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 🙂
An important focus of this year’s summer numeracy program was to increase parent engagement at each site over the course of the three weeks. Each site employed a variety of creative strategies to ensure that families were learning together at summer numeracy. In addition to daily greeting and sharing of success stories with parents upon pick up and drop off, our teaching team concentrated on three main strategies for parent engagement, as discussed below.
In order to create an open and welcoming atmosphere, parents were invited to attend an open house during the second and third week of math camp. This gave parents the opportunity to interact with the teaching team in a low-stress environment over coffee and muffins, and to participate in some of their children’s favourite math centres. It was a wonderful opportunity for our Mathletes to share some of their key learnings with family, and they took great pride in showing off their STEM creations (i.e. boats, bridges).
During these open houses, one of the most popular activities was the shared reading centre, which boasted a wide array of math-related picture books. For me, this reinforced the power and value of connecting math and literature (see previous blog post on this topic). The students were excited to share a story with family, and there was meaningful math talk occurring as a result!
Second open house
2. Travelling math bags
We also wanted to extend summer numeracy learning to the home environment, so we put together Travelling math bags for each family (siblings took home one for the household). Depending on the strengths and needs of the recipient(s), the math bags were composed of a variety of potential items, including:
math question (see picture)
deck of cards with specific game outlined
math-related picture book
As attendance for some of our Mathletes was inconsistent, this was not a successful strategy for every student. Also, since we were sending these math bags home daily, the biggest challenge was changing up the contents of the bags during the day so that the Mathletes could have new (or at least slightly different) activities for each night. However, we did receive positive feedback from multiple parents who appreciated the activities included in the math bags. They gained ideas for how they could practice and improve numeracy skills with their children at home- often in a surprisingly fun way!
Travelling math bag
3. Take-home math kits
The travelling math bags were only intended for use during the course of the summer numeracy camp, yet we wanted our Mathletes to be able to extend their learning throughout the summer. Thanks to some excellent budgeting and generous funding, we were able to save enough to put together take-home math kits for each of our Mathletes that we distributed on the last day of camp. These kits consisted of:
deck of cards
While not all directly related to math learning (I’m pretty sure the Smarties were all devoured immediately 🙂 ), these items also served as reminders of some of the activities we carried out as a group. A simple deck of cards and a pair dice can provide so many opportunities for math learning in a family setting (see here and here for ideas). Ideally, we hope that our students use the tools provided in their take-home math kits to continue to build upon what they learned during the summer numeracy program!
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!
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!
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
1 small bottle of glue
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.
Testing triangle strength
Components of a bridge
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…
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
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?
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
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!
Littlest Mathlete wowing us all with his patterning!
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
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
Egg carton piece (single egg portion)
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
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!
Number talks were developed as a teaching strategy to increase student understanding of the numerical relationships that serve as the foundation for math rules, which we often just memorize without truly understanding. Number talks strive to helps students compute with accuracy, efficiency and flexibility. These number talks are characterized by purposeful classroom discussions about carefully selected computation problems that prompt a larger discussion about mathematical thinking. As described by Parrish (2010), “[t]he problems ina number talk are designed to elicit specific strategies that focus on number relationships and number theory.” These short sessions do not replace ongoing math curriculum, but rather take place alongside it. Their overall goal is computational fluency.
Each number talk begins with the teacher sharing a math problem with the class. Students are given time to mentally figure out the problem. The teacher then provides opportunities for students to share their answers and strategies (i.e. mathematical thinking). Number talk problems can be presented in a small-group or whole-group environment. The entire classroom number talk generally lasts 5 to 15 minutes, and gives students the opportunity to share, examine and defend different strategies and solutions.
While this strategy can look slightly different from class to class, number talks typically:
Form part of the daily classroom routine
Take place in a cohesive classroom community and safe environment (e.g. a culture of acceptance).
Include opportunities for students to think before checking solutions
Rely on open-ended problems of varying difficulty
Involve interaction with classmates and the teacher
Improve students’ self-correction skills
While I did not have the opportunity to test out this strategy with my own students, I was able to observe how a number talk functions during a math PLC held at my host school for junior teachers. The presenting teacher explained how number talks could serve as an activation or introduction at the start of every class by using adding, subtracting, multiplication or division problems. The sample number talk used the following problems:
5 x 5
5 x 50
5 x 49
This quick set of problems moved from more friendly to less friendly numbers, and at each stage participants were instructed to show a ‘thumbs up’ if they had an answer. Some key guiding/prompting questions when discussing different solutions and strategies included:
How did you know that?
Did anyone have a different strategy?
Could you check your answer using a different strategy?
Does that make sense?
The teacher accepted each student’s strategy and then went through them all to determine which were reasonable and which were not. This was a quick and efficient strategy for engaging students in purposeful math talk while developing problem-solving and computation skills. It would be a valuable addition to the daily routine in any classroom!
There is nothing worse than watching a student struggle with a math problem because they’re not sure where to start or misinterpreted the question. Enter the CUBES math strategy!
C- Circle the key numbers.
It is important for students to recognize that sometimes there are numbers given in a question that are irrelevant to the problem itself. By circling only the key numbers, the students are forced to pause and think about which numbers they actually require in order to answer the question.
U- Underline the question.
Similarly, problems can sometimes be overly wordy and the students lose track of what they’re being asked to do. The students gain a clearer understanding of what the problem is actually asking them to do by finding and underlining the question- and they become aware if they have no idea what the question is at all!
B- Box any math “action” words.
For this step of the strategy, students have a tendency to box all the math words rather than the math action words, so it is important to explain this distinction. As the question will not always use explicit math action words (e.g. multiply, add, explain), the students need to also be looking for implicit math action words (e.g. how many? how far? how long?).
E- Evaluate (what steps should I take?)
Organization of thinking is half the battle for some students. If the students can identify the steps they need to take to answer the question before they begin to tackle it, it can alleviate frustration later on. By generating a plan of action, students often feel better prepared to attempt the problem and can sometimes identify errors in their proposed solution before carrying it out.
S- Solve and Check (Does my answer make sense? How can I double check?)
While solving the problem is obviously important, checking the answer using a different strategy is equally as important for students (i.e. can you solve the problem a different way and still get the same answer?). Not only does this give students confidence in their answers, it enables them to identify and fix their errors and double check that their answer makes sense.
Try implementing this strategy by reviewing a problem that your students have already solved so they can focus on the CUBES strategy itself. You could also put up a poster describing the CUBES strategy in your classroom, and distribute mini CUBES handouts (as pictured above) for students to glue in their math notebooks. Happy problem-solving!