Archive for the ‘intermediate’ Category

summer professional reading: Teaching Mathematical Thinking

Posted on: July 25th, 2017 by jnovakowski No Comments

IMG_6362Teaching Mathematical Thinking: Tasks & Questions to Strengthen Practices and Processes

by Marian Small

foreward by Linda Dacey

published by Teachers College Press, 2017

 

 

In this recently published book, well known math educator and author Marian Small highlights an important aspect of the discipline of mathematics – the thinking practices and processes that are “the doing of mathematics” when engaging in mathematical problems and learning content.

For those wanting clear examples of practices such as mathematical modeling, structure and argument are – the author clearly defines these with examples from across grade bands (K-2, 3-5 and 6-8).

For each practice/process, the author includes:

1) a definition with examples

2) where that practice/process is seen in K-8 mathematics

3) examples of problems, across grade bands, that might bring out that practice/process, often with examples of student responses

4) assessment questions for the educator to use to help notice and reflect on the students’ use of the practice/process

5) a short summary

I can’t think of another book that makes such careful nods to the Canadian mathematics education landscape. Although the focus is on the eight American Common Core standards for mathematical practice, the author connects these to our mathematical processes/competencies in Canada (with slight differences in different provinces/regions). Because our Canadian emphasis on visualization and mental math and estimation is not explicit in the American practices, the author has added a final chapter dedicated to these processes.

The problems are chosen to connect to each practice/process but should not be considered practice-specific. There are different types of problems – if you are familiar with Marian Small’s other books, you will understand the type of open-ness, differentiation and complexity built into the problems provided. For each practice/process she provides at least one problem for each grade band and then discusses how students take up the problems, with student examples.

I highly recommend this book. So so many wonderful problems for K-8 students and great information for teachers to help us think about the discipline of mathematics.

~Janice

summer professional reading: Teaching Math with Google Apps

Posted on: July 20th, 2017 by jnovakowski No Comments

IMG_6290Teaching Math with Google Apps: 50 G Suite Activities by Alice Keeler and Diana Herrington

 

Foreward by Jo Boaler

 

Published by Dave Burgess Consulting, Inc. 2017

 

This book opens with a foreward by Jo Boaler, with a call for educators to transform math classes. She references the Forbes list of skills needed for employment such as teamwork, problem solving, communication – all of which she argues can be enhanced through collaboration with technology. She also addresses the issue of “speed” and mathematics and how some students believe they are not “math people” because they are not fast. Boaler explains how the simple submissions of thinking and solutions on a Google form can take away the focus on speed in mathematics.

Authors Alice and Diana have both been math teachers at the high school and college levels. They emphasize the importance of digital tools in reimagining the math class with a focus on collaboration. They outline seven ways to use Google Apps to teach math:

1) Post Directions

2) Watch Students Work

3) Collaboration

4) Shift Students to Higher DOK Levels

5) Students Research

6) Shift to Facilitator

7) Conversations for Depper Understanding

The majority of the book is dedicated to overviewing 50 activities to teach math with Google Apps such as “Small Group Investigation,” “Discuss Strategies,” “Analyze Data Sets” and “Create Geometry Constructions”. The authors suggest asking yourself, “how does this activity make learning better?” Most of the activities use Google Classroom, Google Docs, Google Sheets or Google Slides and provides the advantages of using each format. Also used are Google Search, Google Forms, Google Drive, Google Chrome, Google Drawing, and Google Flights,

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Links to examples and tutorials are provided.  Some key reminders are interspersed throughout this section:

Teach like YouTube and Google exist.

The person doing the work is the person doing the learning.

We are a community of learners and we help each other get better.

The back matter shares examples from classrooms and highlights DOK levels (Depth of Knowledge), the 4 Cs (creativity, critical thinking, communicate and collaborate), mathematical practices for the CCSS and the 5E instructional cycle (engage, explore, explain, elaborate and evaluate).

There are lots of great ideas for tech integration and student collaboration throughout this book. Be mindful that some districts have policies or concerns regarding students having gmail accounts and as Alice has clearly said on Twitter – Outlook and Google apps aren’t really compatible. If having gmail accounts for students is frowned upon, like in my district (Richmond), take some of the ideas from this book and figure out how to make them work with the platforms that you are able to use! That will be one of my goals for the coming year as I see so much opportunity in technology enabling  our secondary students to engage in in-class, cross-class and cross-school collaboration around mathematics.

~Janice

thinking about fractions

Posted on: June 21st, 2017 by jnovakowski No Comments

One of the foundational concepts in grades 3-5 is an understanding of fractions – some of the questions we investigate with students are:

What is a fraction? What makes a fraction a fraction?

How can we order and compare fractions?

When do we use fractions in our daily life?

What different ways can we represent fractions?

What are equivalent fractions?

In our BC mathematics curriculum, the big idea for grade 3 that we guide students to understand by the end of the school year is that “fractions are a type of number that can represent quantities” – this is a significant concept. I often have discussions with older students who have the conception that fractions have something to do with shapes/geometry, possibly because of the models used in school to represent fractional numbers (think circles/pies/pizzas or rectangles/chocolate bars). An intentional focus is to provide opportunities for students to see and represent fractional amounts using different models – area/region, set and linear. In grade 4, students learn about the relationship between fractional and decimal numbers and in grade 5, consolidate their understanding of fractions by working with equivalent fractions. During these investigations, students may begin to see relationships and connections to whole number operations when working with fractions and decimals. We want students to be able to think flexibly with fractions and decimals, just as they do with whole numbers and think about composing and decomposing, benchmark numbers, etc as they consider addition, subtraction, multiplication and division.

This term I have been thinking a lot about fractions with students and teachers. I visited two classes at Homma and a class at Wowk and hosted classes at The Studio at Grauer to investigate fractions including a grade 4 class from Woodward, the grades 3&4 class from Grauer and the Richmond School Program students from Blundell.

The following are some images sharing our investigations. As you scroll through the images, consider:

What do you notice? What do you wonder?

What instructional routines and structures have we used to support students in their understanding of fractions?

What different materials have been provided to create opportunities to think about fractions?

What conceptions do students reveal in their representations of fractions? What might you ask students? How does this information guide where we go next?

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~Janice

The Studio at Grauer

Posted on: June 11th, 2017 by jnovakowski 2 Comments

The Studio at Grauer came to be based on a need we felt existed in our district – a space to engage in professional learning experiences for teachers as well as for a learning environment for students that could be left  “set up”. My office partner, Marie Thom, and I have been talking about this for the last couple of years. The notion of a “pop-up” classroom emerged and Andrew Ferguson, the principal at Grauer, was approached to see if we could use one of the school’s unused classrooms.

December 16 2018 – Room 102

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Room 102 was being used as a storage room until we began our transformation of it in January 2017. Blending Marie’s background in learning environments and my understanding of mathematics teaching and learning, we developed a space focused on mathematics, filled with inspiring materials in a learning environment designed for learners K-Adult. Our goal was to create a flexible, responsive and inclusive learning environment.

Room 102 – January 10 2017

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January 13 2017

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The learning environment is set up to create opportunities for choice, collaboration, personalized learning and inquiry. More specifically for mathematics, our hope was to broaden understanding of what mathematics is and what the teaching and learning of mathematics can be. Often school mathematics is perceived as “arithmetic” and mathematics is a much broader discipline that this. We wanted students and teachers to see math all around them and be inspired to think about mathematics in different ways – to see mathematical ideas in the materials, in pinecones, in buildings and structures, in images of our community, in art, in stories.

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As we discussed how we were going to use the space, we decided to call it The Studio, inspired the notion of an atelier, a studio space used in the early childhood schools of Reggio Emilia.

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Classes from Grauer, as well as visiting classes on “math field trips” visit The Studio to engage in mathematical inquiry. Marie and I take on the role of atelierista, working with the classroom teacher to facilitate learning experiences through different materials in the space. We have intentionally curated both mathematically structured materials like pattern blocks and Cuisenaire rods alongside materials often known as loose parts such as ribbons, gems, rocks, pinecones, etc. We also have art materials available to the students such as paint, clay, charcoal, yarn and wool so that students can express themselves and think using different languages. Students also have access to various tools to support their investigations such as measuring tapes, protractors, grids and ten frames.

The first class to visit The Studio – the grades 3&4 students from Grauer on January 18 2017

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The learning environment is intentionally flexible with choices in seating and tables available for both students and adults. Interestingly, although we have some chairs available, none of the students using The Studio have used them, preferring instead to stand or sit and lie on the carpet or use pillows. We have observed the flow of movement in the space and intentionally have large open spaces for students to move through. Shelves filled with baskets of materials are open and accessible to students. Students can choose the materials they want to use and take them to where they would like to engage.  We took doors off of some cupboards to create more open shelving. All of the furniture, except for four small Ikea open shelves, was found in school storage rooms and thrift shops.

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Provocations are set up on tables for students (and educators) to inspire mathematical thinking and inquiry. Inspired by one of the students’ interests in optical illusions, the grades 3&4 students from Grauer investigated the mathematics embedded in optical illusions. I gathered materials and tools that I hoped would provoke their thinking about optical illusions and the students also accessed and were inspired by other materials in The Studio.

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As we have more classes through The Studio, we have developed documentation to share in the space. Panels, photographs and photobooks are available for students and educators to engage with, to reflect upon and to inspire new experiences.

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One area of pedagogical intention in The Studio has been on noticing, naming and nurturing the Core Competencies and the Mathematics Curricular Competencies from our BC curriculum framework. A focus has been on both communication and creative thinking in mathematics. We intentionally create opportunities for students to engage in different types of communication and to reflect on how they are doing.

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We have had many groups of educators also visit The Studio. Our District Support Team, educators attending our Playful Inquiry professional learning series and teams from schools in our district. Many BC educators involved with our BCAMT Reggio-Inspired Mathematics project have visited the space as well. We have also had visitors from Manitoba and Sweden! We often focus the visits with the questions – what do you notice? what do you wonder?

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We also have a dedicated professional learning library with the teacher resources we recommend around teaching and learning mathematics, the use of loose parts and the importance of the learning environment.

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We had hoped The Studio would inspire similar learning environments in our district but recognize that many schools do not have access to a dedicated room for a studio space – we hope that teachers will be inspired by elements of The Studio for their own classroom learning environments. What has been exciting for Marie and I is that this little project has had a huge ripple effect at Grauer, in our district, and beyond!

~Janice

 

 

creating spaces for playful inquiry: April 2017

Posted on: May 28th, 2017 by jnovakowski No Comments

For our third session of our Creating Spaces for Playful Inquiry dinner series, Richmond educators came together at Grauer Elementary to share and learn together. This year at our sessions we have focused on broad themes or big ideas that cut across curriculum areas and grade levels, beginning with community, then identity and for our third time together this year, we chose to focus on place. Those that attended our Lower Mainland study tour to  the Opal School in Portland created panels reflecting on their experience. Many of our playful inquiry mentors set up either pedagogical provocations or shared provocations they developed to engage their students.

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Four teachers shared their experiences visiting Opal for the first or second time – what had an impact on them and how it is affecting their practice. Thank you to April, Louesa, Laurie and Karen for your thoughtful and passionate presentations!

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Based on feedback from the mentor groups during our January session, Michelle Hikida and I did a short professional learning presentation on playful mathematical inquiry and how we plan around a big idea, use provocations and projects based on students’ interests and curiosities and how we extend and sustain a math inquiry.

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After dinner together, we spent time in our mentor groups, zooming in on an area of interest and sharing and learning from each other.

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We are looking forward to continuing working together next year to support professional learning and building a playful inquiry community across our district.

~Janice

Big Mathematical Ideas for Grades 3-5: year four

Posted on: May 25th, 2017 by jnovakowski No Comments

This after school series is in its fourth year, focused specifically on the foundational mathematical concepts in grades 3-5 such as operations, place value and fractions. This year, we met once a term after school in Jennifer Plummer’s French Immersion grades 3&4 classroom at Homma Elementary (thank you for hosting Jennifer!).

For our first session in the fall, we focused on number sense and operations and the curricular competencies of reasoning and communication. I had recently been to the NCTM Regional Conference in Phoenix and shared a great game that I was reminded of in one of the sessions there.

The Product Gameboard

The Product Gameboard instructions

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At our second session, in January, we focused on financial literacy and mathematical inquiry. I shared the new Pizza Co. game for the iPad from Osmo as well as some other resources and children’s books to support the financial literacy content in the BC Mathematics Curriculum.

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BC K-5 Mathematics Big Ideas

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For our third session in April, teachers in the series requested a focus on assessment, place-based learning and connecting the core and curricular competencies.

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BC K-5 Math Communication

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We explored curricular content and competencies by investigating with power polygons.

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Looking forward to continuing the conversation around big mathematical ideas next year!

~Janice

Vision, Mission and Values Project at Thompson Elementary

Posted on: March 20th, 2017 by jnovakowski

The Richmond School District is embarking on a legacy project, developing new Vision, Mission and Values statements for our district. Teachers were invited to engage in discussions with their students and collect artifacts to contribute to the district process of creating new VMV statements. Two intermediate classes and their teachers from Thompson agreed to do some special filming for a district video for this legacy project.

At the beginning of January, I was able to meet with classroom teachers Lyanne McCaskill (grades 5&6) and Kevin Dimmick (grades 6&7) to plan how this project might unfold. The teachers put a lot of thought into the experiences they wanted to provide for their students. I was fortunate to be able to attend three different learning experiences with the students and to capture the students’ thinking and ideas with photographs and videos.

On the first day, as is the usual routine, the students in both classes entered their rooms to the morning provocation: What do you imagine school could be? They were invited to respond to the question using loose parts. Some students focused on the physical environment while others focused on metaphors and ideas. Each class paused to go to each table group and have those students share what they had done. Students could go back to their spots to revisit their work, connecting to new ideas or inspiration. Students were then asked to reflect on their thinking using a familiar response form.

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In the grades 5&6 class…

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In the grades 6&7 class…

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On the second day, the teachers used the Vision, Mission and Values from the Vancouver Aquarium website to initiate discussion about what Vision, Mission and Values are. In one class, the students were asked to use a familiar response format (Notice, Connect, Wonder) as they discussed and thought about the Aquarium’s VMV statements while in the other class, they used the Vision, Mission, and Values framework and questions as a way to respond.

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In one of the classes, the students sorted different statements they had made in their own reflective writing about VMV – there were very interesting discussions that emerged as students distinguished between what was part of a vision, mission or values statement.

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The students worked in small groups to create charts of words and phrases that connected to Vision, Mission or Values and these were displayed in the classroom for students to read and discuss.

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On our third day together, the teachers began their days as they had the first day – with a morning provocation posted for the students to engage with and loose parts available. The original question of What do you imagine school could be? was now linked to Vision, Mission and Values. The students could choose one or all of the three areas to represent and record their ideas about.

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It was fascinating to watch how the students’ ideas evolved over time and how each class had its own particular culture it terms of what came out in the students’ representations – kindness, inclusion, community, diversity and collaboration were the five big ideas that stood out to me as I listened to and read the students’ contributions to the project.

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An short video compilation of the Thompson VMV experience can be viewed HERE.

What has stuck with me, while spending time in these classrooms is the wisdom of our students. Their lived experiences in different types of learning environments, their understanding of each other, the importance of collaboration and the purpose of schools within a society made my heart sing. Our future is in good hands.

~Janice

creating spaces for playful inquiry: January 2017

Posted on: March 9th, 2017 by jnovakowski

Richmond’s Playful Inquiry Mentors hosted their second dinner event of Creating Spaces for Playful Inquiry series. For this session, we focused on the theme of identity and its connection to the core competencies as well as curricular competencies and content in BC’s curriculum. As sixty Richmond educators joined us in the Grauer multipurpose room, the playful inquiry mentors had set out provocations to engage in.

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A handout of the provocation questions and curricular connections can be downloaded HERE.

Three Richmond teachers (Kelly, Anna and Christy) who visited the Opal School in Portland last June shared how their visit to Opal has inspired their learning environments and their teaching practice.

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Kelly’s presentation on the impact her visit to Opal on her own teaching was summed up in two words – her presentation can be seen HERE.

 

 

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Christy’s presentation on Fine Arts Provocations with her grades 5&6 students can be viewed HERE

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For our professional learning area of focus for the evening, Marie and Hieu shared their thinking about using loose parts as an inclusive practice that particularly supports English Language Learners in our classrooms.

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After dinner together, we broke out into mentor groups to discuss specific areas of interest and to collaborate and plan together.

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Some of the playful inquiry mentors along with some of the participants in this series will be attending a Lower Mainland Study Tour to the Opal School in Portland over spring break and will bring back new sources of inspiration for playful inquiry to share with teachers in Richmond.

~Janice

investigating mathematical big ideas at Hamilton

Posted on: March 9th, 2017 by jnovakowski

In January, I spent some time in the two grades 4 & 5 classrooms at Hamilton Elementary. Coverage was provided to teachers so that they could observe and take part in math lessons in another teacher’s classroom. Teacher were then able to teach this lesson to their own classes, having seen and heard how another class responded and thus, anticipating and planning for their own students. This form of “adapted lesson study” is a common structure we use in professional learning in our district, with time to plan together, observe and discuss and then enact and debrief. The teachers at Hamilton had requested a focus on teaching through the big ideas in the curriculum.

For both classes we focused our planning around these big ideas:

Development of computational fluency and multiplicative thinking requires analysis of patterns and relations in multiplication and division.

Computational fluency and flexibility with numbers extend to operations with larger (multi-digit) numbers.

To focus the students’ thinking, connection-making and our discussions, the question we posed for the students to investigate was: What is the relationship between multiplication and division?

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For each class we began with a game, to activate students’ thinking, get them talking about mathematics and to practice computing multiplication facts. In one class we played Product Gameboard and in the other, the card game Salute.

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IMG_9770After discussing the strategies the students used in each game, a problem was introduced to each class. Both of the problems were taken from the book: Good Uestions for Math Teaching. Using different strategies, I facilities meaning-making of the problems with the students and then the students began to engage in problem-solving. They had an opportunity to “turn and talk” and share their strategies and were encouraged to approach the problem in different ways.

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The students all began with the same problem but could adjust the number of students in the school (in the problem context) they were working with. They used whiteboard to show their different approaches to solving the problem.

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As students shared their solutions and strategies, we asked the students to listen to each other and build on or connecting to each others’ thinking as part of the discourse.

In the second class, a related problem was presented.

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What was interesting to notice as students engaged with this problem is that none of the students paid attention to the “four grades”- it was not required information to work through the problem but would have added an extra layer of complexity and context. We did pause near the end of our time together and this was pointed out, and if I had been with the class the next day, I might have had them re-visit this problem, being mindful of the “four grades” context.

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What students did pay attention to in terms of sense-making for this problem was the types of sports students might be playing and the number of students that would make sense for each team. The students found a context (tennis) that made sense of having one person per team and two per team (doubles). The students shared their different solutions on the large whiteboard which we used as a starting point to compare and contrast their different solutions and strategies and have the students make connections to how both multiplication and division are related and could be used to engage with both of the problems posed to the classes.

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Teaching through the big ideas was also a topic of conversation during an afternoon of Hamilton’s professional development day in January. We will be continuing our conversation at Hamilton’s pro-d day in May and continue to think about ways to nurture ways for students to make connections between mathematical concepts and strategies.

~Janice

making pentominoes

Posted on: February 27th, 2017 by jnovakowski

Many of our schools have the brightly coloured flat plastic pentominoes tucked away in storage cupboards. I have always like pentominoes due to their affordances for puzzles, problem-solving and spatial reasoning.

One thing I’ve noticed is that they have not been particularly appealing to our younger learners and I wondered if it was because of their two-dimensionality. I thought something that was more tactile for them and that they could build with might be more appealing.

When I saw these cubes at the dollar store, my mind went to building pentominoes based on a task I had read about in the Taking Shape book by Joan Moss et al.

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How many different ways can 3 blocks be put together, with all edges and faces flush? How will you know if you have found all the ways? Are two objects congruent if their orientation is different?

With 4 blocks?

With 5 blocks? (pentominoes)

I have done this task with teachers, both as part of our BCAMT Reggio-Inspired Mathematics Collaborative Inquiry Project and as part of a session on the mathematics curricular competencies at our district conference.

Teachers have found this task touches upon so many areas in our curriculum – spatial reasoning, geometry, problem-solving, visualizing, reasoning and analyzing, communicating, etc.

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As an extension to the building task, using the little wood cubes, I glued sets of pentominoes together, using an image of the 12 pentominoes I found online to help me as a guide. I also left lots of blocks not glued to be used for building the different arrangements.

Pro-tip – don’t use liquid white glue on the coloured blocks….the dye runs and you’ll have a mess on your hands and elsewhere!

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One of the unique aspects of pentominoes is that they are able to fit together to form various rectangles. How many different rectangles can you make using some or all of the pentominoes?

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~Janice