Thursday, October 9, 2014

Have Multiple & Factor Confusion?


Kids sometimes find it confusing to differentiate between multiples & factors. Here are a few visuals that I've found helpful:

Multiple: the product of a given number and another number. 

We could show this on a number line:







To find the multiples of 3, we can start with our given number (3) and then multiply it by 2 (6), by 3 (9), by 4 (12), and so on... 

We also see multiples represented by tile (Number Pieces):

1 group of 3 tile = 3
2 groups of 3 tile = 6
3 groups of 3 tile = 9
4 groups of 3 tile = 12







We can also use a visual model to investigate factors.

Factors: numbers that, when multiplied together, result in a product.

One of my favorite ways to do this? Lay out the product with tile (Number Pieces). Let's look at 12:

Use the 12 tile to form as many rectangular arrays as possible. The dimensions produce factors of the number. Here we can see arrays with dimensions of:

1 x 12
2 x 6
3 x 4

(Note: When using this visual model, prime numbers are also easy to distinguish: if it's prime, only one rectangular array--a 1 by the number--is possible.)

Why Do We Care?
My son is currently taking AP Calculus. He just popped in my office, saw what I was doing and said, "For us, finding factors is just one tiny step in a huge process. We do it all the time." Factors are a part of  higher level math! And, at a slightly lower level, students frequently use factors when working with fractions...not to mention (!) multiplication.

Additional Resources
I just created Fold It!...Factors & Multiples, a new set of flap books (Venn Diagrams & Shutter Folds) for students to compare and contrast factors and multiples. When complete, pages make a nice addition to student math journals. Alternate versions (different number combinations as well as blank copies) are included to allow for many uses: differentiation, exit slips, homework, notebook pages, math stations, etc.



The new set is also available as part of a bundled Multiples & Factors Flap Pack that includes the popular Flap Books "Present" Multiples & Factors (pictured right).

As always, new products (& bundles) are introduced with a sale price.

More Ideas
Looking for more ideas? Here are some of my favorites from around the web:
  • Flap Books & Online Games - in this blog entry, I share photos of flap books we made and link to a variety of games on the web
  • Factor & Multiple Anchor Charts with Student-Made Posters from Young Teacher Love - both are awesome.
  • Online Venn Diagram - create your own Venn Diagram to compare factors & multiples (example, right)
  • Multiple Mummy - kids use adding machine tape to make multiple strips and turn their teacher into a mummy.
Tune back for a multiple/factor game that you can play with any size group, 1 to 100!

Tuesday, September 23, 2014

Math Education: It Ain't 1975

It's been 4 months since I last posted on this blog.

4 months. A long time.

But it's been even longer since the 70s. In 1975, I was an elementary student, as were many of today's parents (...or, perhaps, grandparents!) In the past several months I've been doing a lot of thinking about the way the teaching of math has changed and how to communicate with families who were raised with a very, very different math education.

So today I reflect on how math education has changed since I was a schoolchild.

Second Grade, 1975
Yes, that is a poncho!
Might even be wearing culottes/gauchos.

When I was a child...
I thought only some people had math smarts. And then there were the rest of us. Except for a fraction blip in 4th grade, I thought I was one of the smart people. Then I hit high school and it didn't make sense anymore. But I still got an A because our teacher wore 2 hearing aids, didn't take questions, and only required us to turn in one paper as a whole class.

Today's children...
benefit from researchers like Carol Dweck. Children learn that growth mindset means they can become better at math if they work hard and persevere.

When I was a child...
I thought that the correct answer was the only thing that mattered in math.

Today's children...
know that while the correct answer is important, it is also essential for them to be able to communicate the answer and the mathematical thinking behind it; communication skills are essential for twenty-first century work places.

When I was a child...
From Math Vocabulary Cards app
I didn't really use--or even consider--mathematical vocabulary.

Today's children...
use precise mathematical language, symbols, and labeling to create tools for understanding. We've learned that neglectful use of language can lead to misconceptions as students move into upper grades and more complex math topics.

When I was a child...
I thought there was only one way--the standard algorithm--to do a problem. It worked...most of the time...if you could remember the steps. A lot of my peers could not. And we wonder why two-thirds of American adults are math phobic.

Today's children...
learn a variety of mathematical strategies that help them make sense of math. Over time, they are able to select the strategy that is the most efficient for a given situation.

When I was a child...
numbers were the only option. And then, only if you could memorize the steps using the numbers.

Today's children...
use a wide variety of manipulatives (sometimes on an app!), tools, sketches, pictures and models to represent their thinking visually, using concrete items to make sense of the abstract. Math means something because they can model the mathematics behind a problem. They see it and understand it.  3 x 4 is more than a math fact. It's the dimensions on an array that show the area as a product in the number of square units.

When I was a child...
math was a solitary endeavor. Each of us hunkered down to answer our own problems--odd or even or, gulp (!), both--out of the textbook. No talking!

Today's children...
talk about their strategies and solutions. They learn about efficient ways to solve problems as they listen to one another. They prove their thinking and critique the thinking of others, deepening skills in communication while increasing conceptual understanding.

When I was a child...
math didn't necessarily make sense. Math was just math: plug numbers into a standard algorithm and arrive at an answer...hopefully the right one.

Today's children...
know that math makes sense. They look for patterns and structures. They might make sense of 4 x 6 by knowing that it's made up of 2 sets of 2 x 6...  2 x (2 x 6).  (See right.) They know that if you "make 10" a problem is easier to solve. So they might see 9 + 4 as 10 + 3. They can reason that 15 x 13 is (15 x 10) + (15 x 3). Math makes sense.

When I was a child...
math with any complexity whatsoever was done with paper/pencil.

Today's children...
can do a problem like 16 x 12 in their heads. In seconds. Because they can picture it. (See left.)

When I was a child...
it felt as if math was isolated to math class.

Today's children...
know that math is everywhere.




It's not 1975 anymore. And my students (who laugh after hearing that I was taught problems could only be solved one way) would say that it's a good thing, mathematically speaking.



Saturday, May 24, 2014

Parallel & Perpendicular Art


When I noticed that quite a few of my students were confused by the terms "parallel" and "perpendicular," I decided it was time for an end-of-year art activity. I visited my Math - Art Pinterest page and discovered a sweet little project at Math Activities*. Students worked on this during free moments at home and at school.

*You'll notice that we purposely did the lesson a bit different than originally described.


I love the way this turned out. And, yes...they now can tell me what parallel and perpendicular are! ;)

Monday, May 19, 2014

Fraction Arrays: What Do You Notice?


Students made a series of arrays demonstrating fraction multiplication. As a group, they studied the arrays and put them in sequential order by area. This activity alone provided incredible growth opportunity as students discovered mistakes (hey...why is that array bigger than the one next to it but the product is smaller?) and had to occasionally reformulate so that the pattern made sense.

In a subsequent discussion, we shared more observations and looked for additional patterns. We considered similarities and differences in:
  • fraction x fraction multiplication
  • fraction x whole number multiplication
  • whole number x whole number multiplication
Afterwards, students wrote reflections. I posted a sample with our display. I'm so impressed by their insights and understandings. (Click on individual photos for magnification to read student comments.)

Lesson Credit: Bridges in Mathematics, Second Edition, grade 5

Left side.
Middle
Right

Tuesday, May 13, 2014

Number Frames App is Here (Free!)


The Math Learning Center just released another free math app: Number Frames. It's currently available on iTunes and will soon be released in an online version. Can't wait to use this one! Enjoy. :)

Tuesday, May 6, 2014

TPT Sale May 6-7 - extended!


Update: the sale has been extended to the 8th on TPT. Enjoy!

In honor of Teacher Appreciation Week, the sale begins on Tuesday with all of my products offered at reduced prices. Add the TPT promo code for even deeper discounts. Here are a few suggestions for end-of-the-year fun:

Design a Cube City - Students use isometric dot paper to draw 3-D buildings and design their own cube cities.
Poetry & 3-D Art for Every Season - write diamante poems--reviewing nouns, verbs, and adjectives--and create 3-D art displays to brighten any season or holiday.


Skittle Fractions, Estimation & Graphing


Vacation Workstation

Descriptive Poetry & Tissue Painting

Thursday, April 24, 2014

Milk Cap Magic Squares


Ready for some fun? Grab a copy of Ben Franklin and His Magic Squares to read aloud. Stop reading after it says, "Now Ben wondered if he could make the numbers [1-9, in a square] add up to 15..."

Give each student a set of 9 milk caps (I don't EVER throw them away!) to number, 1-9.



Challenge them to make a 3 x 3 array in which every row, column, and diagonal totals 15.


If they think they've found the answer, have them scurry off to write it down! (And hide it from everyone else!)

Stop periodically to talk about strategies. Today we discussed ideas like:

1. Not having all of the biggest numbers--or the smallest numbers--in the same row, column, or diagonal.

2. Setting some of the combinations in #1 and then working around those numbers.

3. Only moving a few caps at a time, especially if you're getting close.

You can also give the online version a try!




Tuesday, April 22, 2014

Student Video Tutorials: Multi-Digit Multiplication


We're excited to share student-produced video tutorials on the multi-digit multiplication strategies that we've been learning. The tutorials were recorded using a free app, Educreations.

As we began, I asked if any of them had ever seen a video tutorial. Grinning, they all said that they had. They were thrilled with the idea of making their own!

After randomly selecting a multi-digit multiplication strategy, they planned presentations, including visuals and text/dialog. This organizer helped guide the process. (Click on the sheet for a free copy.)

https://docs.google.com/document/d/16XdATySZUy6L4-cOReIesmTkbXVqmwwGULmp9ut0FYQ/edit?usp=sharing

Take a peek...
Students plan and write the visuals and the dialogue...




...and practice presentations on mini whiteboards...


This was a powerful way for them to demonstrate what they've learned. As students planned, I heard incredible mathematical conversations.  In one example, while working on the ratio table presentation, a student experimented with several sets of numbers--conversing with classmates about choices--before he came up with an example that demonstrated optimal efficiency.

Enjoy their videos. If you'd care to share, they would love to read your comments!




















The articles in this 3-part series on Teaching Multi-Digit Multiplication include:
1. Multiple Strategies for Multi-Digit Multiplication (introduction)
2. Choosing the BEST Strategy
3. Student Video Tutorials


Friday, April 18, 2014

Explore Volume with Mini Cube Cities


It's April. Time for a little "art break" in math class. In this case, I wanted students to apply what they've learned about volume. I pulled out the isometric drawing paper and we got busy, each student designing a "Mini Metropolis."

I led them through step-by-step practice sheets to learn how to draw cubes and rectangular prisms. Almost immediately I heard, "...this is cool!" followed by "...this is like Minecraft [video game]!"

To make the drawings just a wee bit more 3-D, we learned to do a little shading.



After practice, I gave each City Planner a "Design a Mini Metropolis" sheet and related table. The assignment required them to:

1. draw some buildings with given dimensions, calculating volume
2. draw some buildings with a given volume, calculating dimensions
3. invent some buildings, calculating dimensions & volume



Rulers kept our lines straight.


If we needed help making rectangular prisms with a given volume, centimeter cubes helped.

Aren't the buildings lovely?


We planned for roads, parks, foliage and other metropolis necessities!


Students loved this lesson. Some of them have continued drawing ever since we began this assignment, printing out more isometric dot paper at home. 

If you are interested in purchasing the lesson, you'll find it here.

Tuesday, April 15, 2014

Multi-Digit Multiplication: Choosing the BEST Strategy

Warning: this is a ginormous post, filled with student work. I am feeling just math-teacher-pride-nerdy enough to post it. You've been warned.

I mentioned how my students have learned many strategies for multi-digit multiplication this year. That, in itself, is terrific. But even better? They understand what types of numbers work best with certain strategies. This ability--to determine which strategy is most efficient in a given situation--shows that have an incredible array of mathematical knowledge in their tool boxes.

In the first several examples, students could choose from 20 multiplication expressions, selecting the problem that they thought was a good match for each of the strategies they've learned. They had to:
  1. choose an appropriate problem for the strategy
  2. solve the problem using that strategy
  3. explain why that strategy fit the problem
  4. double check their answer using a second strategy. 
It's important to note that when the year began, some students had zero strategies for multi-digit multiplication. We've come a LOOONNNGGGG way!!

I've added captions to a few examples near the end...stuff you mustn't miss. And then read on...there's more below!
    Strategy: Doubling & Halving
     Strategy: Over Strategy
     Strategy: Using Quarters

     

    Strategy: Four Partial Products
    Strategy: Ratio Table

    Strategy: Standard Algorithm
    Notice that this student deliberately chose more challenging numbers to use on this strategy.
    Cracks me up...I told them that the standard algorithm is the only strategy I was taught. Notice what this kiddo wrote in his explanation...

    After completing the strategy sheets, I compiled a list of the strategies and the problems chosen by students for each one. Students selected the identical expression on some problems (like 99x73 for the over strategy) and had different expressions for others. This led to a great discussion about which strategy was chosen for each, and why.
    Tips for Multi-Digit Multiplication Strategies
    After my students finished the strategy sheets, they made posters--TIPS--for using each strategy...






    I wish you could have heard the discussions that went with both the development and the presentation of these tip sheets. I couldn't believe the amount of higher-level thinking involved.

    Do these students "get" multi-digit multiplication, or what??? A parent told me that her student was about ready to bust a button when he proudly said, "I know TEN ways to solve problems."

    Awesome, awesome work!!!!!! (And thank you to Math Learning Center, Bridges Second Edition, Grade 5, for the strategies and core lessons.)


    The articles in this 3-part series on Teaching Multi-Digit Multiplication include:
    1. Multiple Strategies for Multi-Digit Multiplication (introduction)
    2. Choosing the BEST Strategy
    3. Student Video Tutorials
    Related Posts with Thumbnails
    Blogging tips