We Should Do This More Often

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Our last unit ended with a bang!  Unit 5 Pythagorean Theorem and Volume was opened with act 1 of Dan Meyer’s Taco Cart.  As with most 3 act tasks, this one began with the notice and wonder component. Students’ ideas exceeded what I expected. 




The blue notes the notices, the orange notes the wonders and the black were thoughts added during our discussion about estimations. 

We decided to answer “Who got to the Taco Cart faster?” but said we would come back to answer: are they going at the same speed, was one person running and will they get there at the same time?  The conversation around the estimation became intense. ​​​

​To prove our theories we used anglegs and color tiles to mimick the right triangle created by the path Ben and Dan walked to determine the length of the legs using the length of squares. Even as we were building a student kept repeating, “there has to be a part 2 to this, there just has to be”.  As students concluded the area of the legs combined or the path Dan walked was the same as the area of the square on the hypotenuse or Ben’s route, the excitement grew even more. 

Many thought their estimation of the guys getting to the cart at the same time was correct after the hands on activity. Others held on to the fact that time would play a key role in who got there faster. So I revealed the information for part 2. I love that many students had already developed a rate for the sidewalk to sand speed. Those who shared their conjectures believed the rate was 2 to 1. From the provided information it was determined it was 2.5 to 1. 

“I know you are not about to do this to us.”

Then I decided to press the pause button on the task. After students recorded the speed and distances provided on their recording sheet, I instructed them to put the papers in their porfolios to which a student exclaimed, “I know you are not about to do this to us!” Can you say completely hooked?!

We finished the day completing a practice task from Hands-on Standards:


The next day, students came in asking, “Are we doing Taco Cart today?!” Each time I told them, “No, but I promise we will finish it this week.”  The suspense grew and grew. Everyday they came in asking the same question. We worked through a few more concept development tasks from the GA Math Frameworks before returning to Taco Cart on Friday. 

Students had an opportunity to use everything they learned during the week along with the information obtained in part 2 to work out the problem and develop a conclusion. Some students remained stuck in estimation mode (which was disappointing) 


Others focused on using their understanding of the Pythagorean Theorem to form their conclusion but did not factor the time element:


While others were able to make the connections:


This was the best part:


After the screaming subsided, a young lady shouted, “that was intense. We should do this more often!”  

Musical Chairs

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We’re nearing the end of our unit on exponents and scientific notation. This means it’s time to prepare for our common assessment. What better way to do this than with a game of musical chairs. 👍🏾

Last year, I had the pleasure of attending a session at GCTM conducted by educators from Hart County, Georgia. They discussed engaging ways to implement practice in the math classroom; perfect for me as practice was my area of instructional weakness when I taught 7th grade two years ago. 

Musical chairs was one of the activities presented where you setup chairs in traditional musical chairs style. In each chair, a question is placed face down. The music is played, I used Keep Your Head Up, Good to be Alive and Run, and students circle the chairs. My 8th graders were gitty and circled the chairs in suspense of the music stopping. When the music stopped, the quickly sat down and began solving the problems in their seat. Once finished I would check their answer and offer feedback. We repeated this process over and over until class ended. 

Students played as if someone could be eliminated, one student continuously asked, “how do people get out?!”  I never answered mainly because I had not thought about that aspect. Students answered about 6 questions from their study guide, received immediate feedback and it would’ve been more if time had allowed. Everyone was engaged and excited. Everyone worked to answer the questions. 

Hindsight, take time to answer questions myself before the game to avoid solving them mentally during the game. Develop a way for students to get “out” but keep them engaged in the game. 

So I Tried Clothesline Number lines

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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

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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

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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

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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

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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.