Fractions: Area Models

Over the past two weeks, we've been reviewing how fractions can represent part of a whole.

We've been representing unit fractions (fractions with a numerator of 1) and benchmark fractions (commonly used/familiar fractions such as 1/2, 1/4, 3/4) using 4x6 arrays as an area model.

In class, we've been looking at these models and having discussions to prove whether or not they are the fraction we're looking for (ex: 1/3 or NOT 1/3?).  Often, students think fractions have to "look" a certain way, but we've seen that fractions can be the same size (area), but different shapes.  Some students are even starting to make connections to division/multiplication as inverse to help them decide a fraction's area.  We'll continue to build on this strategy after Thanksgiving Break as we investigate fractions of sets and whether or not a fraction of one whole (or set) is always the same size as the fraction of another whole (or set).

If you've ever been asked to share something with a sibling (or friend), then you know how important it is to be able to compare fractions!

This year, I've shared a few stories about my brother, Andrew, and myself when we were kids.  Like most siblings, we argued, occasionally got in trouble together, and tried to trick each other.

(Coincidentally, my brother picture-mailed me this recently.  Notice how he chose to send a cute picture of himself and his big brown eyes that always make our mom, who also has brown eyes and is the youngest child like he is,  forget all about whatever he did and a picture of me on the verge of a tattle!)

Keeping peace in families and friendships...just one reason why being able to compare fractions is important in real life outside of school!

Multi-Digit Multiplication Strategy: Expanded Form Algorithm

Today I introduced our first multiplication algorithm.  Using an algorithm to perform a math operation simply means that so long as you follow certain steps correctly, the right answer will always result.  When using algorithms for some operations (subtraction, division), the order of the steps that are followed matters.  However, when adding and multiplying, the order of combining numbers does not affect the final answer (Commutative Property).  One of our students, Nick, proved this today when he "accidentally" found the the partial products in a different order than I modeled, but still came up with the same final product.  I chose to model the same order as the traditional multiplication algorithm, for the sake of easing the transition into using it.  That's one great thing about how different math instruction is these days--students going beyond a "one size fits all" strategy and truly understanding what they're doing and why it works.

The expanded form algorithm is closely related to last week's strategy, Partial Products.  Since our students were very successful last week with breaking apart both factors by place value and finding partial products, this week I've given our students the choice to use (or choose not to use) an array to model what's happening.

This week, we started writing problems vertically to prepare for using the traditional multiplication algorithm.  It's VERY IMPORTANT to make sure the digits are neatly-written, straight above each other (just like when adding and subtracting vertically, you might get the wrong answer if your digits aren't properly lined up, starting with the ones place).

I used color-coding to familiarize students with the order of the traditional multiplication algorithm:

First, find the ones place of the second factor (1).
Find the ones place of the first factor (4).
Find the partial product (1 x 4 = 4); record under problem.

Second, look back at the ones place of the second factor (1).
Find the tens place of the first factor (3). (It's very important for your student recognize and verbalize that this is not just 3, but 3 tens, or 30!)
Find the partial product (1 x 30 = 30); record under problem.

Third, find the tens place of the second factor (5, remember this is 5 tens, or 50!).
Look back at the ones place of the first factor (4).
Find the partial product (50 x 4 = 200); record under problem.

Last, find the tens place of the second factor (5, remember this is 5 tens, or 50!).
Look back at the tens place of the first factor (3, remember this is 3 tens, or 30!).
Find the partial product (50 x 30 = 1500); record under problem.

Are your partial products neatly lined up by place value?  Now combine the partial products by adding!  Draw a box around your final product.  Students may also use a calculator to check their final products.  In class, we indicate this by putting a check mark next to the final product after confirming this is the correct answer using a calculator.

I'm also making color-coding optional on Home Learning.  For many visual learners, this will help them see and remember the steps.  I also used to color to help students understand what they're doing (combining the red parts of a factor with the yellow parts of another factor to make the orange parts, the partial products.  I did the same having students combine the red parts of a factor and blue parts of another factor to make the purple parts, partial products). 

Verbalizing the steps using expanded form (telling the value of each digit) and using color as a visual will help the students develop greater number sense, which is the foundation for all math!

This will also lead to a much deeper understanding of the traditional multiplication algorithm.

Thank You, Soldiers!

Thank You, Soldiers


This poem will be added to our POETRY folders and discussed. Students will view this video and reading and discussing the poem. Students will also be writing about the emotions that they, as readers, feel when reading this poem -- this is call the MOOD.

Multi-Digit Multiplication Strategy: Partial Products

This week, students will be taking a familiar strategy, Break Apart Factor(s) by Place Value, and applying it to two-digit by two-digit multiplication problems.  Though the order of finding and combining partial products will not change the final answer (Commutative Property), I used the first four colors of the rainbow to help students figure out the partial products in the same order as the standard multiplication algorithm (the way most of us learned to multiply multi-digit numbers...red is the first partial product, orange is the second, etc).  I'm hoping this will make the transition easier when we get there.  :-)  I'm also encouraging the students to use a calculator to check their final answer after they've worked out the problem using this strategy.