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Category: Geometry

What Do You Notice? poster – Fractions on the Geoboard

What Do You Notice? poster – Fractions on the Geoboard

I just hosted my sixth fabulous Family Math Night event of the school year and with it came another new What Do You Notice? poster.  Here are the deets:
Title:  Fractions on the Geoboard
Skills

K-2:  shapes, counting, area

3-5:  congruency, area, fractions, equivalent fractions

Background Information: This is similar to the Squares and More Squares poster.  Like that one, this was designed around fractions but at a higher level.  The dots for each square represent the dots on a regular geoboard.  Each square is made up of 5 by 5 dots.  If lines were drawn connecting each of the dots, the larger square would show 16 small squares (4 x 4 small squares).  Each of these small squares represents one square unit.

I divided each of the large squares into fourths.  The third and fifth squares on the poster are the easiest representations of fourths.  It is also easy to see equivalent fractions such as ½ = 2/4.

Upper elementary students can figure out the area of the sections by counting the square units.  Just like in the Squares and More Squares poster, I deliberately made different shapes with the same area because sometime students will think that because the shape is different, the area must be different, as well.

Some students may notice congruency.  For example, in the first square, two of the sections are identical (congruent) even though the orientations are different.

Here’s what it looked like at the event:

Family Math Night Collaborative Project: Space Invaders

Family Math Night Collaborative Project: Space Invaders

I’m excited to share with you my latest Family Math Night Collaborative Project: Space Invaders. Here’s a photo of the final result. (There are actually 3 aliens to choose from in the lesson plan. This is alien #1).

Here’s some of the background information I include in the lesson plan:

In 1978, Tomohiro Nishikado, a Japanese video game developer, released his video game Space Invaders. It was such a popular game that it helped catapult video gaming into a global industry. The pixelated aliens in the game became a popular icon.

Pixels are small single-colored squares that make up images in computer graphics. These pixels are displayed as a bitmap, a rectangular matrix of dots. These pixels, sometimes called dots, are each assigned a specific color and are arranged along the horizontal axis (x-coordinate) and vertical axis (y-coordinate) of the matrix.

Computer graphics have come a long way in the last decade and look much more sophisticated today than they did back in 1978. But back when graphics were first being designed on computers, they had a “boxy” look. That’s because the screen displays (screen resolutions) were not as good as they are today.

Note: For the purpose of this activity, each pixel does not need to be represented by a single color.

Some of you may know that I always put together a video of my collaborative projects describing in detail how to do the activity and offering additional tips.  I’ve included the video below for you.

If you’re not familiar with my Family Math Night Collaborative Projects, take a peek at some of the previous ones I’ve done.  I include one of these projects at each of my Family Math Night events and leave it behind as a “gift” to the school for inviting me to host an event.  They’re often one of the most popular stations at the event.

What Do You Notice? Descriptions

What Do You Notice? Descriptions

It’s been on my TO DO list for a long time.  Years, in fact.  But I’ve finally checked it off and I’m thrilled with the results.

As many of you know, for each of my Family Math Night events I include a What Do You Notice? poster.  These posters are designed to get kids and parents thinking about math on a deeper level.  Although each poster has been included on our website, there has never been details…until now.  I’ve taken each of the posters and written a thorough description of the math involved.  I’ve also included the specific skills by grade span, K-2 and 3-5, and given several examples of student responses.

Here’s an example:

What Do You Notice? Rectangular Arrays

 

Skills
Primary students (K-2): shapes, counting, repeated addition, area

Intermediate students (3-5): classifying quadrilaterals, area model of multiplication, multiplication, prime, composite, and square numbers

Mathematical Background
For this one, I represented each number as a rectangular array. I also color-coded the arrays hoping that students would notice the relationship between the colors and the arrays that went with them. Notice how the orange arrays are square numbers. The red arrays are our prime numbers. Then I used blue for arrays that were non-square and had a length of two. Purple was non-square with a length of three.

I was also hoping that students would notice that some numbers were represented by more than one array (composite numbers). Prime numbers had only one array. Note: ‘1’ is not a prime number which is why I colored it orange – the color of the square numbers.

This activity does a nice job of visually reinforcing the area model of multiplication: L x W = A

Sample Student Responses

“The long grid before the square and/or rectangle grid(s) have the same number of blocks.”
“None of them are the same.”
“There is a color pattern.”
“Some numbers are on there more than once.”

The idea is to make it easier for you to include these posters at your events.  But it’s not just limited to Family Math Night events.  Teachers have written me about how they include them in their classroom learning and some are even displaying them in the school hallways to give students something to think about as they walk the halls.

However you decide to use them, I know they’ll help your students explore math in new and exciting ways.

The Power of the Estimation Jar

The Power of the Estimation Jar

We often think of math as the exact-answer subject.  But the kind of math that we do most often during the day doesn’t require an exact answer.  We use this particular math skill when we need to figure out how much time we need to get ready in the morning.  Or whether we have enough gas in the car to get to work. Or whether $50 is enough to cover the items in our shopping cart.

The math skill we use the most is, of course, estimation. And estimating accurately requires a high level of math.  That’s because it’s abstract which means we need to tap into our number sense and reasoning skills.

One way to provide our students with opportunities to work on their estimation skills is during computation practice.  Instead of diving right in to figure out 15 x 12, have students come up with an estimate…about what the answer will be. In fact, periodically I ask students NOT to determine the exact answer and, instead, have them turn in their work with only their estimates recorded.  This is hard for them to do in the beginning because they are so used to working out arithmetic problems, but they soon learn the value in thinking about the problem first.

A fun way to get students to work on their estimating skills is through the estimation jar.  I’ve included two of my estimation videos below.  The first video describes using the estimation jar in the classroom as a way to develop, not only estimation skills, but place value and number sense, as well.

The second video is filled with tips on setting up your estimation table at your Family Math Night event.  It includes something I’ve been adding to my estimation tables recently – the use of a referent.

You’ll find in both videos that there is a heavy emphasis on getting students to think about and make sense of numbers.  I discovered an example of this in action one day while cleaning up after a Family Math Night event.  It was such a powerful example of number sense that I’m now including “thinking” paper at my estimation stations.  If you missed the newsletter where I describe this priceless find, check it out here.  And click here to get the pdf of the thinking paper I’m now using.

A Twist on the Estimation Jar – Classroom Version

Setting up the Estimation Table at your Family Math Night event

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Family Math Night What Do You Notice? Poster

Family Math Night What Do You Notice? Poster

So for this What Do You Notice? poster, I decided to tie in rectangular arrays with prime and composite numbers.  That said, whatever math-y thing students notice is totally acceptable.  For example, one student noticed that the “buildings” had square windows.  Great.  That’s a little bit of geometry.  Another student noticed that each set of colored rectangles included the same number of squares.  Again, great, as that required some counting and comparison.  Just like the student who noticed that there are more yellow squares than any other color square.

As far as the prime and composite numbers…yep, we got that covered, too!  Here’s how David described it on his post-it, “The first 3 are composite, the last 2 are prime.  There are different arrays for each number.”  And he’s right.  The composite numbers have more than one rectangular array while the prime numbers only have one array.

But here’s an observation I totally didn’t notice.  “The columns go one higher, one lower.”  And, indeed, they do!

What Do You Notice