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:

1. Representing numbers – how might children use these tools to represent 7? 3? How do they know?
2. 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?
3. 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.

Now try to think through these activities using 5 and 10 Frames and linking cubes to show numbers. How is this the same and different than using a rekenrek? There are a number of games and activities involving dot cards and 10 frames that can emphasize 10.

You can find out more on the Building Math Minds Rekenrek activities site.

### Subitizing

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.

Try ThisBuilding Math Minds has a great site for subitizing games. You can find some ideas in this Evergreen Games Overview.

### 100s Chart

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:

1. Place value blocks or other counters, such as coffee stir sticks.
2. Arrow Cards
3. Find the digit on the 100’s chart

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

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.

### Math Swat

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!

### Interactive Manipulatives  – ICT Math

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.

Math fluency is the ability to perform mathematical operations quickly and accurately. Math automaticity with basic facts is part of fluency. John Munroe (2011)  indicates that we as learners have a finite amount of working memory. It is important that student working memory is available to learn new math concepts, solve complex problems and think creatively in mathematics, rather than being used to recall basic math facts.

So, how do we promote automaticity of basic math facts without endless worksheets, mad minutes and text-book assignments? Math games, puzzles and routines related to grade level concepts allow for flexibility in thinking, practice and student engagement and fun. The following documents can help us connect practice with curricula:

# Math Routines

There has been a significant amount of research in the area of mathematics routines to enhance learning, building automaticity and fluency. There are a number of key resources that are helpful to teachers from grades 1 to 10.

## Productive Mathematics Discussion

Margaret Smith (2011) has created a structure for planning for and implementing classroom discussion in a mathematics classroom. Discussion and sharing mathematical thinking is the key to most mathematics routines.

This sequence of thinking can be applied to a teaching strategy, Learner Generated Examples (Crawley, 2010), to create a powerful way of sharing student thinking in mathematics.

### Learner Generated Examples:

Brian Crawley, a teacher from Saskatchewan, did extensive research on learner-generated examples. Overlaying the Five Practices, you create a cycle that can be applied to many number routines:

Move through this cycle three times. This allows students to push beyond the knowledge that they find easy to access and move to more and more complex ideas.

## Examples of Math Routines

There are several powerful math routines gathered from many Math Routine Resources that can be adapted to different concepts over grade levels. The following is a short list:

The key to math routines influencing math fluency in your classroom is to choose appropriate concepts and numbers related to your curriculum. For example, in the routine “Today’s Number”, if you are in early primary, you might choose a number between 1 and 10. If you are teaching ideas around skip counting, perhaps choosing the number 6. If you are working on perfect squares, perhaps choosing a number like 36.

# Math Puzzles

Math puzzles allow learners to explore mathematical ideas and practice thinking flexibly. Well-designed puzzles are engaging and logical, with most of the time focussing on solving the puzzle mathematically. As with routines, puzzles can be adapted to the range of numbers and concepts appropriate for a given grade level.

### Kakooma

• Examples and explanation are found on Greg Tang’s Kakooma Resources website
• Kakooma helps students build automaticity with addition and multiplication.
• There are a variety of number puzzles, with a diverse range of numbers and volume of questions to solve.

### Two-Ways

• Also created by Dr. Grayson Wheatley, Math Two-Ways are found in his Developing Mathematical Fluency resource.
• Examples:
• You can also use a blank Two-Ways sheet to create your own or have students create them for others to practice.

# Math Games

When choosing and setting up games in your classroom, along with considering the math concepts you are emphasizing, it is also important to consider student grouping and classroom norms.

By creating homogeneous groupings, it is possible that some groups will use tools to help them calculate or identify numbers, such as:

In heterogeneous math game groupings, it potentially frustrating for both the stronger student and student who needs more time to compute or recall math facts.

There are many excellent math games sites that include Math online games. As well, there are many games that require few materials and are engaging for students.

Of special note are Alphashapes and Alphashapes Games that can be played to understand mathematics language.

The key to finding and using math games in your classroom is to know the math that you would like to build fluency in, and then search for those concepts. There are literally thousands of great sites that you can access both online and paper copy games.