If you’ve never seen me talk in a large group you’ll have to imagine the energy, cadence and passion you see in my ShadowCon talk brought to this longer form.

Thanks to the Twitter Math Camp organizing committee for inviting me to talk, and to the whole community for supporting my work over the years. It means a lot; I hope to return the favor many times over.

—

**Lisa Henry’s introduction:**

Christopher Danielson teaches and writes in Minnesota. You may know him through his documentation of his children’s mathematical antics on Talking Math with Your Kids, through his exhortations not to share bad “Common Core” homework assignments on Facebook, or through his shapes book *Which One Doesn’t Belong?*

Of course you may not know him at all. But if you do, you’ll recognize that he holds little sacred besides the responsibility we take on in this profession to foster the growth of young minds.

He is currently working on a teacher guide for *Which One Doesn’t Belong?* to be published by Stenhouse in the spring. He encourages each and every one of you to promote the heck out of *Common Core Math for Parents for Dummies*. Most of his time this summer is devoted to bringing Math On-A-Stick—a new event to support children and caregivers in informal math activities—to the Minnesota State Fair, and he continues to work with Desmos on developing online networked classroom activities such as Polygraph and Function Carnival.

—

This is a very American talk about teaching. From what I’ve learned about teaching in other countries with robust educational systems—Singapore, Finland, Japan, Germany, and so on—the U.S. is unique in its tradition of sink-or-swim for teachers.

We equip new teachers with a modest set of tools and experiences, and we say *Do the best you can with what you’ve got!*

At the policy level, we understand that this is a disaster. But in the American fashion we try to legislate and standardize our way to improvement. We issue pacing guides and measure fidelity to adopted texts. And of course we measure teacher quality by testing students in an effort to standardize learning.

In this sense, the message of my talk today is a very American one. My message is this: *Find what you love. Do more of that.*

Viewed one way, this is advice to teachers trying to survive and to serve their students well in the era of NCLB and high-stakes testing. (There are probably several such folks in our midst today.)

And that will be a valuable takeaway, but it’s not the heart of my talk. The heart of my talk is more forward-looking and hopeful.

Sending minimally prepared teachers into the field, leaving them to figure it out on their own, and then evaluating whether they have—these things we do well in this country. If you believe that quality comes from sorting out the bad apples, then we’ve built a good machine for this, and the major impediment to improving it is the unions. I assume that this is familiar rhetoric.

What we don’t do well is orient and induct teachers to a community of professionals. We don’t structure our communities to draw on the diverse strengths and passions of its members. This is something that I understand those other nations I mentioned do much better than we do.

Community.

For me, the group assembled here, together with the ones who would be here if they could, and many more of the teaching professionals we interact with—whether regularly or sporadically—for me, this group is a community. A community that grows and changes in response to the contributions of its members. It’s not a community that agrees on everything—no community can while remaining honest, open and vibrant. Instead, disagreements offer healthy opportunities for the community and its individual members to grow.

So the hopeful vision in my message (Find what you love. Do more of that.) is that in identifying where *your *heart is in this profession, you can strengthen your voice and focus your efforts as you contribute to and help shape this community. What our larger American educational system does poorly—foster a professional community that grows and responds to the diverse strengths of its members—the MTBoS does quite well.

So I put two big questions in front of you:

What do you love?

How can you incorporate more of that in what you do? (In your classroom and in your community)

When I ask, *What do you love?* I don’t want to hear that you love…

Rectangles or stats or 3-Act Lessons or Spirals or Technology or Groups.

I want you to dig deeper. Those things embody what you really love. Whatever you are truly passionate about is bigger than these things. If you can say *why* you love rectangles or stats or whatever, you’ll be closer to the kind of thing I have in mind.

So now I’ll tell you what I love and how that—in the context of what I have to do—helps to guide my teaching and my contributions to our community.

This is so nerdy. I really hope this is a safe space for this.

I love ambiguity.

The spaces between the certainties are much more interesting to me than the certainties themselves. Ambiguity can provoke wonder, surprise, reflection and clarification.

I’ll share with you how I incorporate this love in the work I do in the classroom and in our community. Let’s start with a video.

This video lacks ambiguity.

Children are smarter than this.

Children can handle ambiguity, which is what I have found so appealing about the *Which One Doesn’t Belong* framework. (Megan Franke and Terry Wyberg mentions)

In case you aren’t familiar with it, I’ll give you the rap I give kids.

[insert]

Let me tell you what children say in response to this richer set of shapes.

[do that]

As they talk about these things, I summarize, paraphrase, probe, review and restate. By the time we’re done, we have a list of properties of shapes. Which of these properties are they *supposed* to use in deciding which one doesn’t belong? Which of these properties are important? Which ones matter? It depends.

Here’s another set of shapes.

Sometimes ambiguity comes from studying new objects for which we don’t have a repertoire of vocabulary.

Which properties are important? Which ones matter?

We don’t know yet when we’re looking at a new class of objects. But when we do agree that a property is important, we can name it.

So all that vocabulary which is associated with geometry—that vocabulary isn’t important on its own. Instead it points to important properties, or to properties which *somebody* has deemed important.

(A non-math example: at some point, it became clear that this growing collection of math teachers online needed naming. The community’s historians are still working on the lineage of the phrase *mathtwitterblogosphere* and the abbreviation *mtbos*, The point is that the name exists because there was a thing that needed naming)

Here is what kids do with the ambiguity of the spirals.

[say what that is]

One last WODB example.

The shape in the lower right is the one that provokes discussion. (get to vertices)

Returning to the theme of what you have to do…

In my capacity as a College Algebra teacher, I have to teach rational functions.

Lots of rules and vocabulary and certainty are associated with standard textbook treatments of rational functions. There are asymptotes (vertical, horizontal, oblique…), rules to go along with locating these. There are zeroes and intercepts and symmetries and on and on…

So I played Polygraph: Rationals with my students.

The design of Polygraph is that it puts you in the position of needing to describe to somebody else what you see before you have a shared vocabulary for it—as with WODB. But now *importance* has an implicit definition. A property is important if it’s useful for distinguishing between functions, and for helping your partner to do that too.

At the beginning of the game ,we shuffle the functions to suggest to you that location in the grid isn’t important. But maybe you think it is. So you ask about it. And you’re likely to get burned.

Orientation, by the way, is important in this context. The difference between 1/x and -1/x is one of orientation.

So orientation doesn’t matter in plane geometry, but it does matter in coordinate algebra.

And the graph on the left is a function, but the one on the right isn’t.

Orientation matters.

But these are both squares.

(Tangentially related…How far can you turn a parabola and have it still be a function?)

I’ll finish with an example from the community. Meg Craig cares about kindness and empathy. She wrote about this on her blog recently. She urged us all to remember that *We as teachers are all trying, to the best of our ability, to have students reach the best of their ability. *** **If you haven’t read this post, you need to.

That kind of thing has a lasting impact on our community. Just the other day, I got worked up about fences.

There is so much wrong here. Unambiguously wrong, and I called it out. Then I was thinking later that day. *OK. That was angry Triangleman. What could kinder gentler Triangleman do?*

Well, what do I do in my classroom? What do I do at home with my children? In both of these contexts, I educate patiently. I accept people for where they are and I try to help them see new perspectives; to think differently about things than they do right now.

What does that look like on Twitter? I don’t know. It really is such a great medium for ranting. But I do know that the reason I’m asking myself this is that MathyMeg brought her strengths and passions to our community.

I told you to find what you love, and to do more of that. I told you that I love ambiguity. Maybe you’re the opposite of me. Maybe what you love is certainty in mathematics. How can you help your students appreciate and understand the unique nature of mathematical truth (different from all other disciplines)?

Maybe your heart is with the beauty of geometric forms, or the rhythmic regularity of patterning. Maybe you love how statistics can inform us as we strive to make equitable decisions in an unjust world.

Truth, beauty, regularity, fairness…

Each of these is a more important grounding for your classroom perspective than a pacing guide or textbook sequencing. But none of them is antithetical to these either. What you love can be found in what you have to do.

So my message to you is simple. Name what you love—be explicit about what makes your mathematical heart sing; what resonates in the depth of your teacher soul—and look for it in every corner of your professional life. Share with your students and share it with us, your colleagues and your community.

]]>Naturally I bought some.

Tabitha (8 years old) asked—as she does in these scenarios which occur with great frequency—*Are you just buying that because it’s mathy?*

Yes, sweetie. Yes I am.

But how to put them to use?

After many rejected ideas, here’s my favorite.

Here are the contents of one pack.

That’s 5, 2, 9, 1, 3, 2, 4, 3, 9. Their sum is 38.

I’m setting the over/under on the sum of the next pack at 41. Do you want the over or the under? Why?

Play along with your questions and answers in the comments.

I’ll open the pack on Wednesday, May 27.

Tagged: probability, snacks ]]>

I want to tell you about a vision of a beautiful thing, and I want to ask you to help make it happen.

*Math on a Stick logo by Emily Bremner Forbes, who makes beautiful things. Many thanks, Emily!*

*Math on a Stick* will be an annual event at the Minnesota State Fair (*12 days of fun ending Labor Day!*) that engages young children (4—10 years old) and their caregivers in informal mathematics activity and conversation using the Fair as a context.

- Parents will push children on a protractor swing so that together they can notice the angles and fractions of a circle the children travel through.
- Parents and children will use beautiful tiles to make shapes and intriguing patterns.
- They will comb the fairgrounds looking for groups of many different sizes, asking questions such
*How many mini donuts are in a bag?, How many sides does the Agriculture-Horticulture building have?*and*Why is it so hard to find a group of 17?* - They will notice the rotational and reflection symmetry in a wide variety of plants and flowers, then copy these symmetries by making a paper flower to take home.

Math on a Stick has four components:

- The Math-y Midway
- The Garden of Symmetry
- The Number Game
- Visiting mathematicians and mathematical artists.

Find out more about each of these below.

The major question now is whether Math on a Stick happens for the first time this year or next. The organizing body is the Minnesota Council of Teachers of Mathematics. The Math Forum is by our side. Max Ray and Annie Fetter from the Math Forum plan to come to Minnesota to help run the event. The Minnesota State Fair and Minnesota State Fair Foundation love the idea. We just need to convince all parties that it is possible to pull this off in the coming three months, and we need to locate the funding to make it happen.

We’ll need help with three things:

- Volunteer hours this summer, before the Fair
- Volunteer hours during the Fair
- Funding

Of course I expect that most who heed this call will hail from the great state of Minnesota, but I encourage others to consider scheduling a visit. This will be a wonderful event, and the Minnesota State Fair is truly a grand spectacle.

Before the Fair, we’ll need help finding and creating the things that will make the event go.

During the Fair, we’ll need help staffing the event. It runs 9 a.m. to 9 p.m. August 27—Sept. 7. We’ll have have about four shifts a day and we’ll require multiple people staffing each shift.

If we get Math on a Stick up and running this summer, one of our first orders of business will be to establish our volunteer website. Please check your summer calendars, pencil us in, and keep an eye on this blog for more information.

If you (or someone you know, or an organization you are involved with) are in a position to help fund Math on a Stick, get in touch with the Minnesota State Fair Foundation to let them know you’d like to help make this happen. Our overall budget is on the order of $20,000.

Here are specifics on the four components of Math on a Stick.

The major activity at Math on a Stick is The Number Game. Adapted for math from the Alphabet Forest’s Word Game, children and parents are challenged to find groups of every size 1—20 at the fair. Examples: A corn dog has 1 stick, a cow has 4 legs, the Ferris Wheel has 20 carts.

Players receive a form they carry with them around the fair to record their findings, and can return with a completed form to claim a ribbon. Additionally, players can email, tweet, and post to Instagram, their Number Game fair photos. These are curated by Math on a Stick volunteers and posted to a public display that resets each day so that collectively State Fair attendees recreate daily a new visual answer guide to the Number Game.

A protractor swingset, tables with fun tessellating tiles, and images from *Which One Doesn’t Belong? *and a (forthcoming) counting book to play with and discuss.

Flowers are grown in planters along a path. As you walk from one end of the path to the other, you pass flowers with increasingly complex symmetry. Grasses (with one line of symmetry) are near one end. Irises are a bit further along (with three rotational symmetries), and sunflowers are near the far end (with MANY symmetries). Visitors to the Garden of Symmetry are invited to carry a tool consisting of two small mirrors taped together to investigate symmetries in the garden and the interpretive signage.

An activity area is set aside for a daily visit from a mathematician or mathematical artist. Each provides engaging, hands-on math activities during a scheduled period each day. We will draw upon talent from Minnesota, as well as nationally (budget allowing).

For full details on the event, have a look at our Math on a Stick white paper.

Hit me in the comments with any questions you have.

Get in touch with me through the About/Contact page on this blog.

Please help us build this thing. It’s going to be great!

Tagged: math on a stick, minnesota state fair, Minnesota State Fair Foundation, state fair ]]>

Last week I was at two schools: Dowling in Minneapolis and Echo Park in Burnville. I talked with one kindergarten class, three first grade classes and four second grade classes. I have learned a lot.

In particular…

Young children find composing and decomposing shapes to be much more compelling than adults tend to. They nearly all saw the bottom-right figure here as being a square and four circles. Adults *can* see that, of course, but we are more likely to think “not a polygon”.

On that note, I am now quite certain that we spend way too much time having young children sort polygons from non-polygons. That bottom-right shape has many more interesting properties than that of *not being a polygon*.

For example, a class of second graders on Friday were variously split on the number of “corners” that shape has. Is it 0, 4 or 8? Second graders can understand each other’s arguments for and against these possibilities.

These arguments can lead to the reason that mathematicians use *vertex* instead of *corner*. “What exactly is a vertex?” is a much richer and meatier mathematical question than “How many vertices does this shape have?” But if that latter question only comes up with respect to convex polygons, then it is unproblematic and not interesting for very long.

So imagine for just a moment that the lower-right figure has 8 vertices (and it wouldn’t be too difficult, I now believe, to get a classroom full of second graders to agree to this perspective, whether it agrees with the textbook definition of *vertex* or not).

Now kids can work on stating exactly what makes a *vertex*.

And what makes a vertex is going to be awfully close to what makes a point of non-differentiability (large point at apex of figure below).

I’m telling you: in twenty minutes with second graders, we can get very close to investigating things that are challenging for calculus students to describe. My point is that second graders are ready to do some real mathematics, and that sorting polygons from non-polygons is not the road to it.

Other things I found interesting:

• When kids give us something close to the answer we expect, it is easy to fool ourselves into thinking they understand. Example: on the page below, one boy said about the lower left shape that “if you tip your head, it’s a square.” A couple minutes later, it occurred to me that there might be more to the story. I asked whether the shape *is* *a square* when your head isn’t tipped, or whether it only *becomes* *a square *when you tip your head. He confirmed that it’s the latter.

• Another second grade class was unanimous that the one in the lower right doesn’t belong because it’s not a square. When I asked “is the lower left now a square, or does it only become a square when you tip it?” the class was evenly split. This was surprising to both me and the classroom teacher.

• *Diamondness* is entirely dependent on orientation in the mind of a K—2 student.

• The 1:1 correspondence of sides of sides to vertices in polygons is not at all obvious to young children. I sort of knew this but saw it come up again and again in our work.

• A first grader said that the spirals below didn’t belong with all the other shapes we had seen that day because “you can’t color them in”.

Even the unshaded ones that had come before *could have been colored in*, you see. These spirals you cannot color in even if you try. What a brilliant and intuitive way into talking about closed figures—those that can be colored in.

Tagged: elementary math, geometry, polygons, shapes ]]>

I created a shapes book for all ages. The digital version is free for now. Details are in this post over at TMWYK.

Tagged: book, geometry, shapes ]]>

- Lines
- Parabolas
- Rational functions
- Hexagons

The hexagons will be familiar to long-time readers of this blog.

I have run the parabolas version in College Algebra, and the hexagons version in my Ed Tech course. It was a huge hit both times—lots of conversation happened both electronically and out loud in the classroom. It’s a ton of fun.

I am especially pleased with the rational functions version. It makes for challenging work—even among the mathematically astute Team Desmos in recent trial runs.

Read the Desmos blog post on the matter if you like.

Tagged: desmos, functions, game, hexagons, polygraph ]]>

Griffin (10 years old, 5th grade) had been doing some multiplication in class today. Somehow his scratch paper ended up on our couch.

Here is one thing I saw.

Naturally I wanted to ask the boy about it. He consented.

Me:I see you were multiplying 37 by 22 here.

Griffin(10 years old): Yeah. But I got it wrong so I did it again with the lattice.

Me:How did you know you got it wrong?

G: I put it in the answer box and it was wrong.

It turns out they were doing some online exercises. There is an electronic scratchpad, which he found awkward to use with a mouse (duh), plus his teacher wanted to be able to see their work, so was encouraging paper and pencil work anyway.

I was really hoping he would say that 37 times 22 *has* to be a lot bigger than 202. Alas he did not.

Anyway, back to the conversation.

Me:OK. Now 37 times 2 isn’t 101. But let’s imagine that’s right for now. We’ll come back to that.

G: Wait. That’s supposed to be 37 times 2? I though you just multiplied that by that, and that by that.

He indicated 7 times 2, and then 3 times the same 2 as he spoke.

Me:Yes. But when you do that, you’ll get the same thing as 37 times 2.

A brief moment of silence hung between us.

Me:Whatis37 times 2?

G: Well….74.

Let us pause to reflect here.

This boy can think about numbers. He got 37 times 2 faster in his head than I would have with pencil and paper. But when he uses the standard algorithm that all goes out the window in favor of the steps.

THE STEPS WIN, PEOPLE!

The steps trump thinking. The steps trump number sense.

The steps triumph over all.

Back to the conversation.

Me:Yes. 74. Good. I like that you thought that out. Let’s go back to imagining that 101 is right for a moment. Then the next thing you did was multiply 37 by this 2, right?

I gestured to the 2 in the tens place.

G: Yes.

Me:But that’s not really a 2.

G: Oh. Yeah.

Me:That’s a 20. Twotens.

G: Yeah.

Me:So it would be 101 tens.

G: Yeah.

I know this reads like I was dragging him through the line of reasoning, but I assure you that this is ground he knows well. I leading him along a well known path that he didn’t realize he was on, not dragging him trailing behind me through new territory. We had other things to discuss. Bedtime was approaching. We needed to move on.

Me:Now. We both know that 37 times 2 isn’t 101. Let’s look at how that goes. You multiplied 7 by 2, right?

G: Yup. That’s 14.

Me:So you write the 4 and carry the 1.

G: That’s what I did.

Me:mmmm?

G: Oh. I wrote the one

Me:and carried the 4. Yeah. If you had done it the other way around, you’d have the 4 there [indicating the units place], and then 3 times 2 plus 1.

G: Seven.

Me:Yeah. So there’s your 74.

This place value error was consistent in his work on this page.

Let me be clear: this error will be easy to fix. I have no fears that my boy will be unable to multiply in his adolescence or adult life. Indeed, once he knew that he had wrong answers (because the computer told him so), he went back to his favorite algorithm—the lattice—and got correct answers.

I am not worried about this boy. He is and he will be fine.

But I want to point out…I *need* to point out that this is exactly the outcome you should expect when you go about teaching standard algorithms.

If you wonder why your kids (whether your offspring, your students, or both) are not *thinking *about the math they are doing, it is because the algorithms we (you) teach them are designed so that people do not have to think. That is why they are efficient.

If you want kids who get right answers without thinking, then go ahead and keep focusing on those steps. Griffin gets right answer with the lattice algorithm, and I have every confidence that I can train him to get right answers with the standard algorithm too.

But we should not kid ourselves that we are teaching mathematical thinking along the way. Griffin turned off part of his brain (the part that gets 37 times 2 quickly) in order to follow a set of steps that didn’t make sense to him.

And we shouldn’t kid ourselves that this is only an issue in the elementary grades when kids are learning arithmetic.

**Algebra**. The quadratic formula is an algorithm. Every algebra student memorizes it. How it relates to inverses, though? Totally obfuscated. See, we don’t have kids *find *inverses of quadratics because those inverses are not functions; they are relations. If we *did *have kids find inverses of quadratics, they could think about the relationship between the quadratic formula:

and the formula for the inverse relation of the general form of a quadratic:

**Calculus**. So many formulas (algorithms) that force students not to think about the underlying relationships. If we wanted students to really think about rates of change (which are what Calculus is really about), we might have them develop a theory of secant lines and finite differences before we do limits and tangent lines. We might have Calculus students do tasks such as Sweet Tooth from Mathalicious (free throughout October!). There, students think about *marginal enjoyment *and *total enjoyment*.

**On and on**.

This is pervasive in mathematics teaching.

The results are mistaken for the content.

So we teach kids to get results. And we inadvertently teach them not to use what they know about the content—not to look for new things to know. Not to question or wonder or connect.

I’m telling you, though, that it doesn’t have to be this way.

Consider the case of *Talking Math with Your Kids*. There we have reports from around the country of parents and children talking about the ideas of mathematics, not the procedures.

Consider the case of Kristin (@MathMinds on Twitter), a fifth grade teacher, and her student “Billy”. Billy made an unusual claim about even and odd numbers. She followed up, she shared, we discussed on Twitter. Pretty soon, teachers around the country were engaged in thinking about whether Billy would call 3.0 *even* or *odd*.

But standard algorithms don’t teach any of that. They teach children to get answers. They teach children not to think.

I have read about it. I have thought about it. And tonight I saw it in my very own home.

Tagged: 10 years old, algorithms, Griffin, lattice, mathematical thinking, multiplication, standard algorithm ]]>

You can either shed a tear or do something about it. (or both)

If you choose the latter, join me and a whole bunch of others at EdCamp Math and Science MN.

It is free. It takes place Friday, October 17 (during MEA weekend).

See you there.

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One consequence of this is that I am getting daily emails from people who read the piece and feel moved to comment. I do believe the Internet ought to facilitate dialogue. So I have been replying to these emails.

Sometimes, people leave a wrong email address in the contact form and they bounce back. So, in a show of good faith, I share with you a recent email and my reply. Perhaps Gavin will come back to the blog and read my reply. Perhaps he will not.

Anyway, here goes.

**Gavin** writes:

I do not know if you failed to do your research, but the number line is clearly part of Common Core, for instance:

“CCSS.Math.Content.6.NS.C.6

Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.”I am not sure how to comment on the article because you have banned commenting on them. I do not have a twitter and therefore cannot join the disucssion there. I hope that you’re not intentionally trying to cast a positive light on Common Core but are instead trying to give an unbiased account of it.

**I** reply:

Thank you for taking the time to read, and to write.

I just want to clarify that my claim was not that number lines do not appear in the Common Core. They do appear there, as you point out with your citation. You are completely correct.

But if I remember correctly, the worksheet in question was a second grade worksheet. My claim was this: “There is nothing in the Common Core State Standards that requires students to use number lines to perform multi-digit subtraction.”

I stand by that claim. Even the number line standard you cite in sixth grade doesn’t reference the number line as a way to understand multi-digit subtraction. Instead the spirit of that standard is to use the number line as a way to represent negative numbers (such as -9 or -1/2), and then to understand the coordinate plane. Simply put, if students are going to graph functions in algebra, they will need to work with number lines in earlier grades.

As for the comments thing…I was saddened to have to turn them off for that 5 reasons post. But I am committed to maintaining a reasoned and productive tone on this blog. The comments (both pro- and con- on the Common Core) were spiraling out of control and I simply did not have the time to manage them. It seems clear to me that people are able to comment on the piece as it gets shared on Facebook, but I don’t have access to the comments on other people’s shares so I cannot speak to their quality, and I am not responsible for them in the way I am when they are on my blog.

Finally, you can search my blog for “Common Core” and find that I have made some rather pointed critiques of some specific standards in the Common Core—including engaging and arguing with Bill McCallum (a Common Core author) on matters involving rates, ratios and unit rates. All on the record, and you would be welcome to join the conversation in comments on those posts. I have no interest in promoting CCSS. I do have an interest in making sure that critiques are honest and fair.

Best wishes and thanks again for writing.

Christopher

Tagged: ccss, ccssm, common core ]]>

Audience is parents, and this may appear in the title (*Common Core Math for Parents For Dummies* is the working title). It goes for the big picture in each of the grade levels, K—8.

The *For Dummies *format is pretty rigid but there will be no mistaking authorship. A few sample section headings (and the grades where they will appear) to whet your appetite:

1st grade.Saying bye-bye to key words

1st grade.Understanding the importance of ten

2nd grade.Why units matter

2nd grade.Place value

2nd grade.More about place value

2nd grade.Seriously. Place value.

4th grade.Multiplication: Whatisit and why not just memorize the facts?

5th grade.Standard algorithms: Doing things “the old-fashioned way”?

6th grade.Dividing fractions—More fun than you’d think!

6th grade.Area: It all goes back to rectangles

8th grade.Congruence and similarity: Two kinds of sameness

Catch you all later. I have some writing to do!

I’ll keep you posted.

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