Wednesday, June 22, 2016

Multiplication Strategies: x2, x4, x8

Over the past week, my son and I have made tremendous progress with multiplication. (Intro post here.) Each day, we add to his fluency toolbox by looking at specific strategies. The (related) strategies for 2s, 4s, and 8s, have been especially fruitful. Let's look at why...

2s...Dare to DOUBLE!
Twos are easy-peasy. Just a matter of doubling. We can see an example in this array.

If you multiply something by 2, you only need to double. Instead of 1 group of 6, you have 2 groups of 6; you just double 6.

In Bridges in Mathematics, the strategy for multiplying by 4s is called Double-Double. It's easy to see why.

We already doubled when we multiplied by 2. To go from 2x a number to 4x a number, we double. So we double, then double again.

6 is doubled to 12 (x2)
12 is doubled to 24 (x4)

I bet you can guess what's coming next!

We call the strategy for 8s Double-Double-Double.
For 8x, we double 3 times:

6 is doubled to 12 (x2)
12 is doubled to 24 (x4)
24 is doubled to 48 (x8)

Can you see it in the model?

This is not a multiplication "trick" but rather a strategy with meaning behind it. Children need to see the visual model and understand what "Double-Double-Double" means. Once they understand the concept, they can apply it in wonderful ways.

I asked my son (just developing fluency with single digit multiplication) to consider these problems.

8 x 15 = ?

He doubled 15 and got 30. He doubled 30 (60). And doubled once more to get 120. So 8 x 15 = 120.

4 x 13 = ?

Double 13 to get 26. Double 26 to get 52. (Of course he then wanted to keep going and figure out 8 x 13. Double 52 and get 104!)

25 x 8 = ?

50, 100, 200, done! This problem was also a great opportunity to talk about another strategy. Do you know what it is? Leave your ideas in the comments below to start a RICH exchange.

Hope you're having a double dose of summer fun!

Little Girl Graphic from:
Number Frames (free app) from:

Friday, June 17, 2016

Multiplication Fluency: Summer Practice

Ready for a start-to-summer quiz?
Question: What happens when the math coach's child begins the summer by taking a multiplication fluency assessment in which he answers 20 problems in 4.5 minutes when the fluency guideline is 20 problems in 1 minute?

Answer: Summer math!  (Don't you wish you lived at my house?)
In case anyone else is in a similar predicament, here are a few resources to get you started...

First, "fluency" does not equate memorization. If you're interested in the difference between "by memory" and "memorization," check out this article. Fluency means accurate, efficient, and flexible mathematical thinking. Think about reading fluency. A fluent reader is not just fast. 120 words-per-mind counts for nothing without comprehension. Fluent readers AND mathematicians are  accurate, efficient, and flexible.

Although every child needs to master all three areas, he may demonstrate challenges in one area over the others. In our case, flexibility is an issue. Although my child knows some strategies for working with multiplication, it doesn't appear to be something that's been emphasized in his education. To that end, we are working to increase his strategy toolbox.

I pulled the Multiplication & Division Discussion Cards from Opening Eyes to Mathematics. (Cards are located on pp. 32-35 in this pdf, free from The Math Learning Center.) We flip through several cards a day and talk about what strategies could be used to solve a problem. For example:

What multiplication expression is represented here? (8 x 8)

How could we look at pieces of this array to help us solve the problem? Maybe we could see it as two parts: 8 x 5 and 8 x 3.

So 8 x 8 = (8 x 5) + (8 x 3) = 40 + 24 = 64

Or maybe you see it as two groups of 4 x 8:

What about this one? Could you use what you already know about 10 x 7 to help you figure out
9 x 7?

Strategies become critical when you get to larger multiplication problems, so this is definitely something we want to work on now.

Although fluency doesn't equate speed, it is generally expected that students be able to complete 20 problems in 1 minute to meet fluency standards. With the strategies in hand, I plan to assess his progress using the free online program, Xtra Math. I would not use this in isolation as I don't want to overemphasize speed, but when combined with visual models and strategies, it's a reasonable way for  both of us to track his progress.

Tuesday, March 15, 2016

Standard Algorithm For Subtraction: Sometimes Yes, Sometimes NO!!!!

Introductory note: For the past year, I've been working as a K-5 Math Coach. Not surprisingly, I have learned a lot. I hope the following blog post expresses just a bit of the wonder of the past year...

I developed a new appreciation for the power of strategies taught in Bridges when a third grader approached me for help on a worksheet he received in his (non-Bridges) classroom. The “Zero-Concept” worksheet included 36 problems with multi-digit subtraction, intended for practice with borrowing across zeros, solely using the standard algorithm.

Although this was the intent (and yes, the way many of us were taught!), it quickly became obvious that several other strategies might produce more efficient results. The third grade standard 3.NBT.2 specifically calls for this:

“Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.”

One of the key words, “algorithms” is plural for a reason. We want students to develop fluency defined by accuracy, efficiency, and flexibility. In this standard, students demonstrate fluency using multiple, flexible strategies--strategies selected because of their strength with a particular set of numbers.

The very first problem, 420-115, seemed a good candidate for Partial Place Value Splitting, one of several strategies explored in Bridges.

The student could mentally solve the problem using this strategy; he was surprised by how easy it was to break the subtrahend into manageable pieces and then subtract.

Another problem, 200-189, seemed ideal for Finding the Difference.

Again, once he understood the strategy, the problem was easy to solve mentally. In comparison, the standard algorithm was very complex and inefficient, leaving a lot of room for error.

The Removal Strategy (using a Number Line) worked for 500-333. He also noted that this could be done mentally using Partial Place Value Splitting, taking away 300, then 30, then 3.

Once again, borrowing across multiple zeros seemed unnecessarily complex with a high possibility of error.

A problem like 703-187 became a prime candidate for Constant Difference. Here it's illustrated on a number line:

He agreed that it was far easier to solve 716-200 than 703-187. And it's so simple to get there. Just add 13 to both the minuend and the subtrahend.

Looking over the worksheet we noted that while the standard algorithm might be an efficient method for a handful of problems, for the majority it was not. But perhaps the most surprising to my student: the number of problems that could be completely solved with mental math, using one of the above strategies.

If we think of fluency in terms of accurate, efficient, and flexible thinking, students are best served when they have a variety of strategies from which to choose. By the time we were done, my young friend heartily agreed!

Saturday, August 29, 2015

Fly on the Math Teacher's Wall: Personalized Math Notebook Covers

Here's an idea for getting your math year off to a great start...

When I posted about Math Journals & Notebooks, I mentioned that I loved the idea of having children make covers relating math to themselves as Courtney shares at A Middle School Survival Guide. Ideally, I'd begin the lesson by reading aloud a book that relates math to everyday life. (I mention several suggestions here.)

When I first considered what might go on a cover, I didn't have a lot of ideas. I just thought about # of siblings or children, year of birth or age, height or weight, etc. But the more I considered, the more ideas multiplied! I'd definitely want to do this as a brainstorming activity with students rather than giving them a list. See what your collective brain energy can come up with! How is math related to our daily life? Here are some of the things we thought of:
  • time you wake up/go to sleep
  • # of favorite ____________ (sports, colors, hobbies)
  • # of years _____________ (teaching, being a student, playing an instrument or sport)
  • time each day that you ___________ (exercise, go to school, watch tv, read, play video games)
  • # of _____________ that you own (pets, video games, books)
  • # of years until you (finish school, turn 21, want to get married or have kids)
  • cost of your favorite (restaurant meal, soda, candy bar)
  • amount you spent per week on (lunch, snacks, coffee)
The possibilities are endless!

These covers then become a fabulous jumping-off point for PROBLEM SOLVING.

After students finish their covers, have them generate several problems on 3x5" notecards that use the information they created. For example, on my cover, I posted the following:

I went ahead and wrote my problem on the cover itself, but would have students write on cards. My question, "How many hours do I sleep each night? Each week?" could then be posed to other students. In the classroom, I could put my cover under the document camera and ask students to answer the question posed on my card(s). They could then share a variety of strategies for solving the problem. In a homeschool setting, children could write problems for siblings or parents to solve. Problems could be written at a wide variety of levels, making them grade and age appropriate.

At the Northwest Math Conference I went to a workshop entitled, "Taking the Numb Out of Numbers" by Don Fraser (Ontario, Canada). He began by telling the group of 30 of us, "Did you know that in a group of 23 or 24 there is a 50% chance that at least two people in the group will have the same birthday?" He then gave us a graph showing us the probability of sharing the same birthday in groups of varying sizes. In a group our size--30 people--the likelihood was 70%. We graphed the days/months for birthdays in the room. Interestingly enough, none of us shared the same birthday...we were in the 30%. After looking at the data, Don asked us to come up with problem solving questions--real life questions--based on the information we'd collected. It was amazing to see how many questions we could generate, at all different levels of mathematical knowledge and proficiency.

Don encouraged us to begin each day by reading a "story" and having kids make up a question/word problem. Going back to the math notebook covers, imagine the possibilities if you put ONE child's notebook cover up each day and asked kids to generate questions from the "stories" found there. The problem solving possibilities are endless!

Do your students make personalized math notebook covers? What interesting stats have they included? Comment below with your stories and then visit Mrs. Balius and read what she has to say about setting up daily math routines!!! :)

Saturday, February 21, 2015

Fly on the Math Teacher's Wall: Squashing Fraction Misconceptions

The irony is not lost on me. Fractions, the math concept I most struggled with in elementary school, is now one of my favorites to teach. In this Blog Hop, my math blogging friends and I will be exploring fraction misconceptions. Here we go...

After years of operating with whole numbers, it's new territory to see fractions and understand the what numerator and denominator mean. What do each of those numbers really mean?

In this example, we'll look at an egg carton. First, we'll consider what the whole this case the whole is the entire egg carton.

Look at the following examples and ask yourself:

1. What does the string show?
2. What do the tile show?


The string shows how many parts our whole is divided into (our denominator) and the tile show how many of those parts have been filled (our numerator.)

A common misconception results when students look at the pieces in the model without taking the meaning of numerator/denominator into consideration. For example, in the first photo above, a student might say that they have 6 pieces, so it's 6/2. Most students, however, can readily tell you that the top example shows one-half, so a bit of probing (Where do you see the 1 in 1/2? Where do you see the 2 in 1/2?) helps to reestablish context.

In a similar example, I've heard students struggle with the question, "What fraction of a dollar is a nickel?"

Many students will answer "one-fifth" because they are thinking of 5 cents; if it has a 5 in it, it must be 1/5.

I like to pull out Money Value Pieces and again revisit the concept of numerator and denominator. We first talk about what our "whole" is: 100 cents.

I might ask, "What does 1/5 look like on our model?" Since we've explored numerator and denominator, they know that the whole would be broken into five portions:

It doesn't take long for someone to say, "One-fifth of a dollar is 20 cents!" (They can check this using the dime piece, a ten strip.) Then, using the nickel model, they explore how many pieces it would take to cover the dollar. "Twenty! So a nickel is 1/20 of a dollar!"

Students need many opportunities to explore the concepts of numerator and denominator using a variety of manipulatives and visual models. (More love2learn2day examples here.) I ask them to record their thinking in a variety of venues: math notebooks, class anchor charts, and video productions. In this ScreenChomp example, you'll hear a pair of students explain the meaning of numerator and denominator; notice that they use more than one visual to explain their thinking.

To continue on the Fraction Misconceptions Blog Hop, please visit my friend Jamie at Miss Math Dork!

Sunday, January 4, 2015

Measuring & Graphing with an Amaryllis

Our family received an Amaryllis kit from Grandma for Christmas. Today, the kids and I planted the bulb and got ready for a little measuring & graphing activity. Want to join us? Here's how:

1. Purchase an Amaryllis kit from a local store. In winter, they are widely available.

2. Plant the bulb according to package directions. (If yours, like ours, arrives with a hard disk of "plant medium," you might want to have a discussion about how much the peat changes by volume after water is added.)

3. Place an anchor in the soil to support your rulers. We used chopsticks.

4. Tape the ruler to the anchor so that it aligns with the top of the bulb's neck.

5. Measure. Our bulb already had green growth, albeit at a weird angle. We just measured straight across at the top so as not to break the plant. I told the boys that we'd measure our plant the same way I measure the tippy top! My 12yo is measuring in centimeters and had to scale a blank graph to go with his estimate for ultimate growth. My 8yo is measuring in inches, to the nearest half inch.

Here are a variety of options for graphing amaryllis growth.

***This activity was created to say THANK YOU for your support this past year. I appreciate you!

Tuesday, December 23, 2014

iHeart Math Holiday Hop

Happy MATH-y Holidays! I'm hosting the final day of the iHeart Math Holiday Hop. The entire advent calendar (see bottom of post) has now been unlocked; you can now go back and download freebies--23!--from ALL of my math blogging friends. Some of the activities are seasonal and you'll want to use them when you return to school in January. Others can be saved until next Christmas. Some, like mine, can be used any day of the year!

Stocking Stuffer #1 - Giving Back: Favorite Math Books & Games

It's no secret that I'm obsessed with math-related children's literature. Even if your Christmas shopping is finished, you still have plenty of time to grab the gigantic list and head to the library to check out a stack of favorite books to share with a child.

Since we are a big game family, I've also compiled a list of favorite Math Toys, Gifts, and Games. Chances are, some of these are in your closet. Why not pull them out to enjoy over holiday break? In the interest of minimalizing, you could pass around a set of games amongst friends.

In addition, this free Math Game List handout can be passed along to teachers, parents, and homeschoolers who are interested in enriching children's math experiences at home through simple card and board games. This list, which also includes a few game-like activities, could be used in a variety of ways:
  • letter home to families at end of school year
  • math center ideas for school or homeschool
  • early-finishers list
  • a checklist for summer fun

    Stocking Stuffer #2 - Math Tip: Gingerbread Math

    Here's a timely online activity for young mathematicians. On Topmarks Maths: Gingerbread Man Game, students can choose from the following:
    • matching written numbers to dots
    • ordering #s of dots by quantity
    • counting
    • sequencing numbers
    • counting with one-to-one correspondence
    Teachers can use the full-screen function and project the activities for transition times. So fun!

    You know you have a hit when you demonstrate the new homework assignment and the class collectively says, "Oooooooh, COOL!"

    Since we've been using clocks as a model for learning fractions, I thought it might be fun to make Flip this case, mini books in which fractions appear to move, getting either bigger or smaller (depending upon the order in which you compile the pages.)

    The pdf comes with 3 pages of "clock friendly" fractions. (23 fractions with an extra blank one, just in case.) At right, you see several samples of Flip Book cards.  The assignment asks students to:
    1. Color each given fraction. The cards come with numbers and blank clocks.
    2. Cut out the cards.
    3. Sequence the cards. They are purposely printed out of order. The set includes equivalent fractions that must be placed sequentially.
    4. Staple into a Flip Book.
    The pdf is FREE in my Teachers Pay Teachers  and Teachers Notebook Stores. The pdf was revised to improve "flip quality." Grab your revision if you downloaded before then! ;) The clocks are now on the right side, so the opposite of what you see in the video below.

    One of my students made a little stop action video to demonstrate...AKA, the "Separatists' evil clock plans..."

    Visit all 23 of these math bloggers for fabulous tips and freebies! Just click on the calendar squares to link to individual blogs. Happy Holidays!

    Image Map

    Thursday, December 11, 2014

    Fairy Tale Favorites!

    Fairy tales invite students to think about short stories in new, exciting, and creative ways. Here are some of our favorite activities that focus on character in grades 4 and up.

    Poetry & Art: Character
    I invite students to create diamante poems that either show the changes in a single character or contrast two characters. After writing a poem, students create background displays. Here's how...

    First, think of a prominent feature from the book that is fairly easy to define in an chunky outline, like a beanstalk (right) or stars (left.) Then, make shape templates out of cardboard scraps and use teeny pieces of tape to mount them on background paper.

    Spray them with tempera paint that has been watered down just slightly...enough so it will go through a spray bottle. (Test this ahead of time and remind students that less is more. Too much spray will give you "lake effect poetry.")

    When dry, remove templates and write the poem. A couple colored construction paper cut-outs help the artwork to "pop" and give it more depth. I use this tempera spray art for a variety of poetry-art projects. Students always love it!

    Character Study Sheets

    "Character analysis"'s not a phrase that brings students running. But what if you change it up a bit? How about scoring characters on report cards? It's so much fun to consider...

    What grade might you give Red Riding Hood for "follows instructions?"
    How about Cinderella's stepsisters? What grades might they receive for "gets along well with others?"

    Character Report Card

    Character: Adjective Scale
    Throw in a few adjectives--and their opposites--and you've got the makings of more creative character analysis.

    On a scale of 1-10, how might you rate each of the three pigs for lazy versus industrious behavior? How would Hansel and Gretel score on impulse control? The possibilities are endless...

    Casting Characters
    Finally, what if you got to cast characters in a movie production? Can you describe each character and cast a famous actor or actress in the roles?

    I love teaching short story through folk and fairy tales! Read more about our unit adventures here and here.

    You can also preview the character analysis sheets here.

    Thursday, December 4, 2014

    Christmas Art & Poetry: Winter Art from the Heart

    I spent the morning with an incredible group of (almost 30!) second graders, creating winter scenes.

    In preparation for their scenes--and for poetry writing--we brainstormed nouns, verbs, and adjectives that they might find in a book with the subject of "winter." They came up with fabulous lists.

    I am tickled with how these turned out. On another day, they'll complete poems to go on the back of each winter scene. Here are a few to share...

    Wisemen, angel, Mary & Joseph

    Notice the "naughty" list?

    Love the reindeer.

    Wreaths on the fence are a nice touch.

    So much detail. Love the colored birds.

    Now THAT is an angel! (And what a Rudolph!)

    Check out the Grinch!

    Frozen comes to life! The kids started singing spontaneous carols--and the Frozen tunes--while they worked!

    Penguin habitat!

    Tree is all decorated.

    Angel AND Santa!

    Displays for a class of 30 kids takes up a lot of counter space! :)
    You'll find the updated lesson here. (If you've already purchased it, please download your free revision with winter scenes.)

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