Measurement and Geometry
Measurement and Geometry are related concepts that fall under what previous curricula called Shape & Space. Throughout K-7, the big ideas all share the foundational concept of the ability to describe, measure, and compare spatial relationships. This key concept is a critical part of numeracy as our learners develop spatial sense.
In Primary grades students identify, describe, build, and sort 2-D shapes and 3-D objects by exploring attributes and recognizing similarities and differences. As they go through the Intermediate grades students learn to classify shapes by their attributes, including learning vocabulary relevant to each type of shape or object. Our visible world is full of shapes and objects that our learners experience every day.
Many of these geometrical concepts then connect to number concepts through exploring measurement. Over K-7 students measure and compare length, area, volume, capacity, mass, time, and angles. Students begin developing the concepts by measuring common attributes through comparison. They then learn to appreciate the value of direct measurement, at first using non-standard units and then standard metric units. Indirect measurements are figured out by using direct measurements, for example, using dimensions to determine an area.
Beginning in Grade 4 with symmetry, students also develop spatial sense with transformations. In Grades 5-7 students identify and construct transformations using slides (translations), flips (reflections), and turns (rotations).
As students explore measurement and geometry, there are many opportunities to connect to students’ lives, community, culture, and place. With these experiences we are honouring the following First Peoples Principle of Learning: Learning is holistic, reflexive, reflective, experiential, and relational (focused on connectedness, on reciprocal relationships, and a sense of place).
As we learn about key concepts in measurement and geometry, we will also be developing many curricular competencies. Three that we have chosen to focus on in our designing of lesson ideas are:
- Estimate reasonably
- Visualize to explore mathematical concepts
- Use mathematical vocabulary and language to contribute to mathematical discussions
Although these three curricular competencies have been highlighted, there will be many opportunities to develop many curricular competencies during the investigation of measurement and geometry.
Learning Story for Kindergarten: Measurement and Probability
The concepts of measurement and geometry are experienced by children in their daily lives, during play, and in more formal math learning experiences. In Kindergarten, children develop the language of attributes (quantity, corners, sides, straight/curved) when building and creating with shapes, and the language of comparison when measuring.
Many children may come to Kindergarten being able to identify basic shapes such as a square, circle, and triangle. What is more of a focus in Kindergarten is investigating the shapes and their attributes and using specific vocabulary and language to describe shapes. Building and creating with shapes creates opportunities for children to use this math vocabulary and language. Both 2D shape and 3D objects are explored through attributes. For example, a can of soup may be held up or passed around for the children to feel the curved edges and surface. In Kindergarten students are not expected to know a can is a cylinder although this vocabulary may be introduced within the investigations. Students might say, “The can is smooth and curves around,” or “It rolls when it is on its side.”
For measurement, comparative thinking and language is the focus in Kindergarten. Children might exclaim noticings such as, “I am taller than you,” or “That’s heavier than me.” Children at this age often want to build the longest train or the highest tower during their play which is a good opportunity to nudge mathematical vocabulary and language. Along with concepts of linear measurement, (length, height, width, distance), children in Kindergarten learn about the beginning ideas of mass (heavier than, lighter than) and capacity (holds more, holds less) through direct comparison of materials and objects.
The concepts of each of geometry and measurement may be introduced during a focused week of study and then woven throughout and connected to math and other areas of learning throughout the year. For example, positionality and shape is often explored when patterning in math and in visual arts, and there are many opportunities to measure in areas of learning such as science and ADST.
Key Concepts
Single Attributes of Shapes
Attributes such as size, quantity, straight/curved sides are used to sort, describe and create with both 2D and 3D shapes and objects
Direct Comparative Measurement
Two or more materials or objects are compared with relation to length, mass or capacity, using a baseline to compare against
Key Measurement and Geometry Concept 1: Single Attributes of Shapes
Overview
The learning focus for geometry in Kindergarten is investigating shapes and their attributes and using specific vocabulary and language to describe shapes. Building and creating with shapes creates opportunities for children to use this math vocabulary and language. Both 2D shape and 3D objects are explored through attributes. Attributes for 2D shapes include small, large, short, long, wide, narrow, numbers of sides, number of corners, types of lines/sides such as straight or curvy. Many children will come to kindergarten being able to identify a square, circle or triangle but may not know the general attributes for all squares, circles and triangles. It is okay to use the correct mathematical names for shapes in Kindergarten but it is not the focus for learning or assessment. Likewise, Kindergarten students are also learning about the attributes of 3D shapes and objects such as round, boxy, pointy, flat, stands up, rolls, slides etc, but are not required to identify them by name (sphere, cube, prism, etc). Instead, students are playing with and investigating shapes, composing and decomposing shapes, and creating structures and designs with shapes. Spatial reasoning and the understanding of movement and position within space is something young children naturally do in their play, whether with blocks, small world play or gross motor play outdoors.
Geometry Foundations:
Students will come into Kindergarten with informal experiences with geometry (shape and space) from home and pre-K learning contexts such as:
- sorting materials by attributes (colours, size, shape)
- play that involves positionality involving concepts such as in front of, above, below, next to, through (use of blocks, maps, trains, cars, small world play, etc)
- block play that involves stacking, rolling, sliding, and balancing different three-dimensional shapes and items.
- puzzles, drawing, and apps involving two-dimensional shapes
Foundational, supporting concepts and related competencies that are needed to develop this grade level concept:
- No necessary concepts needed as students are just beginning to formalize geometry concepts in Kindergarten.
- One thing to be mindful of is that children may come to Kindergarten being able to “identify” shapes and name basic shapes like a square, circle, triangle but they may not be able to describe the attributes of the shapes or know what makes a square a square for example. One way to nudge their thinking about this is, for example, to share multiple different three-sided shapes in different orientations, not just the typical equilateral triangle with one vertex pointing up.
Progression:
Kindergarten students are connecting their informal learning about shapes with a focus on describing the attributes of 2D and 3D shapes. You may note a range in students’ geometric understanding and spatial reasoning:
Students may say shapes are the same because they have some similarities (ie square sides) but are different in size, number of sides/corners, etc.
- Students can match shapes such as placing a shape on a template or fitting a shape into a puzzle.
- Students will begin to understand orientation and rotate/move a shape to match a presented shape/prototype.
- Students begin to generalize and identify what some shapes are such as squares, triangles and circles from informally hearing these in their discussions with adults, siblings, peers and exposure to media.
- Students begin using classifying and categorizing thinking and can sort and compare shapes by attributes.
- Students will make connections and find things in the world that are similar to geometric shapes such as a plate being like a circle or a ball being round.
- As students are developing mathematical vocabulary and language, they may use gesturing or “air drawing” to show their understanding of attributes and shapes.
- Spatial reasoning is developing with students aware of 2D-3D relationships, position and movement in space, different perspectives and an informal understanding of balance and symmetry.
- Students will begin to use comparative thinking and be able to explain how two shapes or objects are the same and different, using mathematical vocabulary and language.
Sample Week at a Glance:
This week plan focused on a general introduction to the idea of shapes and space in Kindergarten, introducing both 2D and 3D geometric shapes and objects/items. Through children’s investigation and play, spatial reasoning and position in space are also explored.
If you use “soft starts” to begin the day, you could provide different geometry-focused materials on each table invitation such as:
- Pattern blocks
- Tangrams
- Unit blocks (small size)
- Puzzles
- Lego
- Pentominoes
Learning focus: Introduction to concept of geometric shapes and their attributes.
Before
Share two images from the Same But Different website such as the following:
Ask students to share what they notice about how the two shape are the same (pink colour, four corners, four sides, straight sides) and how they are different (shape on left has two long sides and two short sides, shape on right has all sides the same length). Students may have other ideas as well. This is an opportunity to hear what language and understanding students have to describe the attributes of shapes.
During
Invite students to make (compose and decompose) shapes out of different materials as well as math materials. On tables, provide collections of materials that allow for creating 2D shapes (straight and curved sides) such as popsicle sticks, chopsticks, dowels, twigs, pipe cleaners, or pieces of ribbon and yarn as well as math materials such as pattern blocks, tangram pieces or colour tiles. Provide these prompts to support students’ investigations: What different shapes can you make? How would you describe them? Some students may also choose to create 3D shapes with the materials.
After
Invite students to share some of the shapes they created and describe them by their attributes (small, long, straight sides, number of sides etc). If created, choose a 2D shape and a 3D shape and ask students to share how they are the same and how they are different. If needed, provide language such as “2D shapes are flat” and “3D shapes take up space.” Also, based on observations, if a child makes a “squiggle” with a ribbon for example and calls it a shape, you could share this with the group and explain that in math, something is considered a shape if it is “closed”.
Learning focus: Noticing and describing attributes of shapes in the environment.
Before
Share some of the illustrations and text about shapes in nature from the book Shapes and Patterns in Nature by Stepanka Sekaninova. Jana Sedlackova, and Magdalena Konecna, the book Shapes All Around by Kate Riggs and Laetitia Devernay, or share some images of trees or plants from the local area. Elicit noticing and discussion with prompts like: How would you describe the shapes? What do you notice about them? How are natural shapes and “math shapes” the same and how are they different?
During
Take a shape walk outdoors either around your school park or in the community. Ask students to look for different types of shapes and record their findings on clipboards by drawing and labeling, tracing around the items on paper or whiteboards, or by taking photographs. Pause for some discussions such as discussing the difference between 2D and 3D and natural/organic vs geometric/math/human-made shapes. Students might say things like, “it’s like a triangle” or “those leaves have points” or “that tree is like a tube” or “that long branch is curvy” to describe different attributes of the shapes they find.
After
Invite students to share a shape they found with a classmate and describe it in three ways (focused on attributes).
As a follow-up to this lesson, you could also create a class book of the students’ drawings or photographs.
Learning focus: 2D Shapes are made of lines (straight and curved) and are closed.
Before
Sitting together as a group, provide each student with a pipe cleaner and ask them what different types of lines they can make. Record their different lines, such as squiggle, wavy, straight, curved, zigzag, on a whiteboard or chart. Now invite them to take one of their lines and twist the ends together to create a closed shape. Have students look at each other’s shapes and discuss what they notice about the different shapes.
Optional: Read When A Line Bends, A Shape Begins by Rhonda Growler Greene. Provide each student with a pipe cleaner to bend and make into closed shapes as you read the story.
During
Indoors option: Provide materials such as pipe cleaners, popsicle sticks, yarn and ribbon for students to make different shapes (2D and 3D) with.
Outdoors option: Go for a walk outdoors and find natural materials such as sticks, twigs, leaves etc to make different shapes (2D and 3D) with.
After
Have students use an iPad camera or similar tool to take photos of the shapes they have created and share on a screen for all the students to discuss and compare or have students do a “gallery walk” to compare the shape/s they created to those created by their classmates. Prompt students with the question: How are the shapes the same and how are they different?
Learning focus: Connecting shapes and Indigenous design elements.
Before
Share a piece of art by a local Indigenous artist. Ask the students to describe the shapes they see. Highlight design elements/shapes of one of the local Indigenous cultures in your area. For example, Coast Salish art includes three significant shapes (more info here). Ask students what they notice about the shapes and what connections they are making.
During
Invite students to draw, paint, or create a design. Provide a range of materials such as pencils, crayons, pencil crayons, chalk pastels, watercolour or tempera paint, and surfaces to draw and create on. Loose parts and math materials can also be provided as a choice. You may include specific criteria such as “create a design with at least three different shapes.” Have students leave them in place for other children to view.
After
Share the piece of art again, highlighting the shapes that had been discussed. Invite students to do a “gallery walk” of the students’ creations and to look out for the significant shapes that had been discussed.
Learning focus: Using positional language and spatial reasoning to describe 3D structures.
Before
At each table group/small group of students, provide a small collection of 3D blocks and boxes and containers (invite families to send in clean boxes etc from their recycling bins). Prompt students to explore the materials with: What shapes can you find in these objects? What can you do with these objects? How could you sort these objects?
Have students share their findings.
Optional: Read Circle Over Berry by Carter Higgins if you feel the students need some vocabulary and positional language to use in their investigations.
During
Using the materials from the group investigation, along with other classroom materials, invite students to build and describe a structure. You might need to provide an example of a structure like a tower, bridge or building. As you join small groups of students while they are building, nudge their mathematical thinking by asking them to describe the objects they are using and positional language to describe their placement, for example “The little blocks are around the tall block with a curved block on the top of it.” Invite students to record their structure on paper with drawing and labeling. You may need to support students’ drawing by having them think about what lines/shapes they will draw for each object. Leave the structures in place.
After
Collect the students’ drawings of their structures and give them out to the students to try and match the 2D drawing with the 3D structure. Choose one structure to highlight for group discussion and prompt students: What do you notice? What shapes live here?
Noticing what students are able to do and what they know about shapes this week will determine your next steps. You may note that students may need more time to develop the specific mathematical language needed to describe and compare the attributes of shapes and this can be woven into math play centres and math read alouds with picture books throughout the school year.
Suggestions for Assessment
Most assessment of these concepts will be through observing and listening. Create a chart of related learning standards where you can make notes of your observations and how students are discussing these ideas.
By the end of Kindergarten, students will be able to:
- Create and describe 2D shapes by the quantity of sides or corners or whether they have curved or straight sides.
- Find and describe 3D shapes by the quantity of edges or corners/vertices or whether they have curved or straight faces.
- Use positional language (such as beside, on top of, under) to describe the orientation of compositions of shapes and structures built with objects.
- “The tall block is under the curvy one.”
Suggested Links and Resources
Same But Different Math
https://www.samebutdifferentmath.com/geometry
Which One Doesn’t Belong
BC Reggio-Inspired Mathematics Project: Investigating Shapes
https://bit.ly/BCRIM-InvestigatingShapes
Coast Salish Design Elements/Shapes
https://legacy.uvic.ca/gallery/salishcurriculum/coast-salish-design-elements/
Northwest Coast Art: Basic Formline Elements and Shapes
Children’s Books
Shapes and Patterns in Nature by Stepanka Sekaninova. Jana Sedlackova, and Magdalena Konecna
Shapes All Around by Kate Riggs and Laetitia Devernay
When A Line Bends, A Shape Begins by Rhonda Gowler Greene
Circle Under Berry by Carter Higgins
Key Measurement and Geometry Concept 2: Direct Comparative Measurement
Overview
In Kindergarten, direct comparison is the measurement skills being developed focusing on attributes of length, capacity and mass. Children measure through direct comparison of materials and objects and develop specific mathematical vocabulary to describe their comparisons. When ordering a set of nesting dolls from shortest to tallest, children might be heard using language such as short, shortest and tall, taller and taller. When measuring objects’ linear attributes (length, width, height, distance), children learn to line up the objects against a baseline so they can compare the objects directly. In Kindergarten, children may have an intuitive sense of what containers will hold more or what objects are heavier than others. These understandings about capacity and mass can be developed through play and investigations with materials and objects. Vocabulary and language development occurs alongside play with objects and the use of concrete materials. Gestures may support students’ language and conceptual development as they learn to measure length, capacity and mass using objects and personal referents such as their hands or feet.
Measurement Foundations:
Students will come into Kindergarten with informal experiences with measurement from home and pre-K learning contexts such as:
- comparing their size, weight and height to their family members
- using general measurement language such as big and small
- playing with cups and containers and filling them up with sand, water, etc
- being weighed and measured with tools such as scales and measuring tapes at home or the doctor’s office
- an understanding that they are getting taller as they get older
- there are different tools for measuring such as measuring cups, scales, rulers, etc
- that quantity/numbers are associated with measurements
Foundational, supporting concepts and related competencies that are needed to develop this grade level concept:
- No necessary concepts needed as students are just beginning to formalize measurement concepts in Kindergarten.
- Children will generally come into Kindergarten using very general language about measurement and use words such as big and small to describe the size of something without paying specific attention to the attribute being measured.
Progression:
Kindergarten students are connecting their informal learning about size and measuring to formal experiences with direct comparative measurement.
-Students make simple comparisons of length intuitively but may not recognize length, capacity or mass as distinct attributes (separate from general size).
-Students begin to identify length/height/width/distance as an attribute.
-Students will physically align two or more objects to determine which is longer or if they are the same length. At this stage, students will begin using math specific vocabulary for their comparative measurements such as long, longer, and longest.
-Students begin to identify capacity and mass as attributes of objects but will be developing their strategies in skills in comparing the capacity of containers or the mass of objects.
-Students begin to use approximations or comparative estimates such as “I think that box will hold more books than this one.”
-Students may begin to “quantify” measurements in a playful way, by counting, or by using tools and this skill will be developed in grade 1.
Sample Week at a Glance
This Kindergarten week plan focused on a general introduction to the concept of measurement by direct comparison.
If you use “soft starts” to begin the day, you could provide different measurement-focused materials and tools on each table invitation such as:
- Basket of ribbons, toy snakes/dinosaurs or other collections to order by length, width or height
- Balance scales and objects to compare mass
- Different boxes and containers and objects to compare what can fit in the boxes/containers (capacity)
Learning focus: Direct comparison of two objects for linear measurement and development of mathematical vocabulary.
Before
Open the book Actual Size by Steve Jenkins to the page with the gorilla’s hand. Invite a few students up to compare the size of their hand to the gorilla’s hand. Prompt with the question: What could you say about how your hand compares to the gorilla’s hand? Support the students with using more specific language than bigger and smaller and help them to focus on length and width using terms such as longer, shorter, and wider.
During
Invite students to choose an object from around the classroom and then find something that is taller than the item, about the same height as the item, and something that is shorter than the item. Repeat with new items. Students can record their findings on a clipboard or small whiteboard to share.
After
Bring the students together on the carpet and show them a nesting doll (or other similar toy that has graduated sizes). Open the doll and line up all the different dolls from shortest to tallest. Ask students to name/describe each doll using mathematical vocabulary such as short, shortest, tall, taller tallest. Ask the students to consider what other parts of the dolls they could measure/compare.
Learning focus: Direct comparison of two containers and estimating then comparing which one holds more.
Before
Read the story What Will fit? by Grace Lin to introduce the context of filling a container. Share a reusable shopping bag or basket you use and have students share their ideas about how much or what it could hold.
During
On students’ tables or in centres,, provide a collection of small containers, bowls or cups. Invite students to choose two items and estimate which one will hold more. Using water, sand, glass gems or other small material, students can fill up on container and then pour it into another container to check their estimates. Repeat with different containers and materials.
After
Gather students together and invite them to share what they found out about measuring capacity/how much something holds. Ask them to share any surprises or challenges they had.
Learning focus: Direct comparison of two objects and estimating then comparing which one weighs more.
Before
Read either Just a Little Bit by Ann Tompert or Balancing Act by Ellen Stoll Walsh. You might need to provide the terms see-saw or teeter-totter for the students. Invite the students to share their thinking about what happened in the stories and what is needed for both sides to be “the same”.
During
Provide students with balance scales, blocks (flat ramp and fulcrum) or pieces of playdough with thick tongue depressor like popsicle sticks to make their own see-saw/teeter-totters. Have students choose items in the classroom and then use the balancing tools to investigate their mass by finding something that is about the same weight as their item, something heavier and something lighter. Ask the students to think about how they can record their findings and provide choices of notebooks, clipboards with paper, mini whiteboards or digital tools for them to record and share their findings.
After
Gather children together on the carpet and invite them to bring their item and their recorded findings to share. Invite students to turn and talk with a partner and use mathematical language to describe what they did and what they found out.
Learning focus: Practice of measuring skills and use of specific mathematical vocabulary and language to show understanding of direct comparative measurement.
Before
Go on a short math walk outdoors and invite students to notice things that they could measure. Invite them to find something that is taller or heavier than them and something that is shorter or lighter.
During
Collect items outdoors for students to compare and measure such as twigs, leaves, cones, shells, rocks, grasses, etc. In partners or triads, invite students to compare and order their collection by attribute. Watch for (and remind if necessary) students using a baseline (linear measurement) or referent (capacity/mass) for comparison and applying mathematical language as they engage in the task.
After
Have students walk around to see each others’ collections and practice using mathematical language to describe the collection and how it is ordered.
Learning focus: Practice of measuring skills and use of specific mathematical vocabulary and language to show understanding of direct comparative measurement.
Before
Share an image from Estimation 180 or a similar image and invite students to estimate which container holds more and to explain their reasoning. (In Kindergarten, we are not focused on quantifying their estimate or using standard units).
During
Math Centres
Students can choose from the following centres to practice measuring in different ways or use tasks/materials that students in your class are familiar with:
- Collections: provide collections of cones, shells, buttons, ribbons, toys, etc and invite students to compare, measure and order the items.
- Cups and Containers: Using tubs of water or sand, invite students to investigate the capacity of different cups, containers, bowls and spoons.
- Balance Scales: Have students use balance scales to weigh and compare different items in the classroom.
- Hand measuring: Have students trace and cut out the shape of their hand (can do in partners to help each other with the tracing). Use their hand cut out to compare and measure different objects in the classroom. This can be bridged to grade 1 learning learning standards that involve using non-standardized units for measuring.
After
Gather students together and invite students to share with a partner: what they did, what they learned/practiced and what their goal is for their math learning (related to measurement). Invite a few students to share their goals and set a class goal for what is needed next for their learning.
After this week of introducing measurement concepts, these concepts can be further developed during math centres, outdoor play, and woven into other areas of learning such as science, visual arts, physical education, and ADST.
Suggestions for Assessment
Most assessment of these concepts will be through observing and listening. Create a chart of related learning standards where you can make notes of your observations and how students are discussing these ideas.
By the end of Kindergarten, students will be able to:
- Compare two objects and can determine which one is longer or taller.
- Estimate and compare if one item is heavier or lighter than another.
- Estimate and compare how many objects two different containers will hold.
Suggested Links and Resources
Estimation 180
Children’s Books
Measuring Penny by Loreen Leedy
Just a Little Bit by Ann Tompert
Balancing Act by Ellen Stoll Walsh
Actual Size by Steve Jenkins
What Will fit? by Grace Lin
Just How Long Can a Long String Be? by Keith Baker
Too Small Tyson by JaNay Brown-Wood
Great Estimations by Bruce Goldstone