What were many of us MOST looking forward to for Christmas? Food, Fun and Family… Fun in our family over the years has been board games, cards and dice games. And then new health orders came to Saskatchewan and that hope seemed to be dashed… BUT… there was a huge ‘aha’ when I was asked to design and facilitate a “Let’s Play: Math Games Online workshop” this past week. I put my research hat on and found some online platforms for play. There are SO many cool fun platforms that are FREE to everyone. There will be Cribbage for Christmas after all!!

I have always believed that my confidence and competence in math came from the games I played as a child. Cribbage with my Grandpa and Great Grandpa, Monopoly with my friend Tommy, Go Fish with my sister, 31 with my Mom and Dad… So, how might we play these games online with Jori and Michael in Martensville, Erin in Comox, and Adam in Saskatoon? And better yet, how do we play them with the kids’ cousins Paige in Halifax, Amanda and Christian in Winnipeg, and then there is Auntie Sandra in Melfort and Grandma Carol down in Mesa this Christmas? My family is FAR too large to list everyone here, but you get the idea. Here are some games and platforms that might bring some cheer to your homes and families over the holidays.

Each of the following does NOT require an account or sign in. For each, you:

Start a game.

Copy the link and send it out by text or email to your game friends.

Set up a phone call or video call so that you can talk to everyone in the game. (Did you know that your iPhone can create a group call up to 5 people??) Or use Zoom, Google Meet, Facebook Rooms, What’s App… SO many possibilities. If two of you are in the same house on your own devices, consider either using your phone on speaker, or each of you use headphones/ear buds so you don’t get sound feedback.

Each of these is designed for each player to have their own device – they seem to work on computers or tablet/iPad or smart phone.

Addition, subtraction, multiplication and division are foundational
skills that are applied to many mathematical concepts. Often, when we are hoping
for student automaticity and fluency in numbers, number operations are what we
are talking about.

Mathematical Models

Models are the way we are representing numbers so that we
can do number operations. There are a number of different models that are
helpful to students understanding number operations.

Regardless of what number operation we are talking about, it
is important that children are able to break numbers into parts.

Friendly Numbers – children are often able to understand number
operations with ‘friendly’ numbers like 2, 5, and 10. Breaking a 7 into a and a 2 allows us to use number facts that are
more familiar.

Place Value Partitioning – when we are working with multi-digit
numbers, it is helpful for us to break numbers up into the values of their
digits – for example, 327 is 300 + 20 + 7.

Number Operation Strategies

There are many different strategies that children use to
perform number operations. A misconception is that all children need to know
and use all strategies. It is important for us to expose children to different
strategies through classroom discussion and routines such as number talks and
number strings. When combined with Margaret Smith’s ideas around Orchestrating
Classroom Discussion, we can set a task for students and

Predict what strategies they might use. Order
these from least to most complex.

Observe students doing mathematical tasks –
using white boards allows us to see their thinking. We can then identify
different strategies being used.

Have students share their thinking in an order
from least to most complex. This should not include every child sharing for
every task. A small handful of children sharing in a logical order can help
students understand the next more complex solution. In this way, children are
being exposed to other strategies, will be able to understand those that are
close to their own, and increase the sophistication of their thinking.

Strategies

Connection to Addition

Connection to Multiplication

Counting: This is a common strategy when one of the numbers is small.

Addition by counting or counting on from one number. Ex: 25 + 7 = 25, 26, 27, 28, 29, 30, 31, 32.

Skip counting by one of the numbers being multiplied. 9 x 5 = 9, 18, 27, 36, 45

Decomposing Numbers: breaking numbers apart.

Adding friendly numbers. Ex: when you need to add 12, breaking it into +10 and then +2 more.

Making 10. Ex: when adding 5 + 7, recognizing that 5 + 5 = 10, and so it is 10 + 2 more = 12.

Breaking one or both numbers into place value. Ex: 23 + 47 is 20 + 40; 3 + 7

Multiplying friendly numbers. Ex: when you need to multiply by 6, break it into x 5 and 1 more.

Partial Products: Breaking one or both numbers into place value. Ex: 23 x 47 is (20 + 3) x (40 + 7)

Compensation: this is very common when a number is close to 10.

Rounding one of the numbers to a friendly number, then compensating the answer at the end for the difference. Ex: 36 + 9 is close to 36 + 10, subtract 1. Ex: 36 + 11 is close to 36 + 10, add 1.

Rounding one of the numbers to a friendly number, then compensating the answer at the end for the difference. Ex: 99 x 5 is close to 100 x 5, subtract 5 Ex: 101 x 5 is close to 100 x , add 5

Double/Half

Recognizing that 4 + 4 is double 4, or 8. Recognizing that 4 + 3 is almost double 4, subtract 1.

Recognizing that 5 x a number is the same as ½ of 10 x a number. Ex: 9 x 5 is half of 9 x 10 = 45

Standard Algorithm

Traditional algorithm, symbolic regrouping.

Traditional algorithm, symbolic regrouping.

A Bridge between Addition and Multiplication: Doubles

Doubles are one way to think about adding a number to itself, as well as the start to multiplicative thinking.

Doubles are an important bridge between adding and multiplication.

Addition is the bringing together of two or more numbers, or
quantities to make a new total.

Sometimes, when we add numbers, the total in a given place
value is more than 10. This means that we need to regroup, or carry, a digit to
the next place. There is a great explanation of regrouping for addition and
subtraction on Study.com.

Subtraction is the opposite operation to addition. For each
set of three numbers, there are two subtraction and one addition number facts.
These are called fact families. For example:

As we move from single digit to multi-digit addition and
subtraction, it is important that we maintain place value, and continue to move
through the concrete to abstract continuum.

A helpful progression for teaching addition and subtraction can be found on the Math Smarts site.

This is the first structure that we introduce
children to.

It builds on the understanding of addition but
in the context of equal sized groups.

Rectangular Array/Area Model

This is often the second representation of multiplication introduced It is useful to show the commutative property that 3 x 4 = 4 x 3 = 12

Number Line

A number line can represent skip counting
visually.

Scaling

Scaling is the most abstract structure, as it
cannot be understood through counting.

Scaling is frequently used in everyday life when
comparing quantities or measuring.

Single Digit Multiplication Facts

Multiplication facts should be introduced and mastered by
relating to existing knowledge. If students are stuck in a ‘counting’ stage –
either by ones or skip-counting to know their single-digit multiplication
facts, it is important that they understand strategies beyond counting before
they practice. Counting is a dangerous
stage for students, as they can get stuck in this inefficient and often
inaccurate stage. Students should not
move to multi-digit multiplication before they understand multiplication
strategies for single-digit multiplication.

It is important that students understand the
commutative property 2 x 4 = 8 and 4 x 2 = 8.

2 x 4 should be related to the addition fact 4 +
4 = 8, or double 4.

Using a multiplication table as a visual
structure is helpful to see patterns in multiplication facts.

In an equal grouping (quotition) question, the total number
are known, and the size of each group is known.

The
unknown is how many groups there are.

Equal Sharing

In an equal sharing (partition) question, the total number
are known, and the number of groups is known.

The
unknown is how many are in each group.

Number Line

Ratio

This is a comparison of the scale of two quantities and is
often referred to as scale factor. This is a difficult concept as you can’t
subtract to find the ratio.

Division Facts

Relate division
facts back to multiplication facts families:

Ex) 6 x 8 = 48

8 x 6 = 48

48 ÷ 6 = 8

48 ÷ 8 = 6

Once students have understanding and fluency with single digit multiplication and division fact families they are ready to move on to multi-digit fact families.

So What do Students DO with Number Operations?

Simple computation is not enough for children to experience.
They need to have opportunities to explore and wonder about numbers and how
they work together. Regardless of the routine or task, children should be
encouraged to use different concrete and pictorial models to show their
thinking.

Number strings can help children see the pattern
in number operations. They are helpful for children to see the pattern in
number operations, which is the foundation for algebraic thinking.

Patterns are everywhere. Exploring and identifying patterns can help children understand our number system, operations, spatial understanding and the foundations of algebra. Mathematics is the study of patterns and exploring them through play can begin mathematical and algebraic thinking in early years. Click here for a downloadable version of this post.

There are several big ideas related to patterns:

Patterns exist and occur regularly in the
natural and man-made world.

Patterns can be recognized, extended and
generalized using words and symbols.

The same pattern can be found in many different
forms – physical objects, sounds, movements and symbols.

The progression of patterns through Saskatchewan Curricula:

When viewing patterns, it is useful to know the following terms:

Element – an action, object, sound or symbol that
is part of a sequence.

Core – the shortest string of elements that
repeats.

Pattern – a sequence of elements that has a
repeating core.

Children will develop their ability to recognize and
manipulate patterns differently. Some children will move through the following progression:

Exploring patterns also gives children practice and exposure
to other mathematical ideas, including:

Counting and cardinality – counting the number of
items in the unit of a repeating pattern, or how many items are added in an increasing
pattern.

Adding and subtracting – generalizing about an increasing
or decreasing pattern – how many more or less.

Position and spatial properties – which element
comes next, which element is between two others, reversing order of elements.

How might you teach patterns?

As with many mathematical ideas in early years, it is
important to connect ideas. Learning is not linear! It is important that
children use physical materials from their environment to build and explore
patterns rather than relying on drawing and colouring patterns. Buttons, toys,
linking cubes and natural materials can all be used to create patterns.

The Measured
Mom has a list of fun ways to engage young children in exploring patterns. It
is fun to take children outside. Megan Zeni describes how you might have
children explore Patterns
Outside and in Nature.

Repeating Patterns

Repeating patterns can be introduced using concrete objects,
sounds, body movements or symbols. Exploring with a variety of materials can
help children identify what is
creating a pattern.

Pattern Strips can be made using any shape or object. Students can work independently or in groups to copy the pattern on a strip using real objects. These patterns can then be extended. Watch whether they are copying each element separately or if they have identified the core of the pattern and are able to place all of the elements of the core at the same time. This might look like:

If the pattern is red/blue/red/blue – children will
place the red and blue at the same time.

A significant step in understanding patterns is when
children are able to identify that the same pattern exists even when the
materials are different. Using some type of symbol, children are able to code a
pattern and compare it to other patterns. If they choose to code the pattern
using the alphabet, they might describe it as A-B-A-B or A-A-B-A-A-B. An
extension with pattern strips is to create the same pattern with different
materials.

You can give each group a set of different
pattern strips, and they find which strips are showing the same pattern.

Children can work in groups, one child is the
pattern caller. They choose 3-4 pattern strips and lay them face up on their
table. They then ‘secretly’ choose one of the strips and calls out the pattern
code. Their group members try to identify which strip is being read.

Growing Patterns

In Saskatchewan, children begin to explore increasing patterns
in grade 2, and decreasing patterns starting in grade 3. The beginning of understanding growing
patterns is for children to experience building them with concrete objects.

It is important for children to record their observations. A
table can help students record the number for each step in the pattern. Using a
table, students can predict how many items are needed to create a certain step
in the pattern.

Patterns with Numbers

Number patterns are woven
throughout our number system, how we perform operations and the ways we represent
numbers. John Van de Walle and LouAnn H. Lovin (Teaching Student-Centered Mathematics K-3, 2006) have created a
number pattern activity that has students identify how a number string continues
by identifying the pattern present.

Skip Counting

Skip
counting is an excellent source of patterns. We often limit skip counting to
small numbers like 2, 3 or 5. We also often start skip counting at 0. Children
can explore the patterns that are created when we skip count by larger numbers,
changing the start number. It is a great idea to use a calculator, so students don’t
get bogged down in computation!

If we start at 7 and skip count by 5’s, what
pattern do we see?

7, 12, 17, 22, 27, 32…

If we start at 7 and skip count by 55’s, what
pattern do we see?

7, 62, 117, 172, 227, 282, 337, …

What do we notice about these two patterns?

Patterns in the Hundreds Chart

Start and Jump on the 100s Chart

Using a
hundreds chart, have students colour in the pattern created by one of the Start
and Jump Numbers sequences given. If different students represent different
patterns, what do they notice?

How do patterns change when

the start number changes?

The jump number changes?

Which skip count numbers create columns?

Let’s Build the 100’s Chart

Using a pocket 100s chart or interactive 100’s
chart. Place the following number cards in the pockets:

4, 10, 17, 32, 48

Gather students so they can see the 100’s chart
easily.

Hold up a number related to one of those in the
chart, such as 18. Ask “who would like to place this number?”. Explain how you
know where to put it.

Choose numbers to place based on the number
concepts you are working on:

If you are working on adding 10, choose numbers
that emphasize that concept.

If you are working on skip counting, choose
numbers that emphasize what you are counting by.

Game: Arrow Clues

Clues can be created on cards or written large
enough for all players to play the same clues.

Arrow clues can look like:

Differentiation:

Students can play with or without a 100s chart
to refer to.

Have students
describe the impact of each of the types of arrows on the VALUE of the number.

Missing Number Puzzles

Using the patterns in the 100’s chart, children can figure out the missing numbers when only a part of the 100s chart is provided.

Place value and number sense are foundational concepts on which others build over the years in mathematics. Some of the big ideas within place value include:

Concept Progression Over Time

In Saskatchewan, our curriculum identifies the following ideas:

In Kindergarten, children learn that counting tells us how many. The whole numbers are in a particular order and there are patterns in the way we say them that help us remember their order.

In Grade 1, children understand place value in individual numbers – they look at 17 as a quantity. We can compare and order numbers.

In Grade 2, children understand that the value of the digit depends on its location or place.

In Grade 3, children consolidate their understanding that the place determines a number’s value.

Ideas for Teaching Place Value

Rekenreks, 5 and 10 Frames

Number sense is a foundation of place value. Relating
numbers to ‘friendly’ 5 and 10 are key ideas that can move children past
counting.

Try This – Use a
rekenrek to show the following:

Representing numbers – how might children use
these tools to represent 7? 3? How do they know?

Quick flash – flash a number of beads on a
rekenrek and have children tell you what the number is. How do they know this
is the number? Are they counting? Or comparing to the ‘friendly’ 5 or 10?

Model numbers in a number string – showing 4,
then 5, then 6. Some children will see the pattern of 1 less than 5, 5, and 1
more than 5. You can then repeat with 3, 5, and 7.

Subitizing is a foundational skill and occurs when children
know that a number of objects is present without counting. Subitizing can occur
with random displays of objects or dots, or patterned dots like you would see
on a dice, dominoes or ten frames.

The hundreds chart is an important tool for children to see
patterns in our number system. There are a number of games and activities that
you can try to emphasize different math ideas.

Try This – There are a number of blogs and vlogs that teachers have created to highlight the 100’s chart. Buggy and Buddy does a good job curating ideas from a number of sources. You can also have children try to find the missing numbers on a 100’s chart to emphasize the patterns in our number system.

Base 10 Blocks

Based 10 blocks are a foundational manipulative to help
children understand our number system.

Try This – Go to Hand2Mind website and scroll down to view the lessons provided. These are organized by grade band so that you can find what might fit your students best. Use the base 10 blocks provided to try to work through some of these lessons

Place Value Misconceptions

Misconceptions can be created by a mis-applied pattern, or
incomplete understanding of number concepts. The following are some place value
misconceptions that occur in Early Years, and some possible instructional strategies
to address them.

Misconception: A number is a number, and does not represent a bundle of 10, 100, 1000 etc. objects regardless of its position in a number.

Example: 1 means one, so when it
is placed in a number 17, it still represents one rather than 10.

What to do about it? Use the concrete to abstract continuum to
represent 17:

Place value blocks or other counters, such as coffee stir sticks.

Misconception: Students represent numbers after 100 as they sound.

Example: Students think that the number after 100 is 1001, then 1002, 1003, etc.

What to do about it: Use a chart that goes beyond 100, have children fill in the next numbers after 100.

Misconception: The student orders numbers based on the value of the digits, instead of place value.

Example: 67>103 because 6 and
7 are bigger than 1 and 2.

What to do about it:Have students represent numbers using base 10 blocks and then write out expressions using > and < when comparing.

What to do about it:Have students show numbers on a number line to see which numbers are further from zero to the right.

Misconception: The student struggles with the teen numbers, as they are different from the pattern in other decades.

Example: Students may say “eleventeen”
or may not understand that 16 is ten and six. They may also think that sixteen
is 61 because we say the number six first.

What to do about it: Christina, The Recovering Traditionalist, has curated a number of games and ideas for addressing how to teach the teens.

Having Fun with Math

Mathematics should be playful, and there are a number of
games that can build fluency in mathematics.

Combo-10

This game allows students to see how numbers fit together to
make 10 using domino-like game pieces. It is for groups of 2 – 4 players.

Try This – Play with at least two people or groups. Each group needs 1 set of dominoes. Lay them face down. Each person/group draws 7. The rest are the draw pile.

The player with the highest double (or most dots if there are no doubles drawn) plays first. A piece can be played if the number of dots on one side of the domino adds to 10 with a domino on the table. Doubles can be laid sideways, allowing more arms to grow.

A wild card is a domino whose dots add to 10. If you play a wild card, you can play twice.

Snap

Snap is a game
played with linking cubes. Each pair receives 10 linking cubes. Players may
want to start with the cubes in a stack, alternating colours:

Try This – One player has a stack of 10 cubes behind their back. ‘Snap
off’ part of the stack and show the part that is remaining to your partner.

The partner tries to guess how many were snapped off and hidden from view. The unknown part is revealed.

Variations:

Using more or fewer blocks in the stack.

Breaking the 10 cubes apart and hiding some of them underneath an opaque glass or container.

Race to 100

The goal of this game
is to get to 100 first without going over.

Try This – Play
the Game

Each player starts at 1. The first player uses a spinner or dice to generate a number. They can move up the 100s chart by their number of tens or ones until one player gets to 100 without going first.

Variations:

Each player gets 6 turns. The closest to 100 without going over wins.

Continue playing until a player lands exactly on 100. If the roll takes them over 100, they lose that turn.

Flyswatter math combines the fun of moving and slapping with
the chance to learn number recognition and solving math problems.

Creating the game board: The game board can be as small or
as large as you would like and include the number range and type of numbers
that you are working with in your classroom.

Try This – Play a
Game with two lines of players. Each line has their own swatter.

Counting: swat the numbers in order – in either direction.

Number recognition: say the number and have learners swat the correct symbol.

Counting and 1:1 correspondence: give a number of counters, blocks, etc – they count and then swat the number.

Addition or subtraction facts: give the fact, swat the correct sum.

Addition facts: give the sum and one addend, swat the missing part.

Skip counting: swat the numbers as they count by 2s, 5s, etc.

Using Technology in Mathematics

Technology can be used to enhance mathematics in a number of
different ways:

Place Value Online Games

As you know, not all online games are created equally!
Sometimes, they are just online worksheet with little engagement. Sheppard
Software is a site that encourage practice through play, including flexible
thinking about place value.

Try This – Try playing one of the place value games, Underline Digit Value, on Sheppard Software.

Interactive Whiteboards

These whiteboards all allow you and students to share
thinking. They can include audio, pictures, and mark ups. Some apps are free,
while others require a subscription.

Try This – Log into one of the interactive whiteboards below that you have not used before. Use the username and password provided on the sticky note!

These interactive manipulatives can be used to explore math
ideas. These tools are web-based and do not require a log in or download.

Try This – Go to the Arrow Cards tool in the “Teaching Tools” at ICT Math. You can show the value of numbers using arrow cards along with either rek-n-reks or base 10 blocks simultaneously. Show the value for 3299. What happens when you add one more ones digit?

QR Code Scavenger Hunt

This teaching idea comes from Kristin Kennedy and is available free on Teachers Pay Teachers. It would be relatively easy to create your own based on this idea.