So I Tried Clothesline Number lines


I was introduced to clothesline number lines through the #MTBoS.  I inquired more about it and was provided these resources (1, 2, and 3) by @mr_stadel.  After doing some research and making sense of the instructional idea for myself, I decided to try it.

We’re focusing on scientific notation, not just converting and operations but ordering and comparing.  On my first try with clothesline number lines I provided each student with a card.  I asked half of the class to record any number in standard form and the other half of the class write any number in scientific notation.  The goal was to order the numbers from least to greatest.

Students took about 90 seconds to place their numbers on the number line, sliding cards and shifting cards where they felt necessary.  As a whole group, we took a step back to analyze the cards and one student said aloud, “Which end is the least and which is greatest?”  I though to myself, great thinking and we briefly discussed how an empty number line can start and end with any number.  From that part on, while in the heat of the moment, I thought the lesson was an epic fail.  Students remained focus only on the order of the number in scientific notation and did not consider how they compared to the numbers in standard form.  As I tried to draw their attention to this oversight, I noticed some students began to check out of our discussion.  So I summarized our discoveries and moved on to the next activity.

Later as I reflected upon this activity, I noticed the beauty in the oversight.  Although students were able to convert between the two ways to represent large and small quantities, they had not yet been able to apply this understanding.  I also realized students did not have a full understanding of the magnitude of the number when written in scientific notation.  Perfect, I knew exactly what I needed to target!

On the high of the benefits of using clothesline number lines, I quickly developed a plan to present this within a professional learning session I was slated to conducted for other math teachers within my district.  This time, the focus concept would be expressions and equations.

Teachers said their biggest take-aways were:

  • ability to assess students’ number sense when using variables
  • ability to discuss equivalent expressions using the number line
  • connecting kinesthetic number lines to the clothes line number line
  • making the connection to solving equations using the clothesline number line

Many teachers committed to trying this idea within their classrooms within the upcoming week.

Put My Strategy Up There


I have a habit of not saying anything a kid can say.  So when students present a strategy, even if it is the ultimate strategy I wanted to present, I give the student credit.  Here are a few examples of thoughts students have provided.  We can it their strategy and refer to their suggestions often.



It has become such a habit, when a student stated her strategy for converting a large number from scientific notation to standard form she yelled, “Put my strategy up there, I want my own strategy!”  And so I did.

Having Fun with Scientific Notation Part 2


I can’t really say students traditionally struggle when using operations with scientific notation.  Some of my students were apprehensive about the concept as we transitioned from converting to computing.  Again the saving grace was beginning with the conceptual using base ten blocks.

Students were able to see the quantities being added having the same blocks like when we were adding (6 x 10^3) + (3 x 10^3) or related blocks like with (6 x 10^3) + (3 x 10^2).  The discussion of regrouping blocks to add unlike blocks went over well allowing a student to conclude this:


We used a combination of concrete manipulatives and virtual manipulatives.

By the end of the lesson a student stated, “Is it really that easy?  I thought scientific notation would be hard!”

Having Fun with Scientific Notation Part 1


I’ve been enjoying my year teaching Introduction to Algebra.  This portion of Unit 3 isn’t any different as we are working with scientific notation.  Last year, I wrote a post about how teachers have traditionally discussed this concept in this post and this post.  Therefore, I wanted to do my very best at presenting this concept through the lens of place value understanding.

Our first activity was connecting base ten blocks, powers of 10 and decimal notation or standard form through the context of a number line.  Hindsight, this would have been a great bridge to implementing clothesline number lines.  What we actually did was create a human number line.  As we transitioned from concrete to representational, we constantly referred back to our physical number line and the equivalent values.

Looking at the numerical representation, students were able to identify patterns among the powers of 10 and exponents.  Beginning with base ten blocks made the transition to converting between decimal notation and scientific notation very simple.  We looked at 3 cubes and determined not only did they represent 3,000 but also 10^3 three times or 3 x 10^3.   We tried a few examples using the blocks we actually had before extending it out to the millions.

By now you may be thinking, how did we discuss exacting how to convert a number in standard form to scientific notation.  Of course the topic of the decimal point moving can up, mainly because students heard this terminology within their science class.  I used a draw place value chart to demonstrate how the digits shift and the decimal stays put.  Later, I was introduced with this site which helps to illustrate my point.  During our discussions, I would explicitly state when the digits shifted and how many spaces and connect it to a pattern previously identified by a student.

Another benefit of the base ten blocks:

3 flats is 3 x 10^2

6 flats is 6 x 10^2

9 flats is 9 x 10^2

10 flats is 10 x 10^2 which students were able to determine was equivalent to 1 x 10^3.  We were able to conclude is scientific notation is written with a whole number coefficient less than 10.  Students had a conceptual understanding of why it is a whole number less than 10.

Using Ms. Pacman to Introduce Transformations


If you aren’t familiar with Robert Kaplisnky’s Ms. Pacman, you may want to take a moment to read through the lesson

Before we began our math discussions about transformations, I had students work in their table groups at their vertical whiteboards to describe Ms. Pacman’s movements in the initial video.  All but one group used terms such as slow or “in different directions” or right angles. When I inquired about their descriptions by asking, “how would you describe her movements other than slow” or 90 degree angles (to which I contributed our previous discussions about angles), only a few could produce directional movements. 

I was at a lost of how students would get  from here to describing transformations which was the goal of the lesson. I was bailed out by this group’s description:

As they looked around at other groups descriptions, they erased what they had and wrote slow and in different directions. I quickly asked them to rerecord their original thinking and proceeded to ask about Ms. Pacman’s movement on each pathway drawn. To which they responded up, down, left or right. 

I called the rest of the class’ attention to this group’s thinking. Other groups imitated the pathway requesting to see the video again to ensure they were accurately drawing the path.  This group described her horizontal movement as east and west and her vertical movement as north and south.  After seeing the thinking of surrounding groups, they added more explanations to their board (pictured above).

This group identified the right angles as places Ms. Pacman turned. 

We came back together as a whole group to discuss our layers of thinking. 

Layer 1: Identifying the movement. During a running of the video, a student (without my prompting) came to the board and traced the path used by Ms. Pacman.  I called on the group who I first identified to label her movements by sliding right, left, up and down. We labeled the path with the initials of the direction she moved. I asked the group who used cardinal directions to share their thinking of the path. I asked the class if we could say right or east, left or west, up or north, down or south. They agreed so I labeled the path using the initials of the cardinal directions. We used this video to determine if we correctly identified her slides. 

Layer 2: identifying turns. I asked group 1 why they circled all of the right angles on their path. They explained the right angles are the places where she changed position. One of the group members asked if that was called a rotation, as she had learned about transformation in her Connections class. Someone else blurted out she was turning. We replayed the original video and students shouted turn each time she rounded a right angle. One student asked if she flipped instead of turned. 

Enter layer 3: We briefly discussed what it would look like if she had flipped instead of turned. One student offered the synonym mirrored. We replayed the video and concluded she flipped or mirrored once at the very beginning. 

Layer 4: Summarizing. We summarized our lesson by putting our conclusions on an anchor chart. 

We discussed the moves or transformations made in order. I began by using the language the students stated in their explanations. Then I attached the formalize math language to each. For example, in recording the example of reflection, I drew a representation of Ms. Pacman flipped or mirrored and stated, “this is what we call a reflection”.  Although dilation was not a part of this lessons, we extended our discussions by briefly connecting Pixels, an Adam Sandler movie where Pacman is enlarged or dilated. This anchor chart was hung in the room as a reference for math language and understanding of the four transformations based on this context. 

Intentional Assessment


We have been slowing going through the Formative Instructional Practices developed by the Georgia DOE for the past 3 years. This year, we are focusing in on the student self assessment and feedback modules. I’ve dabbled in student self assessment in the past like here and here

This year, I’m trying to be more intentional with making student self assessment a part of our habit of mind. What this looks like is school mandated content maps, rubric scoring (rubric based on Dane Ehlert’s rubric found here)of assignments and this:

Each morning, within the morning message, I encourage students to review the learning target(s) and move their picture accordingly. Twice a week, we revisit the content map and rate our understanding of the learning targets. I’m very intentional about sharing the true purpose of the formative assessments we complete and the reason homework is assigned. 

Remember, my goal is to make this a habit of mind for all students. Has every student moved a picture? Not yet. Does every student take the content map seriously? Not yet. Does every student complete their *homework? Not yet. But I will continue the conversation until every student feels like Nate who while working independently got up quietly from his seat and moved his picture. Or like Dayane who asked, “If I understand this learning target but not the other, can I still move my picture?” 

*a note about homework. Personally and as a parent I hate homework. For as long as I can remember it’s just been busy work which causes strife and tears. When I assign it, students have several days to complete enough to where they feel they had enough practice. I differentiate homework based on who needs additional practice with what. Or I differentiate based on student choice. 

Starting My Unit with Desmos


One of my goals this year has been to establish a context for learning at the beginning of each unit (or subunit). In my district, unit 2 for Introduction to Algebra is Transformations. My 8th grade team decided to start with angles, which changed my plan a bit. Did I panic and complain?  Goodness no! (That was for all the Pete the Cat fans 😉). I went to Desmos and looked for a lesson on angles. 

Day one of angles we went to the computer lab to partake in Lines, Transversals and Angles, which was my students’ first experience with Desmos and mine with a large group of students. They were so engaged and engrossed in the activity it was difficult to slow them enough to discuss the overlays used to explain placement of dots to identify congruent angles. 

Over two days, students were able to make sense of angle relationships through the use of this activity and I had a Birdseye view of their thinking. I loved how the system captures the information for me to return to later. I used my formative assessment data collection sheet (not pictured) and recorded where individual students were based on the expectations of the learning target. 

After this bit of exploration, I conducted guided instruction focusing on the characteristics students identified during the investigation. This was a great springboard into the angles discussion going from identifying angle pairs to using their characteristics to find missing angle measures and will now lead us into triangle measures. 

This subunit included activities such as Transversals, Tape and Stickies, Angle pair flash cards, a word wall game called It’s on the Word Wall, a project and a formative assessment using Plickers. As we transition to triangles, I wish I had taken the time to return to this Desmos lesson and compare what students know and understand after looking at these concepts in different ways. 
*It’s on the Word Wall rules can be found here