Code a City!

A favourite small world play activity that the children enjoy is our 'Code a City' challenge. This math invitation requires just a few materials - a city template, blocks, dice, and mini vehicles. The 'Skyscrapers' book is a perfect nonfiction resource to supplement the learning.

Children select a template. They can use one or multiple dice depending on their ability. Roll and add the pips on the dice together. 

Write the corresponding number in a space on the template.

Build a tower with that many blocks.

When your city is complete add mini vehicles for some small world play!

Printable Subitizing Cards

We use these subitizing cards for many games in our classroom. This past week we invited children to play a 'roll and record' game. Using one die or two dice, children would roll, subitize or add, and cover that number on their card using a holiday trinket. Click on the image of the card to download your own copy of the four subitizing cards!

Easy St. Patrick's Day Math Ideas

St. Patrick's Day is a fun holiday that helps jump start the spring season! Here are a few easy to plan math activities for children!

1. Shamrock Subitizing- Write different numerals in the center of each shamrock and encourage children to place dominoes with the corresponding number of pips in each leaf.

 

2. Estimation/Counting Jars - Fill jars with a variety of different seasonal loose parts. Encourage children to estimate how many objects are in each jar. They can then use a variety of math tools (e.g., number grid, ten frame, hundreds chart) to count the total objects. These can then be recorded on a paper for comparison.

3. Rainbow Numbers - Provide children with a variety of numbers and encourage them to explore these in different ways (e.g., sorting by colour, ordering, create number combinations).

4. Colour and Coin Sorting - Recycled marker caps can become 'pieces' of the rainbow. Add fun play coins and seasonal pots to the tray and encourage children to sort, count, and pattern with the loose parts.

5. Roll and Record - Here is a fun template that can be used by children to support many math explorations! Children can roll and subitize a die and colour in the corresponding number of shamrocks. Two children can each roll and colour in shamrocks using their own colour and try to get 4 in a row. Children can colour the shamrocks in using different colours to create patterns. The possibilities for this template are endless!

Get the free printable here: Shamrock Printable

 

6. Which One Doesn't Belong? -  Show children pictures of different shamrocks and ask them to articulate which picture does not belong. 

Get the free printable here: WODB Shamrocks

7. Gold Coin Count - Add dice, coins, number lines and writing materials to this game board and encourage children to create and play their own treasure games!

Get the free printable here: Gold Coin Count

8. Find the Gold Coding - Draw the path needed to get from home to the pot of gold! Use arrows to display direction of movement.

Get the free printable here: Gold Coding Mat 

 

 

 

Subitizing Game Printable

Subitizing is the ability to instantly recognize the number of objects without actually counting them.

 

This week we practised subitizing using a number cube and this subitizing grid. Children rolled the number cube and dabbed a corresponding square on the grid. There are so many possibilities for how else children can extend this game. It's perfect for small group math work, a math center or independent practice during whole group time. Visit the link to download a copy of the grid. Enjoy!

Algebra in Kindergarten? Absolutely!

 "Pure mathematics is, in its way, the poetry of logical ideas."

Albert Einstein

 

When you think about algebra you might have memories of sitting in a high school math class, searching for unknown values in linear and quadratic equations. Those long ago math courses may seem far removed from today's kindergarten classrooms but did you know that it is essential for educators to promote algebraic reasoning in early childhood education?

I devoured the most recent issue of Young Children (Volume 77, Number 3), especially the article Promoting Algebraic Reasoning in the Early Years by Lindsey Perry. 

In her work Perry advocates for algebra in early math programs that explore two main ideas: composition and decomposition of numbers and properties of operations. According to Perry (2022) algebraic reasoning "involves seeing and describing patterns and relationships between quantities that may be unknown...which builds upon students' understanding of patterns and relationships with known quantities and values" (pg. 17). Perry posits that if children can observe and describe number relationships they can begin to symbolically represent relationships between numbers. 

Composition and Decomposition of Numbers

When children compose they understand that a number can be put together using its parts (e.g., 5 plus 5 equals 10). Decomposition is the opposite where a number can be broken apart in different ways (e.g., 10 can be broken into 8 and 2 or 7 and 3). When children compose and decompose numbers they understand how to manipulate numbers in different ways, which helps them become flexible when solving calculations. For example mentally adding 68 + 22 can become easier when children realize that the ones values total 10 and then add this to the tens value (10 + 60 + 20). Adding 68 plus 22 is the same as adding 10 plus 60 plus 20 but the second strategy may be easier to mentally calculate for many people.

Properties of Operations

Properties of operations encourage children to work flexibly with numbers in order to recognize and manipulate their relationships. This helps them simply calculations in order to more efficiently and accurately solve them. For example the order of addends (numbers added together) does not matter in order to arrive at a sum. 

a + b = b + a 

6 + 4 = 4 + a therefore a must be 6.

The commutative property applies to addition and multiplication. The order of numbers can be switched and it does not change the answer of the operation.

2 + 7 = 9 and 7 + 2 = 9

4 x 5 = 20 and 5 x 4 = 20

The inversion property states that all integers have an inverse number that when added equal zero. 

3 + (-3) = 0

Although complicated young children can play with inversion when they become interested in, and work flexibly with equations.

3 + 2 - 2 = 3

So how can early childhood educators encourage children to participate in activities that promote early algebra? Here are some simple activities that can be used regularly to build children's confidence, ability and interest in number sense.

Equation Line

 

Provide children with a variety of subitizing cards and math symbols (addition sign, subtraction sign, equal sign). Encourage children to arrange the cards in different ways in order to create equivalencies.

 Make 5 (or 10)

Show children a total number of cubes (starting with 5 and then 10 is helpful). Hide the cubes behind your back and remove some. Show children the remaining number of cubes and encourage them to calculate how many are hiding.

Singing Songs with a Five/Ten Frame

When singing popular songs and finger plays with children (e.g., 5 Little Monkeys, 10 in the Bed) add a five or ten frame as a visual and manipulate the number of counters in the frame to match the number being sang. 

Counting Beads

   

A string of counting beads can easily be made using two colours of wooden beads secured on a pipe cleaner. Encourage children to use these when playing number games or engaging in number talks.

Domino Sort

Provide a mat for children (here a foam shamrock has been used but any shape will work). Write numbers on individual mats. Encourage children to sort dominoes and match their quantities to the mats in order to represent the many different dot arrangements possible for each number.

Roll a Ring

 

Seasonal rings are a fun tool to use in math games. Provide children with dice and encourage them to roll and add (or subtract) the numbers. Children can then wear the corresponding number of rings on their fingers. If two players play the game, they can each roll and wear rings and then compare hands to see who has more or less.

Name Equations

 

We enjoy representing children's names with boxes and encouraging them to think about the number, size and shape of the letters. These boxes are also fun to represent at equations so that children can play with their names and integrate a bit of math into literacy.

Calendar-based Number Talks

Morning message is a great time to encourage a daily number talk. We often represent the date in different ways (e.g., dice faces, dominoes, tallies, frames) and then encourage children to calculate the number by paying careful attention to the representations and operation signs used.

Which One Is Wrong?

 

Another favourite number talk is 'Which One is Wrong'. Different equations are displayed and children are challenged to explore each one using manipulatives (e.g., cubes, bead strings) to find the incorrect one. 

What other activities do you use to help children with early algebra? Let's connect on social media @McLennan1977!

Most Requested Presentations

Hello everyone! I have been getting many requests for presentations lately. I love working directly with educators, especially in live workshops. There is something magical about helping people (re)discover a love of math. Due to Covid restrictions and my busy schedule, presentations can sometimes be difficult to plan. So as requested, I'm sharing my three most popular presentations here for those who are interested. I'd love to chat about what you are doing in your districts, schools and classrooms! Send me a message or tweet/instagram me @McLennan1977 and we can connect!

 

Books to continue the learning:

 

Embracing Math: Cultivating a Mindset for Exploring & Learning

 

Kindercoding Unplugged: Screen-Free Activities for Beginners

 

Joyful Math: Invitations to Play and Explore in the Early Childhood Classroom

What does the Equal Sign Really Mean?

A few years ago I was presented with this question at a workshop and asked to consider how a group of students would respond. After some discussion our group thought that children might recognize the third as incorrect. I was surprised when the presenter shared that most children, regardless of grade or age, think that the third example is right and the rest are incorrect. Why is that?

Most children associate the equal sign (=) with the word 'answer', so they look for traditional algebraic representations. This is why despite the third statement being incorrect, it looks like something most children regularly see and use in math (addend plus addend equals sum) so they assume it is correct. Example one, two and four don't look typical for many children, so they aren't sure about them and assume they are incorrect.

Mathematical equations must have an equal sign indicating that two expressions have the same value.

10 is the same (or equal to) 10.
3 + 7 and 8 + 2 both equal 10.
10 is the same as 9 + 1.

Therefore providing children introductions to many different ways of writing equations, and tangible, hands-on experience with the idea of equality is very important. Despite their age young children are capable of using complex math in context, including the proper terminology for symbols like the equal sign. This can be done in many ways in kindergarten. Over the course of the last year we have been on a journey to help children understand the equal sign. This blog post outlines some of what we have done in order to achieve success.

Using Proper Terminology in Math Discussion and Discourse

In our classroom we start each morning with a number talk during our morning message. I was curious to see what my students knew about the equal sign, and how they would describe their thinking mathematically. I asked the following question:

The first few times we reviewed this prompt children replied by telling me that the equal sign meant 'answer'. They knew that four added to one was the same as five, but they could not articulate this clearly. I knew that much practice was needed to help children look at numbers in new ways, explore the idea of equal amounts and equality, and play with equations in different ways. After a few months exploring these throughout many whole and small group conversations and math invitations, I again asked children the same question. They responded by saying:

"Four plus one is the same as five. They are both five."
"Both sides are the same. They are equal. It's like if I gave you four and one cookies and I had five. We would both have the same. It would be fair."
"Each side is the same as the other."
"It's like this." (child holds up one hand and shows five fingers and then holds up the other hand with five fingers. "Each hand has five fingers. They are the same."

In our classroom we explored the following activities many times and in many different contexts to help build this algebraic understanding. Much of this work happened before we even looked at a written equation with numbers and symbols.

Building on Mirrors

We offered children mini wooden cubes on mirrors. As they built towers children realized that the reflections of their creations were doubling the total amount that they used. This was a rich opportunity to discuss the idea of equal (e.g., "The number of blocks used in your tower is equal to the blocks in its reflection.") and doubling (e.g., "We can double the number of blocks you used in your tower to calculate the total number of blocks used.").

       

Subitizing Match

Understanding that there are many different ways to represent the same, or equal amounts, is a foundational number sense strategy. Not only does this help children become more accurate and confident when identifying/calculating/comparing sets of objects, it provides an opportunity to show equal amounts (e.g., five tallies = five dots = the number 5).

We used subitizing cards and encouraged children to match them to random numbers written on a chart...

         

 ...and also played games where children were encouraged to find similarities between number representations. In this game children were asked to find at least three different dot arrangements to represent the same number.

      

Creating equal number strings is also helpful. Sometimes we will use the date as a number prompt and ask the children to explore representations in different ways. In the following picture we showed children three different ways of arranging 8 hearts and asked them to create equations based on what they saw. 

After exploring the arrangements the children shared the following:

4 = 4 

4 + 4 = 8

2 + 2 + 2 + 2 = 8
8 + 0 = 8
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 8

Sharing at Centers

Play dough is one of the most popular centers in our classroom and despite making a double batch each week, sharing equally among the large number of friends tends to be problematic. Usually the first child to visit the center grabs the entire big ball of dough, and reluctantly tears off small amounts to the children who slowly make their way there. "That's not fair! You have more than me!" is frequently heard at the table. As educators we felt this would be an excellent, real life situation to help children think about equality and equal amounts in a meaningful context. In addition to supporting and scaffolding this directly at the center, we encouraged children to think about fair, equal amounts by adding plates and laminated photos of each child to the center.    

             

We also added tools like cutters and asked children how they could divide the play dough into equal amounts. "How do you know it's equal?" was an interesting conversation starter and the children's ideas for equality were interesting (e.g., "We could see if it fits in the same container.", "We could measure it with a scale to see if it weighs the same.")  

Real Life Math Problems

Inspired by the children's problem solving at the play dough center, we used our morning message to ask children deeper, more complex questions regarding equality and fair sharing. Because they love seeing themselves on the morning message, it was effective to ask how four children might share six cookies equally. 

 

The children saw the cookies as two groups of three, and then split the three cookies in three ways. We used arrows to represent what it was they were saying. After some conversation and use of real props they also recognized that three halves were equal to one whole and one half of a cookie, helping us delve in early fraction work.

1/2 + 1/2 + 1/2 = 1 + 1/2 

  Building Equals

The children love to create with pattern blocks and are especially skilled at designing intricate tessellations. Wanting to introduce the concept of rotational symmetry we created eight equal sections using tape on the carpet. After building the children recognized that the blocks used in each section were equal to, or the same, as the other seven. This activity was also made available during outdoor play by placing tape directly on the pavement and offering children a basket of pattern blocks.  

  Equation Clothesline

Another equality invitation offered to children for exploration was the equation clothesline. This consisted of a string hung between two posts, clothespins, and subitizing cards. Children were able to represent balanced equations by finding different representations of numbers and pinning them on the clothesline. Equations could be simple (as shown in the photo where the number 2 = 2 dots on a five frame) or complex by adding addition or subtraction signs on each side of the equation (1 + 1 = 3 - 1). This provided multiple entry points into the activity with a way of differentiating it for children's needs and interests. 

Visual Representation of an Equation

In order to help children move towards thinking in algebraic representations we provided a visual 'scale' along with numbers written on sticky notes. Children were invited to try and create a balanced equation by first placing numbers on the scale and then adding the equal sign (also written on a sticky note) to the visual. We did not write the equal sign directly on the paper because we also had 'greater than' and 'less than' signs offered on stickies in order to differentiate the activity for children who were working at that level. 

Equation Sort

Another activity to help children practise identifying correct equations is the 'equation sort'. In this activity we presented children with a number of equations and asked them to sort them as 'correct' or 'incorrect'. Children had to justify their choices by using math language to describe their thinking. A great extension is to offer children blank strips and encourage them to create their own correct and incorrect equations and place them under the corresponding categories.

Greater Than, Less Than

Now that children are comfortable with the concept of equality, we are exploring other relationships that numbers have with one another. Inequality is something that children have expressed an interest it. We are starting our explorations by using the language 'less than' and 'greater than' in contexts to describe sets of objects, and creating the  <  and  >  symbols to show these relationships. Cubes and craft sticks are an easy invitation to try out.

It will be interesting to see where this journey continues. Please feel free to share your explorations and activities with equality and inequality in the comments below, or tweet us your ideas and feedback at @McLennan1977.