I held it together for most the period with the fifth graders today. And then the last 4 minutes happened. When I started to say the few things I wanted to say to them before they left my class (gulp, forever), I started to cry. Words could not describe how special this group of students are to me, in part because I have had them for two years. I remember the first day with them two years ago. We sat in the hallway between their classrooms, played a game to teach them that math did not always have a clear-cut answer. They were young 4th graders at the time. To think of how much they have grown not only in math but overall as young adults is incredible. This class kept me on my toes and helped me grow as an educator. We had rich discussions, questioned one another and questioned ourselves and we took our math out into the real world. We designed, created, built, tested, proved. We took our math and our learning very seriously. But that did not stop us from having loads of fun. My favorite moments in this class was hooking them from the beginning with our mission and task for the day. Once I introduced it to them, I would see the smiles and hear chuckles as they excitedly awaited starting. This energy, joy and love of learning started from that very first day to our very last day together. Just yesterday, I started math class with diapers on each desk and asked my students what they thought we would be doing today. They sure didn't know but they were surely interested. In honor of Father's Day, we were doing a cost analysis of cloth vs. disposable diapers. "What does it cost to supply a new baby with diapers? Do you know how much it cost for your parents to take care of you?" Laughs all over the room. Getting them to be interested and be excited about math is my favorite part. In one short year, we have: designed a prototype, gauge potential customer interest, analyzed projected profits, predicted wildlife population using Goldfish and ratios and proportions, scaled models, presented on "Shark Tank," asked, tested and designed our own statistical questions, analyzed the nutritional value in fast food, measured the dangers of texting and driving, and used mathematics to see how unrealistic Barbies truly are (new, see pictures below). It's all in a year's work, and what an amazing year it has been. Families, thank you from the bottom of my heart for trusting in me to teach your child. I am excited when I think of their growth and I am eager to see them continue to grow throughout their years here at Riverdale. Although I will not be your child's math teacher next year, I still hold them dear to my heart as I hold all of you. Thank you for your unwavering support, your trust and your enthusiasm. I encourage you (and your child) to reach out should you have any questions or need support in any way. Have a wonderful summer, and thank you again for everything this year. I will miss you and your child dearly! Truly, Vi Tamargo The latest math room adventures
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Exploring with Polystrips to learn about triangle construction Gallery Walk of peer rules for triangle construction. | Math 6X is working away in our fun, very exploratory unit, Shapes and Designs! The students are having a blast as we set out daily with a geometric question we are anxious to solve. As we explore with our shapes and mathematical tools, we look for patterns, observations and rules that might help us explain the world around us, and the beauty of geometric. This week, students worked with polystrips to answer two big questions: 1) What side length combinations make it possible to construct a unique triangle? 2) What are the minimum rules/conditions can you give someone that would allow them to construct a unique triangle? We used our polystrips to try to make different triangles. For example, students tried to make a triangle that is 8 cm, 9 cm and 2 cm. They realized that this cannot work. When they tried making a 8 cm, 8 cm and 4 cm triangle, it worked. What we really wanted to uncover, though, was why some combinations worked and why some didn't. These theories because a point of discussion and the discussion provided the "fire" for more testing and observations. I loved watching them working away, trying to figure out the mathematical relationships that the greatest mathematicians uncovered years and years ago. "C'mon mathematicians! Let's figure this out!" Yes, I do like to heckle in class. But it was so exciting - we were mathematicians in our own lab! I loved watching them come up with theories, refine their theories and support them with evidence. It is truly a joy to watch this group work. Having had them now for almost two full years, I have seen that growth and I have gotten to know them not only as students but as human beings. It's been a lot of work to get them to a place where they understand and value collaboration, the importance of listening and building off of one another's ideas, but I truly think we are getting there! Check out the photos of them in action. Here are some questions you might ask them at home: 1) Why is it that a triangle is so "rigid," meaning that each side length combination can only build one unique triangle? 2) What type of information would you need to give someone in order for them to build this unique triangle? 3) If you know three angles of a triangle, is that enough information to draw the triangle? Why or why not? |
Working together to uncover mathematical patterns and relationships
We recently completed our unit, Comparing and Scaling, where we explored how ratios and proportions can be used to compare measured quantities. We considered, when is it best to use percents, rates, ratios, and proportions to make comparisons - as opposed to using addition and subtraction relationships? We developed strategies for writing and solving proportions all the while making the connection to past strategies: ratio/rate tables, equivalent fractions, tape diagrams, scaling up and down. In applying our skills, we used unit rates to compare costs. We extended our work to percents of a whole, imagining how much we would make as salespeople at a store making commission, researching the markups on basic items (water, clothing, movie theatre popcorn, etc.).
We even used our knowledge to determine, "How do we measure with mathematics how dangerous texting and driving is?" This task really shocked students! We timed them typing a basic response to a text message question and applied their rate (words per second) to measure the distance one would drive at that rate. We also analyzed statistics regarding teenager-related vs. adult-related car accidents to figure out if teenagers were truly more dangerous drivers than adults. I can definitely say that our students realize how dangerous texting and driving is - what a good message a few years before they start driving themselves! This is a bunch that loves to apply what they are learning to solve real-world problems, and I continue to embrace these opportunities in class to make their learning relevant. They tackle any challenge I bring their way, and they respond with wonderment and awe, one of the habits we have been working on as a school.
If you are looking for ways to bring real-world application into your work with your child at home, don't forget to check out my website on classroom central. Happy weekend!
We even used our knowledge to determine, "How do we measure with mathematics how dangerous texting and driving is?" This task really shocked students! We timed them typing a basic response to a text message question and applied their rate (words per second) to measure the distance one would drive at that rate. We also analyzed statistics regarding teenager-related vs. adult-related car accidents to figure out if teenagers were truly more dangerous drivers than adults. I can definitely say that our students realize how dangerous texting and driving is - what a good message a few years before they start driving themselves! This is a bunch that loves to apply what they are learning to solve real-world problems, and I continue to embrace these opportunities in class to make their learning relevant. They tackle any challenge I bring their way, and they respond with wonderment and awe, one of the habits we have been working on as a school.
If you are looking for ways to bring real-world application into your work with your child at home, don't forget to check out my website on classroom central. Happy weekend!
"This is called a mini-project. How ironic."
To wrap up our last unit on similarity, I gave students the task of taking a real-life object and shrinking or enlarging it by a scale factor of their own choosing. They took this opportunity to literally blow up the objects in gigantic proportions. Throughout the few days we worked on the project in class, the class laughed about how ironic it was that this was called a "mini-project" when there was nothing mini about it at all. I love math humor.
I was thoroughly impressed how students attended to precision during this task. They worked so diligently and carefully to measure each part of their object. They calculated how to enlarge the figure by their chosen scale factor, and worked carefully to draw the image of their figure accurately. As they worked, they solidified an understanding of what changes and what stays the same when a figure is enlarged or reduced.
Here are some questions you might pose to reinforce their work in this unit:
To wrap up our last unit on similarity, I gave students the task of taking a real-life object and shrinking or enlarging it by a scale factor of their own choosing. They took this opportunity to literally blow up the objects in gigantic proportions. Throughout the few days we worked on the project in class, the class laughed about how ironic it was that this was called a "mini-project" when there was nothing mini about it at all. I love math humor.
I was thoroughly impressed how students attended to precision during this task. They worked so diligently and carefully to measure each part of their object. They calculated how to enlarge the figure by their chosen scale factor, and worked carefully to draw the image of their figure accurately. As they worked, they solidified an understanding of what changes and what stays the same when a figure is enlarged or reduced.
Here are some questions you might pose to reinforce their work in this unit:
- How do you know that two objects are similar?
- When scaling a figure, what stays the same? What changes?
- How is the area affected when a figure is scaled?
In our current unit, "Stretching and Shrinking" students explore and deepen their understanding of what it means for two figures to be similar. Through exploring with rubber-bands to create similar figures, relating similarity to enlarging and reducing figures in a copy machine and then creating similar family members for animated characters, the Wumps, students developed an understanding of similarity as enlarging or reducing a figure by the same scale factor. The started to distinguish the difference between how the word similarity is used in every day life to discuss people or things that look kind of alike, and the mathematical definition of similarity. In math, we need much more than just "they look the same" to call them similar. Creating similar family figures for Mug Wump, students started to see how a figure stretched or reduced by length or width only does not produce a similar figure to Mug. This week, we apply our newfound knowledge to take a polygon and create a rep-tile of it, or a similar figure to it. In our work, we not only need to apply our current knowledge but our past knowledge working with area, perimeter, quadrilaterals and non-quadrilaterals as well as measuring length. This habit of mind encourages us to make deep connections beyond the current unit, finding relationships between mathematical skills and domains. Here are some questions i encourage you to ask your child:
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Well, we did it! It was a TOUGH unit but students took the challenge and they soared! We wrapped up 2015 with Shark Tank presentations. I was quite impressed with their presentations - their engaging introductions and their knowledge of the numbers behind their products. The final presentation, though, was just a culmination of the hard work they have been doing all along. When I printed their business report for them, I was actually stunned by the solid 16 pages of work they had produced, not to mention their business spreadsheet and presentation. And along they way, they were revising, refining and constantly improving their work. Along the way, we faced challenges - this was no easy unit. We learned through it together and it was very much a lesson in overcoming struggle and continuous growth.
Here is a link to the Shark Tank videos. All their pictures are here, too, but scroll down for the videos. Lesson learned: I will check the sound quality next year. So sorry!
I loved watching their creativity develop through this project. It could have been taken in so many directions and I love that about it! We'll continue to deepen our understanding of variables and patterns in a future unit working more formally on linear relationships. Here are some questions you might ask your child at home:
1. What factors in your business are dependent on other variables?
2. How are equations useful for businesses? What are the advantages and disadvantages of equations over tables and graphs?
3. How do constant relationships appear on a graph?
4. What can you tell about a situation based on a graph? A table? An equation?
I'm so proud of our students and I am looking forward to working with them the second half of this year. For now, let's all take a break and celebrate a well-deserved two weeks. Thank you for all your support of our students and of me. Have a wonderful holiday with your family!
Here is a link to the Shark Tank videos. All their pictures are here, too, but scroll down for the videos. Lesson learned: I will check the sound quality next year. So sorry!
I loved watching their creativity develop through this project. It could have been taken in so many directions and I love that about it! We'll continue to deepen our understanding of variables and patterns in a future unit working more formally on linear relationships. Here are some questions you might ask your child at home:
1. What factors in your business are dependent on other variables?
2. How are equations useful for businesses? What are the advantages and disadvantages of equations over tables and graphs?
3. How do constant relationships appear on a graph?
4. What can you tell about a situation based on a graph? A table? An equation?
I'm so proud of our students and I am looking forward to working with them the second half of this year. For now, let's all take a break and celebrate a well-deserved two weeks. Thank you for all your support of our students and of me. Have a wonderful holiday with your family!
We're off and away! This has been a BIG unit for our 6X students. In our Variables and Patterns unit, 6X students have created a unique product/business, conducted market research to determine customer interest at different price points, analyze potential income at their self-selected price, and examined expenses and profit over a period of time. To extend our work even more, we learned how a spreadsheet can provide a "snapshot" of our income, expenses and profit. We learned how to graph our data, and analyzed how we can use equations and formulas on the computer to look at sales over time.
I'm so impressed by our students' resilience and willingness to be challenged in this unit. This is a tough unit in itself but they have embraced the opportunity to apply their learning through their business project. The project certainly provided with ample practice in modeling with mathematics (Common Core mathematical practice) as we examined how tables, equations and graphs tell a story about relationships between variables. Having to apply their learned mathematical skills to their business also encouraged them to see how important it is to be accurate with their work, as errors mean decreased profit and imprecise reports - investors would not be happy about this!
I encourage you to ask your child about their project - they are creative, thoughtful and impressive! Here are some questions you might ask them:
I'm so impressed by our students' resilience and willingness to be challenged in this unit. This is a tough unit in itself but they have embraced the opportunity to apply their learning through their business project. The project certainly provided with ample practice in modeling with mathematics (Common Core mathematical practice) as we examined how tables, equations and graphs tell a story about relationships between variables. Having to apply their learned mathematical skills to their business also encouraged them to see how important it is to be accurate with their work, as errors mean decreased profit and imprecise reports - investors would not be happy about this!
I encourage you to ask your child about their project - they are creative, thoughtful and impressive! Here are some questions you might ask them:
- How did you come up with your price point?
- How can you use your market research data to project the number of people who would be interested in Portland? What about in the United States?
- How can you use algebraic equations to help you predict how many customers you will need to make an income of $1000? How do you determine profit?
I am loving our current unit, Variables and Patterns. In Math 6X, we are looking at relationships that are dependent on another factor (total distance driven over a period of time, temperature throughout the year, etc.). We examine graphs, tables and written words and wonder, "What story does this tell us about the situation and relationship between these factors? How does 'change' show up on a graph? How is it different than 'change' in a table? Or in a story?" I love how the start of this unit provides so many opportunities for students to employ Habits of Mind: thinking flexibly and thinking about thinking. Students analyzed, for example, a table of total distance traveled over nine hours to determine what factors contribute to the varying speeds. Students had to reason based on the data what the average speed was for most of the trip if they traveled a certain distance and traveled in a van for the last hour. As we worked, we found ourselves using the eraser cap more than usual. Even more, we had to accept the "discomfort" that came with not knowing the exact answer but believing in our own conjectures based on the data. We practiced listening with empathy as we worked through all of the possibilities for the situation, building off of the work and ideas of each other.
As we continue to examine variability in tables and graphs, students are applying their knowledge to a "business' of their own making. Students are currently in the process of designing their own item to sell and conducting market research to determine how different price points affect the number of potential customers they will have. Students will organize their data in a table and graph, and analyze their data. Students will use mathematical evidence and reasoning to decide on a final price point and present their work to the class. As we continue our investigation of variables and patterns, students will apply their knowledge to their business, examining potential income, expenses and completing a business report to better understand their potential profit. They will present their business in a "Shark Tank" presentation. I can't wait to see their work develop throughout this unit!
I encourage you to ask your child about the progress of their work. They tend to be so independent in their work but they have so many ideas to share, and certainly will benefit from your feedback!
As we continue to examine variability in tables and graphs, students are applying their knowledge to a "business' of their own making. Students are currently in the process of designing their own item to sell and conducting market research to determine how different price points affect the number of potential customers they will have. Students will organize their data in a table and graph, and analyze their data. Students will use mathematical evidence and reasoning to decide on a final price point and present their work to the class. As we continue our investigation of variables and patterns, students will apply their knowledge to their business, examining potential income, expenses and completing a business report to better understand their potential profit. They will present their business in a "Shark Tank" presentation. I can't wait to see their work develop throughout this unit!
I encourage you to ask your child about the progress of their work. They tend to be so independent in their work but they have so many ideas to share, and certainly will benefit from your feedback!
- What change did you notice with the number of potential customers as your price point increased? How does this change appear in your table? What will this change look like on a graph?
- How did you decide your price point? What mathematical evidence did you use (from your market research) to support your choice?
- Based on your data, if you sold your item at X price, what percentage of people do you think would be interested?
Math 6X continues moving forward and I'm so proud of their intellectual and social growth as a group. We truly have a wonderful dynamic in the class. We've been working hard on Listening with Empathy, a skill that is tough in our nonstop world. I researched a variety of rubrics used in colleges and universities to measure classwork and participation, and created my own inspired by what I found and our focus on Habits of Mind. They seemed to embrace the fact that I expected this higher level of discourse from them and evaluated them on it. We talked in our class about how contributing to mathematical discourse looks like:
This unit, we've extended our number line to learn and observe negative numbers. We've looked at relationships with adding and subtracting integers, finding patterns that led us to the algorithms for computing with them. Integer chips and the number line have helped us immensely with making sense of what's actually happening when we add and subtract integers. We started to incorporate Schoology for mathematical discussions assigned as homework, and I loved seeing them making use of structure and critiquing the reasoning of others.
They are embracing their unit project - budgeting for their finances with school, a part-time job and all the expenses that come along with being a 6th grader. :) They have to maintain a balanced budget using a variety of tools (GoogleSpreadsheet, Excel, mobile apps, etc.) and figure out if, after a month, will they be in the red or will they have a little bit extra in the end? Here are some of your children's (many) funny questions to and comments for me:
Some questions you might ask your children at home:
- Building and enhancing mathematical discussions by actively listening and making connections to what others are saying
- Listening and thinking that demonstrates deep engagement with the math
- Taking in what others share and building off of ideas because it enhances all of our learning
This unit, we've extended our number line to learn and observe negative numbers. We've looked at relationships with adding and subtracting integers, finding patterns that led us to the algorithms for computing with them. Integer chips and the number line have helped us immensely with making sense of what's actually happening when we add and subtract integers. We started to incorporate Schoology for mathematical discussions assigned as homework, and I loved seeing them making use of structure and critiquing the reasoning of others.
They are embracing their unit project - budgeting for their finances with school, a part-time job and all the expenses that come along with being a 6th grader. :) They have to maintain a balanced budget using a variety of tools (GoogleSpreadsheet, Excel, mobile apps, etc.) and figure out if, after a month, will they be in the red or will they have a little bit extra in the end? Here are some of your children's (many) funny questions to and comments for me:
- If I only eat a third of the steak, do I have to pay for the whole thing?
- Oh no! We're going on a trip next week! Do I have to pay for my airplane ticket?
- I'm just going to starve this month so I can have money left at the end of the month!
- They seriously make me laugh, but you have to appreciate their enthusiasm. I can't wait to share these projects with you.
Some questions you might ask your children at home:
- How does a number line or integer chips help us understand why subtracting a negative is the same thing as adding?
- What mathematical reasoning can we apply to determine what expression will give us the largest value?
-5280 - 800 or 5280 + (-800)
Allowance Negotiations, Body Proportions and the Search for Dividing Fractions in the Real World
9/25/2015
We wrapped up our first unit, Let's Be Rational, this Friday! It was a great start to the year and I'm SO excited for our first 7th grade book, Accentuate the Negative. In the previous unit, students tackled a number of skills: reviewing estimating with fractions, multiplying fractions and extending their knowledge to computing the quotient of fractions. We used the Connected Mathematics curriculum to make solid estimates, draw efficient models to better understand the relationship between multiplication and division and to lead us finally to the algorithm for dividing fractions. Throughout our work this unit, we also had to interpret a proposed allowance (set by our lovely parents) to figure out how much money was actually allocated to each category (i.e. 1/5 of your allowance was reserved for emergencies, with 1/2 of that being for school supply emergencies, etc.). The students loved this real-world context and even had the chance to readjust their budget for "re-negotiations."
At the end of the unit, we started to build in expressions and equations. I took this opportunity to bridge our algebraic work with biology, art and measurement. While there are many variations to human body proportions, there are certain "standard ranges" that can be used to relate body parts. For example, did you know that the length of your head is 1/8 of your height? In Math 6X, we used these ideas to write equations relating parts and to figure out an unknown length given another.
Students also finished their unit presentations on dividing fractions. Students used a variety of technology (Movenote, GoogleSlides, ShowMe app) to demonstrate how dividing fractions can solve a student-designed mathematical dilemma. I was so inspired by their creativity and enthusiasm! Send me an email for the private link to their presentations!
At home, here are some questions you might want to ask your child:
At the end of the unit, we started to build in expressions and equations. I took this opportunity to bridge our algebraic work with biology, art and measurement. While there are many variations to human body proportions, there are certain "standard ranges" that can be used to relate body parts. For example, did you know that the length of your head is 1/8 of your height? In Math 6X, we used these ideas to write equations relating parts and to figure out an unknown length given another.
Students also finished their unit presentations on dividing fractions. Students used a variety of technology (Movenote, GoogleSlides, ShowMe app) to demonstrate how dividing fractions can solve a student-designed mathematical dilemma. I was so inspired by their creativity and enthusiasm! Send me an email for the private link to their presentations!
At home, here are some questions you might want to ask your child:
- Why does the algorithm for dividing fractions work? How is "2 divided by 1/2" the same as "2 x 2?"
- What strategies can you use to make estimating the product or quotient of a fractional problem easier?
- How can fact families help us solve multiplication and division problems more efficiently?
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