterweight, and I look forward to more experimentation across grades K-5. And again, thanks to Math = Love for the inspiration!

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Courtesy of Sarah Carter, a pair of high yield geometry routines I experimented with last month with our fifth graders: Quick Build I flashed this image on the SmartBoard for a few seconds, then took it down: Using snap cubes, students had to build the structure from memory: After all had built the structure, I asked them to share how they had visualized it in their minds, and their responses were quite varied.

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Pentomino Puzzles | Mr. Orr is a Geek.com

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Pentomino Puzzles | Mr. Orr is a Geek.com

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A Hands-on activity is blended with interactive technology. Students begin with puzzles but ultimately create algebraic expressions and solve equations.

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SlamDunkMath: Analyzing a Bike Rim in 5 Days

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SlamDunkMath: Analyzing a Bike Rim in 5 Days

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At Glebe Collegiate we have tried hard to spiral our MHF 4U course (called advanced functions) over the past 4-5 years. It really has been a collective effort within our department. You can find a whole bunch of information about it here thanks to Janice Bernstein. I posted the whiteboard notes and an early spiraled version of the course here.

We are trying hard BUT I feel like the course is still missing that activity based feeling that I have managed to create in my grade 10 applied course (for example my lesson on T-Shirts). It just feels like we still tell them the things we would like them to discover or experience.

Also I VALUE the mathematical processes in the curriculum document. So.....I started thinking about how I could get my students to discover and experience more of the Trigonometry in the course. One of the activities we have been doing is radian rulers from plates. Here and here is what it looked like.

This is my attempt at creating an exploratory activity 1) to discover and understand radians, 2) to develop the relationship between arclength, radius and the angle measure in radians 3) to understand angular velocity and 4) to develop an understanding of the relationship between the angle rotated and the height of a point on the bicycle rim.

BIKE RIMS
First I went and scavenged some bike rims from a local bike store and cut out all the spokes. Needed 10 but only found 7 so I had to use three tires as well so that each group of 3 students had their own rim or tire.
DAY 1 Creating a radian ruler for their bike rim.

Students were put into random groups of three.( I randomize students by giving them a card from 1-10 which matches the pods of three desks and whiteboards (4 by 8 ft for each group) around my room-these numbers never change all semester just the students at those desks each activity). Verbal instructions ( no paper-throughout all 5 days).

1) Tape the number of their group onto their rim or tire.

2) Create a paper ruler that has the length of the circumference of your rim or tire. Measure that length and figure out the radius of your rim (on whiteboards). Measure the diameter roughly to check.

3) Mark your ruler using a red marker in radius units. HUH? If your radius was 10 cm you would put a mark every 10 cm and mark them 1, 2, 3 etc until you got to the end.

As this was happening I circulated and started having discussions with groups about how many they got. Why? Students started making connections that it was a little more than 6 because the circumference is 2pi*r. Once all were at this point we had a full class discussion that we could put 2pi at the end of the ruler and that was equivalent to 6.28 (about).

4) Next we talked about it if we folded it in half (pi) and in half again and half again. I asked groups to do this and crease their rulers ( be precise ) and then mark the creases in terms of pi. I actually hesitated with the colour and of course a student hollered " Purple for pi". Me "of course".

5) Next we talked about if we folded in half, then in thirds (like you would fold a letter for an envelope) and then in half-we would get 12 pieces - so 2pi divided into 12 parts - each part would be pi/6. Students labelled these in green. Of course some were all ready marked because they were equivalent to some of the angles from the previous folds.

6) I then had groups put a decimal value at every crease (radius units).

7) Lastly they were instructed to put the degree value in orange. Student "Orange for the degree sign". Enough said!

Here is the final ruler with the rim for one group.

8) Lastly we wrapped our rulers back around our rims and talked about the angle and the number of radii around the rim.

9) Clean up time-hung the rulers for tomorrow.

DAY 2 Circular Radian Number Lines

This idea came to me after seeing this tweet from @CHaugneland

So this required some set up. I wanted each group to have a circular number line that they could hang angles in radians on. Here is a pano of the class as they were doing this part of the activity.

Obviously there were many comments about the hanging rims around the room as students entered class.

1) The class discussed that we were going to use these as circular number lines and that they would hook cards of different angles in the right spot relative to each other and then check their placements using their radian ruler from yesterday.

2) I had prepared 4 sets of cards.

First set had positive angles only with multiples of pi/4 between 0 and 2pi. So pi/4, pi/2, 3pi/4, pi, 5pi/4, 3pi/2, 7pi/4 and 2pi. I also included some that were equivalent with a higher denominator, for example 6pi/8.

Second set had positive and negative angles with multiples of pi/4 between -2pi and +2pi. This time I included equivalent ones that were positive and negative, for example +pi/4 and -7pi/4. And again some that were equivalent with a higher denominator.

Third set had positive and negative angles with multiples of pi/3 between -2pi and +2pi with equivalent angles.

Fourth set had positive and negative angles with multiples of pi/6 between -2pi and +2pi with equivalent angles.

For the fifth set I had groups make a set of cards and place them based on instructions that I gave them. For example

1. decimal values between -6.28 and +6.28.

2. between -4pi and -2pi or 2pi and 4pi with multiples of pi/4

3. between 0 and 2pi with multiples of pi/8

4. etc

3) The card sets were placed around the room on the ledges of my white boards. A group would pick up a set, place them on their circular number line and then check placements with their radian ruler. Groups then checked in with me-I typically drilled them and asked lots of questions. Groups then returned the card set and picked up the next one. Repeat.

Here are some pictures.

Some thoughts:

There were many groups that wanted to know where to place zero and what direction was positive? negative?

Checking with the radian ruler was tricky but doable.

Mathematical discussions and productive struggle were through the roof.

High level of engagement.

A colleague thought I was crazy because of the work required for just this one day. It was well worth the experience as students participated whole-heartily in rich mathematical discussions. Would I do this again-ABSOLUTELY.

Arclength = Radius *Angle in radians

Believe it or not it took some thought for me to clue into this. Plot Arclength(y) versus Angle in radians(x). The slope of the line will be the radius of their tire, and the equation will fall out. Here is how it went down.

1) Same groups as the last two days. Get your bike rim, your radian ruler, whiteboard marker and a measuring tape.

2) Measure the arclength for a particular angle in radians and make a table of values.

As I was demonstrating this by wrapping the radian ruler around the tire and the tape measure around the tire (which was difficult-see picture) this conversation ensued.

Student asks and states "Do we really have to wrap both around each time? Couldn't we just lay it flat and measure - it is the same thing".

Other student " As long as we realize that when we measure the distance on the (radian) ruler we are really measuring the arclength around the rim"

3) Make a table of values and graph the data. Find the slope of the line. Find the equation of the line using technology (or algebraically) (or both)

4) Calculate the radius of your rim again. Circumference = 2* pi * radius

5) At this point I asked if there was any connections between the slope of the line and the radius of the bike rim. We went around the room and each group yelled out their slope followed by the radius of their tire. AHA the slope of the line was the radius. And there you have it

Arclength = Radius *Angle in radians

6) We briefly worked on an application.

Some samples of groups whiteboards at the end of the period.
DAY 4 PART ONE

Distance A Point On The Wheel Travels

1) We grabbed a bike rim and the entire class went out into the hall. I asked some students to time how long the rim rolled for while I counted the number of revolutions. We picked a starting point. One of the students marked where we stopped the wheel. I rolled the wheel and I counted revolutions while some students timed the roll until I yelled stop.

5 revolutions of a wheel with a radius of 25.9 cm in 8.68 seconds

2) Groups were sent back into the classroom to work on whiteboards and find the total distance travelled by the point on the rim. I also asked them for the angular velocity in radians per second. I also asked them to figure out the total distance travelled by the point on the rim using the angular velocity. Here is the work of one group for the data given above.
For a different rim
7 revolutions of a wheel with a radius of 26.1 cm in 10.11 seconds

3) Once all groups had done the calculations we debriefed. All groups went out in the hall and measured the distance we had marked earlier. The calculations were right on.

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At Glebe Collegiate we have tried hard to spiral our MHF 4U course (called advanced functions) over the past 4-5 years. It really has been a collective effort within our department. You can find a whole bunch of information about it here thanks to Janice Bernstein.

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Two Ways To Integrate Problem-Based Learning In A Unit (And Another To Avoid) - Robert Kaplinsky - Glenrock Consulting, LLC

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Two Ways To Integrate Problem-Based Learning In A Unit (And Another To Avoid) - Robert Kaplinsky - Glenrock Consulting, LLC

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dy/dan » Blog Archive » Creating a Need for Coordinate Parentheses & Combining Like Terms

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dy/dan » Blog Archive » Creating a Need for Coordinate Parentheses & Combining Like Terms

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Our first approach in preparing a new lesson is often to ask, "Where does this skill apply in the world of work or in the world outside the classroom?" There may well be a great answer for some skills, but this strategy generalizes very poorly to lots of mathematics. So instead, I try first to ask myself, "Why did we invent this skill? How does this skill resolve the limits of older skills? If this skill is aspirin, then what is the headache and how do I create it?"

Two examples from my recent past.

Why did we invent the skill of combining like terms in an expression? Why not leave the terms uncombined? Maybe the terms are fine! Why disturb the terms?

One reason to combine like terms is that it's easier to perform operations on the terms when they're combined. So let's put students in a place to experience that use:

Put it on an opener. The expression simplifies to x ², giving students an enormous incentive to learn to combine like terms before evaluating.

[I'm grateful to Annie Forest for bringing the example to mind. She also adds a context, if that's what you're into.]

When students first learn to graph points, the parentheses are the first convention they throw out the window. And it's hard to blame them. If I told you to graph the point 2, 5, would you need the parentheses to know the point I'm talking about? No.

So why did mathematicians invent parentheses? What purpose do they serve, assuming that purpose isn't "tormenting middle school students thousands of years in the future."

It turns out that, while it's very easy to graph a single point with or without parentheses, graphing lots of points becomes very difficult without the parentheses. So let's put students in a place to experience that need:

Graph the coordinates:

-2, 3, 5, -2, 8, 1, -4, 0, -10, 4, -7, -3, -2, 7, 2, -5, -3

You can't even easily tell if there are an even number of numbers!

[My thanks to various workshop participants for helping me understand this.]

The need for combining like terms is Harel's need for computation and the need for parentheses is Harel's need for communication. I can't recommend his paper enough in which he outlines five needs for all of mathematics.

My point isn't that we should avoid real-world or job-world applications of mathematics. My point is that for some mathematics that is actually impossible. But that doesn't mean the mathematics was invented arbitrarily or for no reason or for malicious reasons. There was a need.

Math sometimes feels purposeless to students, a bunch of rules invented by people who wanted to make children miserable thousands of years in the future. We can put students in a place to experience those purposes instead.

We explored these ideas in a summer series.

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Creating a Need for Coordinate Parentheses & Combining Like Terms

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Our first approach in preparing a new lesson is often to ask, "Where does this skill apply in the world of work or in the world outside the classroom?" There may well be a great answer for some skills, but this strategy generalizes very poorly to lots of mathematics. So...

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Exit 10A: The Standard Formerly Known As 4.2.3D.2.a

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Exit 10A: The Standard Formerly Known As 4.2.3D.2.a

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From the Common Core Math standards:

For more than a decade, research studies of mathematics education in high-performing countries have concluded that mathematics education in the United States must become substantially more focused and coherent in order to improve mathematics achievement in this country. To deliver on this promise, the mathematics standards are designed to address the problem of a curriculum that is "a mile wide and an inch deep."

Agree with the conclusions or not, there's no doubt that promise has been kept, at least at the elementary level here in the Garden State. If you don't believe me, compare the old grade 3 standards to the new ones. Many of the skills and concepts I taught as a third grade teacher are no longer there. One that did survive the purge was a confounding little number we grew to know quite well as 4.2.3D.2.a. It has changed its name to 3.MD.B.4, but retained all of its deviousness. Can you guess what it is?

If you guessed measuring to the nearest quarter inch, you are correct!

Seems not much has changed since my third grade days. Despite our best efforts, which this year included counting by 1/4s in counting circles, the kids had enough trouble with 3.MD.4 to keep the folks over at Math Mistakes busy for weeks. I'll propose the following reasons; feel free to add your own:

Kids count 3 hash marks between each inch and express their response in thirds.

Kids do not know how to write fractions and mixed numbers correctly.

Kids think that when they are asked to measure something to the nearest 1/4 inch, their answer has to include a fourth.

Kids are unable to connect what they know about fractions and number lines to the actual ruler.

As I talked it through the issue with our grade 3 team, I decided to throw out an idea I had been turning over in my mind. The teachers felt it had potential, and when I got back to my room I started working up a model:

What might happen if, when asking a kid to measure something to the nearest quarter inch, you gave the kid the square inch and quarter inch pieces and said something like,
"Use these to measure. If you need the little quarter inch pieces, use them. If not, then don't."
Would they do this?

We decided to find out. Theresa grabbed a student who had difficulty with the skill on the assessment. I gave her the square inch and quarter inch pieces and let her give it a try.

From here it's a natural move to...

...and then finally:

We scaffold many concepts with manipulatives. We use base ten blocks to help children understand place value concepts and regrouping, and tiles to help children compare and work with fractions. Why should measurement be any different? Rulers need scaffolds too!
Did we beat 3.MD.B.4? The jury's still out. We need to get the square and quarter inch pieces into the hands of the teachers, then into the hands of their students. We need to redouble our efforts at counting by 1/4s, connect that work to number lines and rulers, and make sure the kids have an understanding of how to both say and write the fractions and mixed numbers. If we do all that, I think we can give 3.MD.B.4 a run for its money. It may even have to change its name again!

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The Standard Formerly Known As 4.2.3D.2.a

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From the Common Core Math standards: For more than a decade, research studies of mathematics education in high-performing countries have concluded that mathematics education in the United States must become substantially more focused and coherent in order to improve mathematics achievement in this country.

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First Grade Math Fight... Fractions, Proportional Reasoning, and Algebra, Oh my! | Elementary Math Addict

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First Grade Math Fight... Fractions, Proportional Reasoning, and Algebra, Oh my! | Elementary Math Addict

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Okay, so I knew this "Math Fight" was coming. (Thank you, John Stevens) The standard I was going after was 1.G.3

Although, this is the performance standard (a very simple standard at first glance) that students need to be able to do, my mathematical goal was different. Further down (like about 3 pages later) in the CA math framework, I find this beauty of a mathematical goal: (Mathematical goal meaning - What is it about math you want students to understand?)

"Students need experiences with different-sized circles and rectangles to recognize that when they cut something into two equal pieces, each piece will equal one half of its original whole (1.G.3). Children should recognize that the halves of two different wholes are not necessarily the same size. They should also reason that decomposing equal shares into more equal shares results in smaller equal shares." Van de Walle also chimes in about early fraction work and said, "Given children's experiences with fairly sharing items among family and friends, sharing tasks are a good place to begin the development of fractions. The sharing tasks allow children to develop concepts of fractions from an activity that makes sense to them, rather than having the structure imposed on them." (Teaching Student-Centered Mathematics K-2)

So I wondered, "What can I do to create a little controversy?" Then I had an idea. Yes, it concerns cookies... Older grade teachers don't check out yet, this lesson is not limited to the idea of parts of a whole. I began double planning teacher questions and some anticipated responses.

Fast forward to today's lesson... I am wearing a leopard apron.

Kids kept asking me why I was wearing an apron. I told them I had been up all night baking cookies, which was true. So, I called two students up to the front of the room to be my helpers. I told the kids I had two halves of cookies. I asked them if it would be fair if I gave them each half of a cookie. The cookies are hidden in a paper bag. The class all shouts out, "Yes!" So I give half of a small cookie to Amore and half of a huge cookie to Shawn.
The uproar was on the border of out of control, but it's in the name of math, so... we kept going. This argument begged a student created definition of what half is and what half is not.

Don't worry, this is only the launch of the lesson. I know from listening to upper grade teachers and reading Van de Walle that students have a difficult time understanding fractions as numbers and not only parts of a whole. So I stole an idea that Andrew Stadel shared and used it in a different way. (BTW - this is an example of how elementary, even primary, can steal ideas from middle and high school teachers!) I put zero and one on a string number line and asked the kids where half would go. They partner talked their disagreements, but in the end it came out that half should be equidistant between zero and one.
I chose to record the language of half in words. See the last sentence of the pic below.

Still not done.
Time for a little estimation...
I held up one small cookie and one extra large cookie and asked the kids to choose a just right brave estimate for the question, "How many small cookies equal a large cookie?" Students wrote their estimations on their desk with a marker.

Partner talk...
Then we laid out small cookies on the large cookie to see how many it would take. It turned out to be 10 so we added 2 - 10 to the existing number line.

Then I strung up another number line directly above it and hung zero up above the other zero. This number line represents the large cookies.

"Boys and Girls, if the yellow number line represents the small cookies and the pink represents the large cookies, where will the one go on the large cookie number line?"

Kids whispered to each other. Brave ones called out, "Above the 10!"

Kids, "Because it takes ten cookies to equal one big cookie."

Hmm "Well, if that's true then where would one half of a large cookie go on the number line?"

Kids, "Above the 5! Half way between the zero and the ten!"

Kids, "Because 5 is half of ten. 5+5 is 10."

Push: "If half goes here, where would half of half go?"
Kids disagreed on this one. Some thought between half and one, but after some kids convincing each other, and a demonstration of breaking half into half they came to consensus that half of half is half way between zero and one half. Kids created some examples and non examples of half and divided half into half again.
Tomorrow students will reflect on what they know so far about halves and wholes in their math journals.
The next goal will be: "They should also reason that decomposing equal shares into more equal shares results in smaller equal shares."
Thinking - What is the pattern and/or relationship between the number of parts and the size of the parts. We will continue to work on equal sharing tasks as Van de Walle suggested.

I'm wondering how shallow my student's understanding would be if I didn't read the whole framework, and just glanced at the standard.
This lesson planted seeds for fractional reasoning, proportional reasoning and algebraic understanding. I'm sure I'll continue to chip away at this lesson, but I think we are off to a great start.

P.S. Of course, the kids had to have cookies!

Thanks Matt Vaudry and John Stevens for the fun inspiration!

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First Grade Math Fight... Fractions, Proportional Reasoning, and Algebra, Oh my!

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Okay, so I knew this "Math Fight" was coming. (Thank you, John Stevens) The standard I was going after was 1.G.3 Although, this is the performance standard (a very simple standard at first glance) that students need to be able to do, my mathematical goal was different.

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This is a common sight in many of the classrooms I visit:

Students are taught to use these strategiesin order to help them make sense of what they read... in reading class.

Every math teacher knows that many students have trouble solving word problems. Looking at these displays day in and day out got me thinking: might these active reading strategies be useful in math class? Curious to find out, I enlisted Shannon and her third grade class in a little experiment.
In your typical series of lessons on solving number stories, at least as presented in our curriculum, the kids are given the following:

They are provided with some information:

And then asked to solve some related number stories:

We decided to try something different. We provided the kids with the animal information, and asked them to use their active reading strategies:

It's really quite similar to noticing and wondering:

Next, we asked the kids to write questions for their classmates to solve.

Shannon and I vetted them, and threw in some of our own. We typed them up and taped them to index cards. Students got to choose which problems they wanted to solve, and each had a sheet where they had to record both an active reading strategy and a solution:

I tried it out with a fourth grader who has particular problems with number stories.

Some comments, reflections, and questions:

Treating number stories as texts similar to those encountered in reading class could be one way to demystify these often intimidating and challenging tasks.

This strategy can help kids with the first step in solving number stories: making sense of the problem.

One advantage was that the kids didn't have to be taught these strategies. They already knew them.

Integrating reading and math builds curricular coherence. It's second nature for them to "post-it" up their books. Why not their number stories? Why can't math have what reading's having?

I haven't followed up, although I'd like to revisit this activity with Shannon's class before the year is out. There's a lot happening in the MTBoS surrounding the issue of math and literacy. How do we help kids make sense of number stories? What strategies can we provide to help them make sense of text-heavy problem solving scenarios? Noticing and wondering and numberless word problems are two that have been offered up and that have proven benefits. Could this be another? Fire away.

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This is a common sight in many of the classrooms I visit: Students are taught to use these strategiesin order to help them make sense of what they read... in reading class. Every math teacher knows that many students have trouble solving word problems. Looking at these displays day in and day out got me thinking: might these active reading strategies be useful in math class?

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Revolutionize the first 10-15 minutes of your math class! (and beyond) | CollectEdNY

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Revolutionize the first 10-15 minutes of your math class! (and beyond) | CollectEdNY

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You could also file this under community, stupid.

Early on, the goal was learning how to teach. Over the course of the last school year I realized it's time to start paying attention to who I teach.

It starts slowly. A kid tells you their favorite video game. They tell you about the GameBoy they owned when they were younger. You talk about the one you owned at the same time when you weren't as young. A class who laughs at how quaint and ancient the 90s were. One tells you about a hobby they think is embarrassing. Another is going to be the first in the family to graduate high school. The first admitted to college. The first to graduate college. Next they want to know what college is like. Can I actually do it? It sounds so hard. My sister tried to go to college and had to drop out. Mister, by the way, how do you get a driver's license?

A kid tells you how many siblings they have. You find out what their parents do for a living. Another mentions their parents divorce. A parent is deceased. A parent went to jail. They've been in jail. Adopted. Abandoned. Undocumented. Almost pregnant. Pregnant. Gay. Self-harming. Abused. Hospitalized.

Hundreds of them. All of them sharing very personal things with you. All of them assigning some minor or major role in their life to you. Some are just students who had a good time. Others needing you to fill in as a father figure. Most well-adjusted, happy teenagers. Many with major life events that changed them markedly. Many more with common struggles but completely different outcomes. Seniors from years ago who not only graduated college but claim you as their inspiration. One delivers office passes in April. In June you're at the funeral.

This is supposed to be a job? It's just work?

This human aspect is just so hard to convey to the people I know in other work environments. The scale is just so different. My previous job involved juggling the needs of like, 5 people. Now it's 300.

Effectively transferring math content to these kids is an amazing part of my day. Kids constantly impress me with their cleverness and eagerness to figure out what's going on. My number goal is to make our time together their favorite part of the day. In the varied classes I've taught, it was incredibly rare to find a kid who lacked a desire to learn something from me, math or otherwise.

I will always love teaching them math. But what has improved that the most is learning more about my students as people. Summer Camp is only marginally about the content. It's about furthering the relationships. It is very hard some days to deal with the hard truth they drop in these conversations. At the same time a kid tells you how much they appreciate your advice, or the letter of recommendation, or the extra time for a project and it makes your week. It puts pressure on me to do right by them. To be prepared. To have something worthwhile to share and respect their time. I will gladly invest the time planning and grading if it means continuing to have a positive experience together.

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You could also file this under community, stupid. Early on, the goal was learning how to teach. Over the course of the last school year I realized it's time to start paying attention to who I teach. It starts slowly. A kid tells you their favorite video game.

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Fractions on a Number Line | Marilyn Burns Math Blog

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Fractions on a Number Line | Marilyn Burns Math Blog

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Wednesday, March 16th, 2016

The fourth graders I've been working with had been studying fractions for about a week before I taught this lesson. They were using fraction pieces that they'd each cut for a beginning set of fractions-1 whole, 1/2, 1/4, 1/8, and 1/16. Magnetic versions of their fraction pieces were especially useful for this lesson. (Information about these Fraction Kits appears at the end of the blog.)

To prepare for the lesson, I cut a length of string about a yard and a half long and tied four knots in it, spacing them 15 inches apart. I chose this distance between knots to match the length of the blue whole strip they each had and the magnetic whole strip I had.

I talked with the class about how knotted strings were used as early measuring tools, before we had rulers and tape measures. I taped the string on the board and asked them to think about it as a number line. I labeled the knots 0, 1, 2, and 3.

Then I asked the students to watch as I moved my finger along the string from the point marked 0 to the point marked 1. I said, "Tell me to stop when you think I'm halfway between these two knots, halfway between zero and one." I moved my finger slowly, tracing the string, until they shouted, "Stop!"

We checked with the red 1/2 strip to confirm. When I asked them how I could label this point, they said it was 1/2. I wrote 1/2 on the board under the point..

Then I did the same to locate a point halfway between 0 and 1/2. We checked the distance from zero to the point with the one-fourth fraction piece, and I labeled the point 1/4. I repeated to label 3/4 and 1 1/2.

I then said something like, "So here I used fractions to label points on a number line, just as we used zero, one, two, and three. So I think we can say that fractions are also numbers." I'm not sure they all were completely convinced, but there were nods and no rebuttals.

Next I talked with them about different names for the same fractions. I reminded them that they call me "Ms. Burns" but that some people call me "Marilyn." We talked about other fractions we could use as labels for 1/2. As students made suggestions, I listed them on the board under 1/2 on the number line and verified them when possible with fraction pieces.

Then I wrote 50/100 on the list, for which we didn't have fraction pieces. The students seemed clear that it was equal to 1/2. They offered other fractions that they thought were also equal to 1/2, and I added them to the list. I asked them to talk with a partner and come up with a statement that explains how you'd know by looking at a fraction if it was another label for 1/2.

There were two views. Jaleya said, "A fraction is equal to one-half if two times the numerator equals the denominator." (Actually this was her third iteration-she stumbled a few times and each time I asked her to try again until she could deliver her thought clearly.) Alisha had a different view. She said, "A fraction is the same as one-half if you can add the numerator to itself and that gives the denominator."

I then asked them for different numbers we could write as labels for 1, and then for 1 1/2.

When Jhoany offered 24/16 for 1 1/2, most of the students seemed confused. I wrote Jhoany's suggestion on the board as a conjecture to return to at another time. I did this to give his classmates time to think about his idea.

The day after the lesson, on a prompt from Sara, his regular teacher, Jhoany wrote a letter to me about why he thought 24/16 was equal to 1/2. Here's what he wrote:

I'm not in class every day, so I sent Sara a response for her to share with Jhoany. I'm curious about how he responds. Stay tuned!

Learning about fractions, as with all new ideas, takes time. Students benefit from exploring with concrete materials, having experiences to express how they are reasoning, and listening to the ideas of others. They need time to revise, cement, and extend their thinking. I plan to return to this experience with fractions on a number line as the students learn more about comparing and ordering fractions.

About the Fraction Kit
The Fraction Kit is the single most effective manipulative material that I rely on for teaching fractions. It's been a mainstay of my teaching repertoire for years and I've used it with students in grades 3 through 8, in whole-class instruction and in small groups when providing intervention. The Fraction Kit is featured in various of my resources-including About Teaching Mathematics (p. 422-426), in Fractions A, B, and C; and Teaching Arithmetic: Lessons for Introducing Fractions. It's possible to cut the fraction strips from construction paper and make the number cubes for the games, but it's also possible to purchase a Fraction Kit that has materials for 30 students.

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The fourth graders I'm working with on a regular basis are learning about fractions. During a class conversation, one student declared, "Fractions aren't numbers." Most of the others in the class agreed. I tried to help with the misunderstanding by teaching a lesson about placing fractions on a number line.

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My job encouraging teachers to explore and experiment with some of the defining MTBoS routines is made easier when there's an explicit connection between those routines and what their students are asked to do on our program's assessments. When those stars align, I feel we're close to accomplishing real change.
So why was I heartened to see the Everyday Math second grade mid-year assessment several weeks ago? Let's take a look:

MTBoS Routine: Which One Doesn't Belong?

Mid-Year Assessment: Question 2

Hey! What's this? Which one doesn't belong? I'd never before seen an item like this on one of our assessments. There were a variety of responses:

MTBoS Activity: Counting Circles

Mid-Year Assessment: Question 6

Completing counts and noticing patterns are reinforced by the counting circle routine.

The kids are starting to understand how number lines are constructed, especially how the numbers need to be spaced at regular intervals.
OK, we still have a ways to go...

...but now we know who needs more time hanging out on the clothesline.

Three MTBoS routines, three aligned assessment questions.
And I almost forgot about this item, which showed up in a grade 3 journal several months ago:

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My job encouraging teachers to explore and experiment with some of the defining MTBoS routines is made easier when there's an explicit connection between those routines and what their students are asked to do on our program's assessments. When those stars align, I feel we're close to accomplishing real change.

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Teaching the Crossed Chord Theorem | Continuous Everywhere but Differentiable Nowhere

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Teaching the Crossed Chord Theorem | Continuous Everywhere but Differentiable Nowhere

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Continuous Everywhere but Differentiable Nowhere

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At the end of the past summer (oh how long ago that was! glorious days of freedom!), I hunkered down at a coffeeshop with @jacehan in preparation for this school year. I was fixing up some of the things we did last year in geometry. One thing I wasn't pleased with was how we taught the crossed chord theorem...

So I created a totally new approach. Instead of having students discover the theorem, I would work backwards. Here is the TL;DR from my last post - after I had created the activity.

The TL;DR version: students investigate all quadrilaterals where the diagonals satisfy the property that ac=bd. Students are guided to make a conjecture which we as teachers know will be wrong. Then we show a counter-example to blow their conjecture up. And them bam: they have to try again. Using geogebra and some more encouragement, students discover that all cyclic quadrilaterals satisfy ac=bd. And so the circle emerges out of this investigation of quadrilaterals and diagonals. This is, then, the crossed chord theorem. Which students got at by investigating quadrilaterals. Weird. Now they are in a prime place for wondering why the circle shows up. Proof time!

When I shared the activity, I got a couple suggestions from @k8nowak and @bowenkerins and so I modified it with a single tweak which made it oh so much more powerful. In this post, I will talk about my experience implementing the activity, as well as share the modification I made. However I entreat you to read the original post as I'm not going to outline everything again! So go read! Okay? Okay.

(1) First off, the change. I made a change to the very last question on the sheet... Instead of having students look for blermions "in the wild," I had them fix three points and find a bunch of different locations for the fourth point (so the quadrilateral would be a blermion).

[docx: 01 Crossed Diagonals [new]]

At the end of class, I had students fill out a google form with their possible fourth points location.

(2) In class, when students were filling out the conjecture in #4, I saw a number of interesting conversations happening. Their conjectures were essentially: (a) all blermions need at least one pair of parallel sides, (b) all blermions have supplementary adjacent angles, and (c) a blermion has opposite angles supplementary. Most students didn't find any kites that were blermions, which is why they came up with conjectures (a) and (b). But when the few students who found blermion kites said this to the class, we realized that (a) and (b) couldn't hold anymore. But conjecture (c) was a possibility still.

Now to be clear, I was expecting conjectures (a) and (b). I was floored when not one but two groups out of five wanted to persevere and find a good conjecture, and used geogebra to measure angles. It was awesome. And it led to a great discussion later on. More on that later.

(3) I asked kids why I had put question #5 on the sheet... what might have been my motivation? I liked asking that question and having groups discuss, because they all recognized that by only looking at "nice" shapes (which, granted, I asked them to do), they could only make limited conjectures. And as soon as they see the blermion in #5, most conjectures would go out the window. The point? To show students that all quadrilaterals aren't "nice."

(4) The moment when students saw all their group's data together in #6... Well, two of my groups got to this point in class. It was ... incredible. Kids had their minds blown. Something totally unexpected happened.

For the other groups, I shared the class data (from the data they entered in the spreadsheet):

Holy cow! It is so beautiful! All possible fourth points of a blermion seem to lie on a circle!

(5) Students then wrote a conjecture, and we said if the conjecture were true, we'd suspect (from the Always Sometimes Never questions in #3) that all squares, rectangles, and isosceles trapezoids could be inscribed in a circle. And we discussed how we were going to prove the opposite: If you have a cyclic quadrilateral (yes, I introduced that term!), that product theorem thingie (a)(c)=(b)(d) holds with the diagonals. Okay, we were a bit more formal, but that was the crux of things.

(6) Before proving that, I wanted to exploit the conjecture (c). I had students prove that all cyclic quadrilaterals had opposite angles that were supplementary. They struggled a bit with this, but once they had their insight, BOOM. (They used the inscribed-central angle theorem thingie - a central angle is half its corresponding inscribed angle).

(7) Then I left students to prove the crossed chord theorem. I gave them this sheet:

[docx: 01 Crossed Diagonals (proof)]

Almost all kids got to the point where they recognized two pairs of similar triangles. And they recognized that if they could prove one pair of similar triangles were congruent, they could set up a proportion and be done! But the problem was proving the triangles similar. Almost all groups got stuck here - and even though I said: you're almost there! Think about the inscribed-central angle theorem! - they couldn't progress. I didn't do a great job of knowing what to say next.

What I did was show them this (which they had created earlier). For some reason, this did not work for them as a hint.

In the future, what I should do is just highlight an arc for them... and say "this arc can guide you!"

Maybe that will work better?

However, eventually all groups got the proof.

(8) At this point, I had students start solving problems. Two with quadrilaterals and two with chords.

Again, I asked them why I included had the second two types of questions, and had the discuss in groups. They recognized that the theorem didn't need to be stated with cyclic quadrilaterals... Instead it held if we are talking about two line segments in a circle (at that point, I introduced and defined the terminology "chord"). Then I had students write the theorem we had proven without reference to the quadrilateral, and we went around and shared and critiqued the wording.

***

I don't always love the stuff I come up with. Sometimes it flops. Sometimes it's pretty good. But rarely do I think it's so awesome that I would give it the stamp of "highly recommended." This gets that from me. It is interactive, there is a moment when kids's minds are blown, and it ties together so many interesting ideas.

For context, I did this after we did our unit on similarity. We then proved that the base angles of isosceles triangles were congruent. We then used that to prove the inscribed-central angle conjecture (download here: A Conjecture about Inscribed Angles). And then this. It flows so nicely.

***

Yesterday, as we were wrapping this all up, I said to my kids:

"This thing we just proved about circles and chords... this is at the top of a mountain... this theorem is based on lots and lots of other things. If I gave you a bunch of circles with two intersecting chords in it at the beginning of the year, and said, give me some conjectures about this, I doubt you would ever have stumbled upon this... or if you did, it would have taken you a long time and it would have been an accidental discovery. You still wouldn't have known why it was true. But now you have built so much mathematics throughout the year that this wasn't an insurmountable feat. What ideas did this theorem lie on?"

And we even had more. We had to know a lot to get there. But wow, it might have seemed impossible at the start of the year, but it was totally doable with all the tools we've put in our toolbelts. And how wonderful and inspirational is that?!
Update: Here is a post about an extension I did on this - involving merblions.

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At the end of the past summer (oh how long ago that was! glorious days of freedom!), I hunkered down at a coffeeshop with @jacehan in preparation for this school year. I was fixing up some of the things we did last year in geometry. One thing I wasn't pleased with was how we taught...

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Why Are You Using That Problem? | Robert Kaplinsky - Glenrock Consulting, LLC

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Why Are You Using That Problem? | Robert Kaplinsky - Glenrock Consulting, LLC

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McSquared: Rethinking How We See Mistakes

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McSquared: Rethinking How We See Mistakes

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I had a flashlight moment recently.

I was helping a student to learn how to balance chemical equations. I had done a few examples, and then I had her try one. Part of the procedure calls for a certain amount of trial and error; you try a number in one part of the equation, then you track how it fans out in the rest of the equation. She hesitated for such a long time, that suddenly I realized what was paralyzing her. She thought she was already supposed to know what the right "guess" was. I told her, put any number, fully expect it to be wrong, and then we'll use it to get the right one. Her response was immediate, she just put down a number, and it was beautifully wrong, because even though I think she had intended it as a "Here you go I told you how stupid I am" moment of self-pity, she immediately said "Oh wait, no, this would be better." And it was. It wasn't right, but it was better. Her wrong answer pointed her toward a better one. It seemed like she had needed permission to jump in with a mistake, before she could even experience the unlocking that happened microseconds later.

This has led me to believe that, at least sometimes, we should be calling those "mistakes" something else. Anything that leads to illumination isn't a "mis"-anything, it's progress. So from now on, we're changing our attitude towards mistakes in my class, starting with what we're calling them - flashlights. If the growth mindset movement is correct, then what we call things affects how we see them and react to them, both on an emotional level and a cognitive one. I think "flashlight" is more positive, and, more importantly, it's more accurate. Those flashlights are our guides. They show us where the gaps are between what was taught and what was learned. (read more about these gaps in Dylan Wiliam's "Embedded Formative Assessment".)

Here are a few other ideas I've had about changing my own class's attitude toward mistakes.

When a student makes a mistake, I'm going to praise them at least as much as when they don't make a mistake. I want them to know that at that moment, they are straight up legit teaching someone something. I've been saying "I'm so glad you said that!" or "That's the best mistake I've seen today!" or "I was hoping someone would make that mistake!" That last one I hope makes them feel like they're my secret accomplices in teaching. It also creates a sort of suspense in my class, like everyone's waiting for that magic mistake to happen, the same as if it's a jackpot they're all trying to hit. Because that's what it's going to feel like when they hit it.

We can legitimize mistakes as learning opportunities if we not only talk about what the mistake was, but where it came from. Because mistakes almost always have some truth in them. For example, when kids distribute incorrectly like this: 3(2×4 x) = 3×2×3×4 x, it helps to say to them - I know why you did that, you were thinking 3(2 + 4 x), which would be 3×2 + 3×4 x. I think it's a relief for them to know that the way they think has some grain of logic to it, at least enough so that another person can backtrack with them to where attention is needed.

If I HAVE to use the word mistake, then I'll use an adjective like beautiful, glorious, or brilliant before it, because I don't want mistake to be a bad word - I want it to be a sign that thinking is happening, neurons are firing, lost souls are finding their way. Those are all beautiful and glorious things to happen in class, and I want as many of them as I can get.

I've also been thanking my students for their mistakes, because they're doing some heavy lifting for us all. For example, the other day I asked if log 3 + log 5 could be replaced with a single logarithm using a log property. One student said no, because they didn't have the same base. Flashlight! She thought the 3 and the 5 were the bases. Not only did this show me that at least one person was looking for the base in the wrong place, but was also not aware that the unwritten base was 10. Two things learned in one shot because of her, so this was a double flashlight, and I thanked her. Later, another student thought that:

would lead to xz = y. Flashlight! At least one person was cross multiplying instead of doing fraction multiplication. They learned when that works, and when it doesn't, and I learned that I am so not ever going to use cross-multiplication ever again. Thanks, kid!

I'm thinking it would be nice if the other kids thanked them too. I'm not sure if I'll actually get them to, because that would probably be a bit forced. But the way I see it, the kid making a glorious mistake right away in class, as soon as we've done something new, is doing everyone a favour. Everyone else can now avoid making it later, when they're all alone. If that happens, they'll either not have any idea that they've made one, or they won't have anyone to help them straighten it out. Much better that it happens when we're all there.

This is a big one, and it's probably not going to be popular, even with me. I think our kids need us to not just SAY it's ok to make mistakes, we need to BE okay with them.

Embracing my own mistakes: Because I'm not just talking about their mistakes, I'm talking about mine too. The ones I'm so careful not to ever let them see me doing. That's why whenever I have to figure something out in front of them I get so totally flustered that I usually say, "Ahem, well kids, I don't want to take class time to do this, I'll ahem figure it out later and get back to you. Move along now, nothing to see here." And I thereby give them the message "Mistakes are great! For you that is, not me. For me they're embarrassing, humiliating, and scary and they never happen anyway so yeah." I need to face it, enlist input, and maybe even get help, for example ask "Why do I doubt my answer is right? What kind of answer would make more sense? Up until where did you get the same thing?"

Problem solving on the fly: When a teacher only ever shows students how to solve a problem that they (the teacher) already know how to solve, that's great, and of course I do that, but we're really being disingenuous if it stops there. We're showing them a nice sequence of rules, being fluidly followed by a calm, confident person who is already in possession of the answer, and who therefore couldn't be any less like them when they're solving a new problem. I think we need to give our kids the chance to watch how we handle something that is truly new, something in which we truly have no idea what to do first. And show them how we're not afraid of that feeling, that nobody needs to be afraid of that feeling, because everybody feels that feeling! Let's get stopped in our tracks together, try stuff together, mess up, go back, sleep on it, admit we're secretly looking for the answer key...you know, normal life.

But if a mistake happens during a test, it's very bad, right? We all know what marks do to kids, and how they absolutely halt all learning, whether the marks are good or bad. So yes, of course, I don't want mistakes happening then, and nobody does. But they will happen, at any time, so I'd rather get the kids armed with strategies to detect and fix them. And most importantly, remain calm.

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I had a flashlight moment recently. I was helping a student to learn how to balance chemical equations. I had done a few examples, and then I had her try one. Part of the procedure calls for a certain amount of trial and error; you try a number in one part of the equation, then you track how it fans out in the rest of the equation.

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Following Learning: Five regular routines

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Following Learning: Five regular routines

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The idea is that you can justify the choice of any of the four items as the odd one out.

Students write three numbers on a blank number line on their whiteboard, one too big, one too small and as close as you can ( Graham Fletcher has some ; these can be extended into whole lesson activities which are great modelling activities - and then of course there's our regular, estimation180 ).

See, think, wonder

Last year it was notice and wonder, but I like the differentiation of notice into see and think, and feel it's worth distinguishing the two. (Both are brilliant though!) I'm reading Making Thinking Visible, and this is one of their routines.

The point is just to present an image and see what people see, preferably recording it all.

I've also adapted this to be what equations do you see, often presenting an image of Cuisenaire rods:

I've blogged about this before. The idea is to count from different starting points, with different jumps. We do it round in a circle. Then we stop, and think about what the number three or four jumps on would be. Students share how they know.

I haven't used these yet, but this is a great idea, and Nat Banting is developing a really useful site with lots of images: fractiontalks.com - watch this space!

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The idea is that you can justify the choice of any of the four items as the odd one out. Students write three numbers on a blank number line on their whiteboard, one too big, one too small and as close as you can ( Graham Fletcher has some ; these can be extended into whole lesson activities which are great modelling activities - and then of course there's our regular, estimation180 ).

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Well week three of explore math twitter blogosphere. And I am late. Anyhow.......

Here is my better question that I like to explore with my students:

"What are the criteria for a good question?"

I do this lesson with my grade 10 applied students every semester. I have written about the details of this lesson which are here.

Here are the posters over the years that have come out of this activity.

Lots of criteria that have shown up over the years.

Co-Creating criteria for what makes a good question with your class, early in the course, can cause your students to ask better questions throughout your course.

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Well week three of explore math twitter blogosphere. And I am late. Anyhow....... Here is my better question that I like to explore with my students: "What are the criteria for a good question?" I do this lesson with my grade 10 applied students every semester.

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Better Questions – Two Truths & One Lie | Mr. Orr is a Geek.com

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Better Questions – Two Truths & One Lie | Mr. Orr is a Geek.com