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Blank Build it Draw it Write it template

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Blank Build it Draw it Write it template

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13. Spatial Structuring Assessment -DS, white

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12. Day 1 Formative Assessment - white, SS

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10. MathematicalThinkingProcess - color copy

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Day 1 Models for Multiplication & Division

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Day 1 Models for Multiplication & Division

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Day 1 Multiplication & Division Problem Types

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Day 1 Multiplication & Division Problem Types

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4. Multiplication and division situations - white

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3. BDW Problem types - color copy, white, SS, l

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0. Open-Middle-Worksheet (white, DS, BW)

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100 Questions that Promote Math Discourse

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100 Questions that Promote Math Discourse

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Ready: 100 questions that promote Mathematical Discourse

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100 Questions to Promote Mathematical Discourse by Ready author Dr. Gladis Kersaint. A free printable with 100 questions to promote mathematical thinking and encourage classroom discourse.

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Graham Fletcher Blog

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Last month I posted The Progression of Multiplication hoping that a couple of friends and parents would find it helpful. Well, here's my stab at the progression of division. I understand that there's lots of ways to model division and this is only ONE of them.

Understanding the vertical progression of mathematics is really important in the conceptual development of everyone's understanding.

Hope it helps.

Stay thirsty my friends!

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Last month I posted The Progression of Multiplication hoping that a couple of friends and parents would find it helpful. Well, here's my stab at the progression of division. I understand th...

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Graham Fletcher blog

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There’s a lot happening in elementary mathematics and it’s really tough for teachers to keep up with it all…let alone parents and administrators. I keep reminding myself that the …

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Jason Zimba blog

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Since I've been doing Saturday School with my kids, I've continued to appreciate how intricate is the structure of the addition facts as they play out in the curriculum. Right now, my younger is learning some of the facts from memory and, for some others, learning the strategy of making ten.

At this stage of learning, making ten is a great strategy for a problem like 8 + 5. But the strategy doesn't help you in a problem like 12 + 3. To solve that problem, at this stage of learning, I'd say you want to

appreciate the place value structure as 10 + 2 + 3,

know from memory that 2 + 3 = 5, and also

understand the meaning of teen numbers so that 10 + 5 = 15.

And in general, each region of the a + b map has its own story. This graphic shows the map with different regions color-coded.

Here is a key to the colors:

I use this map to orient myself toward my kids' learning as they progress toward fluency with addition facts and knowing them from memory. For example, the image below shows the worksheet I made for today's Saturday School. (Downloadable PDF here.) You'll see that the worksheet has 36 sums for practice-precisely the 36 sums that are coded yellow above. This is an intense practice day for making ten. Over the past few Saturdays, we established that the partners of 10 (magenta) were down cold, that there was fluency within 10 (red), and that the structure of the teen numbers was well understood (green). These are the key prerequisites for making ten. (Another is the ability to use properties of addition where helpful.)

Previous posts about Saturday School: 1, 2

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Since I've been doing Saturday School with my kids, I've continued to appreciate how intricate is the structure of the addition facts as they play out in the curriculum. Right now, my younger is learning some of the facts from memory and, for some others, learning the strategy of making ten.

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Multiplication, Division, etc;

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Illuminations Task

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K-5 Math Teaching Resources is a leading New York City based math consultancy firm. Our mission is to transform the teaching and learning of mathematics, improving outcomes for all students in elementary and middle school. Click on a tab below for information on our research-based professional development and customizable consulting services. Our results-driven elementary and middle school consultants are committed to working side-by-side with administrators, coaches and teachers to build exceptional quality math instruction and curriculum for schools and school districts, transforming students' math education and achievement.

On this website you will find mathematics to support standards-aligned math instruction. Use the grade level tabs at the top of this page to download sample resources for each grade level from Kindergarten through 5th Grade or visit our EBook Store to purchase a complete set.

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Research based resources and professional learning opportunities to transform math teaching and learning communities.

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3-Acts Lessons | Questioning My Metacognition

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3-Acts Lessons | Questioning My Metacognition

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Check out Dan's Blog or watch Math Class Needs a Makeover

Kindergarten Recording Sheet
Want more of the good stuff? 3-Act Recording Sheet
Dan Meyer's (6-12) Robert Kaplinsky' (K-12) Mike Wiernicki (K-8)

Andrew Stadel (6-12) Dane Ehlert (6-12) Kyle Pearce (3-12)

Jon Orr (6-12) Catherine Castillo(K-5) Kendra Lomax (K-3)

Kristen Acosta (K-5)

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Check out Dan's Blog or watch Math Class Needs a Makeover Kindergarten Recording Sheet Want more of the good stuff?

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Table Toppers

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Most of the LEARN NC Content has been archived using the Wayback Machine. Instructions for how to access that content is below, including links to frequently requested materials.

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Dan Meyer asked: What is the cheapest way to package up these candies?? What is the first question that comes to your mind?

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Khan Academy Video

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The problem with dividing any number by zero

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Sal explains why there cannot be a single number that is the result of dividing by zero.

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Part 1: Dividing By Smaller and Smaller Numbers
by a High School Math Teacher

Suppose you've got a pizza. A nice charcoal-cooked one from New Haven, or an oven-hot Chicago deep-dish, or even one of those organic San Francisco artisan pies that somehow make artichoke hearts seem like they belong on a pizza. And, generous soul that you are, you've decided to share.

How many people can you feed if everybody gets half of a pizza (a hearty helping)?

And how many can you feed if everybody gets 1/10 of a pizza (a cheesy snack)?

1 pizza ÷ 0.1 pizzas per person = 10 people.

And how many if everybody gets 1/100 of a pizza (a bite-sized morsel)?

1 pizza ÷ 0.01 pizzas per person = 100 people.

And how many can you feed if each gets 1/1000 of a pizza (a crumb with a dab of sauce)?

1 pizza ÷ 0.001 pizzas per person = 1000 people.

The smaller the slice you give to each person, the more people you can feed. Or, more abstractly: the smaller the number you divide by, the bigger the result.

Now, take it one step further: What if each person gets 0% of a pizza?

How many people can you feed? Well, there's no limit, because you're not actually feeding them anything. If the earth's seven billion people all show up at your door, asking for their share of pizza, you can say "No problem!" because "their share of pizza" amounts to nothing at all. Add another seven billion, and you'd say the same thing. How many people can you feed? There's no answer.

When you divide a number by 0, there's no single answer. To divide is to break something into piles of a certain size. And breaking something into piles of size zero just doesn't make sense.
Part 2: "Inverse of Multiplication"
by a Math PhD Candidate
While she was washing dishes, I asked my fiancee why you can't divide by zero. Her off-the-top-of-her-head answer was more concise than mine. (In my defense, I get the dishes cleaner than she does.)

When you divide by a number - let's say 4 - you're asking, "How many times can 4 go into the number?" So:

But when you divide by 0, you're asking, "How many times can 0 go into the number?" And no matter how many zeroes you add, 0 + 0 + 0 + 0 ... will never equal 12. So 12 ÷ 0 is undefined.
Part 3: "Inverse of Multiplication" Redux
by an Elementary-Level Math Specialist

I then ran both of these explanations by my sister Jenna, a K-8 math specialist. She liked Taryn's answer, and gave her own even more concise version.

Division is the inverse of multiplication. So when you divide 12 by 4, you're saying: "What times 4 gives you 12?"

Therefore, dividing by zero is like asking, "What times 0 gives you 12?" There's obviously no answer, since any multiple of 0 will be 0.

Part 4: Tying it All Together
by a Professor (my dad)

At dinner with my father James (a professor of Operations Research), I asked him to explain why you can't divide by zero. He gave an explanation pretty similar to mine, and then summarized the relative merits of the two approaches quite nicely.

The Taryn/Jenna explanation, he said, cuts to the chase, and will satisfy a wider (and younger) audience. It starts by saying, "Well, here's what division is," and then showing that the concept makes no sense when applied to zero.

The Ben/James explanation, meanwhile, is valuable because it doesn't cut to the chase. It connects the question "Can you divide by zero?" to other ideas (limits and asymptotic behavior), and gets more to the conceptual heart of the problem.

Anyway, there you have it. Four math professionals, two basic explanations, and one more blog adding its voice to the din of answers on this subject.

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Suppose you've got a pizza. A nice charcoal-cooked one from New Haven, or an oven-hot Chicago deep-dish, or even one of those organic San Francisco artisan pies that somehow make artichoke hearts seem like they belong on a pizza. And, generous soul that you are, you've decided to share.

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Math Coach / Fact Fluency for Mutliplication

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Math Coach / Fact Fluency for Mutliplication

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Math Coach / Fact Fluency for Mutliplication

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Use this applet to determine the volume of a box by filling it with cubes, rows of cubes, or layers of cubes. Determine the surface area as well by using the box's net.

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K-5 Math Teaching Resources

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K-5 Math Teaching Resources

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Read aloud

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Read aloud

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K-5 Math Teaching Resources is a leading New York City based math consultancy firm. Our mission is to transform the teaching and learning of mathematics, improving outcomes for all students in elementary and middle school. Click on a tab below for information on our research-based professional development and customizable consulting services. Our results-driven elementary and middle school consultants are committed to working side-by-side with administrators, coaches and teachers to build exceptional quality math instruction and curriculum for schools and school districts, transforming students' math education and achievement.

On this website you will find mathematics to support standards-aligned math instruction. Use the grade level tabs at the top of this page to download sample resources for each grade level from Kindergarten through 5th Grade or visit our EBook Store to purchase a complete set.

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Research based resources and professional learning opportunities to transform math teaching and learning communities.

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Area Model for Division

Visualize solutions to multi-digit division problems by modeling them with virtual manipulatives. This interactive exercise focuses on calculating quotients by working with area models and visualizing the distributive property of division. This resource is part of the Math at the Core: Middle School collection.

See this activity on Annenberg Learner's professional development course for teachers.

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Visualize solutions to multi-digit division problems by modeling them with virtual manipulatives. This interactive exercise focuses on calculating quotients by working with area models and visualizing the distributive property of division. This resource is part of the Math at the Core: Middle School collection.

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The Multiplier Effect

In this segment from Cyberchase, the CyberSquad develops the new Digifizz toy, which sends a tiny burst of sparkles shooting into the air each time it is used. With the toy gaining popularity, the CyberSquad must figure out how many Digifizz toys to produce in order to meet the demand. Using what they call "The Multiplier Effect" they find out that the number of Digifizz toys they must make can be calculated by multiplying the number of students in each class by the product of the number of classes and the number of schools. It turns out to be a lot of Digifizz!

Learn more about Cyberchase.

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In this segment from Cyberchase, the CyberSquad develops the new Digifizz toy, which sends a tiny burst of sparkles shooting into the air each time it is used. With the toy gaining popularity, the CyberSquad must figure out how many Digifizz toys to produce in order to meet the demand.

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Multiplying Bigger Numbers Video (Cyberchase)

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Multiplying Bigger Numbers Video (Cyberchase)

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Multiplying Bigger Numbers

In this video segment from Cyberchase, every time Delete sneezes he creates three clones of himself. The CyberSquad uses their knowledge of multiplication to figure out how many clones exist after Delete sneezes 103 times.

Learn more about Cyberchase.

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In this video segment from Cyberchase, every time Delete sneezes he creates three clones of himself. The CyberSquad uses their knowledge of multiplication to figure out how many clones exist after Delete sneezes 103 times. Learn more about Cyberchase.

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NBT Multiplication Progression Teacher Module

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NBT Multiplication Progression Teacher Module

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Number & Operations in Base 10: 5.NBT.B.5 | Tennessee Department of Education

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In this segment, discover how students attach meaning to the standard multiplication algorithm for whole numbers. Fluently multiply multi-digit whole numbers using the standard algorithm.

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Representing Volume of Right Rectangular Prisms with Unit Cubes

Use your spatial reasoning skills to calculate the volume when given a 3D figure. This video focuses on a right rectangular prism and shows you how unit blocks can help you visualize, then calculate the volume. This video was submitted through the Innovation Math Challenge, a contest open to professional and nonprofessional producers.

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Representing Volume of Right Rectangular Prisms with Unit Cubes

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Use your spatial reasoning skills to calculate the volume when given a 3D figure. This video focuses on a right rectangular prism and shows you how unit blocks can help you visualize, then calculate the volume. This video was submitted through the Innovation Math Challenge, a contest open to professional and nonprofessional producers.

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Multiplying with Standard Algorithm - Donut Video

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Multiplying with Standard Algorithm - Donut Video

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Multidigit Multiplication with Standard Algorithm: The Donut Doze Off

In this Cyberchase video, Bianca has a new job decorating fresh-baked donuts. Her supervisor asked her to decorate 12 trays of donuts, but Bianca thought he meant 12 donuts. When she discovers her mistake, she uses multiplication strategies and an understanding of repeated addition to discover the true scope of her job. In the accompanying classroom activity, students play a game against the teacher in which students get practice multiplying two-digit numbers. This resource is part of the Math at the Core: Middle School Collection.

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Multidigit Multiplication with Standard Algorithm: The Donut Doze Off

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In this Cyberchase video, Bianca has a new job decorating fresh-baked donuts. Her supervisor asked her to decorate 12 trays of donuts, but Bianca thought he meant 12 donuts. When she discovers her mistake, she uses multiplication strategies and an understanding of repeated addition to discover the true scope of her job.

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Sharing Halloween Candy Video (Cyberchase)

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Sharing Halloween Candy Video (Cyberchase)

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Sharing Halloween Candy | Cyberchase

In this video segment from Cyberchase, Buzz, Delete, and Harold are trying to share 18 pieces of candy equally among themselves. Jackie helps them by demonstrating how to divide the candies equally by distributing the candies in rounds.

Learn more about Cyberchase.

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In this video segment from Cyberchase, Buzz, Delete, and Harold are trying to share 18 pieces of candy equally among themselves. Jackie helps them by demonstrating how to divide the candies equally by distributing the candies in rounds. Learn more about Cyberchase.

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Mr Elementary Math provides math ideas, math strategies and math resources for K -5 teachers.

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multiplication vertical with and without trading

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fractions to decimals to percentages conversions

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decimals to percentages to fractions conversions

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percentages to decimals to fractions conversions

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Maths Charts || Math Posters || Free, printable || by Jenny Eather

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Over 250 free printable maths reference charts for interactive whiteboards, classroom displays, math walls, student handouts, homework help, concept introduction and consolidation and other math reference needs. © Jenny Eather 2012. Mathematical areas covered include Numbers, Operations on Numbers, Fractions, Decimals, Percent and Percentages, Ratios and Rates, Beginning Algebra, Data and Statistics, Probability, Geometry, Measurement, Time and Money.

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Introduction to division with remainders

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A Squadron of Bugs: Introducing Division with Remainders with a Book

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Use this applet to illustrate division and remainders by the division of food to aliens, dinosaurs, penguins and more.

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A Brief Ode to Blank Paper - The Stenhouse Blog

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A Brief Ode to Blank Paper - The Stenhouse Blog

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Prescriptive worksheets can often cheat children out of real thinking and understanding. Read how Tracy Zager encourages her own children to explore and communicate their mathematical thinking through writing-on simple blank paper. And then hop over to her blog to read how she continually makes math a natural, fun, and important part of her family's conversation. A Brief Ode to Blank Paper
Maya (7) brought home a folder full of completed math worksheets yesterday, which put me in a funk. First, there was the bugs problem.

I couldn't decide which part of this problem bothered me most. Was it the ridiculous premise? I mean, come on. Was it that the problem was, yet again, a multiple-choice question? Was it the stubborn insistence on drawing bugs with the wrong number of legs? Was it that students have no room to work, but the publisher took plenty of room for cutesie drawings? Harumph.

As awful as the bugs problem was, the page that really upset me was this one:

Sigh. A few issues:

The strategies are named wrong, and made to look more complicated than they are.

There are better and other ways to solve this problem.

Students are left with 2″ to do math on this entire page, which is barely enough room to do the standard algorithm. There's no chance they'll try out one of the other strategies with no space to work.

Once again, we're turning useful, general computation strategies into prescriptive algorithms. Breaking the numbers up by place value and adding the partial sums becomes: Step 1) Add hundreds...

I'm not the first math teacher to notice that prescriptive worksheets are a problem. Kamii, the CGI group, and others have all written about it. Yesterday, though, I was upset as a parent because Maya has stellar, flexible mental math skills, and her instinct to think is being undermined by this curriculum. I asked her why she'd opted for the standard algorithm on all the problems, and she said, "Those other strategies are too confusing." I covered up her solution to 597 + 122 and asked her to solve it mentally.

"Well, I'd give 3 to the 597 to make it 600. Then 600 + 122 is 722. I'd take the 3 back, so it's 719."

I did the same thing for 209 + 376:

"200 + 300 is 500. 500 + 70 is 570. 9 + 6 is 15. 570 + 15 is 585."

I pointed to the top of the page and said, "You just used this strategy. You broke the numbers up into place value parts, and then added each part together, starting with the biggest part."

Her jaw dropped.

And my mind clicked. She has made NO connection between the mental math strategies she uses with fluency and all this junk on the worksheets. The reason? She's never been given the chance to record her own thinking at school.

I think I've decided what one of my bigger problems with this curriculum is: they never use blank paper. They never write 209 + 376 at the top of a big piece of paper and let kids have at it. The kids never get a chance to wrestle with keeping track of their thinking or figure out organizational strategies. All math problems are either on worksheets or educational technology. The kids just don't write enough.

So now I know what to do with Maya at home. We're going to spend some time with blank paper, where she has to work out how to write down what she does in her head. She needs to make mistakes, lose track, not be able to follow her own thinking, and then ultimately figure out ways that make sense. She needs to be able to write down her thinking so that she and her mathematical community can follow it. I'm on it.

Two hours later, after dinner, Daphne (5) got us started. Our dining room chairs have decorative nailheads, and the kids are forever running their fingers over them and counting them. Daphne said, "Someday, I'm going to get out a math journal and count all these nailheads and write it down so I know how many there are." Before I knew it, she was off! Someday turned out to be right then. Both kids got in the game.

Daphne was incredibly excited to count AND write down her results. Check them out:

If you want to understand her notation, take a peek at the short videos:

The power of blank paper, baby.

Perhaps inspired by all of this discussion and my venting, Maya asked if she could get a piece of blank paper when she did her homework, which is truly a counter-cultural act with this curriculum. "Of course!" I nearly sang.

She created this number line and used it to solve the final problem.

When I asked her about it, I pointed out that she didn't just answer the question by saying, "The red one." She wrote about the problem more generally: "I used a number line and found that anything less than 350 would fit and 270 is less so red paint is less than 350!"

She said, "Well, when I wrote it myself I thought about it more."

Precisely my point.
This post first appeared on Tracy's blog.
Entry Filed under: math

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Prescriptive worksheets can often cheat children out of real thinking and understanding. Read how Tracy Zager encourages her own children to explore and communicate their mathematical thinking through writing-on simple blank paper. And then hop over to her blog to read how she continually makes math a natural, fun, and important part of her family's conversation.

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Dividing Numbers into Two Factors and Building Arrays to Represent Each Factorization

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This article looks at how models support mathematical thinking about numbers and the number system

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Operations & Algebraic Thinking | Open Middle

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Operations & Algebraic Thinking | Open Middle

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Operations & Algebraic Thinking | Open Middle

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Two Kinds of Division | Mathematics | Classroom Resources | PBS LearningMedia

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Two Kinds of Division | Mathematics | Classroom Resources | PBS LearningMedia

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Two Kinds of Division

Learn about the two different types of division problems. This video focuses on partitive and quotative models of division and when they're each useful.

This video was submitted through the Innovation Math Challenge, a contest open to professional and nonprofessional producers and is part of the Math at the Core: Middle School collection.

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Learn about the two different types of division problems. This video focuses on partitive and quotative models of division and when they're each useful. This video was submitted through the Innovation Math Challenge, a contest open to professional and nonprofessional producers and is part of the Math at the Core: Middle School collection.