As math teachers, we spend much of our time planning for students who need extra time or support to understand mathematics at our grade level. We also worry about how we might create appropriate challenge for students who have a strong foundation in math and are ready to move ahead before their classmates are ready. The last thing we want is for our most able mathematics students to get bored and disengage. Enrichment seems like a good option, but how do we do that?
Why? Goals
Enrichment activies are meant to allow students to
engage in rich tasks and activities at grade level
stay with their peers
replace foundational math tasks that other students require to build understanding
reinforce mathematical thinking skills so that students are even more able to engage in grade-level math
When? Timing
There are at least two opportunities for enrichment within your instructional sequence.
When we pause to help students review and build readiness for grade-level instruction. This might be in the form of a concept review, reteach, or responsive station.
Enrichment tasks can be an option for students who do not need to review or practice foundational skills.
Enrichment tasks that are loosely based on the same concepts as are in the review will help students be even more ready for grade-level instruction.
When we are teaching grade-level concepts and the majority of students require more time to complete tasks.
Enrichment tasks can be an option for students who are done work earlier than their peers.
Enrichment tasks that are loosely based on the grade-level concept, but allow for creative thinking and application can allow students to see mathematics in a creative and contextual way.
What? Characteristics
Enrichment tasks take on all different forms. Some common characteristics might be:
Engaging
Hands-on
Applied to the Real World
Creative
Games
Puzzles
Math Outside of Curriculum
Connecting Math Concepts
Social Justice topics
You can access hundreds of curated resources in this Enrichment Task Google Folder. Feel free to share, download, and use these resources.
Who? Target Students
All students might be able to engage in Enrichment tasks at some point in the year. Different students have different strengths. Enrichment is not exclusive to gifted students.
How? Classroom Structures
A station or enrichment corner are relatively simple to set up. One thing to consider is what tasks to have available at what time. A suggestion is that the tasks in the enrichment station should be loosely related to the math concept being experienced by the rest of the class. This ensures that the students doing enrichment tasks are going to be even more able to complete grade-level math tasks. When enrichment tasks are completely unrelated to the math students will experience next, we are inadvertently creating a time gap in student learning and may actually contribute to lower achievement.
Our school year seems to race by faster and faster every year. A worry that we have is that our students might not be ready for their next concept, next grade, or next step in their education journey. Some of our hopes while year planning would be to:
Prioritize concepts that are foundational to the next school year.
Estimate time so that higher priority outcomes have more time.
Order concepts logically so that math ideas build within a grade.
Cluster outcomes that help kids understand the connections between math ideas.
A prioritized and sequenced math year plan is not a pacing guide. Rather, it is a roadmap that helps you, as teacher, know where you are going next, and provides an estimate for how long to spend on a concept. A prioritized and sequenced year plan should be revisited several times through the year to see where you are at, what might need to change, and what your students need next.
Creating a Prioritized and Sequenced Math Year Plan
Prioritize Outcomes
To prioritize outcomes in our grade level, it is important to know what concepts lead directly into next year’s math curriculum. Some tools that help you do this are Curricular Through Lines. It is most helpful to use the document that shows YOUR grade as well as the grade level after you:
Highlight outcomes YELLOW if they lead directly into next year. For example, multiples and factors in Grade 6 (N6.2) leads directly into adding and subtracting fractions (N7.5) and divisbility rules (N7.1).
Do not highlight an outcome if it does not lead directly into next year. For example, numbers greater than 1000000 in Grade 6 (N6.1) does not lead into any outcome in Grade 7.
Go back to your yellow outcomes. In a given year, you might want to have 6-7 outcomes that are highest priority (GREEN). If you have more than 6-7 outcomes highlighted yellow, which of those would be most important to emphasize? You might want to have a discussion with the next grade teacher to help you determine this.
At the end of this process, you might have:
6-7 HIGHEST priority outcomes.
some MEDIUM priority outcomes.
some LOWEST priority outcomes.
Cluster Outcomes
Some of our curricula have several outcomes that would be much more effective if they are taught together. For example, in Grade 7, there are several patterns and relations outcomes and a shape and space outcome that are easier to teach if you put them together:
P7.1 – Relationships between tables of values, graphs, and linear relations
P7.2 – Understanding equations and expressions
P7.3 – Solving one and two step equations with whole numbers
P7.4 – Solving one and two step equations with integers
SS7.4 – Ordered pairs and the Cartesian Plane
When you look at these holistically, it might make sense to cluster these outcomes into:
P7.3 and P7.4 – Solving one and two step equations
P7.1, P7.2, SS7.4 – Representing linear equations as a graph, table of values, equations, words, and pictures/manipulatives.
Once you cluster your outcomes in this way, they act as a single unit within your year plan.
Sequence Outcome Clusters
There are several things to consider when you are creating your outcome sequence:
How might you start the school year? What are they ready for after summer break?
If you follow the order of curriculum, we would begin with place value. While this might be logical at early grades, it can sometimes be daunting for students coming back after summer.
Consider starting with a topic that has lots of hands-on opportunities that can help students understand what numbers and shapes are. This will help get them ready for place value later on.
Example: Consider starting with a graphing and data outcome so that students can use numbers as they create axes for graphs and explore how big those numbers are within a real-life context.
What outcomes are pre-skills for other outcomes in your grade?
There are several examples where one outcome logically comes before another one.
Example: In grade 2, it makes sense that numbers to 100 would be taught before adding and subtracting numbers to 100.
Sometimes we might think something is a preskill that is not. For example, in grade 3 we might think that we need to do addition and subtraction before we do multiplication and division. Because multiplication in grade 3 is limited to 5 x 5, we don’t need to be able to add large numbers before we multiply. In fact, the pre-skills for multiplication are:
knowing numbers to 25.
skip counting by 2, 5.
decomposing numbers.
How might you end the school year? What is best suited for May and June instruction?
Spring is a great time to go outside and extend your classroom. When you consider what math fits into outdoor experiences, you might want to plan for those concepts in the spring.
Example: Grade 3 data and graphing could be based on finding things in the natural environment, creating concrete graphs, pictographs, tallies, and graphs representing what you find.
Create Time Guidelines
The time you spend on each topic is determined mostly by how many highest priority outcomes or outcome clusters you have identified and how many days of mathematics you have in a school year. If you have math every day, you can count on having approximately 150 days of math, as there are always concerts, field trips, and other things that impact instructional time.
Estimate the amount of time per outcome (or outcome cluster):
Highest Priority Outcome – 22 to 25 days
Lowest Priority Outcome – 2 to 5 days
Medium Priority Outcome – 5 to 10 days
Estimate the total number of days you might need for all of your prioritized outcomes.
Is this approximately equal to your instructional time?
Do you need to shorten timelines? Lengthen them?
Plot your units out onto a school year calendar.
Do outcomes begin and end at logical times? What might need to shift for holidays and report cards?
Sample Year Plans
Sample year plans are not meant to show you a ‘right’ answer, but rather to be an example of what a prioritized and sequenced year plan might look like.
It is important to revisit and revise your year plan through the year. Student needs and unexpected disruptions require adjusting along the way. Things to consider when you are reflecting on your year plan are:
Where did I think we would be in our sequence right now? Where are we actually at?
When I look ahead to the end of the year, do I need to adjust the amount of time I am spending on the units I have left?
Are there some of my prioritized outcomes that I need to deprioritize to give us time on what is MOST important?
Might I need to shift some of my lower and medium priority outcomes to stations or centers?
Are there some outcomes that I might connect to other curricula? Examples include:
Some shape and space outcomes moved to Art (transformations, area, 3D objects)
Some statistics outcomes moved to Social Studies (measures of central tendency, graphing)
It is important to keep your year plan current so that can make informed decisions throughout your year. Reflecting regularly can help you build a strong foundation for your students.
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.
Terry Johanson Consulting 