Author Jacqueline Davies' contribution to this year's Battle of the Books, The Lemonade War, gives kids many opportunities to consider the math involved in running a business. I'm excited to discover a chapter book that makes business, economics and math exciting. But it's much more than that...

Summary: Siblings--Evan, a 4th grader, and Jessie, his younger sister who is skipping ahead to his grade--compete to see who can generate the most money selling lemonade before the end of summer break. Evan struggles with feeling inferior to Jessie who is very smart in math. She, on the other hand, finds it hard to understand people (her symptoms read like a child with Asperger's, perhaps?), and wishes that her brother would help her navigate the scary world of entering 4th grade.

Although the math and business content is phenomenal, perhaps more importantly, the story illuminates the very complex feelings of two siblings who deeply love one another, all while grappling with the realities of their own strengths and weaknesses. This would be a great book to read aloud and discuss.
Recommended age: 9 & up

Merry Christmas! I wish you an abundance of love, merriment, and time spent with friends and family. Math Monday will resume next week. Hopefully, today you'll be counting presents, naps or peanut butter balls instead. :)

Order and compare fractions on a number line with these simple materials from the Dollar Store.

Materials Needed:
clothespins (pack of 36 for $1)
hanger
Sharpie

Directions:
1. Use a fine Sharpie to label clothespins with the fractions you wish to explore.
2. Place clothespins in a Math Station or Workbox, along with a hanger.

3. Ask kids to clip the clothespins to the hanger--the "number line"--in order. Differentiate with varying numbers of clothespins. Change the range of the number line by removing larger or smaller numbers. For example, the number line could include clothespins from 0-1 or from 0-1/2. When students find equivalent fractions, they may clip them to fractions of the same value in a vertical line.

Additional Option: You could number or letter the back of the clothespins so that students could check their own work. (Label a set from A-Z, for example.)

Clothespin Fractions provide a powerful model for discussion. My ten-year-old son, an eager test subject, jumped right in to order the clothespins. His dad checked answers. I wish I'd recorded the ensuing dialogue. My husband, a guy with a ton of math smarts, tried to figure out the answers using percents. Since this is something he regularly uses at work to create documents, he felt like it was an efficient method. In contrast, I relied on common denominators. My husband helped my son to make a couple order changes but finally admitted that in one case, my son's original answer was actually correct. (Talk about making a kid's day!) We then talked about how my method was easier the merits of using percents vs. common denominators. I felt a fraction wee bit smug.

But more importantly, we all had a fun, productive Math Talk!

Want more ideas for teaching fractions? (One of my favorite topics to teach!) Click here! :)

CCSS Standards in Grades 3-4:

CCSS.3.NF.A.3a Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.

CCSS.3.NF.A.3b Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model.

CCSS.4.NF.A.1 Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

CCSS.4.NF.A.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2.

Teaching angle measurement?* Although it's tempting to put a protractor straight into the hands of students, with a little creativity, pattern blocks, and some simple movements you can build a foundation for the concept of angle measurement. This week, we've been challenged by a fun, free set of lessons by The Math Learning Center, C3 Circles & Angles.

Our first challenge, in Activity 1, was to measure all angles in a set of pattern blocks using nothing more than this:

My normally brilliant student looked at me like I had lost my mind. How in the world could you measure the angles of a pattern block with that??? We first talked about what we knew:

1. Straight line = 180 degrees
2. Perpendicular line = 90 degrees
3. Orange square pattern block has to have 90 degree angles because it's a square. It also fits in the corner of the drawing above.

But what about all the rest of the shapes? How could you figure their angles?

After a time, we discovered that pattern blocks could be combined to form a straight angle of 180 degrees. In the example below, we could see that 3 triangles (or in the illustration, 1 triangle + 1 rhombus) formed a straight angle. Therefore, 180/3 = 60 degrees...or 1 green triangle has angles of 60 degrees each since it's an equilateral triangle. In a similar fashion, we made combinations of pattern blocks until we knew the angles of all the pattern blocks. Pattern blocks now became a tool for additional measurement opportunities.

In Activity 2, we measured "Human Angles," first considering real-world connections: a sister who had to have physical therapy for a knee injury, a brother whose extreme joint mobility allows him to do circus-like feats of movement,...even a non-human connection--an owl--and it's circular head movement. But at first it wasn't easy for us to see human movement in degrees.

We added a number of visual tools: still photos, craft sticks, and moveable paper models. The following video captures some of our thoughts as we explored angles...one step degree at a time.

You can grab a free set of the angle lessons by looking for C3 Circles & Angles. What do you do to bring angle measurement to life?

This lesson includes CCSS Standards: CCSS 4.MD.C.5 Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement CCSS 4.MD.C.7 Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.

I'm currently reading Stephen Wilbers book, Keys to Great Writing. In it, he talks about 4 myths that limit a person's writing potential. As I read them, I thought about--what else?--MATH. Here are his myths with a little mathematical editing...

Four Myths that Limit Your Potential (or your student's) as a WriterMathematician:

1. Myth #1: Only people with natural ability can learn to writedo math well.

Reflection: I spent most of my life believing this. Without question, some people have mathematical gifts, just as others are skilled in athletics or the arts. But given the right environment, the rest of us can prosper mathematically. We would never tell someone that only people with natural ability can learn to read or walk or drive a car well.

2. Myth #2: People who are good in math and sciencereading and writing are inherently incapable of using languagedoing math effectively.

Reflection: English major here. Long-time lover of children's literature. Now hugely benefitting from visual math models that I never learned as a child. Give me a good piece of math-related children's literature and scary math topics will turn on a dime.

3. Myth #3: Achieving writingmathematical competence is a matter of learning to avoid errors.

Reflection: Errors are often our best teachers. Ask Edison.

4. Myth #4: Learning to writedo math well is easy if you just learn the right tricks.

Reflection: This myth causes havoc, leading us to believe, among other things, that

When I watch finger multiplication, for instance, I'm impressed from a "boy, do you ever have a great party trick!" viewpoint. I am NOT impressed mathematically. But if I ask you what 6 x 7 is...and you explain several ways to calculate the answer...now THAT is impressive!

Question to Ponder: Which math myths most limit your potential?

As I've mentioned, when I was a 4th grader, fractions were a big, hairy monster in my otherwise happy little mathematical world.

I'm still trying to heal the fractional parts of my 4th grader self. Over the last few years I've made progress:

Ten + years ago: fractions made me slightly nauseous, rendering me unable to think.

Five years ago: although no longer causing nausea and palpitations, fractions did require a rather strong antiperspirant

Today: while deoderant still comes in handy, a little voice in my head now does a mini "wahoo" when I hear someone utter the word "fractions."

So whenever I see a clever lesson, one that helps fractions to make some visual sense, I take note. A free lesson from The Tutor House, Pizza Pie and the Number Line Freebie, helps kids to see the relationship between the circle "pizza" fractions we use so often and the fractions on a number line. Since the number line fraction model is such an integral part of the Common Core State Standards, this is a particularly valuable tool.

Credit: Clip Art of the 1/2 fraction man (above) is provided by Phillip Martin. His site has GREAT, free math clip art!

As many bloggers pause to honor and remember the community of Sandy Hook Elementary, I want to suggest one article, "Helping Students Through Tragedy."

I first wrote about Equilibrio, Architecto, and Cliko, one of my all-time favorite math purchases, several years ago. In the original blog entry you'll find a more thorough description of the product itself. Basically, it's a set of geometric blocks that are used to build structures that require balance or an eye for 3-D (depending on which book you're using.) My kids began using the series at ages 4 and 7. Now ages 6 and 10, I'm amazed at how the activities continue to challenge them.

Having recently purchased another set of blocks, a friend's visit inspired an "Equilibrio Competition." They opened the book so both could see the structure and then raced to see who could successfully build it first. When one finished, he helped the other to figure it out.

This would make such a good addition to geometry, whether at home or school, providing immense opportunity for differentiation. It screams "Math Station!"

Disclaimer: I bought my own set--two sets of blocks and all 3 books--and have no contact with the company that produces it. If you use the Amazon link to buy your own, Grace and Hope (foster care for kids in China) will make a few cents (at no cost to you.)

Math & Literature: Explore Doubling, uses two books, One Grain of Rice and The King's Chessboard, depicting a rich royalty figure who, feeling indebted to a commoner (a poor village girl, a wiseman), agrees to pay the "paltry sum" of a single grain of rice...to be doubled each day for a period of days (a month, the squares on a chessboard.) These "paired books" could be compared and contrasted to consider any number of concepts--character, setting, plot...and, yes, math!

The free download includes a template for a comparison flapbook as well as a problem solving worksheet in which students consider which is the better deal: one dollar, or a penny doubled each day for one week.

Please grab a copy! And thank you for looking!

Referenced Common Core State Standards: CCSS 4.OA.C.5, 4.MD.A, 4.NBT

A math friend recently sent me a link to some wonderful videos that walk through visual models on dividing with fractions. If you are an adult who memorized the rule for dividing fractions--invert and multiply--but had absolutely no idea of the reason behind the rule -OR- if you are a teacher (homeschool or public!) and are trying to help your kiddos understand this concept, you'll love these videos.

Republishing one of my favorite posts... It's a good time of year to cozy up with a cup of hot chocolate and a good movie (with math!) :)

A couple years ago I began using video clips in my continuing education classes for teachers. Why? Short movie excerpts have the powerful ability to:

illustrate a point

incite discussion

provide a humorous "jumping off point" for a particular concept

pose "real life" problem solving

capture students' attention

address a variety of student learning styles

present a wide variety of mathematical strands

help students to consider what they know and what they have yet to learn

Many, many movies contain mathematical clips. Some are best used with advanced math students while others are perfect for younger children.

Consider this clip from the movie, Father of the Bride. Mr. Banks is frustrated by the fact that hotdog buns come in packs of 12 while hotdogs come in packs of 8. Pose the question to students:

How many packs of each would he need to buy to make the hotdogs and buns come out even?

Here's a clip from Ma and Pa Kettle. I love this clip because it illustrates how important it is for students to explain not only WHAT the answer is but also HOW they got an answer. Can your student explain where Ma/Pa went wrong in each proof?

I also enjoy math problems from game shows. Students of all ages like to feel smarter than television competition. (Why else are game shows so successful?) Here's an excerpt from Who Wants to Be a Millionaire in which actress Patricia Heaton struggles to answer the following question:

If a Euro is $1.50, five Euros is worth what?

A. Thirty quarters
B. Fifty dimes
C. Seventy nickels
D. Ninety pennies

Many more online resources are available on math in the movies:

Math and the Movies: Two teachers list many movie excerpts and provide free pdf lessons to go with each.

Mathematics in Movies: A Harvard professor has collected an extensive collection of movie titles. He includes the movie clips in a library on his site.

Simpsons Math: Math lessons using episodes from The Simpsons.

Countdown: using technology of Quicktime to develop math skills.

LivingMathForum: Julie compiled a list of recommended math video learning materials from the Living Math Forum Yahoo group.

Mathematics Goes to the Movies: titles and short descriptions of over 500 movies and TV episodes that contain mathematics. (Thanks, Sue, for alerting me to this site!)

A few movie titles that are totally oriented around math and may be worth purchasing to add to your own math library:

And, of course, any of the Cyberchase episodes. (My kids have learned an incredible amount of math from watching this PBS series. You can also play on-line games and see some excerpts at the PBS kids website.)

Lastly, on YouTube you can see Norton Juster's The Dot and the Line... Enjoy! :)

You can print FREE full size math practice books for grades K-5. Seriously...

Bridges Practice Books
Using the above link, click on "Practice Books" for the grade level you desire. The consumable books are available for purchase, but you can download the FULL set of blacklines (pdf files) for FREE.

Written to address the NCTM Focal Points, Bridges Practice Books provide activities and worksheets for additional skill review, informal paper-and-pencil assessment, preparation for standardized testing, and differentiated instruction. These books may be used with any math program to provide practice with skills targeted for mastery at each grade level.

The blacklines include the answer keys. Also, each book is indexed by skill so it's easy to find multi-digit addition & subtraction, fractions, decimals, problem solving, etc...

Grade K Skills: Numbers to 30, comparing and ordering sets, skip counting, early addition and subtraction, story problems.

Grade 1 Skills: Numbers to 100 and beyond, number patterns, place value, facts to 10, money, time, graphing, and problem solving.

Grade 2 Skills: Numbers to 1,000, skip counting and number patterns, facts to 18, place value, double-digit computation, money, time, and problem solving.

Grade 3 Skills: Numbers to 10,000, multi-digit addition and subtraction, multiplication and division concepts, fractions, equations, perimeter, time, money, and problem solving.

Grade 4 Skills: Multiplication and division facts, multi-digit addition, subtraction and multiplication, fractions and decimals, patterns and equations, area and perimeter, data analysis, and problem solving.

Grade 5 Skills: Multiplication and division facts, multi-digit addition, subtraction and multiplication, fractions and decimals, patterns and equations, area and perimeter, data analysis, and problem solving.

Having great fun with a kindergarten group. Yesterday we thought about what makes 5. We did a little subitizing and made use of the Math Learning Center's free app, Number Rack, that is also available as an online application for those who don't have a mobile device. Since I was a portable classroom (working in the hallway), I used my iPad. The kindergarteners were so impressed...until I said it wasn't mine! Here's one of the activities we did:

First, I showed them my virtual Number Rack and explained that today we'd only be using the red beads.

I asked them how many red beads I had.

Then I changed the configuration and asked again.

And again. "How many red beads?"

And again. "How many red beads?"

And again. "How many red beads?"

Eventually most of them started laughing. "There's always FIVE BEADS!!!" After they straightened me out, I explained the next step.

First, I would hide some red beads on the right side of the screen under a flap. I'd set it up like this, but they wouldn't see any red beads at all, at least at first...

Because I'd make it even trickier. I would cover the beads on the left with my hand, move my hand to flash them a REALLY QUICK glimpse of what was hiding, and then cover it up again. They'd have to tell me what was hiding on the main (left) part of the screen based on their quick view (subitizing.)

After they guessed, I'd show them what was there. Then they could try to figure out how many red beads were hiding on the right side of the screen and we'd check by pulling down the ? flap to check.

I learned a lot about students' number sense. Especially "Dan's." At first, I thought Dan had the concept of making five. When 2 beads were seen, he said that 3 were hidden. But then when 3 were shown, he said that 4 were hidden. When 4 were shown, he insisted that 5 were hidden. Later I worked with him individually using a 5 frame. We have a bit more work to do!

Try out the Number Rack. Do a little subitizing. And "make five" math! (Or make five math friends!)