More Clothesline Math

I’m having a bit of a hard time shutting work down for the Christmas break, so I thought I would see if I could make up these other primary clothesline cards that I have been pondering.

They are a bit addictive… as I make more cards, I keep thinking about more cards I could make…

This set has benchmarks of 0, 5, 10, 15, 20, and then uses ten frames and dominoes.  I added the numbers from 11-20 with the ten frames and then mixed them up to make addition cards with the ten frames.  I did all the make tens and the doubles and some random other combinations.  The dominoes have all the doubles and all the make tens and then some near doubles and some other random combinations…

Does anyone want to try them out?  I have no printer at home, so can’t test with my own kiddos and we are on holidays until January now (woohoo!).  If you have a chance to try them, I’d love to hear how it goes!  Feedback and suggestions welcome 🙂

See my original post about primary clothesline cards here.

Download the original primary clothesline cards here.

Download the new card set here.

 

 

 

Advertisements

Number Clothesline Test Run

We (our district’s 2 Intermediate Math Liaisons) are doing a “Getting Started with Number Talks” workshop on our ProD day this upcoming Monday.  This is the last Number Talks workshop we will offer this year, and we have decided to open it to K-10 teachers.  We have been getting quite a few requests for Number Talks in primary and secondary grades, so we thought we would stretch ourselves.

For part of our workshop, we are going to showcase the Number Clothesline as a nice complement for Number Talks.  In the past, we have done this activity with fractions, decimals, and percentage cards (see this post), but we wanted to try to differentiate a bit for our primary and secondary attendees.  I have been intrigued by the idea of using a double clothesline for algebra concepts and so for our secondary extension, we are going to try this activity from Andrew Stadel’s Estimation 180 website.

I was trying to imagine what a beginning clothesline might look like for our early primary students, and so I made up some cards with numbers, dots, ten frames and fingers (after reading this fabulous article by Jo Boaler).  I thought I would do a test run this weekend with my own kiddos (Kindergarten and Grade 2).

We set up the double number line in the living room and this is how it went down…

img_1775
We started with the numbers all mixed up and I asked them to help me fix it…
img_1776
A standoff…   (My oldest son (blue shirt) doesn’t have dirt all over his face… we forgot it was moustache day at school today to celebrate the end of Mo-vember, so he took it upon himself to draw one on.)
img_1777
Using the 5 card to measure out how much space needs to be left for the missing 3 and 4 cards.
img_1778
Watching big brother make enough space for the missing cards.
img_1779
My little guy was pretty pumped about the hand cards.  Took him a bit to figure out where “neuf” would go on the bottom if there was no “neuf” up top (french immersion is working!).
img_1782
A 15!!!!
img_1783
Big guy using the 10 card to help him figure out how far over the 15 should go.  Little guy was pretty stumped by the “zero” hand for a while – he couldn’t really tell what number it was supposed to be – kept guessing it was a four.
img_1785
A 20!! Not enough room – we’ll put it way over here!
img_1787
Another “neuf!”  We already have one!  They decided to start doubling them up.

Overall, a pretty successful test run.  A few thoughts…

  • They both enjoyed the activity, especially the different types of pictures.
  • They were both disappointed that there were no numbers between 10 and 20 for the bottom number line.  I can’t really do that with fingers, but maybe I will make ten frame cards up to 20 and look for domino pictures (at least up to 15’s… do dominos go up to double 20’s?).
  • As usual, I am impressed with how kids solve problems when they are left to themselves to figure out what makes sense.  They didn’t need me to help them with anything – they figured out how to make the right spaces, what to do about missing numbers, what to do about double numbers etc. all on their own.  Another good reminder that sometimes the most powerful teaching is to set the stage carefully, ask good questions and then stand back and let the kids do the thinking!!

If you would like to have these cards for your class, you can download them below.  I will update this file if I manage to make some more ten frame or domino cards, but for now they only have numbers 1-10 in the dots, hands and ten frames.

Downloadable Primary Clothesline Cards

 

Number Talks Meet Fractions

We had our fourth meeting of our “Making Number Talks Matter” book club last night.  Our focus for this meeting was on Fractions.  We spoke a little bit about decimals and percents, but we spent most of our time looking at how we can support our students in developing conceptual understandings of fractions.

This blog post is meant to serve as a recap for those who were there, a fill-in for those who couldn’t make it, and a record for anyone else who is interested!

We started off our meeting with our usual conversation about how things are going in classrooms with number talks.  Some participants shared their ideas for how they are keeping track of student thinking.  One teacher has tested out incorporating our student self-assessment and another has been using her document camera instead of the whiteboard to record student thinking – she then has a record of strategies being used with student names attached to help inform her Number Talks planning.  It is so inspiring to hear about how excited students are about participating in Number Talks.  I hope you will all continue to carve out time in your class for them!

We then looked at a “Fractions on the Number Line” activity as a group.  We used a double number line for this activity.  We placed the benchmarks of 0, 1/2 and 1 on the top number line and then each participant had a number to place on the line.  First, we had teachers talk in groups to order the numbers at their table and then one-by-one the tables came up to put their numbers on the bottom number line, re-arranging as necessary to make it make sense.

img_1757
The benchmarks (once again a photo re-enactment)
img_1758
The Double Number Line (imagine the cards are hanging on the wall on two pieces of yarn)

From this activity, we moved on to talking about the BIG IDEAS for fractional thinking in the new Grade 4-7 curriculum.  We used this quote from the book as our jumping-off point:

…for success in high school, there is no avoiding fractions.

We talked about: what do our students struggle with in terms of fractional thinking? And: what do we want our students to understand about fractions?

Some thoughts that came up:

  • We want our students to understand that the size of the piece changes depending on the size of the whole.  It is possible to have a quarter that is bigger than a half if the two wholes are different.
  • We want our students to understand that fractional pieces have to be the same size but not necessarily the same shape.
  • We want our students to understand that fractions are numbers that exist on the number line.
  • We want to help our students make connections between their existing understanding of number and their understanding of fractions.

We then looked at the BIG IDEAS from the curriculum from Grades 3 – 9: where are our students coming from in primary, and where do we want them to go in secondary?  Now that all the fraction operations have been moved to Grade 8, we have the opportunity to solidify a conceptual understanding of fractions in elementary school so that students are prepared for fraction operations and linking of fractions to algebra in Grades 8 and 9.

From here, the teachers did another activity that connects a visual representation of a fraction to its place on the number line.  (Activity adapted from this blog –  printable download of the activity cards are available).  Teachers coloured in a section of the given square and determined what fractional part they coloured.  They then placed their fraction on the number line again.

fraction-clothesline

As a wrap-up, we briefly reviewed the other three types of Number Talks for fractions that are described in the book: More or less (give a fraction and have students defend whether it is more or less than a half); Closer to 0, Closer to 1/2 or Closer to 1 (give a fraction and have students decide which benchmark it is closer to), and Which is Greater (give two fractions and have students defend which one is greater).

Last but not least, we had a mini “make and take” – teachers took home yarn for a double number line and a package with coloured fraction, decimal and percent cards.  I will update this post with a link to the printable package as soon as I add some improper fractions and mixed numbers to it!  I will also have these packages at our final meeting for people who were unable to join us this week.

Here are a few useful links that we talked about in our meeting:

Number Lines + Fractions

As a grade 4/5 teacher for the last 4 years, I generally find that my students enjoy working with fractions.  They like working with fraction manipulatives and approach visual representations for fractions with relative ease.  Naming and identifying fractions is rarely a problem.  But then we get to comparing and ordering fractions… and it all falls apart.

With my tutoring students (Grades 9-11), fractions are generally a disaster.  They are chugging along fine with whatever they are working on and then… a FRACTION!!!  Reducing fractions and finding common denominators is sometimes OK, but if a fraction is tossed into the middle of an algebraic expression, they don’t know what to do.  There seems to be no recognition that fractions are, in fact, numbers.

I have been doing a lot of thinking this year about how and why fractions seem to be a place of struggle for so many students as they advance through math.

Many of my students treat fractions as a completely new set of numbers, with no connection to whole numbers.  They are obviously not getting the big idea that all numbers are connected and have their own place on the number line.  Part of me wonders if this is because we (I?) over-use certain representations for fractions (pizzas and chocolate bars) and under-use others (number lines).  So, this year, I have really been trying to help students connect what they already know about numbers with the new information that they are learning about fractions.

One activity that I have used a few times this year – very successfully – is a fraction clothesline activity.  This week, I was invited to visit a Grade 4/5 class to introduce decimals by linking them to what the students already knew about fractions (they have been working on fractions for a couple of weeks).  I thought this was a perfect opportunity for me to combine two great things and I set the fraction clothesline up like a number talk (yes, I am obsessed with Number Talks).

I set up the clothesline at the front and we looked at the three benchmark cards I brought: 0, 1/2 and 1.  We hung the 0 and the 1 on the number line and then… NO ONE could tell me where 1/2 went.  Yikes.  This is the part of number talks that still makes me anxious… the WAITING… letting kids think… and HOPING… that SOMEONE… will come up with something to move the conversation forward.  My patience paid off… eventually someone suggested – “well, if it’s one-half, couldn’t we just put it in the middle?”  And, whew… yes.  Yes, we can.  We were rolling again.

Fraction Benchmarks

So, once we had the benchmarks on the line, I handed out a fraction card to each student.  Because we were looking at connecting decimals and fractions, I gave each student a “tenth” – nothing tricky.  Just 1/10 – 9/10.  I asked the students to think for a moment and give me a thumbs up when they were pretty sure they knew where their fraction was supposed to go.  And I waited.  After a while, a few kids (probably about 1/2) had their thumbs up.  So I invited those students who had an idea to come and place their fractions, and others could just watch to see if it helped them figure out where theirs should go.  And… this is what I got:

Fraction Number Line
This photo is actually a re-enactment… I need to get better at taking photos as I go.

Yikes.  Again.  But mistakes are SUCH a valuable opportunity to pinpoint misconceptions. The first student started by defending the location of 9/10.  He explained that because 9/10 was almost one whole, he put it close to the 1.  Whew.  Good start.

Then, the student who placed 1/10 wanted to explain.  She said she placed her fraction there because 10 is bigger than 2, so 1/10 must be bigger than 1/2.  Cool – good explanation, and a good misconception to tackle.  But, this is where I am still working on my “thinking-on-the-spot” skills, and trying to find the balance between “teaching” and letting the students help each other and wrestle with their own thinking.  I ended up drawing some pictures on the whiteboard behind the number line – I went back to the classic “pizza” shape.  I asked the students to help me draw 1/2 behind the 1/2 benchmark card and then 1/10 behind the 1/10 card.  Gasps all over the classroom.  I asked the student if she was still happy with where her card was and – NO, she was not!  She came and moved it to its correct location.  I asked if any other students wanted to move theirs and several more came up to make adjustments (accurately).  I asked the students to explain why they chose to move their cards and they were able to relate the spot on the number line with what the picture would look like.

I then encouraged the students who had not yet placed their cards to come up and find a reasonable spot for them.  This worked well – most of them were able to be successful in the location and were able to explain why.  We also talked about the spacing between fractions on the number line and how it is hard on a clothesline to be exact, but we know that fraction pieces all have to be the same size.

I was pretty happy with this number talk.  The students were pretty confident with ordering fractions with the same denominator, and I thought we got a good start at thinking about fractions with different denominators.  At the end, I gave a few students some “tricky” cards – 0/10, 11/10 and 12/10 and they were able to successfully place and explain these fractions as well.

This lesson really highlighted (again) for me the power of number talks and having opportunities for students to own and explain their thinking.  The real power of number talks is in giving students these types of opportunities on a DAILY basis.  This is what helps them to build up their number sense over time.

What other activities do you use to help students build a broad, connected understanding of fraction concepts?