During my practicum in a grade 8 classroom, my associate teacher shared various techniques for increasing student engagement during math problem-solving. One such technique allowed students to use their own devices to scan QR codes that were posted around the classroom and hallways. By scanning the QR codes, students were able to access multiple different questions and work through them at their own pace. The order of the questions didn’t matter, so students (working in pairs) could disperse and travel freely to the question locations.

While they were working on solving math problems, the simple act of getting students out of their desks and moving between different locations kept them engaged and motivated to work diligently with their partner. *Side note: this class was used to working with visually random groupings, and we often used playing cards to determine groups of 2, 3, or 4 for different activities.

QR code scanning

QR codes in the halls!

QR code details

QR code question and problem solving

This “QR Code Treasure Hunt” functioned best when guidelines were clearly communicated to students before the activity began. For instance, consider the following:

Devices to be used (classroom devices? student devices?)

Availability of QR code reader (app already downloaded on devices?)

Groupings (individual? pairs? small groups? visually random groupings?)

Range in difficulty of questions (simple to increasingly difficult? similar in difficulty?)

Number of questions (length of working time?)

Materials to bring (clipboards/paper/pencil?)

Teacher supervision (monitoring throughout halls?)

Consolidation techniques (select examples? group sharing?)

Overall, the students seemed to appreciate this break from routine and their level of engagement noticeably increased (which was especially obvious during this 8:00- 8:55 AM period)! I will definitely be adding this strategy to my teaching toolbox 🙂

Homeroom, six periods, over a hundred students in a day… While it is infamous for being a difficult year for students, grade 8 can definitely be a whirlwind from the teacher perspective as well! As a second year Teacher Candidate, I am excited to be in a grade 8 math/science placement for this semester.

“The three most important words in education are: relationships, relationships, relationships. Without them, we have nothing.” (Couros, 2015b)

My focus for the first two weeks of school is developing a strong foundation upon which to build positive relationships with students. The video below serves as an excellent reminder for educators to intentionally create learning communities where every child feels heard and valued. In the words of George Couros (2015a), “[w]e need to put ourselves in our student’s shoes before we can create better opportunities for them in our classrooms.” This is a message that has guided me through the first week of classes, and my observations led to three interesting take-aways:

Watch how students interact with one another, and pay special attention to how they choose to divide up when given the opportunity (e.g. Who gravitates towards whom? Which students do not get along? Who is alone?)

Be aware of students with older siblings/family members that you may have taught (e.g. How does this influence their expectations of the course?)

Be purposeful and explicit about your expectations for the year, with particular emphasis on how partner/group work should be completed

While much of the week was dedicated to administrative details, introductory concepts, and establishing routines, we took a few moments with several of our classes to reflect on the coming school year. They were asked to complete one or more of the following sentences on a post-it:

School is important because…

Math is important because…

Science is important because…

In order to be successful this year, I will…

Science is important because…

First week bulletin board

There was some variability in responses, but in general, our students highlighted the importance of school (particularly math and science) in providing future opportunities and contributing to their career paths. They recognized the role of education in increasing their understanding of the world around them and preparing them for daily life (e.g. sports and recreation, job duties, finances, grocery shopping, caring for the environment, etc.). But, perhaps most importantly, many students pointed out the importance of school for building strong social skills and developing positive relationships. This resonated with me, and reinforced my belief that creating healthy, safe and supportive learning environments is the first step in fostering a productive thinking classroom.

Our third and final week at numeracy camp focused on area and perimeter, which we introduced using the picture book Spaghetti and Meatballs for All! A Mathematical Story by Marilyn Burns. It is an engaging story that describes a family reunion, where the arrangement of the tables and chairs is constantly changing as more and more people arrive. The story cleverly delves into the concepts of area and perimeter in an everyday situation such as a family meal.

We read the book aloud to our Mathletes, discussing the differences in seating plans as we followed the storyline. We then used the SMART board to explore the area and perimeter of the different configurations of tables and chairs. For each “seating plan,” we documented the strategies used to find the area and perimeter. After investigating multiple options, the students were able to see the logic in Mrs.Comfort’s original seating plan in the story. This hands-on activity was interesting for the whole group, and our Mathletes particularly enjoyed discussing their favourite meal for family get-togethers!

Read-aloud

Seating plan #1

Seating plan #3

Explaining our thinking

You can find a lesson plan based on this book by Cheryl Rectanus for grades 5/6 here (Math Solutions Professional Development Newsletter). It describes the lesson that Cheryl carried out after reading Spaghetti and Meatballs for All! aloud to the class, and gives some great ideas for prompts and questions that could be asked to deepen the students’ learning.

Activity #2: Area and perimeter art

As an extension to our discussion of area and perimeter, we tasked our Mathletes with creating a piece of artwork out of squares and rectangles on grid paper. We guided them in thinking about the following questions:

What is the area of the spaces you used?

What is the perimeter of your creation?

Which strategy did you use to calculate area and perimeter?

The final products were colourful and creative (see below), and they prompted some great math talk among learners about area and perimeter!

During the last week of math camp, we challenged our Mathletes to use the construction and math skills that they had been practicing to individually and economically build a boat that would float. The parameters of the challenge were simple:

Goal:

Using the materials from the list below, design and construct the least expensive boat possible that will float and carry plastic people on it.

Materials:

Steps of boat construction:

Design and sketch your boat

Decide which materials you will need

Estimate how much you will need of each material

Calculate the approximate cost of your boat materials

Construct boat and adjust cost estimate according to materials actually used

As this was the third week of Math Camp and the students had completed various STEM-based challenges already, they were becoming more efficient at planning, designing and carrying out the process of construction. The added challenge of calculating the cost of their boat was a great differentiation tool, which engaged the older students in particular to minimize their use of resources through unique design. The boat challenge was completed individually, which revealed each Mathlete’s strengths and areas of opportunity more clearly. For example, some students initially constructed ‘rafts’ (i.e. no mast, sail, hull). While this was an economical option, we challenged them to adjust their design so it more closely resembled a boat.

Boat materials and criteria

Boat design

Boat design with materials list

Boat design with materials labeled

Testing different construction designs

Adding style points

Assembling “slotted lumber”

Following design plan

Following design plan

Colour!

Throughout the various steps of their boat construction, students faced many hurdles with regards to design, use of materials, calculation of cost, etc. Yet, the most striking observation from this task was the resiliency and grit demonstrated by our Mathletes as they adopted the ‘Keep Moving Forward‘ mindset and persevered with the task. There was a large variety in the finished products, and many students added colour, decorations and a personal touch that demonstrated immense pride in their boats.

They were very keen to test their creations, so we decided to spend time as a large group floating their boats. One by one, each student placed their boat in the water (they all floated!) and added plastic people figurines until it sank. As a connection to our previous work on patterning, they added people according to the Fibonacci sequence (i.e. 1, 1, 2, 3, 5, 8…) and we recorded how many people each boat held. While some students were initially hesitant to test their boats to the point of sinking, the fun atmosphere and support of their classmates encouraged them to give it a go! We discussed the strengths of their designs and the purpose of minimizing cost (i.e. minimizing use of non-renewable resources). It was a fantastic celebration of their hard work, and each Mathlete received a ‘Boat Building Award’ in recognition of their success!

To kick off the second week at Summer Numeracy Camp, we again wanted to challenge our Mathletes with a team-building exercise that required collaboration and communication: building bridges! We began with a simple question:

“What does good collaboration look, sound and feel like?”

This question generated a great discussion about the skills and attitudes necessary to work well in a team. Given this mutual understanding of what it means to collaborate, we let the students choose their own groups of 3 and each group received the following materials:

100 craft sticks (with 1 elastic)

5 pipe cleaners

10 paperclips

String

1 small bottle of glue

Bowl

The goal of the task was for each team to:

Design and build a bridge to span across a bowl of water.

Test the strength of the bridge (using pebbles)

On the first day, we gave groups a chance to design and begin building the different pieces of their bridges. The teams started by assessing the materials they were given, coming up with a feasible design, and constructing the different components of their bridge. Some teams also recognized the importance of including triangles, while others tried out the strength of the square.

Assessing materials

Design process

Bridge design

Triangles!

Testing triangle strength

Connecting triangles

Components of a bridge

Solid foundation

After leaving their creations to dry overnight, our Mathlete teams continued with their bridge construction the following day. They carried on measuring, testing, and adjusting their designs to figure out how they could be improved. All the teams found something they could adjust or modify to make their bridges sturdier and stronger. We again created some extra shapes for support and let them dry overnight.

Measuring length of bridge

Testing support legs

How much weight can our bridge hold?

How can we make our bridge more level?

On our final day of bridge construction, everything came together beautifully! The students used their resources and demonstrated creativity, perseverance and impressive problem-solving skills to successfully finish their free-standing bridges. During the consolidation, one member from each team explained their design and reasoning to the whole group, and we discussed the differences and similarities among our bridges. As our Mathletes had shown true grit and determination to complete this challenge, we decided to have a bridge celebration and prepared certificates for each participant that highlighted a ‘special mention’ for each group (e.g. positive attitude, perseverance, design and architecture, creative use of materials, problem-solving). They were very proud of their creations, and handing out these certificates was a lovely way to cap off another successful week at math camp!

As environmental inquiry is a large focus of the summer numeracy program, it seemed logical to spend some time working on number patterns- more specifically, identifying number patterns in nature. When thinking about math and nature, the Fibonacci sequence immediately comes to mind. In case you’re not familiar with it, the Fibonacci sequence is as follows:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

Quite simply, it is a series of numbers where each number is the sum of the two numbers before it. For example, the 3 (5th term) is found by adding 1+2. The 5 (6th term) is found by adding 2+3, and so on. The TED Talk below is a great summary of the importance of number patterns in mathematics and highlights some of the wonders of Fibonacci’s sequence (which is FUN and BEAUTIFUL).

Mathematics is not just solving for x, it’s also figuring out WHY. (Arthur Benjamin)

I had previously seen a video of a lesson that centred on the Fibonacci numbers, so I revisited the lesson (here) and was inspired to do something similar with our Mathletes. We carried out a series of activities that formed a week of Fibonacci fun, and some rich learning experiences for our Mathletes (with a little bit of added mystery along the way!).

Activity #1: Fibonacci number sequence

The students worked in pairs for the first activity. Each pair was given an envelope (see picture), but the contents were left a mystery until they opened them up. Inside, they found cue cards with numbers written on them (Fibonacci sequence numbers up to 144). Since they are so accustomed to math centres, some of our Mathletes thought it was an adding game at first (not a bad idea!). Most had the instinct to order them from smallest to largest. Some pairs required extra guidance to get on the right track, and once they had them in order they were instructed to start looking for a pattern. With some prompting for the younger ones, they discovered the pattern rule for the sequence of numbers, and tested it to make sure it applied to the entire set of numbers they were given. Once they were confident in the pattern rule, they could come to me for envelope #2…

Ordering numbers

Fibonacci sequence

Activity #2: Square tiles

For the second activity, the students worked in the same pairs as activity #1. They were given a second envelope that contained square tissue paper tiles in five different colours, and were instructed to see how this could relate to the pattern they just discovered. There were a different number of tiles depending on the colour, but this was not readily apparent for some students. They required prompting to grasp that they had to make different sized squares, and with some guidance they soon discovered that the side length of the squares corresponded with the Fibonacci numbers! For most pairs, they got as far as making squares with the different side lengths (e.g. 1 x1, 1×1, 2×2, 3×3, 5×5). As the various teaching hands were circulating throughout the room, we were able to guide some pairs to assemble their squares into a rectangle.

Creating squares related to Fibonacci sequence

Arranging squares into rectangle

Labelling our work

Both tasks completed

Extension

As some of our older students were finished their squares early on (and even assembled them into a rectangle quite quickly), I challenged them with the following:

Based on the pattern rule that you discovered, what would be the next few numbers in the sequence? How do you know?

Consolidation

We briefly discussed our Fibonacci discovery experience as a whole group, and we discussed the different strategies that the students used to find the pattern rule and create squares of different sizes. As we were continuing with Fibonacci the next day, we left it at that and praised our Mathletes for their excellent inquiry skills!

Activity #3: Spiral

The following day, we started off the lesson by watching a video created by Jo Boaler and her team at YouCubed. This is an excellent video about patterns, the Fibonacci sequence, and where we see patterns in nature (it can be found in the YouCubed week of iMath Day 4 section). The students were very engaged by the examples given in the video (many ooo’s and ahhh’s), and it served as a good consolidation of the activities we had already completed, as well as leading in nicely to our construction of a spiral of squares. I created an exemplar for the students to refer to, and we provided each student with:

Large construction paper

Glue

Assorted pre-cut tissue paper square tiles

We encouraged the students to first pick out the appropriate number of tiles in different colours for each square, and then arrange them on their page before applying glue. Some of our younger Mathletes required a little guidance with creating the pattern, but every student was on task and excited to create their own tribute to Fibonacci! The results were impressive, and we sent our happy campers home with their very own representation of the Fibonacci sequence.

One pattern, many different colours!

Fibonacci fun

Exemplar

Planning rectangle

Teamwork

Littlest Mathlete wowing us all with his patterning!

Fibonacci!

Activity #4: Patterning centres and beading

On our final Fibonacci day, we decided to host a celebration of patterns! We kicked the day off with a variety of patterning math centres (see pictures). One of these centres gave the students a chance to explore pinecones and their associated patterns. Thanks to a gracious donation of a large number of pinecones, our Mathletes were able to work as detectives looking at small, medium and large pinecones. I guided some students incounting the spirals in a clockwise and counter-clockwise direction, trying to find evidence of the Fibonacci sequence. Many were excited for the chance to just hold and manipulate the different types of pinecones, and we were all amazed by the patterns found within!

As our last patterning activity, the students were instructed to create a beaded bracelet that represented the Fibonacci series of numbers. This was the capstone to our week of patterning, and the students were equipped with:

Plastic beads of different colours

White cotton string

Scissors

The results were beautiful, and many of our Mathletes knew the first several numbers of the Fibonacci sequence by heart after creating colour patterns according to these numbers.

Another week of math camp comes to a close, and with it come a few last thoughts about our Fibonacci activities…

Things I would do differently:

Use a sturdier material for square tiles ( I used tissue paper because we had some pre-cut, but it was prone to being accidentally blown around and thus tricky to work with).

Make extension activity clear from the get-go.

Give more guidance for Activity #2 (i.e. tell students they need to use tiles to make squares that have side lengths related to Fibonacci sequence)

Things that worked well:

Working in pairs or small groups (3) for Activities #1 and 2.

Preparing Fibonacci numbers on separate number cardsfor Activity #1 so students could manipulate and move them around.

We had a couple visiting teachers in the room on the day that we carried out the activities with the envelopes, so there were many teaching hands available to guide groups and prompt as needed.

The curious and inquisitive nature of our Mathletes continues to astound me, and their willingness to think outside the box is motivating me to bring my A-game for the remaining week of summer numeracy camp. I can’t wait to see what our final week has in store for us!

As a team building exercise to finish the first week at Summer Math Camp, our Mathletes created simple catapults designed to launch cotton balls. The full description for the catapult design and construction can be found at this Kids Activities blog post.

Each student created their own catapult from the following materials:

7 craft sticks

3 elastics

Egg carton piece (single egg portion)

Cotton ball

Glue

We let the students experiment with how to construct their individual catapults, and provided guidance to those who needed it. The general construction resembled the exemplar below, although some students made adaptations as they saw fit. After testing out their creations, we all traveled down to the gym where students worked in pairs to measure the distance traveled (or height attained) for their cotton ball catapults.

Catapult design- front view

Catapult design- side view

For younger students, it provided the opportunity to practice:

Measuring distance/ height

Recording numbers in a chart

Comparing distances/heights

For the older students, they worked on:

Adapting catapult design to achieve greater distance/height

Adding up the total distance/height achieved over multiple trials

Estimating an average distance/height over a certain number of trials (for more advanced students)

We consolidated this activity by posing questions such as:

What was your longest cotton ball launch?

What was your shortest?

How could you have modified your catapult to launch the cotton ball further/higher?

Are there differences in the catapult designs that make some better at launching cotton balls further, and some better at launching cotton balls higher?

It was amazing to see how engaged the students were during this rich learning task. There were certain students who had been dead-set against anything resembling traditional math throughout the first week; yet even these students were eagerly measuring, adding, and comparing distances for their catapult cotton ball launches. Another great testament to the power of hands-on learning!

I was working with a small group (six) of JK to Gr. 2 students, so I wanted to focus on the different ways that can be used to represent a number. First, we read the book together, stopping at key points to allow the students to think about how each number was being represented. For example, asking:

Is there another way to show seven feet?

How do you know there are seven feet?

I also stopped at a couple points to allow students to predict what the next page will show (e.g. after 7, 10, 20). After finishing the read-aloud, I explained the Mathletes’ task: using cutouts of the different creatures from the book, the students had to show me a certain number of feet by glueing down the cutouts in their math journals. As there was a range of abilities in the group, I created custom targets for each of the six students. For some, they were given the extra challenge of showing me a number without using certain creatures (i.e. without using crab for 10 feet). The Mathletes liked the challenge, and some were getting quite creative in their approach! I think this task lends itself nicely to differentiated instruction, and the students enjoyed mixing and matching the different kinds of feet.

Show me 21 without using a crab!

Show me 31. Show me 17 without using a crab!

Show me 17 and 16

Show me 24

To create the cutouts, I made ten copies of the image below and cut them out. If I were to do it again, I would have the different cutouts in separate little bowls/containers on the table to prevent them from getting mixed up as the students were working. As Marilyn Burns describes, this activity could be scaled up for older grades by incorporating multiplication and equations to show certain numbers. Check out her full blog post on how this book can be used to target different grade levels. A definite must-have for my future teaching library!

As one of our first large-group activities at summer numeracy camp, our Mathletes collaboratively constructed a hundreds chart using a masking tape grid and foam numbers. This was a great way to explore numbers, identify number patterns, and build number sense with the students. As described by Melissa Conklin, some important questions to ask when building the hundreds chart include:

Where does this number belong on the 100’s chart?

How did you know the number belongs there?

Can you see any patterns emerging on the 100’s chart?

After building an entire hundreds chart together, we used an interactive SMART notebook file to identify some number patterns. Students counted by twos, threes, fives, tens, etc., and successfully identified many patterns along with their mistakes or ‘hiccups’! Our Mathletes then worked individually to rebuild the hundreds chart on a piece of grid paper. There were many different strategies being used to efficiently fill in the numbers, and even our youngest students completed their own hundreds chart using a model and prompting. The students were then asked to create a colourful number pattern on their hundreds chart, and many challenged themselves to colour complicated patterns (counting by sixes, nines, etc.).

As a review and consolidation, we also used this riddle to get the students thinking about number patterns in the hundreds chart:

I am thinking of a number between 10 and 100 with a single 9 in it. What might my number be?

The students came up with many answers and explained their thinking, describing how their number fits the three criteria of the riddle!

Hundreds chart construction

Hundreds chart construction

Hundreds chart patterns

Hundreds chart construction

Hundreds chart patterns

Finally, I always like to incorporate a read-aloud if possible to link literature with mathematics. When discussing the hundreds chart, the book One Hundred Hungry Ants by Elinor J. Pinczes is an engaging way to get students thinking about rectangular arrays, multiplication, and factors of one hundred. We consolidated by discussing the different arrangement of rows that the ants tried in order to get to the picnic. At the end of the day, I was impressed by the students’ grasp of the hundreds chart and their progress after completing these fun and interactive activities!

Anchor charts are created with students using chart paper and markers (or a white board/ SMARTboard) to convey the most important or relevant aspects of a concept. This keeps learning readily accessible to the students, creates a visible cue that triggers prior learning, and allows them to make connections to future learning.

Anchor charts in the classroom should:

Communicate the most current and useful learning content

Be created with the students in order to make thinking visible

Be referred to by students and used as tools for new learning

Be neat and organized

Review concepts and recognize future goals

These anchor charts should be posted in designated spaces within the classroom (e.g. clothesline, bulletin board) and rotated regularly so that current learning is represented.

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During practicum, I used this strategy regularly. The students were always involved in creating the anchor chart, as I was usually recording their thinking on chart paper during class discussions. For example, when we were reviewing multiplication (standard, lattice, partial products) and division (standard, partial quotients) strategies, we would solve a problem on chart paper as a class using a specific strategy. This chart paper would then be hung on the math bulletin board for the students to refer to as they were working independently. Anchor charts were also used during language arts to remind students of the characteristics of persuasive writing or the components of the APE strategy, for example. Thanks to associate teachers for some excellent anchor chart samples!