C1. Patterns and Relationships
Specific Expectations
Patterns
C1.1
identify and describe a variety of patterns involving geometric designs, including patterns found in real-life contexts
- geometric designs that can be found in real life:
- nature: turtle shells, fossils, cross-section of a tree trunk
- handicrafts: quilts, knitting, crocheting, smocking, beadwork, weavings, prints
- interior design: wallpaper, floor tiles, carpet, area rugs, hardwood floors, lights
- structures: bridges, ancient pyramids
- codes: Morse code
- cultural practices: henna (mehndi designs), kolam, Celtic knots, Islamic geometric patterns, Hitomezashi stitching, Ankara wax prints
- Human activities, histories, and the natural world are made up of all kinds of patterns and many of them are based on geometric designs.
- Patterns may involve attributes such as colour, shape, texture, thickness, orientation, or material.
Note
- Students can engage in mathematics and patterns through the contexts, cultural histories, and stories of various cultures.
- Patterns do not need to be classified as repeating or otherwise in Grade 2. Instead, focus on the geometric design – are shapes being repeated? Do shapes appear to grow? Do shapes appear to shrink?
Have students collect pictures or make diagrams of geometric patterns that they have found in real life, including those that are personally relevant, and create a class pattern wall. Along with the pictures, have students write in words what they notice about the pattern, such as that the shapes are repeated or that the size of the shape changes. If students are using a “learning log” that they created in Grade 1, they can add new patterns to it.
Provide cross-curricular explorations of geometric designs from different cultures. There are also opportunities to discuss respectful representations of cultural histories and identities. (See the examples given for this expectation.)
C1.2
create and translate patterns using various representations, including shapes and numbers
- creating patterns:
- with materials of different textures, sizes, shapes, colours
- with isometric dot paper, grids, rings:
- with learning tools (e.g., pattern blocks, square tiles):
- translating a sound pattern to a numeric pattern:
- a sound pattern: clap, snap, snap; clap, snap, snap, snap; clap, snap, snap, snap, snap, …
- translated into a number pattern:
- 1 [clap], 2 [snaps]; 1 [clap], 3 [snaps]; 1 [clap], 4 [snaps], …
- translated into a number pattern:
- a sound pattern: clap, snap, snap; clap, snap, snap, snap; clap, snap, snap, snap, snap, …
- translating a word pattern to a numeric pattern:
- a word pattern: blue, orange, orange; blue, orange, orange, orange; blue, orange, orange, orange, orange, ...
- translated into a movement pattern:
- 1 step to the right [blue], 2 steps to the left [orange]; 1 step to the right [blue], 3 steps to the left [orange]; 1 step to the right [blue], 4 steps to the left [orange], ...
- translated into a movement pattern:
- a word pattern: blue, orange, orange; blue, orange, orange, orange; blue, orange, orange, orange, orange, ...
- translating a numeric pattern to a geometric pattern; for example, 5, 10, 15, … can become:
- Sample 1:
- Every group of 5 is 4 squares and 1 triangle:
- Sample 1:
- Sample 2:
- The pattern starts with 5 circles, and 5 more are added each time:
- The same pattern structure can be represented in various ways.
- Patterns can be created by varying a single attribute, or more than one.
- Pattern structures can be generalized.
Note
- Comparing translated patterns highlights the equivalence of their underlying mathematical structure, even though the representations differ.
Demonstrate a pattern of sounds or movements, and ask students to translate the pattern by representing it using numbers and using shapes.
Have students create a pattern using shapes to represent skip counts. They may create a pattern using a repetition of shapes for each count, or they may show the same shape and depict how the amount changes for each count.
For example, they could skip count by 5s:
- Each triangle marks five:
- Ask students questions related to the pattern, such as “How many triangles would you have if the pattern went up to 30?”
- Each additional column marks five:
- Ask students to extend the pattern to position 5. Then ask, “If you kept going, could you reach a position in this pattern that has 34 circles? Why or why not?”
Have students create a pattern to show the equivalence of the values of coins and bills. For example, 1 five-dollar bill is equivalent to 5 loonies, 2 five-dollar bills to 10 loonies, 3 five-dollar bills to 15 loonies, and so on.
C1.3
determine pattern rules and use them to extend patterns, make and justify predictions, and identify missing elements in patterns represented with shapes and numbers
- extending patterns in multiple directions:
- What comes before position 1? What comes after position 16?
- making near and far predictions:
- What shape would be at position 18? What shape would be at position 50?
- identifying missing shapes and numbers:
- What shapes are at positions 9, 10, and 11?
- Patterns can be extended because they are repetitive by nature.
- Pattern rules are generalizations about a pattern, and they can be described in words.
- Patterns can be extended in multiple directions, such as up, down, right, and left.
- To make a near prediction about a pattern is to state or show what a pattern will look like just beyond the given representation of that pattern. The prediction can be verified by extending the pattern.
- To make a far prediction about a pattern is to state or show what a pattern will look like well beyond the given representation of that pattern. Often calculations are needed to make an informed prediction or to verify a prediction.
- To identify missing elements of patterns is to complete a representation for a given pattern by filling in the missing parts.
Note
- In order to extend, predict, or determine missing elements, students need to generalize patterns, using pattern rules.
- Rules should be used to verify predictions and to critically analyse extensions and solutions for missing elements.
Have students expand patterns. For example, what are the next three symbols in the pattern?
turtle, turtle, turtle, turtle, wolf; turtle, turtle, turtle, turtle, wolf; turtle, turtle, turtle, turtle, wolf; turtle, …
Have students extend both ends of a given pattern so that they think about their pattern rule in more than one way. For example, provide students with a pattern made up of a sequence of squares and triangles, and ask them what the next three shapes will be, then ask what three shapes come before position 1:
Ask students to come up with ways to create and extend patterns on a hundreds chart, including using a diagonal. For example, ask them to extend the pattern below in both directions. Ask them to describe and verify their extensions by referring to other numbers on the hundreds chart:
Have students make and test their predictions about patterns to support them in understanding the role of pattern rules in making generalizations. Provide them with a variety of patterns, and have them make near predictions. For example, they could predict and verify the number of tiles that will be in positions 0, 5, and 6 for the pattern below. Also, ask students questions such as “If I had 18 tiles, what position number could I show?”
Have students make far predictions for patterns; for example, what shape is in the 100th position? The 101st position? The 99th position? Students should start making generalizations to make far predictions and communicate this in their justifications. For example, one core has 5 elements, two cores have 10 elements, three cores have 15 elements, and so on.
Provide students with patterns that have missing elements or positions. This will support them in thinking critically about possible pattern rules based on the information they are given. The more information that is missing, the more thinking and possibilities for patterns will emerge. For example:
- Visualize and describe what is behind the rectangle. Explain your thinking.
- What are the missing numbers in these sequences?
- 30, 50, ___, ___, 110, ___, …
- 40, ___, 80, ___, 120, …
C1.4
create and describe patterns to illustrate relationships among whole numbers up to 100
- creating number string patterns to show the relationship between tens and ones:
- when the tens place goes down by 1, the ones place goes up by 10:
73 = | 7 tens | + 3 ones |
73 = | 6 tens | + 13 ones |
73 = | 5 tens | + 23 ones |
73 = | 4 tens | + 33 ones |
73 = | 3 tens | + 43 ones |
73 = | 2 tens | + 53 ones |
73 = | 1 tens | + 63 ones |
73 = | 0 tens | + 73 ones |
- creating number string patterns to show the relationships between addition and subtraction facts in larger numbers:
80 + 7 = 87 |
87 − 7 = 80 |
81 + 6 = 87 |
87 − 6 = 81 |
82 + 5 = 87 |
87 − 5 = 82 |
83 + 4 = 87 |
87 − 4 =83 |
84 + 3 = 87 |
87 − 3 = 84 |
85 + 2 = 87 |
87 − 2 = 85 |
86 + 1 = 87 |
87 − 1 = 86 |
87 + 0 = 87 |
87 − 0 = 87 |
- creating number string patterns to show how number facts involving 7 could be used to add larger numbers:
7 + 0 = 7 |
67 + 0 = 67 |
7 + 1 = 8 |
67 + 1 = 68 |
7 + 2 = 9 |
67 + 2 = 69 |
7 + 3 = 10 |
67 + 3 = 70 |
7 + 4 = 7 + 3 + 1 = 11 |
67 + 4 = 67 + 3 + 1 = 71 |
7 + 5 = 7 + 3 + 2 = 12 |
67 + 5 = 67 + 3 + 2 = 72 |
7 + 6 = 7 + 3 + 3 = 13 |
67 + 6 = 67 + 3 + 3 = 73 |
7 + 7 = 7 + 3 + 4 = 14 |
67 + 7 = 67 + 3 + 4 = 74 |
7 + 8 = 7 + 3 + 5 = 15 |
67 + 8 = 67 + 3 + 5 = 75 |
7 + 9 = 7 + 3 + 6 = 16 |
67 + 9 = 67 + 3 + 6 = 76 |
7 + 10 = 17 |
67 + 10 = 77 |
- Patterns exist in increasing and decreasing numbers based on place value.
Note
- Creating and analysing patterns that involve decomposing numbers will support students in understanding how numbers are related.
- Creating and analysing patterns involving addition and subtraction facts can help students develop fluency with math facts, as well as understand how to maintain equality among expressions.
Provide students with a partial number string pattern based on place value. Have them continue the string to rename a number such as 73 in as many ways as they can, using tens and ones. Support students in recognizing that for each decrease of 1 ten, there is an increase of 10 ones.
73 = |
7 tens |
+ 3 ones |
73 = |
6 tens |
+ 13 ones |
. | . | . |
. | . | . |
. | . | . |
Have students create their own number string patterns to show a mathematical relationship. For example, they may demonstrate how the addition and subtraction facts for 7 can be used to add larger numbers.
7 + 0 = 7 |
67 + 0 = 67 |
7 + 1 = 8 |
67 + 1 = 68 |
7 + 2 = 9 |
67 + 2 = 69 |
7 + 3 = 10 |
67 + 3 = 70 |
7 + 4 = 7 + 3 + 1 = 11 |
67 + 4 = 67 + 3 + 1 = 71 |
7 + 5 = 7 + 3 + 2 = 12 |
67 + 5 = 67 + 3 + 2 = 72 |
7 + 6 = 7 + 3 + 3 = 13 |
67 + 6 = 67 + 3 + 3 = 73 |
7 + 7 = 7 + 3 + 4 = 14 |
67 + 7 = 67 + 3 + 4 = 74 |
7 + 8 = 7 + 3 + 5 = 15 |
67 + 8 = 67 + 3 + 5 = 75 |
7 + 9 = 7 + 3 + 6 = 16 |
67 + 9 = 67 + 3 + 6 = 76 |
7 + 10 = 17 |
67 + 10 = 77 |