Templeton Elementary field day beach ball games

Roadmap to Mathematics: 2nd Grade

[Feature photo (above) by Loren Kerns. (CC BY 2.0 via Flickr)]


A frequently-asked question on homeschooling forums is, “Are my children working at grade level? What do they need to know?”

The Council of the Great City Schools has published a handy 6-page pdf summary of second grade math concepts, with suggestions for how parents can support their children’s learning:

Whether you are a radical unschooler or passionately devoted to your textbook — or, like me, somewhere in between — you can help your children toward these grade-level goals by encouraging them to view mathematics as mental play. Don’t think of the standards as a “to do” list, but as your guide to an adventure of exploration. The key to learning math is to see it the mathematician’s way, as a game of playing with ideas.

The following are excerpts from the roadmap document (along with a couple of extra tips) and links to related posts from the past eight years of playing with math on this blog…

What Your Child Will Learn in 2nd Grade Math

In grade two, students will extend their understanding of place value to the hundreds place. They will use this place value understanding to solve word problems, including those involving length and other units of measure. Students will continue to work on their addition and subtraction skills, quickly and accurately adding and subtracting numbers up through 20 and also working with numbers up through 100. They will also build a foundation for understanding fractions by working with shapes and geometry.

Activities in these areas will include:

  • Measuring lengths of objects in standard units such as inches and centimeters.
  • Solving addition and subtraction word problems involving length or money.
  • Solving addition, subtraction, and comparison word problems using information presented in a bar graph.
  • Writing equations to represent addition of equal numbers. An equation is a mathematical statement that uses numbers and symbols, such as 3 + 3 = 6.


[Photo by geishaboy500 via Flickr (CC BY 2.0).]

Tip: Second grade students should be intimately familiar with the numbers up to 100. Check out the many games and activities in 20+ Things to Do with a Hundred Chart.

Helping Your Child Learn Math

  • Play math games with your child. For example, “I’m thinking of a number. It has 6 tens, but only half that many hundreds. And it has as many ones as how old you are. What is the number?”
  • Have your child create story problems to represent addition, subtraction, and comparisons. For example, “Keisha has 57 cents. She has 14 cents more than Karen has. How much money does Ryan have?”

Self-Portrait: Eden

[Photo by Eden, Janine and Jim. (CC BY 2.0 via Flickr)]

“The way we taught students in the past simply does not prepare them for the higher demands of college and careers today and in the future. Your school and schools throughout the country are working to improve teaching and learning to ensure that all children will graduate high school with the skills they need to be successful.

“In mathematics, this means three major changes. Teachers will concentrate on teaching a more focused set of major math concepts and skills. This will allow students time to master key math concepts and skills in a more organized way throughout the year and from one grade to the next. It will also call for teachers to use rich and challenging math content and to engage students in solving real-world problems in order to inspire greater interest in mathematics.”

Council of the Great City Schools
Parent Roadmaps to the Common Core Standards- Mathematics

Additional Resources

  • For creative ways to build a love for mathematics, follow Moebius Noodles.

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Horseshoes: A Place Value Game

[Feature photo above by Johnmack161 via Wikimedia Commons (CC BY 2.5).]

I first saw place value games on the old PBS Square One TV show (video below). Many teachers have posted versions of the game online, but Snugglenumber by Anna Weltman is by far the cutest variation. Anna kindly gave me permission to use the game in my upcoming Math You Can Play book series, and I added the following variation:



Math Concepts: place value, strategic thinking.
Players: two or more.
Equipment: one deck of playing cards, or a double deck for more than three players.

Separate out the cards numbered ace (one) through nine, plus cards to represent the digit zero. We use the queens (Q is round enough for pretend), but you could also use the tens and just count them as zero.

Shuffle well and deal eleven cards to each player. Arrange your cards in the snugglenumber pattern shown here, one card per blank line, to form numbers that come as close to each target number as you can get it.

Score according to horseshoes rules:

  • Three points for each ringer, or exact hit on the target.
  • One point for each number that is six or less away from the target.
  • If none of the players land in the scoring range for a target number, then score one point for the number closest to that target.

For a quick game, whoever scores the most points wins. Or follow tradition and play additional rounds until one player gets 21 points (40 for championship games) — and you have to win by at least two points over your closest opponent’s score.

But Who’s Counting?

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dividing sausages

Fractions: 1/5 = 1/10 = 1/80 = 1?

[Feature photo is a screen shot from the video “the sausages sharing episode,” see below.]

Fractions: 1/5 = 1/10 = 1/80 = 1?

How in the world can 1/5 be the same as 1/10? Or 1/80 be the same as one whole thing? Such nonsense!

No, not nonsense. This is real-world common sense from a couple of boys faced with a problem just inside the edge of their ability — a problem that stretches them, but that they successfully solve, with a bit of gentle help on vocabulary.

Here’s the problem:

  • How can you divide eight sausages evenly among five people?

Think for a moment about how you (or your child) might solve this puzzle, and then watch the video below.

What Do You Notice?

Continue reading

Natural Math Multiplication Course


This April, the creative people at Moebius Noodles are inviting parents, teachers, playgroup hosts, and math circle leaders to join an open online course about multiplication. My preschool-2nd grade homeschool math group is eager to start!

Each week there will be five activities to help kids learn multiplication by exploring patterns and structure, with adaptations for ages 2-12.

The course starts April 6 and runs for four weeks.

Preliminary Syllabus

Week 1: Introduction.
What is multiplication? Hidden dangers and precursors of math difficulties. From open play to patterns: make your own math. 60 ways to stay creative in math. Our mathematical worries and dreams.

Week 2: Inspired by calculus.
Tree fractals. Substitution fractals. Multiplication towers. Doubling and halving games. Zoom and powers of the Universe.

Week 3: Inspired by algebra.
Factorization diagrams. Mirror books and snowflakes. Combination and chimeras. Spirolaterals and Waldorf stars: drafting by the numbers. MathLexicon.

Week 4: Times tables.
Coloring the monster table. Scavenger hunt: multiplication models and intrinsic facts. Cuisenaire, Montessori, and other arrays. The hidden and exotic patterns. Healthy memorizing.

Sounds like lots of fun!

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fraction pieces

Reblog: A Mathematical Trauma

Feature photo (above) by Jimmie via flickr.

My 8-year-old daughter’s first encounter with improper fractions was a bit more intense than she knew how to handle.

I hope you enjoy this “Throw-back Thursday” blast from the Let’s Play Math! blog archives:

Photo (right) by Old Shoe Woman via Flickr.

Nearing the end of Miquon Blue today, my youngest daughter encountered fractions greater than one. She collapsed on the floor of my bedroom in tears.

The worksheet started innocently enough:

\frac{1}{2} \times 8=\left[ \quad \right]

[Click here to go read the original post.]

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Multiplication Matching Cards

Multiplication Models Card Game

[Poster by Maria Droujkova of NaturalMath.com. This game was originally published as part of the Homeschooling with a Profound Understanding of Fundamental Mathematics Series.]

Homeschooling parents know that one of the biggest challenges for any middle-elementary math student is to master the multiplication facts. It can seem like an unending task to memorize so many facts and be able to pull them out of mental storage in any order on demand.

Too often, we are tempted to stress the rote aspect of such memory work, which makes our children lose their focus on what multiplication really means. Before practicing the times table facts, make sure your student gets plenty of practice recognizing and using the common models for multiplication.

To help your children see what multiplication looks like in real life, explore the multitude of Multiplication Models collected at the Natural Math website. Or try some of the hands-on activities in the Family Multiplication Study.

You may want to pick up this poster and use it for ideas as you play the Tell Me a (Math) Story game. Word problems are important for children learning any new topic in math, because they give children a mental “hook” on which to hang the abstract number concepts.

And for extra practice, you can play my free card game…

Click here to continue reading.


Algebra for (Almost) Any Age


Fawn Nguyen’s Visual Patterns website just keeps getting better and better. Check it out:

In addition to the 115 puzzle patterns (as of this writing), the site features a Gallery page of patterns submitted by students. And under the “Teachers” tab, Fawn shares a form to guide students in thinking their way through to the algebraic formula for a pattern.

How can you use these patterns to develop algebraic thinking with younger students? Mike Lawler and sons demonstrate Pattern #1 in the YouTube video below.

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Things To Do with a Hundred Chart #30


Here’s one more entry for my 20+ Things to Do with a Hundred Chart post, thanks to David Radcliffe in the comments on Monday’s post:

(30) Can you mark ten squares Sudoku-style, so that no two squares share the same row or column? Add up the numbers to get your score. Then try to find a different set of ten Sudoku-style squares. What do you notice? What do you wonder?
[Suggested by David Radcliffe.]

Share Your Ideas

Can you think of anything else we might do with a hundred chart? Add your ideas in the Comments section below, and I’ll add the best ones to our master list.

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Things To Do with a Hundred Chart #29

100chart puzzle

Here’s a new entry for my 20+ Things to Do with a Hundred Chart post:

(29) Blank 100 Grid Number Investigations: Challenge your students to deduce the secret behind each pattern of shaded squares. Then have them make up pattern puzzles of their own.
[Created by Stuart Kay. Free registration required to download pdf printable.]

Share Your Ideas

Can you think of anything else we might do with a hundred chart? Add your ideas in the Comments section below, and I’ll add the best ones to our master list.

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Land or Water 2

Maze Game: Land or Water?

This was a fun activity from Moebius Noodles for our PK-1st grade Homeschool Math in the Park group. The children take turns making a maze and setting a dinosaur inside. Then the other dinosaurs (parents or siblings) try to guess whether their friend is on the land or in the water.

Draw the maze

Player #1

(1) First, draw a big circle on the white board. This is your lake.

(2) With a finger or a bit of cloth, erase a small section of the circle to create the opening for your maze.

(3) Starting at one edge of the opening, draw a random squiggle inside the circle. Make your squiggle end at the other edge of the opening.

Looks like Land

(4) Set your dinosaur anywhere inside the maze.

Player #2

(1) Now it’s your turn to guess. Is the dinosaur standing on the land? Is it swimming in the water?

(2) How will you figure out if you guessed right?

(3) Check by jumping across the lines of the maze. Each jump takes you across a boundary: Splash! (Into the water.) Thump! (Back on the land.) Splash! Thump! … Until you reach the dinosaur inside.

(4) Or go to the maze entrance and walk your dinosaur along the path. Can you find your way?

land or water

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Related posts on Let’s Play Math! blog:


Parents, Teachers: Learn about Teaching Decimals

Many children are confused by decimals. They are convinced 0.48 > 0.6 because 48 is obviously ever so much bigger than 6. Their intuition tells them 0.2 × 0.3 = 0.6 has the clear ring of truth. And they confidently assert that, if you want to multiply a decimal number by 10, all you have to do is add a zero at the end.

What can we do to help our kids understand decimals?

Christopher Danielson (author of Talking Math with Your Kids) will be hosting the Triangleman Decimal Institute, a free, in-depth, online chat for “everyone involved in children’s learning of decimals.” The Institute starts tomorrow, September 30 (sorry for the short notice!), but you can join in the discussion at any time:

Past discussions stay open, so feel free to jump into the course whenever you can. Here is the schedule of “classes”:

Click here to see the TDI topic list →

Talking Math with Your Kids

Danielson-Talking Math

Christopher Danielson, one of my favorite math bloggers, has a new book out that is perfect for parents of preschool and elementary-age children:

It’s a short book with plenty of great stories, advice, and conversation-starters. While Danielson writes directly to parents, the book will also interest grandparents, aunts & uncles, teachers, and anyone else who wants to help children notice and think about math in daily life.

You don’t need special skills to do this. If you can read with your kids, then you can talk math with them. You can support and encourage their developing mathematical minds.

You don’t need to love math. You don’t need to have been particularly successful in school mathematics. You just need to notice when your children are being curious about math, and you need some ideas for turning that curiosity into a conversation.

In nearly all circumstances, our conversations grow organically out of our everyday activity. We have not scheduled “talking math time” in our household. Instead, we talk about these things when it seems natural to do so, when the things we are doing (reading books, making lunch, riding in the car, etc) bump up against important mathematical ideas.

The dialogues in this book are intended to open your eyes to these opportunities in your own family’s life.

— Christopher Danielson
Talking Math with Your Kids

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Related posts on Let’s Play Math! blog:

Multiplying Negative Numbers with Rectangles

I love using rectangles as a model for multiplication. In this video, Mike & son offer a pithy demonstration of WHY a negative number times a negative number has to come out positive:

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Youth Sports Baseball Camp

Quotable: Learning the Math Facts

feature photo above by USAG- Humphreys via flickr (CC BY 2.0)

During off-times, at a long stoplight or in grocery store line, when the kids are restless and ready to argue for the sake of argument, I invite them to play the numbers game.

“Can you tell me how to get to twelve?”

My five year old begins, “You could take two fives and add a two.”

“Take sixty and divide it into five parts,” my nearly-seven year old says.

“You could do two tens and then take away a five and a three,” my younger son adds.

Eventually we run out of options and they begin naming numbers. It’s a simple game that builds up computational fluency, flexible thinking and number sense. I never say, “Can you tell me the transitive properties of numbers?” However, they are understanding that they can play with numbers.

photo by Mike Baird via flickr

photo by Mike Baird via flickr

I didn’t learn the rules of baseball by filling out a packet on baseball facts. Nobody held out a flash card where, in isolation, I recited someone else’s definition of the Infield Fly Rule. I didn’t memorize the rules of balls, strikes, and how to get someone out through a catechism of recitation.

Instead, I played baseball.

John Spencer
Memorizing Math Facts

Conversational Math

The best way for children to build mathematical fluency is through conversation. For more ideas on discussion-based math, check out these posts:

Learning the Math Facts

For more help with learning and practicing the basic arithmetic facts, try these tips and math games:

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Quotable: The Adventure of Learning Math

Math mascot Moby Snoodles

As for mathematics itself, it’s one of the most adventurous endeavors a young child can experience. Mathematics is exotic, even bizarre. It is surprising and unpredictable. And it can be more exciting, scary and dangerous than sailing the high seas!

But most parents and educators don’t present math this way. They just want the children to develop their mathematical skills rather than going for something more nebulous, like the mathematical state of mind.

Children marvel as snowflakes magically become fractals, inviting explorations of infinity, symmetry and recursion. Cookies offer gameplay in combinatorics and calculus. Paint chips come in beautiful gradients, and floor tiles form tessellations. Bedtime routines turn into children’s first algorithms. Cooking, then mashing potatoes (and not the other way around!) humorously introduces commutative property. Noticing and exploring math becomes a lot more interesting, even addictive.

Unlike simplistic math that quickly becomes boring, these deep experiences remain fresh, because they grow together with children’s and parents’ understanding of mathematics.

— Maria Droujkova and Yelena McManaman
Adventurous Math For the Playground Set (Scientific American online)

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Multiplication Matching Cards

PUFM 1.5 Multiplication, Part 2

Poster by Maria Droujkova of NaturalMath.com. In this Homeschooling Math with Profound Understanding (PUFM) Series, we are studying Elementary Mathematics for Teachers and applying its lessons to home education.

Multiplication is taught and explained using three models. Again, it is important for understanding that students see all three models early and often, and learn to use them when solving word problems.

— Thomas H. Parker & Scott J. Baldridge
Elementary Mathematics for Teachers

I hope you are playing the Tell Me a (Math) Story game often, making up word problems for your children and encouraging them to make up some for you. As you play, don’t fall into a rut: Keep the multiplication models from our lesson in mind and use them all. For even greater variety, use the Multiplication Models at NaturalMath.com (or buy the poster) to create your word problems.

Continue reading

New Math by Tom Lehrer

While I was working on the next post in my PUFM Series, I stumbled on an old favorite video. Since I couldn’t think of an excuse to use it in a post about multiplication, I decided to share it today. Enjoy!

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Central City Times Tables

Trouble with Times Tables

[feature photo above by dsb nola via flickr.]

Food for thought:

Imagine that you wanted your children to learn the names of all their cousins, aunts and uncles. But you never actually let them meet or play with them. You just showed them pictures of them, and told them to memorize their names.

Each day you’d have them recite the names, over and over again. You’d say, “OK, this is a picture of your great-aunt Beatrice. Her husband was your great-uncle Earnie. They had three children, your uncles Harpo, Zeppo, and Gummo. Harpo married your aunt Leonie … yadda, yadda, yadda.

— Brian Foley
Times Tables – The Worst Way to Teach Multiplication

On the other hand, if you want your children to develop relationships with the numbers, to learn the math facts naturally, then be sure to tell lots of math stories. And when you are ready to focus on multiplication, be sure to study the patterns and relationships within the times tables.

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Build Mathematical Skills by Delaying Arithmetic, Part 3

To my fellow homeschoolers,

How can our children learn mathematics if we delay teaching formal arithmetic rules? Ask your librarian to help you find some of the wonderful living books about math. Math picture books are great for elementary students. Check your library for the Time-Life “I Love Math” books or the “Young Math Book” series. You’ll be amazed at the advanced topics your children can understand!

Benezet’s students explored their world through measurement, estimation, and mental math. Check out my PUFM Series for mental math thinking strategies that build your child’s understanding of number patterns and relationships.

Grade IV

Still there is no formal instruction in arithmetic.

By means of foot rules and yard sticks, the children are taught the meaning of inch, foot, and yard. They are given much practise in estimating the lengths of various objects in inches, feet, or yards. Each member of the class, for example, is asked to set down on paper his estimate of the height of a certain child, or the width of a window, or the length of the room, and then these estimates are checked by actual measurement.

The children are taught to read the thermometer and are given the significance of 32 degrees, 98.6 degrees, and 212 degrees.

They are introduced to the terms “square inch,” “square foot,” and “square yard” as units of surface measure.

With toy money [or real coins, if available] they are given some practise in making change, in denominations of 5′s only.

All of this work is done mentally. Any problem in making change which cannot be solved without putting figures on paper or on the blackboard is too difficult and is deferred until the children are older.

Toward the end of the year the children will have done a great deal of work in estimating areas, distances, etc., and in checking their estimates by subsequent measuring. The terms “half mile,” “quarter mile,” and “mile” are taught and the children are given an idea of how far these different distances are by actual comparisons or distances measured by automobile speedometer.

The table of time, involving seconds, minutes, and days, is taught before the end of the year. Relation of pounds and ounces is also taught.

— L. P. Benezet
The Teaching of Arithmetic II: The Story of an experiment

Continue reading

Cool Fibonacci Conversion Trick

photo by Muffet via flickr

Maria explains how to use the Fibonacci Numbers to convert distance measurements between miles and kilometers:

P.S.: Congratulations to Maria for her Math Mammoth program being featured in the latest edition of Cathy Duffy’s 100 Top Picks for Homeschool Curriculum! And Home School Buyer’s Co-op has a sale on Cathy Duffy’s book through the end of July.

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Build Mathematical Skills by Delaying Arithmetic, Part 2

To my fellow homeschoolers,

Most young children are not developmentally ready to master abstract, pencil-and-paper rules for manipulating numbers. But they are eager to learn about and explore the world of ideas. Numbers, patterns, and shapes are part of life all around us. As parent-teachers, we have many ways to feed our children’s voracious mental appetites without resorting to workbooks.

To delay formal arithmetic does not mean that we avoid mathematical topics — only that we delay math fact drill and the memorization of procedures. Notice the wide variety of mathematics Benezet’s children explored through books and through their own life experiences:

Grade I

There is no formal instruction in arithmetic. In connection with the use of readers, and as the need for it arises, the children are taught to recognize and read numbers up to 100. This instruction is not concentrated into any particular period or time but comes in incidentally in connection with assignments of the reading lesson or with reference to certain pages of the text.

Meanwhile, the children are given a basic idea of comparison and estimate thru [sic] the understanding of such contrasting words as: more, less; many. few; higher, lower; taller, shorter; earlier, later; narrower, wider; smaller, larger; etc.

As soon as it is practicable the children are taught to keep count of the date upon the calendar. Holidays and birthdays, both of members of the class and their friends and relatives, are noted.

— L. P. Benezet
The Teaching of Arithmetic II: The Story of an experiment

Continue reading

Build Mathematical Skills by Delaying Arithmetic, Part 1

To my fellow homeschoolers,

It’s counter-intuitive, but true: Our children will do better in math if we delay teaching them formal arithmetic skills. In the early years, we need to focus on conversation and reasoning — talking to them about numbers, bugs, patterns, cooking, shapes, dinosaurs, logic, science, gardening, knights, princesses, and whatever else they are interested in.

In the fall of 1929 I made up my mind to try the experiment of abandoning all formal instruction in arithmetic below the seventh grade and concentrating on teaching the children to read, to reason, and to recite – my new Three R’s. And by reciting I did not mean giving back, verbatim, the words of the teacher or of the textbook. I meant speaking the English language.

— L. P. Benezet
The Teaching of Arithmetic I: The Story of an experiment

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How Crazy Can You Make It?

And here is yet more fun from Education Unboxed. This type of page was always one of my my favorites in Miquon Math.


Handmade “How Crazy…?” worksheets are wonderful, but if you want something a tad more polished, I created a printable. The first page has a sample number, and the second is blank so that you can fill in any target:

Add an extra degree of freedom: students can fill in the blanks with equivalent and non-equivalent expressions. Draw lines anchoring the ones that are equivalent to the target number, but leave the non-answers floating in space.

How CRAZY Can You Make It

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Olympic Logic

I love logic puzzles! Nrich Maths offers four fun Olympics Logic puzzles. And be sure to check out the rest of their Nrich Olympics Math as well.

Medals Count

Given the following clues, can you work out the number of gold, silver and bronze medals that France, Italy and Japan got in this international sports competition?

  • Japan has 1 more gold medal, but 3 fewer silver medals, than Italy.
  • France has the most bronze medals (18), but fewest gold medals (7).
  • Each country has at least 6 medals of each type.
  • Italy has 27 medals in total.
  • Italy has 2 more bronze medals than gold medals.
  • The three countries have 38 bronze medals in total.
  • France has twice as many silver medals as Italy has gold medals.

Go to Nrich Maths and try all four puzzles!

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Princess in the Dungeon Game

Yet more fun from Rosie at Education Unboxed. I found these while looking for videos to use in my PUFM Subtraction post. Rosie says:

This is seriously embarrassing and I debated whether to put this video online or not because this is NOT my normal personality, but my girls made up this game and will play it for over an hour and ask for it repeatedly… so I figured someone out there might be able to use it with their kids, too.

If you know me, please don’t ever ask me to do this in public. I will refuse.

Princess in the Dungeon, Part 1 – Fractions with Cuisenaire Rods

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PUFM 1.4 Subtraction

Photo by Martin Thomas via flickr. In this Homeschooling Math with Profound Understanding (PUFM) Series, we are studying Elementary Mathematics for Teachers and applying its lessons to home education.

When adding, we combine two addends to get a sum. For subtraction we are given the sum and one addend and must find the “missing addend”.

— Thomas H. Parker & Scott J. Baldridge
Elementary Mathematics for Teachers

Notice that subtraction is not defined independently of addition. It must be taught along with addition, as an inverse (or mirror-image) operation. The basic question of subtraction is, “What would I have to add to this number, to get that number?”

Inverse operations are a very fundamental idea in mathematics. The inverse of any math operation is whatever will get you back to where you started. In order to fully understand a math operation, you must understand its inverse.

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Multiplication Challenge

Can you explain why the multiplication method in the following video works? How about your upper-elementary or middle school students — can they explain it to you?

Pause the video at 4:30, before he gives the interpretation himself. After you have decided how you would explain it, hit “play” and listen to his explanation.

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Addition Games with Cuisenaire Rods

Education Unboxed has posted some playful addition games for young learners. If your browser has as much trouble displaying Vimeo content as mine does, I’ve included the direct links:

Six is Having a Party! – Math Facts with Cuisenaire Rods

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PUFM 1.3 Addition

Photo by Luis Argerich via flickr. In this Homeschooling Math with Profound Understanding (PUFM) Series, we are studying Elementary Mathematics for Teachers and applying its lessons to home education.

The basic idea of addition is that we are combining similar things. Once again, we meet the counting models from lesson 1.1: sets, measurement, and the numberline. As homeschooling parents, we need to keep our eyes open for a chance to use all of these models — to point them out in the “real world” or to weave them into oral story problems — so our children gain a well-rounded understanding of math.

Addition arises in the set model when we combine two sets, and in the measurement model when we combine objects and measure their total length, weight, etc.

One can also model addition as “steps on the number line”. In this number line model the two summands play different roles: the first specifies our starting point and the second specifies how many steps to take.

— Thomas H. Parker & Scott J. Baldridge
Elementary Mathematics for Teachers

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