Workshop Blog

Teaching Number Operations

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.

Models that Emphasize 10

Models that Emphasize Place Value

Models that Emphasize Patterns

Models that Emphasize Partitioning Number

The Importance of Partitioning Numbers

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

  1. Predict what strategies they might use. Order these from least to most complex.
  2. Observe students doing mathematical tasks – using white boards allows us to see their thinking. We can then identify different strategies being used.
  3. 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.
  • You can read more about teaching doubles here.

Addition and Subtraction

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:

For the numbers 7, 3, 10:

                7 + 3 = 10

                10 – 3 = 7

                10 – 7 = 3

Fact families can be practiced using Number bonds or Missing Part cards.

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.

Multiplication and Division

Conceptual Structures for Multiplication

Repeated Addition

  • 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.

Mental Strategies Continuum

  • Same as (1 facts)
  • Doubles. (2 facts)
  • Doubles and 1 more (3 facts)
  • Double Doubles (4 facts)
  • Tens and fives (10, 5 facts)
  • Relating to tens (9 facts)
  • Remaining facts (6, 7, 8 facts)

Conceptual Structures for Division

Equal Grouping

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.

Some examples of rich interactions include:

Number Talks

  • Number talks promote classroom discussion. Combining number talks with visual or concrete models can help us see what students are thinking.

Number Strings

  • 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.
  • You can see the structure for building number strings here.

SPLAT

  • SPLAT encourages both additive thinking and subitizing. More complex SPLAT lessons are also great for encouraging algebraic thinking with unknowns.

Problem Stories

  • Building problem stories are powerful for children to understand contexts of mathematics in their every day life.
  • Using real objects or pictures encourages children to see math in their environments.

Invitations

Games and Puzzles

  • There are so many games and puzzles that can have children play with number operations.

Open Middle Problems

  • Open middle problems allow for flexible thinking and exploration. You can see a sample here.

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Building Coherent Teams

Michael Fullan (2016) states that coherence is “a shared depth of understanding about the purpose and nature of the work in the minds and actions individually and especially collectively” and is not about specific strategies, frameworks or alignment. So, how might we build coherent teams? How do we determine the ‘right’ actions and de-emphasize actions that are distractions? How might we focus on actions that enhance our collective as well as our autonomy? There are some processes and skills that are helpful.

Positive Communication

To build coherent teams, we need to know and practice communication skills, including paraphrasing and posing questions. A general conversation flow includes:

The Art of Paraphrase

“The purposeful use of paraphrase signals our full attention. It communicates that we understand the teacher’s thoughts, concerns, questions and ideas; or that we are trying to … well-crafted paraphrases align the speaker and responder, establishing understanding and communicating regard. Questions, no matter how well-intentioned, distance by degrees, the asker from the asked.”

(Adapted from Wellman and Lipton)

Things to keep in mind when paraphrasing:

  • Attend fully.
  • Listen with the intention to understand.
  • Capture the essence of the message but in a shorter format.
  • Reflect the essence of voice, tone and gesture.
  • Paraphrase before asking a question.
  • Use the pronoun “you” instead of “I.”

Intentions of Paraphrasing

Well-crafted paraphrases with appropriate pauses trigger more thoughtful responses than questions can alone. Three types of paraphrase, shown in the chart below, widen the range of possible responses. Each type supports relationship and thinking but the paraphrase that shifts the level of abstraction is more likely to create new levels of understanding. Conversations that utilize paraphrasing often move through a pattern of acknowledging to summarizing to abstracting, but there is no right pathway for the conversations.

Types of Questions

Horn and Metler-Armijo (Toolkit for Mentor Practice, 2010) identify three types of questions that are useful for professional conversations:

  • Clarifying questions – are asked to further understanding of the questioner. These types of questions convey that the questioner is actively interested.
  • Probing questions – are asked to have the speaker think more deeply about the concerns, challenges, or actions being taken. These types of questions dig into ideas to move from generalizations to specific ideas.
  • Mediational questions – are “intentionally designed to engage and transform the other person’s thinking and perspective” (Costa and Garmston, 2002). These types of questions are designed to open up and broaden thinking.

Mediational Questions

A special comment on “Why”…

Why questions are part of our everyday language. Why are you late? Why do you not have a pencil? Why are you doing that?

When we are having a conversation that may be emotional or highly charged, a question that begins with “Why” may create a sense of defensiveness. Consider a situation where someone has made a certain decision. Compare the reaction to “Why have you done this action?” vs “What is the impact your decision has had on…?”. A question that begins “What” or “How” is often more thought provoking and has less potential to create a defensive response.

Rectangle: Rounded Corners: Communication skills are key to building coherent teams, as they allow deep conversations to occur amongst team members, provoking thought through active listening.

Liberating Structures

Liberating structures, when used regularly, allow all team members the opportunity to work together to produce solutions, ideas and feel that everyone is contributing to an organization’s next steps. It is possible for every person to generate ideas and lead change.

Integrated~Autonomy

When considering how to best meet the needs of a system and the schools within a system, it is important that we view centralization/standardization and autonomy as both achievable and desirable rather than viewing them as opposite and competing interests. The Integrated~Autonomy liberating structure can help us to:

  • Develop innovative strategies to move forward.
  • Avoid wild swings in policies, programs or structures.
  • Evaluate decisions by asking “are we boosting both Coherence and Autonomy?”.
  • Increase quality of communication between school-based and Increase quality of communication between school-based and central office leaders.

Imagine actions that work towards BOTH increased standardization/centralization and increased Autonomy.

Some Examples

  • Attendance policies and consequences for non-attendance – what policies should be set centrally and which decisions should be made locally?
  • Planning documents required for hand-in/approval – format, timing and requirements for year, unit and lesson plans?
  • Parent conferences and reporting communication – what determined (format, process, content) centrally and what locally?

Structuring the Invitation:

  • Explore the question: Will our purpose be best served by increased local autonomy, including customization and site-based decision-making OR will our purpose be best served by increased coherence, including integration, standardization and centralized decision-making?
  • How might we be more coherent AND more autonomous at the same time?

Troika

This collaborative problem-solving strategy allows for colleagues to share possible solutions in a safe, non-judgmental environment.

In groups of three, learners sit in a triangle facing one another with no table between them. One person is the ‘client’, and the other two are the ‘consultants’.

  1. The client describes their dilemma, barrier or issue for about 2 minutes. The consultants might ask clarifying questions at this time.
  2. The client turns their chair so that their back is to the two consultants. The consultants discuss possible solutions to the client’s issue without any input, affirmation or cues from the client. The client might write down those suggestions that are most helpful. This might last for 2 – 4 minutes.
  3. The client turns around and summarizes what suggestions are most helpful that they might try.

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Assess-Respond-Instruct: Building Math Readiness

Our conversations in mathematics teaching are often centered around the gaps that we observe in student understanding, or how students are not ready to learn the grade level mathematics that we are trying to teach them. When we look at our teaching practices in other subjects, we know that it is important to Activate and Connect Prior Knowledge, and to provide responsive instruction if there are skills that our students are missing. The same holds true in mathematics, but how do we do this in a structured, systematic and efficient way in our classrooms? The Assess-Respond-Instruct framework, developed with and implemented by teachers from across Saskatchewan, does exactly that.

Foundationally, the Assess-Respond-Instruct framework provides opportunities for

  • teachers and students to know whether students have an understanding and fluency with prior knowledge, and
  • filling gaps in knowledge and build fluency, and
  • engage in grade level mathematics.

In order to embark on this way of teaching, some key questions that drive our planning are:

Differentiation vs Modification

A key idea within mathematics is the difference and similarity between differentiation and modification. Working with a school last week, we brainstormed the following key ideas:

Sometimes, a student needs a modified curriculum because they are unable to grasp mathematical concepts. This determination is made with much consultation with the education team, parents, and students. Communication is key between home and school to ensure that parents understand that their child is not working towards grade level outcomes. Rather, they are on a modified curriculum with modified assessment expectations. When a child is working towards a modified curriculum, it should still be differentiated. Students need to experience a variety of modes and strategies to help them achieve their unique learning goals.

The difficulty is when a student or class is inadvertently experiencing a modified curriculum without the pre-thinking and opportunities to engage in grade level mathematics. This might look like a child being identified as ‘not being able’ to add and subtract in grade 4, so they only work on addition and subtraction when their classmates are working towards multiplication and division. In this example, the child is not given an opportunity to engage in grade level outcomes, so the gap in their learning is even larger the following year.

So what is a possible solution? The Assess-Respond-Instruct Framework!

Designing Pre-Assessments

Pre-assessments should focus on mathematics knowledge that students need in order to be ready to engage in new, grade-level instruction.

Content to Pre-assess

We can identify the pre-skills necessary for a new unit of study by mapping curriculum and asking ourselves “What might students know before this grade to help them understand the content at our grade level?”

For example, in Saskatchewan curriculum Grade 6 Saskatchewan Patterns and Relations

  • P6.1 – Extend understanding of patterns and relations in tables of values and graphs.
  • P6.2 – Extend understanding of preservation of equality concretely, pictorially, physically, and symbolically.
  • P6.3 – Extend understanding of patterns and relationships by using expressions and equations involving variables.

In this Grade 6 Saskatchewan example, the blue concepts are grade-level, while the yellow concepts are mathematical ideas that appear in curriculum before Grade 6. By mapping curriculum, you can see that new, grade-level instruction is only one small step beyond what students have experienced in the past.

Analyzing the pre-skills from the example above, we can see that they can be clustered in the following way:

It is important to identify the extent to which students might need to understand a concept. In this Grade 6 example, students need to understand and be fluent in addition with single digit numbers, and subtraction of double digit minus single digit numbers. We would not usually expect students to work with larger numbers when we are solving algebraic equations at this grade level. Even though curriculum has students learn and practice addition and subtraction to 10,000 in Grade 5, we do not need students to use these large numbers in THIS unit of study, so we would not pre-assess or respond to those large numbers.

The content of a pre-assessment for this example unit of study would include:

  • Addition of single digit numbers and subtraction of numbers no larger than 100.
  • Multiplication of single digit numbers and division of numbers no larger than 100.
  • Representing relations, including tables of values and graphs.
  • Solving one step equations, including balance scale representations and missing value equations.

Forms of Pre-assessments

You might have assessment data that you have gathered through school-system pre-assessments, through tools like Pearson’s Numeracy Nets, or you can develop your own simple Pre-assessments.

How might we respond to gaps in pre-skills?

We need to consider both the content and structure that we are using to respond to gaps in understanding. Many teaching innovations focus on one or the other. I would suggest that we need to consider both the content of intervention as well as the process, or structure, that we use to have students interact with that content.

What is Responsive Content? Differentiating Mathematics Intervention

Too often, our mathematics intervention in upper grades involves symbolic practice of a topic that a student is unsure of. Rather than only focussing on symbolic practice, we need to differentiate our intervention – additional practice worksheets are not enough if a student does not understand.

What does differentiation look like in mathematics? If we consider NCTM’s ways of representing algebraic ideas, and the Theory of Multiple Intelligences, a simple way to look at differentiation for every math concept might be:

Whether we are looking at responsive instruction or new instruction, it is important that students are given opportunities to learn new concepts:

  • Concretely and visually
  • Video – this can help auditory learners watch and listen to math concepts
  • Written explanations – a simple and concise description of that mathematical idea
  • Game – a way to interact with peers and have mathematical conversations
  • Practice – to build fluency with foundational math ideas

A planning organizer is helpful in identifying the components that you will have ready for students who need intervention in each topic. The concepts that we focus on for responsive instruction are those identified in our pre-skill analysis of our next unit of instruction. The modes of responsive instruction need to be thought out for each concept, or skill. This provides a robust framework for intervention.

A Classroom Structure – Responsive Stations

There are many ways to structure your classroom to ensure that your students are receiving the instruction that they need. These might include classroom routines that focus on readiness skills, or rotational stations like Daily 3 Math. One innovative structure is Responsive Stations.

Implementation

Once you have the pre-assessment data, students go to those stations that their data on their pre-assessment indicates that they need.

Some helpful organizational hints include:

  • Colour coding your boards helps students know where they are heading.
  • Use a tracking sheet to monitor which pre-skills each student needs to address.
  • Use stickers as rewards to track what stations have been done.
  • Use a short post-assessment to determine that a student understands the content.
  • Use bins of materials at each station to help keep organized.
  • Include an enrichment station for those students who have pre-skills in place. This enrichment station can include games, additional math topics, and ideas such as creating new videos or games based on math concepts.

Once you have provided opportunities for students to be ready for your grade-level instruction, you can then teach new concepts using rich instructional practices that we know help students understand. Through the year, your class will revisit the same pre-skills over time, as many topics repeat as pre-skills throughout curriculum.

If you are interested in learning more about the Assess-Respond-Instruct Framework for building readiness, or would like to bring professional development to your staff in this area, please contact Terry@johansonconsulting.ca

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Teaching Patterns in Early Years

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:

  1. Patterns exist and occur regularly in the natural and man-made world.
  2. Patterns can be recognized, extended and generalized using words and symbols.
  3. 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.

Pattern Match can happen in many forms.

  • 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.

Rectangle: Rounded Corners: What’s Next and Why? Show students five or six numbers from a number pattern. The task for students is to extend the pattern for several more numbers and to explain the rule for generating the pattern. The difficulty of the task depends on the number pattern and the familiarity of students with searching for patterns. Here is a short list of patterns, some easy enough for Kindergarten. 1, 2, 1, 2, 1, 2, … a simple alternating number scheme 1, 2, 2, 3, 3, 3, … each digit repeats according to its value 5, 1, 5, 2, 5, 3, … the courting sequence is interspersed with 5s 2, 4, 6, 8, 10, … even numbers, skip counting by 2s 1, 2, 4, 5, 7, 8, 10, … two counts, then skip one 2, 5, 11, 23, …. double the previous number and add 1 1, 2, 4, 7, 11, 16, …. the number being added increases by 1 2, 12, 22,32, … add 10 Most of the preceding examples also have variations you can try. Make your own! (Van de Walle & Lovin, 2006)

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:
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  • 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.

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Teaching Place Value

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 This – Building 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

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:

Reveal How Many

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

Adapted from https://kidsactivitiesblog.com/

cool math game

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.

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