Online Tools: Math Manipulatives

Having math manipulatives in class is always helpful to show our thinking in different ways and we usually encourage students to use them, but they are more important for use in my role as "teacher". I like to think that I can communicate ideas clearly with my voice, but I know that I am only saying the words that make sense to me. If I show a picture I can allow my students to use their words to share what they see.

The following are free online math manipulatives that you can use with your students. I have seen them be used when doing a PD session by Dr. Theresa Wills. During her classes, she provides the links to the sites that best match the problem she is presenting, encouraging students to go and use them to communicate their thinking. She usually presents 4 tools and encourages students to show their thinking using more than one. Students can screen capture or copy what they have created into a group slide/document, showing their thinking and helping them communicate with their peers.

Math Manipulatives:

Unifix Blocks: A simple drag and drop tool that allows students to create stretches of blocks. They can represent anything from patterns to measurement to ratios.

Cuisenaire Rods: Quickly drag the control to create bars of different sizes. Click and drag them onto the mat. They can be used to represent anything you need.

Ten Frame: The Ten Frame was a new manipulative for me as a MS/HS teachers, but I feel it is important to be aware of all the tools students may have used in the past that they connect with. I can see how this can also be used to help with integers. All students need to do is drag and drop.

Number Line: I use a number line in my class all the time when exploring ratios, rates, integers or just general problem-solving. The tool requires just a few students to start playing with it to see what it can create. Also, the information button provides a clear explanation of all the tools. Have students create their number lines, take a screenshot, and share them with the class.

Algebra Tiles, Base 10 Blocks, Number Lines and More: This site provides lots of visuals tools. By having the blocks and number lines in one location, students can easily combine the tools together to represent what they are wanting to create. There are also backgrounds that can be used that can support algebraic reasoning.

GeoBoard: Just like in person, but now without the hassle of trying to find elastic bands or having the plastic nobs break off. Students can use this to explore various geometric representations.

Fraction Bars: Create your own fractions in either a rectangle or a circle. You decide how many parts it is out of and then add the colours as you need. This is a great tool for students to quickly create the fraction bars they are wanting to represent, make them the same size and compare values.

Pattern Blocks: Never lose the blocks again. Students can pull as many as they need to their board, manipulate the direction and copy/paste to create patterns. There is a protractor that snaps into place to measure any angles. This can use used for patterns, geometry and number sense. There are also backgrounds that have shadows of shapes, allowing students to cover them with the tiles.

Additional Resources:

• Math Learning Centre: Many of the tools listed above are on the site. There are others as well as additional resources to explore
• DiDax Virtual Manipulatives: Some cross over with those in the Math Learning Centre, but there are a few new ones: 120 board, balance boards, spinners and more.
• Math Playground: Many of these require Flash to work which may cause problems with access by students. However, there are lots of options here as well.
• Geogebra Graphing and Calculator: This site provides a geoboard, calculator and graphing software. Students can save their work for later review.
• Desmos Graphing: Students can quickly create a graph, build a table of values or write a function. When finished, they can screenshot the image and add it to their work.
• Desmos Scientific Calculator: Use this as the class scientific calculator, making it easy for everyone to have access and talk about how to use the same device.

Digital Tools: Opening my eyes to online learning

I am now entering week 10 of online learning, and it has been a pivot. I decided to lean into the pivot and use these weeks as an opportunity to explore. In my explorations, I came across Dr. Theresa Wills. She is a professor from George Mason University and has been teaching math online for years. I forget how I found her, but she has opened my eyes to what is possible. Since she has been teaching online for years, her strategies are very well developed and the free daily PD she offers allows anyone to learn from her. I sat and watched many of her recordings and learned the following:

• Teaching is a skill you develop over time and with practice. Teaching online means applying the skills I have developed but in a new medium. I have to now practice these tools, giving myself the time and play to learn them. I can't expect to be good at this unless I practice. 
• There are so many other people who have been doing this for a while and doing it well. There are benefits to this method of learning that I can learn and help my students, now and in the future, if we go back to class. 
• The online teaching community is supportive and caring. I have taken two of Dr. Wills PD sessions and all the teachers I have been grouped with make me feel supported and help me where I am at. 

I recommend that anyone who is teaching online to explore what Dr. Theresa Wills is doing to develop community and collaborative learning in her classes by using Google Slides.

This is the video that hooked me. I hope that it hooks you. 

Goals: 2017-18 School Year

At the half way point through the summer, I found the #MTBoS Sunday Funday prompt of "Goals" to be a nice way to have me start the slow transition back to school mode and back into blogging. Even though I am aware of the benefits of writing my blog, this past year was not a good reflection of that... which leads me to my first goal (and what I hope #MTBoS will help me keep up with).

1. STOP and Reflect in the present: Use my blog as my tool

Reflection is a process that I always have my students do throughout a day/project/unit/hour, but I find that I am not modeling what I feel is an important strategy. Yes, I reflect on my day. Mainly on my bike ride home considering what I will do the next. Or even at the end of a project, considering what I liked, what I would need to change or even complete a tuning protocol. But all of these reflections are after the event has happened, never in the present. I am hoping to use this blog as a means to get a greater sense of the regular pulse of my practice and consider the pulse of my students' overall weeks.

2. Explore Personalization in Greater Depth

I like to think that I provide opportunities for personalization for my students. This happens in their ability to self-select challenge levels, methods of instruction during class time and also to use projects as a means to make the end product their own. What I am still grappling with is the personalization of content. I complete diagnostics at the start of all units, identifying where students are in their understanding/mastery of concepts, but I do not feel that I am using this information fully. I am always challenged in my mind about what going into greater "depth" of a concept looks like, rather than jumping forward to a new one once a student has shown understanding.

3. Getting students to ask Math "I Wonder" questions like we do in Science

I teach Math and Science for Grade 7. In Science, once I have seen students demonstrate the learning goals of the unit, they naturally build on them and ask amazing "I Wonder" questions which drive them to explore further, make greater connections to the content explored and the world around them. I want to find a way to make this routine present in my Math classes as well. I find that my students (and I) do not ask the I Wonder questions, but wait to be presented with a problem to solve. I am hoping that exploring Personalization will help support this, but I also know it is a mindset that I need to change in myself first. As a lover of math, I have never asked my own "I wonder" questions but instead waited for fun problems to solve. 

Patent Laws, Areas of Shapes and Launching into Structures - June is on its way!

It is weird to think that May is already here and June is just around the corner. However, when I feel the energy in my classroom, it is evident that these Grade 7 students are ready to move and get going into action. This week the goal was to have a short launch into their activity and then send them loose to explore, try and play. The learning goals for this week were:

• Through understanding forces, we can prevent failure of a structure
• Understand area is the space taken up by a 2D object
• Explore the relationship between finding the area of a rectangle to the creation of a formula for the area of a triangle

Science:

This week was the wrap up of the solar oven project since we cooked our s'mores on Friday. With the twist of having students decide if they should make their results of experiments for public or private use, I wanted to take time to explore how this is applied in the world outside our classroom through patents. I decided to have a discussion period where students would understand and question how patents can motivate as well as make it difficult to innovate. 

To launch, I used the Poll Everywhere app to have the students share anonymously how they felt about making the results of their solar oven experiments for others to use. Through the different questions it was identified that:

• In this project, students made all of the experiments public for others to us, but if there was the option of charging their peers a chocolate bar to use their ideas they would have done so. 
• If they knew that not everyone in the class had a chocolate bar to spend, there was a slight shift to making it public but the majority still remained keeping it private and charging. 
• When asked what they think causes people to want to create inventions, the highest ranked choice was because they had a problem to solve, but also the idea of financial rewards appeared. 

After discussing the results of our class data, I walked them through the process of getting a patent and outlining the time and investment of money that is required. We searched on the international database of patents to see if an idea we had for our solar oven was already created. They found this challenging and time consuming. They also saw how inventions that are patented show how to make and produce the invention to everyone online, which they saw as a way to spark ideas in others...if others could find the patent. 

We then took time to ask why inventions had not been created to solve problems such as clean water, when we have made great improvements in cell phone technology. By talking through different world views, they were able to identify that the financial investment required to develop a product would then be gained back by selling the invention. However, if the people needing it have no money to pay, than the inventor would be at a loss of money.

I was really happy that we had this conversation as it allowed the students to see why some of the world's problems have not been solved due to issues that they have never made connections with before. The conversation was enjoyable and worth the period that we spent on it for sure and I would do it again. 

Here is the slide show that was used: https://drive.google.com/open?id=1QVrhWcdUsVrspBD0OUMe9Q_1XdaqrlEZBpfen01sp6o 

To get ready for their upcoming structures unit, they are doing a homework assignment that will let them build and also collapse different structures. It is called Catastrophe and the developers have done a wonderful job of applying the laws of physics to the designs. My students have already started using the terms tension and compression without having learned it in class. 

The assignment they are working on is: https://drive.google.com/open?id=1N3Q3ELicDaoT5lAhIY5IwIIWHlU4slCXvFPrDrjK3m4

Next week is the launch of the structures unit and they will be working with another class. Lots of hands on building and exploration to do as we go into the end of the year!

Math: Finding Area

I was able to participate in a Lesson Study done by the Junior School teachers this week which focused on the development of understanding of area. With a focus on conservation and additivity when finding the area of irregular shapes. During the conversation, the idea that students forget these concepts as they begin to learn the formulas of shapes. With this in mind, I wanted to teach the area of a triangle by starting back at the start of understanding area. 

To do this, I told the students that they would be erasing their brains of all past knowledge and we would be progressing together. We talked about how measurements were originally taken by using body parts, but that different sized people would have different sized hands/feet. Then standardized units of measure were created so that everyone knew what was being talked about. We then explored why they were called units squared. For some students they had not thought of this since their younger years, and only now were able to grasp the concept. It was a short review, but important to understand that it is the total number of square units. 

We when looked at the area of a rectangle and saw how it is an array. You have for example, 10 groups of 6 boxes in the rectangle, so we can use that to do it faster. Students they made the connection that this would relate to the base x height formula they had learned. 

Finally we did a paper cutting exploration to show how two triangles make one rectangle. They understood that if we know how to find the area of a rectangle, than cutting up a triangle so that it fits into a rectangle allows us to use our array understanding to find the area. This was then connected with the formula that we then developed together. 

Once all students felt confident with this task, they were put into groups with their own shape that they needed to create a similar slide show showing how to change the shape back into a rectangle and then use this to develop a formula. Groups were arranged on purpose by processing speed so that they could think through the problem together and no one would be left behind. This was because the goal of the lesson was for them to think deeply about how conservation of area can occur. 

Now, they are creating their slide shows and will present to others in small groups next Wednesday. They will have the slides shared as their own "notes" to refer back on for their homework. This will be a good launch for their individual solving of irregular shapes as well. 

Slide show used: https://drive.google.com/open?id=1AZP9F2b9kqAH7oDlLUaTZ7ARx8j5nGHvK8Xe8yv45L8 

Solar Ovens Week 2, Oral Evaluations and Climate Graph Intro

So, perhaps as predicted the solar oven project too a little longer than predicted. Due to bad weather, we didn't get the best days of testing. This resulted in the exit interviews starting on Thursday rather than Wednesday and will probably enter into next week. However, we are ending today with the big test of the ovens as well as using them to make SMORES! This last part is the surprise to the students.

Here was the week:

Solar Ovens:
Overall a big success. Students continued to think creatively about how to make adaptations to their ovens, but also find the supportive data in their peers experiments. It was interesting how every solar oven experiment was made public for other individuals to use. This was a big change from the start of the project where students were wanting to keep ideas and results private. Next week we are going to be exploring how this decision to apply patents/copyright on research results can be challenging for overall innovation, as well as encourage it.

The new thing this week are the exit interview. Students are completing a verbal evaluation where they need to identify the modification they made, explain how it harnesses the sun's energy and provide evidence from their peers to show that it works. Providing a verbal evaluation has been great for two reasons:

1. Students who are verbally strong now have the chance to share their understanding in a manner that suits their preferred communication style. They come prepared, but can explain further without being concerned about the time it takes to write down their ideas. 
2. Feedback and clarification is automatic. I get very frustrated with written evaluations where the student didn't interpret the question correctly and therefore did not put the right answer down, even though you know they know it. By having it done verbally, I can pose the same question, but then ask clarifying questions afterwards to see if they do understand the material. It is also a great chance, once the evaluation is completed, to provide feedback on both their ability to do a verbal evaluation, but also address the misconceptions or misunderstandings they have about the concept rather than wait 2 weeks to get back their written work. 

Students are coming well prepared which is a big help to me. So far, the results have been quite good as students in this model are better able to share their understanding. If a student is not prepared, they have not been meeting the expectations and have been asked if they would like a chance to re-do or stay with the result. This has also been interesting. 

Materials for this week:

1. Rubric for the exit interviews
2. Homework on renewable resources

Climate Change and Climate Graphs:

The presentations of each of the IPCC's climate graphs were completed this week. Now a bulletin board has been made outside the classroom to show the connections between different graphs and how the data can tell a new story. This has been effective for the students to see the larger picture of climate change, with the supportive data. 

They also took time this week to explore climate graphs (showing temperature and precipitation). This has been very helpful for the Social Studies teacher who used to take a lot of time to explain how to read them. By having it as a part of their homework and an in class activity, the students are learning how to read them with me, but then going to apply the information in their other class. 

Materials for this week:

1. IPCC climate graphs
2. Making connections activity
3. Reading Climate Graphs homework
4. Reading Climate Graphs Homework Check In

Climate Change, Solar Ovens and Exploring Patent Law

Coming to the end of the year, there is always the challenge of how to spark Grade 7 students to keep going as well as create assignments where they can also explore their developing independence. The class had four learning goals this week:

• Science: Identify how heat is transferred and how heat loss can be reduced. 
• Science: Look at trends and evaluate the amount of greenhouse gases accumulated over time and propose courses of action to reduce the amount of greenhouse gases in the atmosphere. 
• Math: Make inferences and convincing arguments using data and the analysis of data from charts, tables and graphs that compare two attributes investigated. 
• Math: identify trends/patterns present to come to a conclusion on an argument

Science: Solar Oven Development

This week was day 2, 3 and 4 for the solar oven project. At this point, students had already determined if they wanted to be in a group, or working alone. Many had already identified the modifications they wanted to make to the "base model" and created experiment planners to test their new ideas. They had also thoroughly read through their peer's lab reports from the individually created heat labs completed over the March Break to find ideas and evidence for possible modifications. 

I really appreciated how the students bought into the hypothetical situation of being in an information bubble. They have only the results from their March Break lab reports to reference as evidence for a design modification. By doing this, students were reading through each others labs with a fine tooth comb. What was the biggest take away was how the students were trying to find ways to use the experimental results of an experiment involving putting an ice cube in a cotton or wool sock to justify a modification to their solar oven. They were able to take the results and apply them to a new situation effectively. However, hearing their discussions over how the data could show their modification showed their deep understanding and also some misperceptions of heat transfer that I had not heard until this point. 

Students who had decided to complete their own experiment are now collecting the results and determining if they want to make the results public for others to use. It was also interesting to see their excited faces during the first trial and seeing the results coming in for each modification. Students were then asked by their peers if they would make the results public or not. It was great to see students challenged with the decision of if their ideas should benefit others or just themselves. We will be wrapping up this conversation next week with an analysis of various situations where the decision of sharing scientific results for others use occurred and the outcomes afterwards. Having Grade 7s debate the purpose of copyright and patent laws will be very interesting now that they have their own personal experience. 

Documents used this week:

1. Solar Oven Student Handout
2. Solar Oven Design Template
3. Solar Oven Experiment Planner
4. Solar Oven Exit Interview Rubric

Math: Reading Graphs With a Focus on Climate Change

Since creating their lab reports from the March Break heat labs, they have developed an understanding of what is required in a proper graph. What we have been focusing on is that each graph has a story to tell, and that this story was was only told because someone had a question. When reading graphs their job is to identify what the question was that was asked and how the graph tells the story of the answer. 

I also wanted students to realize that reading a graph is not difficult, but requires them to slowly decode the information. Even the most confusing graphs can be broken down into parts that they can work to understand independently of each other prior to bringing the ideas together. 

There is a range of abilities in the class for how students can interpret visual data. With the range, some students are still working on reading a line graph while others are ready for a greater challenge. To ensure that all students were being challenged in this area, I wanted to find graphs that were all trying to tell the same story, but in a different way. This way my students, no matter which graph they received to work on, needed to tells its story to the rest of the class so that the big picture of climate change could be understood. 

To do this I went to the IPCC and used their graphs and data that explain climate change. I broke students up to work on a graph that reflected their need for challenge. Upon receiving their graph, they had 10 minutes to work independently to work through the decoding document. Then, after the 10 minutes they joined up with the rest of their group to share what they had decoded as well as where they were stuck. The final goal of the week was to create a slide show presentation to share with the rest of the class. In this presentation they needed to show the graph, explain the graph by decoding it, identify the question that was asked to make the graph, the answer to that question as well as the "why should we care". We have now started the presentations and it is really interesting to see students make connections between the stories their graphs told to the stories being told by other groups graphs.

The next step is that a board is going to be made with all of the graphs. Students will then use string to make a physical link between two graphs (one being their own) and identify what story the two graphs together are able to tell. They will then need to explore what next steps can be made to improve the outcome

Documents used this week:

1. Tuesday Slide Show Presentation
2. IPCC graphs for each group
3. Graph decoding document

Making Math More Like Science

This is my second time teaching Grade 7 Science, and once again I am blown away with the simplicity of the material but how it "blows our brains" as my students explain. As a class we have spent days talking about The Particle Theory. These 5 postulates have caused us to question everything from why does it take longer to make a cold tea than hot to the fact that we are made of the same stuff of dinosaurs. What would seem like 5 simple rules has caused all students in the class to question, explore and expand their thinking.

Having taught math for the majority of my career, I am used to parents and students feeling that the same enrichment and challenge only comes from the exploration of material beyond the curriculum. We have to leave the material of Grade 7 so that a student can become equally engrossed and inquisitive about the material as their peers. However, from observing my Science students I challenge this full heartedly. As we explore the same concepts of Science, each student is driving their thinking forward from their own questions. No new material is being presented to them, but they are using their current knowledge to ask questions and see how they can apply this thinking to the world around them. The difference I see from this and a math class is that math does not encourage student driven questions. There is a sense that an order must occur in the learning of material and that the teacher should guide it. I want to put this thinking in the garbage this year and make it so that math and science are structured in the same manner; the presentation of foundational concepts and the time for students to explore and make connections on their own.

This will be a challenge. I know from personal experience that students are taught to ask questions in science since these exhibit wonder and excitement about the material. Asking questions in math class could have a very different reaction from peers and the teacher. It could indicate an inability to grasp the concept, not able to focus on the prescribed task or "not what we are thinking of at the moment". Students, and parents, are trained to learn, practice and then perform rather than the science of learn, question, explore and perform.

To do this I have made my students first math project to be one where they need to ask their own question where proportional reasoning is the base. How far they push and go with the ideas is up to them. My last blog post goes into more details of the project. However, over this past week I am more convinced that a great exploration and student driven activities are necessary in math classes to develop the math students my high school coworkers are hoping to have come their way. They are wanting to have the thinkers, and not just the calculators.