# data[].attributes.headline
This pin is about problem solving and reasoning with fractions, as well as, investigating the relationships between various parts and wholes. Students also focus on representation of the relative value of fractions. Students use communication skills to clarify their understanding of fraction relationships.

# data[].attributes.subject
This pin is about problem solving and reasoning with fractions, as well as, investigating the relationships between various parts and wholes. Students also focus on representation of the relative value of fractions. Students use communication skills to clarify their understanding of fraction relationships.

# data[].attributes.attachment_link.title
Investigating Fractions with Pattern Blocks

# data[].attributes.body
This website leads to my favorite number sense guru, Andrew Stadel. Stadel talks about some the activities he does during teacher inservice workshops, but they of course are dynamite in the classroom. I love the idea of making a non-static number line using clothes pins...this idea could be used again and again throughout the year. Multiple clothesline could be created to develop number sense with fractions, decimals and much more. Most importantly the number line would aid in creating connections between concepts as students move for unit to unit.

# data[].attributes.attachment_link.readable_text
Dear Chris Shore,

Thanks for introducing Tustin Unified to Clothesline during your awesome professional development workshop!
You, Math Projects Journal, and Clothesline rock ( David Lee Roth style)!

With all of my math heart,
Andrew

*Check out Chris' post on Clothesline ( link coming soon).
**I highly recommend inviting Chris to your district/school for math workshops.

Where do I begin?
I used to have a number line in my old class. But it was static. All of the benchmark numbers were taped to the wall. I used it often, but not often enough.

Flash forward to Chris' workshop last week. He introduced Clothesline using this great quote from Tim McCaffrey:

- Andrew Stadel (@mr_stadel) August 19, 2015

You better believe my ears perked up when I heard "master number sense maker". Check it out!

When I worked with teachers in Irvine today, my ice breaker was asking the teachers,

"How long does Dyson think it should take you to dry your hands with their machine?"

I've never posted this picture online because it's my favorite ice breaker to do with teachers at conferences and workshops. However, I will use the context to illustrate how I introduced Clothesline to about 150 teachers, coaches, administrators today.

As teachers were discussing, I went around and asked

two teachers for a guess that's too low (3 and 5 seconds)

two teachers for a guess that's too high (30 and 40 seconds)

four teachers for a guess that's just right

I used a big black marker to write the numbers on the red papers for the first four teachers with their "wrong" answers. I then had them place the red papers on the 25 feet long clothesline hanging on the side wall. I had the last four teachers write their "just right" guesses on the green slips.

You'll notice the green slips might be hard to read from a distance. I did this on purpose so that we could get the visual effect (from a distance) of placing their green slips accurately on the clothesline using correct spacing (sorry, no pictures), based on where the red slips were originally placed. I reminded the teachers that this is a dynamic number line. You can move the numbers along the clothesline as you please. Please note that the teachers (in this case) were doing all the work, thinking, critiquing, and adjusting. In other words, students should be doing the same in my class as I help facilitate the conversations.

Throughout the rest of the day, I worked with teachers grades 7-12 during three breakout sessions. Therefore, I made a handful of cards for each breakout session to correspond to numbers or expressions relevant to them and their content area. Here's a sample:

I love when Chris used colored papers to focus on the numbers being placed (or in question). Notice the "benchmark" numbers are on white paper.

THE ROPE
The clothesline is 25 feet long. I think this is plenty long. I went to Home Depot and bought a 100 ft. clothesline. I made three cuts to make four lines of 25 feet. I used a flame to burn the ends of the rope so they stay in tact.

THE CLOTHESPINS
I noticed Chris used these to stack equivalent values together vertically. Brilliant. See above for examples that could stack.

THE NUMBERS (or expressions)
NCTM suggests using 3x5 cards, but then you have to use more clothespins, making the number line more static. Chris suggested using strips of paper. Using strips of paper allows the number line to be way more dynamic, allowing the numbers to slide along the clothesline or making it easy to place the numbers or take them off without the use of clothespins.

VARIABLES
Many teachers loved the idea of using variable expressions. Here's how I determined x for each group. I asked the three teachers in possession of the variable expressions to share how long they had been working in the district. For example, three teachers shared 12, 13, and 15 years. Therefore, 40 was the value of our variable, x.

FUTURE USE
Whenever I give a professional development workshop for teachers from now on, I will be using Clothesline. IT'S AWESOME! It is a master number sense maker. If I happen to be at your district or school doing PD, I'll bring a handful of clotheslines to raffle off (or give away). At the end of my last session today, it was awesome to have two excited calculus teachers be extremely thankful for receiving a clothesline. One walked away saying, "I'm going to use this Wednesday." Their first day of school! Calculus!

Last, but not least, test it out at home if you have the chance. My five-year-old son and I had fun this past weekend. He threw me a few surprises.

Clothesline 1
I just tossed up a few numbers on the clothesline for him to first get acquainted with the idea of numbers on the clothesline. "Move the pieces of paper so they make sense to you."

Clothesline 2
My son caught me off guard when he pursued something he was interested in. 1, 2, 3, 6.
" What?"

Clothesline 3
I wanted to see how my son did with spacing the numbers.
"Show me where four and five go."

I'd love to hear about your Clothesline experiences.
Check out Kristin Gray's great post from the other day. I love how much she anticipates student thinking in preparing for a successful Clothesline activity.

Clothesline,
930

# data[].attributes.attachment_link.description
Dear Chris Shore, Thanks for introducing Tustin Unified to Clothesline during your awesome professional development workshop! You, Math Projects Journal, and Clothesline rock ( David Lee Roth style)! With all of my math heart, Andrew *Check out Chris' post on Clothesline ( link coming soon).

# data[].attributes.body
I find the discussion about not teaching invert and multiply poignant as I am right in the middle of teaching a unit about operations with fractions and decimals. As I read I can't help but think all I really want is for my students to jump on board with me and learn the why...instead of the simple how. However, I am struggling to get a majority of my students to buy into understanding, instead of solving for the sake of solving. Everything I read does, however, seem to end up with some algorithm that multiplies, divides, flips and or finds a common denominator and comes to the same answer.  The pin associated with the this text ends up teaching an algorithm but does so after explaining the why. Honestly, it seems a lot like keep, change, flip with window dressing. I suppose what I am saying is, I have tried to build meaning with models and concrete examples but eventually students will have to follow some basic pattern to solve division problems. Perhaps what I have gleaned from reading and searching for pins is that students need a variety of examples and the proper timing of teaching as to develop peoper understanding.

# data[].attributes.attachment_link.readable_text
As elementary teachers, we rarely have the opportunity to explore division of a fraction by a fraction. When we do, it's normally accompanied with Keep-Change-Flip or the saying "Yours is not the reason why, just invert and multiply."

Both are conceptual cripplers.

I've been drafting the 4th installment of the Making Sense Series involving fractions and I'm sharing this post as more of a personal reference should K-C-F make its way round these parts again...and I'm sure it will.
Side note: A while back Fawn and Christopher each shared a post about division of fractions using common denominators. Both posts left lots of math residue and are well worth your time.
Let's start with a model for 3/5 ÷ 1/4.

At some point along the way it becomes inefficient for students to draw models once the conceptual understanding is established. As students represent measurement division of fractions they should be formally recording their thinking.

From here students generate their own algorithm (shortcut). They begin to recognize that they will always get a denominator of "1 whole" so they begin to purposefully leave it out. In doing so, they become more efficient in the procedure of dividing fractions.

Some students begin to eliminate the green and red steps from the above equation because they're seen as repetitive. We've even had one student that "invented" and generalized cross multiplication for division of fractions as they searched for ways to record fewer numbers and symbols.

It looked something like this...

I keep reminding myself that if fractions are the gatekeeper to algebraic reasoning then I need to slow the process down and conceptually understand what's happening. This includes K-C-F.

As students understand the power of creating a whole number denominator they begin to search for more efficient ways to get 1 whole. It's here that they being to explore with equivalent fraction and the idea of using the reciprocal to create a whole.

What I've realized is that aside from complex fractions, the underpinnings of this equation are developed in the elementary grades.

5.NF.3 Interpret a fraction as division of the numerator by the denominator (a÷b = a/b).

4.NF.1-Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b).

3.NF.3.c- Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.

Just like before, students will look to generalize the equation above and find shortcuts. They'll do so by eliminating repetitive steps which would leave them with...

# data[].attributes.attachment_link.description
As elementary teachers, we rarely have the opportunity to explore division of a fraction by a fraction. When we do, it's normally accompanied with Keep-Change-Flip or the saying "Yours is not the reason why, just invert and multiply." Both are conceptual cripplers.

# data[].attributes.body
Thinking about dividing decimals and fractions. This lesson is great as introduction as students can easily find solutions without knowing algorithms, thus creating meaning before being introduced to algorithms.
Follow mini-lesson after the apple exploration...

# data[].attributes.attachment_link.readable_text
Download the Task

How many blocks will it take to level out the balance scale? Estimate

Write an estimate you know is too high. Write an estimate you know is too low.

The weight of the apple. The weight of a single cube
Thanks Robert Kaplinsky for the sequel image.
Click the link here to find more of My 3-Act Lessons

# data[].attributes.attachment_link.description
Act 1 - The Apple Download the Task Watch the Video: How many blocks will it take to level out the balance scale? Estimate Write an estimate you know is too high. Write an es...

# data[].attributes.body
I have used these quick video problems to introduce concepts and get my students to think about solving problems with fractions. Thinking and making concrete connections between fractions and decimals it is helpful to have something they can relate to.  This is good for part to whole thinking and for making connections to decimals, fractions and percentages.

# data[].attributes.attachment_link.readable_text
Download the Task
Watch the Video:

1. How many cups of sugar are in the case of Mountain Dew? Estimate

2. Write an estimate you know is too high. Write an estimate you know is too low.

Mountain Dew Nutrition Facts Relationship between sugar and grams How many cups of sugar
Special thanks to Catherine for sharing this picture (1/7/16)

Click the link here to find more ofMy 3-Act Lessons

# data[].attributes.attachment_link.description
Download the Task Watch the Video: How many cups of sugar are in the case of Mountain Dew? Estimate Write an estimate you know is too high. Write an estimate you know is too low.

# data[].attributes.body
I love Estimation 180 for quick engaging problems that help students build number sense and estimating strategies. Many of the problems presented on the website can be used to build part to whole number sense with fractions as well as crossing over into decimals and percentages.

# data[].attributes.headline
Quick hands-on lesson creating fraction kit models

# data[].attributes.subject
Quick hands-on lesson creating fraction kit models