Snapshot: 7A/7C Data and Representation

The seventh-grade classes began the school year by exploring the different ways data can be represented. We collected three different data sets: birth date, bed time vs. wake-up time, and favorite ice cream flavor, topping, and container. Observations and deductions were made on the first two data sets. Students made astute observations and deduced information about the classes based on the data. They made predictions about how adding the data from 7B might change or re-enforce our observations and deductions. We invited 7B to share their data with us too. We discussed the value of representing data in visual formats and how it can help us find patterns that may not be evident in the raw data form.

For the favorite ice cream flavor, topping, and container data, groups of students were asked to represent the data collected in a graphic manner. Students had to grapple with deciding which data to include, how to sort or organize the data so it was easier to consume, and which tool or strategy would best communicate the information. The diversity of representation was impressive.  A small sampling is located below. Please stop by the upper hallway in the Athena building to take in the rest of the posters. We discussed how the same data set can be distilled in so many different ways. I hoped to illustrate that graphs and data sets can be helpful, but we must be careful to ask questions like: Which data was used? Which data was left out? What is the point of view that is being shared or promoted?

We will practice our observation skills and make sense of patterns and representations throughout the year as we make our way into algebra and algebraic thinking. 

7B Room Design Project

The seventh grade curriculum lends itself to many real-world projects and activities. It is the last concrete and tangible mathematics the students encounter before being thrust into the abstract world of algebra. 

An end-of-the-year project I like to do with the 7s is a room design project. Students measure their bedroom, draw a scale drawing, calculate the surface area of the walls and floor, calculate the amount of paint needed, calculate the amount of flooring needed, calculate the cost of paint and flooring and sales tax, all on a budget. We tackle this project as a series of phases to be completed with many small deadlines to keep them on track. Students get into drawing their blueprints with architectural symbols, picking out paint chip colors, deciding on the finish of the paint, and negotiating how to spend their budget.  It is a practical and real world application of many of the concepts and skills learned in the seventh grade curriculum.

Last year, I had a student independently tape all of his drawing together into a net of his room.  It was brilliant! So this year, I added it to the project and the students liked seeing how their 2D net could be transformed into a 3D representation of their bedroom. I felt like I could imagine standing in their rooms from their scale models. I learned a few things from the construction process that I would change for the future, but it was a great addition to the project. The science teacher and I discussed the possibility of having the students 3D print their rooms next year. I love the idea of cross-curricular integration.  We just need to make the time to plan and execute it next year!

Snapshot: 7B Popcorn Container Challenge

Last Thursday, 7B took their class to the garden for a design challenge. They were given a piece of 8.5 x 11 paper and challenged to construct a rectangular prisms popcorn container with the largest volume. They had access to scissors and tape. Their container was then filled with popcorn. 

There were lots of interesting ideas at the beginning:
-"All the volumes will be the same because we all have the same size piece of paper."
-"A taller container will hold more."
-"A container with the largest surface area on the bottom will be the biggest."

They went to work and as the containers were constructed and volumes calculated and recorded, they were able to refine their hypotheses. We will follow up on class this week with further analysis of their findings. 

Snapshot: 7B Playing Card Proportions Project

 In seventh grade, the students applied their knowledge of ratios, proportions, and scale to a project where they had to produce a scaled version of a regular playing card. The purpose of the project was to have the students apply their understanding of proportions to a real task and for students to develop an appreciation for the feature on tablets and phones where they can just pinch to enlarge and/or shrink a map or image. These are seemingly simple features that require a great deal of mathematics to make functional and accurate. 

I like this project because there are many ways for students to customize their experiences and provide differentiation. They could add difficulty in the value of the card, the suit, the scale factor, and whether or not they did the whole or just part of the card. Artistic ability is not required to succeed on this assignment, but it is an opportunity for those who have an interest in artistic expression to share their interest. It brings a little color and fun to the walls of the classroom!

Snapshot: 7B Rational Numbers

Today, we reviewed our understanding the categories of real numbers. We discussed inclusion and exclusion of different types of numbers at different points in the framework. We used an analogy about which numbers have access to different categories. For example, natural numbers have the all-access pass and fit into all categories of rational numbers. We expanded our conversation to irrational numbers, imaginary numbers and complex numbers. There was lots of curiosity and questions, but many students ended up "with their brain full!" 

7B: Math Challenge 2 - Cyclic Number

Math challenges are designed to push the students on multiple fronts. They need to decipher the problem; they must identify the tools required to solve the problem; they must use the tools correctly; determine if their solution is reasonable; and then figure out how to communicate their thinking on paper and then translate it for an audience. Whew!

I choose math challenges for different purposes each week. Sometimes it will push their problem-solving skills or it might be difficult to document or both! This was a tough math challenge, but I wanted to see what 7B would do when up against something that was hard and that they might not have all the pieces to figure out. Here is the original problem:

The symbols and representation were a hurdle for most students. Those who were able to translate the information in the problem then struggled to figure out how to glean information from the statements. It was great to have students share what they were able to find out and what they could rule out to be true. Everyone who came up to share added more information to the pool of knowledge. I was really impressed with how students were able to give credit to peers and their efforts, especially those who did not solve the problem, but advanced our understanding of the problem. 

The students who were able to complete the problem shared their strategies and key pieces of information that helped them solve the problem. To quote the students, "My mind is blown!" It was satisfying to see the students who solved the problem be able to answer questions about their work and defend their choices. I was particularly impressed with EB2018's ability to find different ways to answer essentially the same question, multiple times. 

We are starting a new tradition with our math challenges this year. After the sharing phase, students are going to reflect on their process, describe what they learned during sharing, and synthesize their experience in a journal entry. I was pleased with our first go around with the reflections. I believe that this is the next step to developing their skills as problem-solvers and communicators. Here are samples of what was written.

Please ask you child to tell you more about this problem and its solution!  

7B: Number Visuals Exploration

The first assignment for 7A was looking at YouCube's Activity, "Number Visuals," from the 1st Inspirational Week of Math in 2015. The class was given the image below to look at and make as many observations as they could. 

The class found all sorts of patterns and relationships between the shapes. Here are some of their findings:

• All the prime numbers are circles.
• Every 4th figure is a group of squares.
• All multiples of three are triangles
• The first row are building blocks for the rest of the chart.
• Starting at 6; when you go down 2 and left 2, you will arrive at a variant of the number.
• You can see the factor of a composite number in the shapes that make up the number.

The group was challenged on who to describe how to move around the chart without using world like diagonal. 

As an extension, the class was charged with figuring out what the 40th figure would look like. The class shared their different ideas (see one student's work above) and they had to defend one of the choices.  

Students are learning to support their ideas and answers with solid evidence and reasoning. This will be an ongoing work in progress, all year long.

Snapshot: 7B Shooting Percentages

As we start off the school year in 6th and 7th grade, the students jump start their brains with an eight-day run of multiplication math facts practice. As a way to add a little fun to necessary skill practice, we let off some steam by doing target practice with our crumpled up mad minute sheets. We keep track of the number of shots made. There was a healthy level of competition on which class did better on the paper shooting. 

This is LO's analysis of the shooting scores for both 6C and 7B. She was methodical and very clear in her process of showing how to convert the fraction into a decimal by division and then converting the decimal into the percent. It is a wonderful example of how to show your thinking and process.

The students were asked to determine which class "won." They came up with several different ways to to do their data analysis. Please ask your child to tell you all the different ways the data was processed!

7A: Tessellated Ceilings of Iranian Mosques

Celling of Hazrate-Masomeh’s mosque in Qom, Iran, all images courtesy of Mehrdad Rasoulifard (@m1rasoulifard) via Colossal

Tessellated Ceilings of Iranian Mosques

Related to our discussion of tessellations, this article came across my feed. The beauty and artistry of these mosques are stunning. Enjoy.

7 Math and Art Connection: M.C. Escher Tessellations

Today, we began exploring how to translate a figure in geometry. We have already looked at line symmetry, rotational symmetry, and reflections. In the course of our discussion, I casually mentioned M.C. Escher's tessellations. I was startled to discover that most of the class had no idea of who M.C.Escher was. I promised the class that I would post a few links so they could explore his work further. 

Resources
M.C. Escher (official site)
Artsy.net: Maurits Cornelius Escher
The Mathematical Art of M.C. Escher
Tessellations.org - Escher Gallery 

Snapshot: 7A Playing Card Proportions

The seventh grade is in the midst of reviewing their understanding of ratios and proportions. They worked with unit rates, did unit conversions in customary and metric units, solved proportions, found percent of change, and applied percents to mark ups and discounts. As we transition into our geometry unit, the class was asked to apply their skills with proportions to make a scale drawing of a playing card. 

Detailed calculations done to scale the playing card.

I like this project because it give the students a sense of how much mathematics is embedded in apps and tools that allow them to proportionally scale images on their devices: click and dragging the corner of an image; pinching or expanding an image on a touch screen; or when a mapping application zooms in or zooms out.


The project allowed for student choice and differentiation. There was a large range of difficulty depending on the card they choose (easy = ace, medium = number cards with increasing difficulty as numbers approached 10, difficult = face card) or the scale factor (easy = 2x, medium = 3.5x, difficult = 1.75x). The project required careful measurement and an understanding of proportional change. 

The students did an outstanding job on the assignment. The scale factor range was 0.5x to 8x. Please stop by the first floor of the sixth street building to check them out in person. 

Snapshot: 7A Expanding our Understanding of Exponents

7A had a lively discussion that helped to expand our understanding of exponents. We began with what we knew about positive whole number exponents with base 10 (blue ink). When we arrived at 10 to the zero power, the class had to figure what made sense. They proposed that the answer could be 0, 1, or 10. We then took arguments for each possible solution. It was rewarding to see their pattern recognition skills come into focus. The two clarifying explanations are in green ink. NSK noticed that the difference in the standard forms was a 9000, 900, 90, and 9 and rationalized that 10^0 had to be 1 for the pattern to continue.  EH noticed that the pattern between values was to divide by 10 so the next number in the pattern was 1. This convinced us that 10^0 power was 1. Then they tackled 10 to the power of negative 1. There was much debate about whether the negative exponent would make the number negative. Logic prevailed and we applied our patterns from before to extrapolate that negative exponents made them fractions. It was invigorating to hear the healthy debate and hypotheses being shared.  

I was excited to hear the new questions that were then generated by these new discoveries:

"Can you have two to the power of 4 to the power of 2? Can powers have power?"

"Can exponents be fractions?"

The doors have been opened to a whole new world. Check back for other musings and discoveries.

Snapshot: 7A Four 4's Challenge

The seventh grade kicked off the school year with the Four 4's challenge. They had to use four 4's to make equations for each number between 0 and 50. They could use the operations add, subtract, multiply, divide, parentheses, exponent, and square roots. They were introduced the concept of factorials (4! = 4 x 3 x 2 x 1 = 24). We found patterns to find families of answers and exercised our understanding of order of operations to find solutions. The even numbers were easier to find and we are now left with the challenge of finding solutions for 31, 33, 37, 39, and 41.  

Can you figure out the missing numbers? 

Snapshot: 7A Variables & Expressions

The seventh graders are grappling with making the leap from concrete numbers to abstract algebraic symbols. This transition is developmental and is nurtured by regular exposure to patterns and looking for connections to the concrete examples. This week, 7A was working out the concept of like algebraic terms. How is x different than an x squared? What does 2x mean? How do we do operations on like and unlike terms?  These the foundation the algebra that is to come.

Snapshot: 7A Number Visuals

The seventh grade started off the year making connections between numbers and visualization of numbers. We studied the image below to find patterns and relationships between numbers. We found patterns of factors, multiples, and prime numbers. We predicted and designed visuals for 36 and 40. What do you notice when you look at these numbers visualized? 

Image credit: https://www.youcubed.org/wim-day-2/