Daily Desmos!

 

This year in Algebra 1 and Algebra 2, students have been using an online graphing calculator at Desmos.com to visualize and explore functions.  Some of our work has included using equations to create pictures with graphs – including points and linear equations for Algebra 1, and quadratics and conics for Algebra 2. 

The editors at Desmos have created an ongoing series called “Daily Desmos.”  Users submit their graphs, and selected entries are posted publicly.  You can find the Daily Desmos challenges online, and you try to re-create the graphs.  They are categorized as basic or advanced – and there are some tricky problems to solve in there. These challenges are not only motivating for students, but they require them to think analytically about how equations and graphs are related.  Students become skilled at completing these challenges by deepening their understanding of changing variables, and how they effect visualizations.

Algebra 2 student David has been working on some original graphs as an extension to his regular class work.  This week, he submitted one of his pieces to the editors at Desmos.com for them to consider for their Daily Desmos series.  The graph was quickly accepted, and posted under their advanced challenge section.  This graph represents David’s deep understanding of graphing and of math as a creative process.  It has been joyful to see how he has pushed himself with this work.  You can check out his graph here: http://dailydesmos.com/2015/05/11/drowning-0-advanced/

Epic Poodle Bungee Jump

8th Grade Honors Algebra just completed a project, in which students used linear equations to predict how many rubber bands it would take to give their poodle a thrilling, yet safe bungee jump from the school balcony.

Students were asked to string together one, two, three four, five and six rubber bands, and then make a prediction based on their measurements.  The object was for their poodle to get as close to the ground as possible, without actually landing on its' head.  They predicted how many they would need for the first balcony, and if they survived the first jump, the poodles made the jump from the second balcony.

Check out the reactions at the 1:20 minute mark and the 1:36 minute mark - not your typical reaction to getting the right answer on a math problem! And look at the celebration from the winning team at 2:23!

Congratulations to our winners, who earned glory and pride -as well as some bonus points on the next quiz!

STUDENT POST: How To Use Algebra Tiles

By Loiza,

In math class we used squares and rectangles to help us solve equations.  Mr Nathaniel tells us that we have to find the area, perimeter, or a solution for the amount of blocks that he lays out. We have to know what each block stands for and that’s the way we make an equation out of the soft squares and rectangles.  Those tools that we are using help us to make an equation, solve the equation by knowing the amount each of them stand for. The blocks make it easier for some people to figure out the answer.  Because its visual, you have to look at the blocks, know what each one stands for, and make the equation that you have to solve using the blocks.

• The small square stands for 1 (because it's 1 long and 1 wide)
• The small rectangle stands for x (because it's 1 long and x wide)
• The big square stands for x² (because it's x long and x wide)
• Blue means positive, and red means it's negative

TEACHER NOTES:

For some kids, using visual logic, or a hands-on approach can help to remove barriers to understanding, and can take some of the intimidation and fear out of learning about difficult concepts.  Using Algebra tiles to teach polynomial multiplication and factoring quadratics help to reinforce an area model of multiplication.  Our recent workshop with renowned math educator Erma Anderson reinforced the idea of moving between multiple representations: from symbolic to pictorial to verbal to tabular and back.  Students are asked to use the tiles to represent ideas, then to illustrate those idea through drawing, and equation solving.  

Visual Thinking: Using our eyes as well as our minds

One way to help capture student interest, and to create better access to math is to express mathematical ideas by using images or visual representations.  In Algebra classes, we have been exploring visual patterns as one lens to identify and study linear, quadratic, and even cubic patterns.

If this image shows steps 1, 2, and 3, how many squares would you predict in step 43? 

These patterns are easy for many students to approach.  Even though they present the same ideas as an equation or a table of values, for many kids, they are much less intimidating.  They also help to cement understanding by asking students to express ideas in multiple ways.

Dividing these shapes up into sections helps us to see the x term, the x squared term, and the constant term in this trinomial pattern.

...and if you think that these are a way to water down the math, try to figure out how many penguins are in the nth term of this friendly looking, but difficult pattern!

The patterns shown here come from a site put together by master teacher Fawn Nguyen.  Check out some more of the patterns she's accumulated, and work with your kids to try to create a general rule for the nth term HERE.

Using our hands as well as our heads

One of the toughest steps for Algebra students to make is from the concrete to the abstract.  In math language, this is often when we try to generalize a result - or to create a rule, which works for a general case rather than just for a specific case.

One example might be multiplication of numbers versus multiplication of variables.  By the time they get to Algebra, students have a good conceptual understanding of 3 times 5, of 46 times 250, or even -32 * 25.  But moving from these concrete examples to an abstract concept like x times y, or 5x+2 times x-5 is often very challenging.

One way to help students to make this leap is by implementing some kinaesthetic activities.  These can help to bridge the gap from the specific to the general and can give students some "muscle memory" to help them remember how to make this leap.  Examples that we have used this year include using algebra tiles to represent variables and numbers, or actually acting out with our bodies some function transformation.

We need to use all of our tools and ideas to assist students in climbing up the ladder of abstraction!

Scholarship and Kindness

At the start of the year, it is important to articulate what we need to have a successful learning environment.  The class is a better place for everyone when students feel that they have a voice in how the classroom will function - and we do include their ideas about how our class will run.

When we collaborated to create our Algebra class learning agreements, there were two themes that kept coming up: the high value that this group places on learning, and on compassion for each other.  We brainstormed all of our ideas with sticky notes, prioritised and categorized these ideas, and put together a poster, which synthesised and summarized these ideas.  You can see our classroom agreements all fall under the umbrella of scholarship, but are rooted in kindness.  Students read over the summary and signed their names to the poster before we hung it up in the classroom.

Algebra Beginnings

Welcome to our AISC Algebra studio!  We'll be using this space to share work throughout the 2014-15 school year.  Check back often to see what we're working on, or sign up to get email updates when we post new content.

Students spent the first weeks of Algebra 1 and Algebra 2 assessing what they know and remember about math, accessing their curiosity, and creating our learning agreements.  We will leverage this curiosity as we get into skill building and Algebra content over the coming months.

During the first few days, students were asked to complete several puzzle challenges.  These tasks helped me to assess their skill and comfort with math operations, to gain some insight into how each student attempts to solve problems, and to see how they reacted to novel tasks.  It also helped to set the stage for group work and cooperative learning.  We definitely need each other to learn - especially in math where we will be sharing our strategies and our thinking often.  Both groups started off with strong thinking and hard work.  There was smoke coming out of the ears of more than one student by the end of week one, but they were asking for more!

David's "Spy" graph

Students will be demonstrating their understanding of Algebra standards through traditional tests and quizzes, but also through project work.  Week two saw the first of these Problem-Based-Assessments (PBA's).  Algebra 2 students spent a few days cementing their understanding of relationships and  functions, and then practiced transforming linear and quadratic functions to create "Math Faces" with a graphing program called Desmos.com.  Our Artist/Mathematicians created original equations, and learned about restricting domain and range to create original works of math-art.  We'll be publishing them here soon, but for now, check out some of their original pieces on our math wall outside of room 112.

For the first PBA in Algebra 1, students showed their creative side by attempting to make every number between 1 and 100 with a project we call the four 4s.  They were asked to use exactly four 4s and any mathematical operations to get to every number up to 100.  This task demonstrates that there are often multiple strategies and solutions to interesting math problems - something students often struggle with.  Again, you can see their progress outside of our classroom.  Our sequel will be some work with five 5s!

We are off to a strong start to the year.  I am fortunate to be a part of the wonderful community at AISC and to work with our group of motivated students!  Please comment freely to let us know your thoughts about our work, and stop back often to check on our progress.